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Dynamic response of concrete beams externally reinforced with carbon fiber reinforced plastic

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Dynamic response of concrete beams externally reinforced with carbon fiber reinforced plastic
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Jerome, David Mark, 1952-
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English
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236 leaves : ill. ; 29 cm.

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Subjects / Keywords:
Beams ( jstor )
Bending ( jstor )
Compressive stress ( jstor )
Damping ( jstor )
Specimens ( jstor )
Steels ( jstor )
Stiffness ( jstor )
Strain gauges ( jstor )
Strain rate ( jstor )
Tensile stress ( jstor )
Aerospace Engineering, Mechanics, and Engineering Science thesis, Ph. D
Composite-reinforced concrete -- Research ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics, and Engineering Science -- UF
Fibrous composites -- Testing ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1996.
Bibliography:
Includes bibliographical references (leaves 170-173).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by David Mark Jerome.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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DYNAMIC RESPONSE OF CONCRETE BEAMS
EXTERNALLY REINFORCED WITH
CARBON FIBER REINFORCED PLASTIC










By

DAVID MARK JEROME


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1996




DYNAMIC RESPONSE OF CONCRETE BEAMS
EXTERNALLY REINFORCED WITH
CARBON FIBER REINFORCED PLASTIC
By
DAVID MARK JEROME
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996


This work is dedicated to my wife Elisabetta Lidia, and our twin sons Matthew Allen and
Nathan Kelley, who were born on November 1, 1993, during the course of this investigation.


ACKNOWLEDGMENTS
The author would like to acknowledge the advice and counsel given to him by his faculty
supervisory committee: Professors E. K. Walsh, D. M. Belk, J. E. Milton and C. S. Anderson.
Special recognition is reserved for Professor C. A. Ross, supervisory committee chair, without
whose advice, mentorship, encouragement, and friendship, this research would not have been
possible.
The author would like to acknowledge the following individuals at the Wright
Laboratory Air Base Technology Branch at Tyndall Air Force Base, Florida, for their
contributions listed below: Mr. Dale W. Wahlstrom for fabrication of the concrete beams and
Split Hopkinson Pressure Bar (SHPB) samples, and for assistance with the compressive and
splitting tension tests on the Forney load frame; Mr. William C. Naylor for strain gage and
instrumentation assistance; Mr. Dean W. Flitzelberger for assistance with the Material Test
System (MTS ) load frame; Mr. Francis W. Barrett III for high speed camera assistance on the
drop weight impact machine; and Mr. Carl R. Hollopeter for maintenance of the drop weight
impact machine and instrumentation assistance. Use of the Forney and MTS load frames and
drop weight impact machine, all of which are owned and supported by the Wright Laboratory
Air Base Technology Branch, Tyndall Air Force Base, Florida, is also gratefully acknowledged.
The author also wishes to acknowledge the assistance of Ms. Cathy A. Rickard for ably
typing a large portion of the manuscript, and Mr. Danny R. Brubaker and Mr. Bruce C. Patterson
for help in preparing several of the figures.
Finally, the author wishes to acknowledge the help and support of the Wright Laboratory
Armament Directorate at Eglin Air Force Base, Florida, during his educational endeavors.


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
KEY TO ABBREVIATIONS vi
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
Objective 1
Background 1
Approach 10
2 EXPERIMENTAL WORK 13
Fabrication of Concrete Beam Specimens 13
Application of the CFRP 15
Surface Preparation 16
Characterization of the Concrete 22
Static Compression Tests 22
Static Splitting Tensile Tests 26
Dynamic Compression Tests 29
Dynamic Splitting Tensile Tests 32
Test Results 37
Static Beam Bending Experiments 39
Description of the MTS Load Frame 39
Instrumentation Used/Measurements Made 42
Test Results 42
Calculation of Fracture Energies 54
Dynamic Beam Bending Experiments 58
Description of the Wyle Laboratories Drop Weight Impact Machine 59
Instrumentation Used 61
Calibration of the Tup 62
Method of Test 73
Interpretation of Test Results 75
Damping Loads 91
Results and Discussion 93
Comparison of Dynamic and Static Test Results 97
IV


3ANALYTICAL MODEL.
103
Section Analysis 103
Region 1 All Elastic 105
Region 2 Cracked Tension Concrete, All Other Elastic 109
Region 3 Cracked Tension Concrete, Inelastic Compression
Concrete, All Other Elastic 112
Comparison to MTS Test Data 115
Determination of Beam Equivalent Mass 121
Single Degree of Freedom Representation of a Beam Subject to a
Half Sine Pulse Impulsive Load 125
Comparison to Drop Weight Impact Test Data 130
4 FINITE ELEMENT METHOD CALCULATIONS 134
Description of the ADINA FEM Computer Code 134
The Concrete Material Model 137
The CFRP Material Model 141
Concrete Beam with CFRP Finite Element Model 141
Dynamic Loading Calculations 142
Results and Comparison to Test Data 144
5 SUMMARY AND DISCUSSION 157
Characterization of the Concrete 158
Static Beam Bending Experiments 159
Dynamic Beam Bending Experiments 160
Static versus Dynamic Beam Bending Experiments 161
Analytical Model 162
Finite Element Method (FEM) Calculations 163
Future Research 164
6 CONCLUSIONS 167
REFERENCES 170
APPENDICES
A SUMMARY OF CONCRETE UNCONFINED COMPRESSIVE
STRENGTHS 174
B MTS LOAD DISPLACEMENT CURVES 177
C SUMMARY OF STATIC THREE POINT BENDING EXPERIMENTS 194
D DYNAMIC BEAM BENDING EXPERIMENTAL DATA TABULAR
SUMMARY 197
E ANALYTICAL MODEL COMPUTER PROGRAM 202
F ADINA INPUT AND PLOT FILES FOR BEAM LW6-43 222
G ADINA INPUT AND PLOT FILES FOR BEAM LW9-66 229
BIOGRAPHICAL SKETCH 236
v


KEY TO ABBREVIATIONS
American Concrete Institute (ACI)
Automatic, Dynamic, Incremental, Nonlinear Analysis (ADINA)
Carbon Fiber Reinforced Plastic (CFRP)
Fiber Reinforced Plastic (FRP)
Finite Element Method (FEM)
Glass Fiber Reinforced Plastic (GFRP)
Linear Voltage Displacement Transducer (LVDT)
Material Test System (MTS)
Methyl Ethyl Ketone (MEK)
Modulus of Rupture (MOR)
Single Degree of Freedom (SDOF)
Split Hopkinson Pressure Bar (SHPB)
Swiss Federal Laboratories for Testing and Research (EMPA)
vi


Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
DYNAMIC RESPONSE OF CONCRETE BEAMS
EXTERNALLY REINFORCED WITH
CARBON FIBER REINFORCED PLASTIC
By
DAVID MARK JEROME
May 1996
Chair: C. Allen Ross
Cochair: James E. Milton
Major Department: Aerospace Engineering, Mechanics, and Engineering Science
A series of 54 laboratory scale concrete beams 3 x 3 x 30 in (7.62 x 7.62 x 76.2 cm) in
size were impulsively loaded to failure in a drop weight impact machine. An additional 16
beams were quasistatically loaded to failure in a load frame. The beams had no internal steel
reinforcement, but instead were externally reinforced on the bottom or tension side of the beams
with one, two, and three ply unidirectional carbon fiber reinforced plastic (CFRP) panels. In
addition, several of the beams were also reinforced on the sides as well as the bottom with three
ply CFRP. The beams were simply supported and loaded at beam midspan.
The lightweight concrete used in the test specimens was characterized via quasistatic and
dynamic compression and splitting tensile tests. In compression, the concrete behaved the same
as normal weight concrete. In tension, however, its behavior was quite different.
The beams sustained dynamic loads with amplitudes up to 10 kips (44.5 kN) and
durations less than 1 millisecond. Measurements including total load, midspan displacement and
strains, and a high speed framing camera (10,000 frames/sec) gave insight into failure
vii


mechanisms. Failure to account for inertia in the loads will result in incorrect calculation of the
beams peak bending load.
The quasistatic bending tests were conducted in load control at 2 lbs/sec (8.9 N/sec).
Both crosshead load and midspan displacement were recorded.
Fracture energies were determined by calculating the areas under the bending load
versus displacement curves. Dynamic fracture energies and peak displacements for the CFRP
reinforced beams were always less than the static values. However, dynamic peak bending loads
were 2-3 times larger than the corresponding static values. This implies that for a given load
rate, a beam has a fixed capacity to absorb energy, dictated by the peak bending load and limited
by displacement, emphasizing the brittle nature of concrete when loaded dynamically.
The beams were analytically idealized as a single degree of freedom (SDOF) system,
subjected to a half sine pulse impulsive load. The SDOF analysis generally overpredicted the
displacement-time behavior of the beams, due to modeling limitations.
Dynamic beam behavior was also studied numerically using the finite element method.
Excellent agreement with experimental evidence was obtained.
viii


CHAPTER 1
INTRODUCTION
Objective
The objective of this investigation was to understand the dynamic behavior of plain
lightweight concrete beams and plain lightweight concrete beams embedded with nylon fibers,
both of which were externally reinforced with carbon fiber reinforced plastic (CFRP) panels.
One, two, and three ply external CFRP strengthening panels were bonded to the bottom or
tension side of laboratory size (3.0 x 3.0 x 30.0 in/7.6 x 7.6 x 76.2 cm) concrete beams. In
addition, three ply CFRP was also applied to the sides as well as the bottom of some of the
beams. A schematic of the externally reinforced beam cross-section is shown in Figure 1. A
total of seventy-two beams were tested statically and dynamically. The experimental and
analytical studies of the static and dynamic response of these beams is the basis for this
dissertation.
Background
Since the mid sixties, existing concrete structures in Europe, South Africa, the US, and
Japan had all been externally post-strengthened by bonding steel plates to the bottom or tension
sides of the beams or slabs. The first actual use of externally applied fibrous composite
materials to enhance the load carrying capability of a concrete structure is attributed to a
researcher in Switzerland in 1989. Kaiser [1] is credited as the first worker to employ carbon
fiber reinforced plastic (CFRP) to post-strengthen concrete beams. Motivation for Kaiser's work
was driven by several factors: difficulties in manipulating the unwieldy steel plates into position
1


2
r
3 in
(7.62 cm)
I
Figure 1. Cross Section of Externally Strengthened Concrete Beam
at the construction site, corrosion at the steel/concrete interface, improper formation of
construction joints, and a limited selection of steel plate sizes. The CFRP material system, with
such desirable material attributes as being lightweight, being very strong (150-320 Ksi/1.034 -
2.206 GPa tensile strength), having outstanding corrosion resistance, and having excellent
fatigue properties, seemed a logical solution to the post-strengthening problem. Kaiser tested
twenty-six beams with a span of 6.56 ft (2 m), and one with a span of 22.96 ft (7 m), both
employing unidirectional CFRP on the bottom or tension side of the beams. He concluded that
the calculation of flexure in reinforced concrete elements post-strengthened with CFRP could be
performed similar to conventionally reinforced concrete structural elements, but that shear
cracking in the concrete could lead to delamination of the CFRP. He also concluded that flexural
cracking did not seem to influence the loading capacity of the beam, and that the addition of the
CFRP on the bottom of the beam led to a finer crack distribution in the beam when compared
with the plain concrete beam. Additionally, he stated that the difference in the thermal


3
expansion coefficients between concrete and the CFRP resulted in stresses at the CFRP/concrete
interface due to changes in temperature, but that after one hundred frost cycles ranging from -13
O O .
to +68 F (-25 to +20 C), no deleterious effects were noted on the load capacity of the beams.
In 1987, Meier and colleagues at the Swiss Federal Laboratories for Materials Testing
and Research (EMPA) conducted feasibility studies on the use of CFRP to rehabilitate the beams
in a bridge; in particular, as cables for cable-stayed and suspension bridges, prestressing tendons,
and other applications where steel repair plates could be replaced with advanced composite
panels [2, 3]. This may have been the genesis of Kaiser's pioneering work. Their more recent
work [4, 5, 6, 7] continues to focus on the large variety of modem materials recently employed
or soon to find application in modern bridge construction, focusing primarily on the use of
fibrous composites. Meier [7] also describes the first use of CFRP to rehabilitate a damaged
bridge in Switzerland in 1991. The Ibach bridge, in the county of Lucerne, was damaged when
construction workers were coring the bottom of a 127.92 ft (39 m) span for a traffic light and
accidentally severed several of the wires in a prestressing tendon in the outer web. The bridge
was successfully repaired with only 13.6 lbs (6.2 kg) of CFRP, compared with the 385 lbs (175
kg) of steel that normally would have been used to repair the bridge.
Triantafillou and Plevris [8] have studied the failure mechanisms of fiber reinforced
plastic (FRP) strengthened concrete beams subjected to a bending load which included: steel
yield-CFRP rupture, steel yield-concrete crushing, compressive failure, and crushing. In
addition to discussing the concept of concrete strengthening with prestressed FRP composite
sheets, they obtained equations describing each failure mechanism and produced diagrams
showing the beam designs for which each failure mechanism is dominant. In collaboration with
colleagues at EMPA in Switzerland, they have also studied hybrid box beams composed of
aluminum and CFRP to meet the requirements of low weight, high strength and stiffness, relative


4
ease of design and fabrication, and high reliability [9], They analyzed the problem of a
minimum weight design, CFRP-reinforced, thin-walled rectangular aluminum section subjected
to given strength and stiffness constraints, and showed that at the optimum design, various local
failure mechanisms occur almost simultaneously, while beam stiffness rarely controls.
Triantafillou and Meier [10] studied the basic mechanics of glass fiber-reinforced plastic (GFRP)
box beams, which combine a layer of concrete in the compression zone and a unidirectional
CFRP laminate in the tension zone. A design methodology for the hybrid sections is also
presented, based on a complete set of stiffness, strength, and ductility design requirements. The
method of strengthening and reinforcing concrete beams using thin FRP sheets as externally
bonded reinforcement, described by Triantafillou and Plevris [8], can be extended if the sheets
are prestressed before they are applied to the concrete surface [11], This prestressing technique
is being considered in the building of new structures, particularly bridges, and the strengthening
of existing ones. An analytical model was developed which predicts the maximum achievable
prestress level, so that the prestressed FRP system does not crack and fail near the two ends upon
release of the prestress force. The flexural behavior of concrete beams prestressed with
externally bonded FRP sheets is discussed, based upon four experiments and their subsequent
analysis. Test results indicate a modest 25 percent improvement in the flexural strength of the
four beams when prestressed FRP panels were used, compared with when the non-prestressed
FRP panels were used.
In China, Shijie and Ruixian [12] have analyzed the adhesive bond characteristics of
GFRPs used to externally reinforce concrete beams. Static loading and fatigue tests were carried
out on GFRP reinforced beams with spans of 13.12 ft (4 m). They concluded that the fatigue life
of a GFRP strengthened beam is approximately three times that of the unreinforced beam.
Applications for repairing two highway bridges were also discussed.


5
At the University of Arizona, Saadatmanesh and Ehsani [13] conducted four tests on 5 ft
(1.52 m) long concrete beams that had been externally reinforced with GFRP plates which were
bonded to the tension side of the beams. One beam was unreinforced, and designated as the
control beam. One #3 steel tensile reinforcement rod was also cast into the beams. The beams
were simply supported and subjected to two concentrated loads symmetrically placed 3 in (7.62
cm) on either side of the midspan. Deflection at the midspan, as well as strains on the surface of
the GFRP plates, were measured as the beams were loaded to failure. Load versus strain and
load versus deflection plots were obtained on all five tests. They concluded that strengthening
concrete beams with epoxy bonded GFRP plates is a feasible way of increasing the load carrying
capacity of existing beams and bridges, and that selection of a suitable epoxy for bonding the
GFRP panels onto the beams is necessary to improve the ultimate capacity of the beam. They
noted that the epoxy adhesive should have sufficient stiffness and strength to transfer the shear
force between the composite plate and the concrete.
Sharif et al. [14] and Ziraba et al. [15], in the Department of Civil Engineering at King
Fahd University of Petroleum and Minerals in Saudi Arabia, experimentally investigated the
repair of initially-loaded reinforced concrete beams with epoxy-bonded fiberglass reinforced
plastic (FRP) plates. The reinforced concrete beams were initially loaded to 85 percent of their
ultimate flexural capacity and subsequently repaired with the epoxy-bonded FRP plates and then
re-loaded to failure. Different repair and plate anchoring schemes were investigated to try and
eliminate premature failure of the FRP plate/bond due to the high concentration of shear stress,
and to promote ductile behavior of the beam. Load-deflection curves for the FRP repaired beams
are presented, and the different failure modes for each beam are discussed. Their results,
consistent with those obtained by other researchers, indicated that the flexural strength of the


6
FRP repaired beams is greater than that of the control beams, and that the ductile behavior of the
FRP repaired beams is inversely proportional to FRP plate thickness.
University of Nevada researchers Gordaninejad et al. [16] focused on the behavior of
composite bridge girders constructed from CFRP sections and concrete slabs. Their study
examined four-point bending of three beams; one plain graphite/epoxy I-beam, and two beams
constructed from concrete slab and graphite/epoxy sections which were adhesively bonded
together. All three beams were one-eighth scale models of bridge girders. Theoretical and
experimental studies were performed on the plain I-beam sections to develop a basic
understanding of the beams flexural behavior. The tests and analyses were then extended to the
composite girders. They concluded that slip at the interface between the concrete slab and the
graphite/epoxy beam had a minor effect on the failure load, but produced a significant reduction
in stiffness of the composite section.
Chajes et al. [17,18] at the University of Delaware tested a series of reinforced concrete
beams in four-point bending to determine the ability of several different types of externally
bonded composite fabric materials to improve the beam's flexural capacity. The different fabric
materials were chosen to allow a variety of fabric stiffnesses and strengths to be studied. The
fabrics used were made of aramid, E-glass, and graphite fibers, and were bonded to the beams
using Sikadur 32, a two-component, high modulus, high strength, construction epoxy. A series
of pull-off tests were run to investigate the bond strength of the adhesive, and it was determined
that a single layer of aramid, a triple layer of E-glass, and a double layer of graphite fabric can be
expected to develop full tensile capacity in approximately 2.0 in (50.8 mm) for both the E-glass
and graphite fabric, and in approximately 3.0 in (76.2 mm) for the aramid fabric. Test results
indicated that the external fabric reinforced beams yielded an increase of approximately 50
percent in both flexural capacity and stiffness. For the beams reinforced with graphite fiber and


7
E-glass fabrics, failures were attributed to fabric tensile failure in the beam's constant moment
section, whereas the beams reinforced with aramid fabric failed due to crushing of the concrete
in compression. An analytical model based on the stress-strain relationships of the concrete,
steel, and composite fabrics was also developed, and comparisons were made with the
experimental results.
Recently, the Japanese have also investigated the use of carbon fiber reinforced plastic to
retrofit several large scale structures in the field [19]. The Japanese work is different from the
European, Saudi, and U.S. approaches in that it makes use of dry fiber CFRP sheets rather than
the pre-preg plates or strips. Two petrochemical industries, the Tonen Corporation and
Mitsubishi Chemical, have aggressively pursued this technology, and have developed proprietary
processes to impregnate the dry CFRP sheets with various epoxy resins. This technique
effectively eliminates the requirement for mechanical or vacuum fixturing when applying the
CFRP sheets to the structure. Unfortunately, the available literature describing their processes
are proprietary and are unavailable through normal journal articles or library access. The only
other work cited were Japanese symposia proceedings, which were also unavailable through
normal library channels.
The U.S. military has also recognized the potential of advanced composite materials to
strengthen and rehabilitate existing concrete structures. In the past, massive concrete protective
structures were constructed to shelter military personnel and equipment from conventional
weapon attacks. Now, as the US military moves into an era in which "mobility" is becoming
more and more important, new materials must be sought out that can be easily transported and
used for rapid construction while still maintaining adequate levels of protection for personnel
and equipment. Composite materials, typically used by the military solely in aerospace
applications, are now being investigated to potentially solve the burgeoning demand for


8
lightweight, durable construction materials. To investigate CFRP's potential for Air Force
applications, Hughes and Strickland [20] at Tyndall Air Force Base in Florida, in conjunction
with faculty members from the University of Florida, Ohio State, and Auburn, have conducted a
series of tests on small concrete beams reinforced with uniaxial CFRP strips on the tension side
of the beams. Initial tests were performed on 1.0 x 1.0 x 12.0 in (2.54 x 2.54 x 30.48 cm) and 2.0
x 2.0 x 12.0 in (5.08 x 5.08 x 30.48 cm) concrete beams in both three and four-point bending
tests. The number of plies in the CFRP reinforcing strip was varied from one to three, and three
different cementitious mixes were used for the beam material. Additionally, CFRP strips were
also applied along the sides of some of the beams. Test results indicated that the failure load was
increased by 4.4 to 9.0 times the baseline failure load when the CFRP strips were applied to the
beams, when compared to plain concrete beams without such strips.
The most recent work to date on concrete beams externally reinforced with CFRP has
been conducted by Chajes et al. [18], Sierakowski et al. [21], and Ross et al. [22], Sierakowski
et al. [21] used a strength of materials approach to develop an analytical model which evaluates
the shift in the neutral axis that occurs in statically loaded concrete beams that have been
externally reinforced with CFRP strips. The shift in the neutral axis of the beam occurs when the
tensile stress in the concrete at the bottom of the beam exceeds the fracture stress of the concrete,
which is roughly 10 percent of the concrete's compressive strength. It is this cracking that causes
a shift in the neutral axis of the beam, and a subsequent change in the beam's planar moment of
inertia. In this analysis, two particular cases were studied. In the first case, the externally
bonded CFRP strips were placed on the bottom or tension side of the beams only, and in the
second case, the strips were placed on the bottom as well as the sides of the beams. The
analytical model is compared to experimental test data gathered on some small scale beams, 3.0
x 3.0 x 30.0 in (7.62 x 7.62 x 76.2 cm ) in size, and some large scale beams, 0.67 x 0.67 x 9.0 ft


9
(0.20 x 0.20 x 2.74 m) in size, both of which were tested in third-point loading in the structures
laboratory owned by the Wright Laboratory Air Base Technology Branch at Tyndall Air Force
Base, Florida. Reasonable agreement between experiment and analysis was obtained. Ross et al.
[22] performed a complete review and analysis of twenty-two experiments that were conducted
in the summer of 1994 also at the Tyndall Air Force Base, Florida, structures laboratory. The
concrete beams were again 0.67 x 0.67 x 9.0 ft (0.20 x 0.20 x 2.74 m), had three ply
unidirectional CFRP on the bottom or tension side of the beams, and were all tested to failure
using a third point loading test frame. The tension steel varied in size from number 3s to number
8s in the beams. Complete section analysis was conducted on the beam cross sections and load-
displacement curves were calculated and compared with the test beams both with and without
CFRP. In an effort to quantify the strengthening effects of the CFRP, an enhancement ratio,
defined as the peak load on a beam with a given steel percentage and CFRP, divided by the peak
load of the same beam with steel only, was obtained and is shown in Figure 2. Results show
considerable enhancement for the beams with the lower tensile steel ratios, and little to no
enhancement for beams with high tensile steel ratios. Ross et al. [22] also concluded that for the
experiments in which concrete beams reinforced with CFRP were tested, the weakest link
appears to be in the shear strength of the adhesive/CFRP interface. When delamination occurs
between the CFRP and adhesive, the tensile strength of the CFRP is not used to its fullest
potential, and the authors recommend further work on improving the bond between the CFRP
and the concrete. The principal conclusion of the study was that a quantifiable strength
enhancement was obtained by the addition of a very thin (0.0175 in/0.4445 mm) strip of CFRP
added to the bottom side of the beams, especially for those beams with tensile steel ratios of one
percent or less.


10
ER.ENHANCEMENT RATIO
PS, PERCENT TENSION STEEL
0 3 4 5 6 7 8
STEEL BAR SIZE
Figure 2. CFRP Enhancement Ratio [22]
Approach
After a thorough and careful review of this literature, it became evident that the approach
for this investigation should consider:
1. A systematic experimental study of the effect of varying the number of plies of
CFRP on the static load carrying capacity of concrete beams.
2. An experimental study of the dynamic behavior of concrete beams both with and
without external CFRP reinforcing panels.
3. Development of a comprehensive analytical model which attempts to predict not
only the static load-displacement behavior of the beams, but the dynamic
displacement-time behavior as well.


11
4. Finite element method (FEM) calculations of the dynamic response of beams
both with and without external CFRP reinforcing panels.
In order to carry out a systematic experimental study of the effect of varying the number
of plies of the CFRP on both the static and dynamic load carrying capacity of the beams, the
approach was to test laboratory scale concrete beams with none, one, two, and three plies of
unidirectional CFRP to failure both statically, using a MTS load frame, and dynamically, using
a drop weight test machine. All tests were conducted in center-point loading mode, as shown in
Figure 3. Eighteen static and fifty-four dynamic beam bending experiments were attempted,
based on the test matrix shown in Table 1 below.
Figure 3. Center or Three Point Bending Mode Used in Static and Dynamic Tests
It has already been well-established that concrete, although brittle, is a rate sensitive
material both in tension and compression. In order to thoroughly characterize the concrete used
in the beam test samples, the approach was to conduct a series of static and dynamic


12
Table 1. Test Matrix for Static and Dynamic Beam Bending Experiments
Dynamic Tests
Static
Number of
Number of
Tests
Iterations
Beams
Beam
* Drop Height # 1 Drop Height #2 Drop Height #3
BO
X X
X X
3
12
B1
X X
X X
3
12
B2
X X
X X
3
12
B3
X X
X X
3
12
B4
X X
X X
3
12
B5
X X
X X
3
12
Total Number Beams
72
Ksyl
BO
Plain Concrete Beam
B1
Concrete Beam with 1 ply
CFRP
on Bottom of Beam
B2
Concrete Beam with 2 ply
CFRP
on Bottom of Beam
B3
Concrete Beam with 3 ply
CFRP
on Bottom of Beam
B4
Concrete Beam with 3 ply
CFRP
on Bottom and Sides of Beam
B5
Fibrous Concrete Beam with 3 ply CFRP on Bottom of Beam
compression and splitting tension tests. Standard compression tests on concrete cylinders were
also required to ensure that the static unconfined compressive strength f c was known for each
beam. This attribute is very important, since other material parameters which were required for
subsequent analyses and modeling are typically derived from it, such as the secant modulus and
modulus of rupture. This was the approach taken for the research.


CHAPTER 2
EXPERIMENTAL WORK
Fabrication Of Concrete Beam Specimens
A total of 72 lightweight concrete beams were prepared for the experimental portion of
the research. The beam size chosen was 3.0 x 3.0 x 30.0 in ( 7.62 x 7.62 x 76.2 cm), and the
aggregate used was a lightweight aggregate with the trade name Solite which reduced the
overall density of the concrete by about 20 percent. The aggregate used in these beams was
sieved to pass a 0.375 in ( 0.925 cm) sieve, but was retained on a 0.25 in (0.635 cm) sieve. No
tension steel reinforcement was used in any of the beams, due to their small size. Each batch of
concrete beams were cast in lots of eight, with four 4.0 in (10.16 cm) diameter by 8.0 in (20.32
cm) long cylinders cast at the same time as the beams for subsequent material property testing.
The mix proportions in Table 2 were used for each lot of eight beams and four cylinders cast.
Table 2. Lightweight Concrete Mix Proportions
Type 1 Portland Cement
35.25 lbs (16.02 kg)
Concrete Sand
85.63 lbs (38.92 kg)
Lightweight Aggregate
69.38 lbs (31.54 kg)
Water
19.00 lbs (8.64 kg)
13


14
A 5.6 percent moisture content in the aggregate was chosen as the standard. The
aggregate was soaked in water for one day and then allowed to drain off overnight prior to
combining it with the other mix constituents to facilitate the mixing process. If the moisture
content of the mix turned out to be too high, the amounts of aggregate and water were then
adjusted to account for the difference. This made a workable mix that flowed well on the
vibrator, with no measurable slump. The unit weight for all of the beams cast was 118 lbs/ft3
(1892.7 kg/m3). Table 3 lists the beam designators and the dates on which those lots of beams
were cast.
Table 3. Beam Designator and Cast Schedule
BF.AM DESIGNATOR
DATE CAST
LW1-1 through LW1-8
16 November 1994
LW2-9 through LW2-16
29 November 1994
LW3-17 through LW3-24
6 December 1994
LW4-25 through LW4-32
14 December 1994
LW5-33 through LW5-40
17 January 1995
LW6-41 through LW6-48
18 January 1995
LW7-49 through LW7-56
23 February 1995
LW8-57 through LW8-64
28 February 1995
LW9-65 through LW9-72
2 March 1995
LW10-73 through LW10-80
7 March 1995
LW11-81 through LW 11-88
9 March 1995


15
Additionally, another half batch of concrete was cast into one 12.0 x 12.0 x 6.0 in (30.48
x 30.48 x 15.24 cm) box to make samples for Split Hopkinson Pressure Bar (SHPB) testing
along with two additional 4.0 in (10.16 cm) diameter by 8.0 in (20.32 cm) long cylinders, and
one additional 6.0 in (15.24 cm) diameter by 12.0 in (30.48 cm) long cylinder for subsequent
concrete characterization testing. These samples were cast on 29 November 1994. For the last
two sets of beams, labeled LWF10 and LWF11, three pounds of nylon fibers per cubic yard (1.78
kg/m3) were added to the mix to make two sets of fibrous concrete beams.
After casting, each beam was initially cured underwater for 28 days, and then cured in
air until it was subsequently tested. The minimum recommended time for concrete test
specimens between fabrication and conducting any type of dynamic testing is 60 days; for these
beams the minimum time turned out to be 90 days. This is to allow time for the concrete to
achieve sufficient strength, since its strength increases with time.
Application of the CFRP
Once the concrete beams had sufficiently dried after the 28 day water cure, the process
of bonding the CFRP to the beams could begin. The one, two, and three ply CFRP panels were
supplied by the Wright Laboratory Materials Directorate, Wright Patterson Air Force Base,
Ohio. The panels were all of the same material, an AS4C/1919 graphite epoxy. The C
designation denotes a commercial grade of AS4 graphite epoxy, and the 1919 denotes a 250 F
(121.1 C) cure. Table 4 contains the relevant material properties of AS4C/1919 CFRP.
All of the CFRP panels came as pre-cut panels 3.0 x 30.0 in ( 7.62 x 76.2 cm) in size.
The cured ply thicknesses of the one, two, and three ply panels were individually measured with
a micrometer; the other material properties were provided by the Wright Laboratory Materials
Directorate.


16
Table 4. Pre-Preg CFRP Material Properties
0 Tensile Strength
320 ksi (2206.9 MPa)
O
0 Tensile Modulus
20 x 106 psi (137.9 GPa)
Fiber Volume
60 percent
Cured Ply Thickness
one ply
0.0085 in (0.2159 mm)
two ply
0.0140 in (0.3556 mm)
three ply
0.0195 in (0.4953 mm)
The adhesive used to bond the CFRP panels to the concrete beams was a thixotropic,
modified, two part epoxy engineering adhesive manufactured by the Dexter Corporation in
Seabrook, New Flampshire with the trade name Hysol This adhesive was recommended for its
toughness, flexibility, and efficacy in bonding dissimilar materials. The adhesive has excellent
O O
peel and lap shear strengths, and has a pot-life of 40 50 minutes at 77 F (28.9 C) for 0.55 lbs
(250 gms) of mixed adhesive. The two part epoxy was mixed in equal parts (by either weight or
volume since the densities of the two parts were the same) and mixed thoroughly until the off-
white Part A and black Part B were a uniform gray in color. Some heat buildup or exotherm was
noticed during the mixing process, but since less than 1 cup of Part A and Part B were ever
mixed at one time, no excessive exotherm or heat buildup ever developed.
Surface Preparation
As with any bonding or adhesive process, proper surface preparation of the bonded
materials is a necessity. The bonding surfaces should be clean, dry, and properly prepared. The


17
bonded parts should be held in intimate contact until the adhesive is set. It was not necessary to
O
maintain fixturing for the entire adhesive cure schedule (3 days @ 77 F) but only until handling
strength is achieved. The manufacturer considers handling strength to be the same as tensile lap
0
shear strength. Handling strength of 750 psi (5.2 MPa) is achieved in 6 8 hours @11 V
O
(28.9 C). Figure 4 is the manufacturer's graph of tensile lap shear strength versus time at room
O
temperature 77 F.
TIME AT ROOM TEMPERATURE, DAYS
. (K)
Figure 4. Tensile Lap Shear Strength (psi) versus Time (days) for Dexter Hysol Adhesive
To properly prepare the concrete and CFRP surfaces to be bonded, all grease, oil, and
foreign particles had to be removed from each surface. As with most high performance
engineering adhesives, this step is critical, since for good wetting, the surface to be bonded
should have a higher surface tension than the adhesive. Therefore, the adhesive manufacturer's


18
recommendations were closely followed to ensure proper surface preparation of the concrete and
CFRP. For the CFRP, the following surface preparation steps were taken:
1. Use peel ply side of CFRP for bonding surface.
2. Degrease surface with methyl ethyl ketone (MEK).
3. Lightly abrade surface with medium grit emery paper. Take extreme care to
avoid exposing the carbon reinforcing fibers. (This is particularly important for
the one ply CFRP panels since they are only 0.0085 in (0.2159 mm) thick).
4. Repeat step 2; degrease surface with methyl ethyl ketone (MEK).
Since the concrete beams were new, recently cast test specimens, it was extremely
important to make sure that the de molding release agent, which was either 10W-30 motor oil
or WD-40 lubricant, was completely removed from the bottoms and sides of the beams. The
following procedure was used to prepare the surfaces of the concrete beams:
1. Degrease the surface to be bonded with acetone.
2. Repeat step 1, if excessive de-molding oil is noted on the exterior surface.
3. Mechanically scarify (abrade) the surface with a grinding wheel.
4. Repeat step 1, degrease the surface with acetone.
5. Remove dust and concrete particulates with a stiff bristle whisk broom.
The cleaned concrete and CFRP surfaces should be bonded as soon as possible after the surface
preparation procedures have been accomplished, and the bonding procedure itself should be
performed in a room separate from the room or area in which the cleaning and surface
preparation procedures are accomplished.
After the above surface preparation steps were completed, the CFRP strengthening
panels were ready to be bonded. Six beams and six CFRP panels could be reasonably handled
and bonded at one time. One third of a cup of Part A and one third of a cup of Part B Hysol


19
adhesive was sufficient to produce a 0.030 in (0.762 cm) bondline between the CFRP and the
beam for six specimens. After the Hysol was spread on the surface with a wide-bladed putty
knife as shown in Figure 5(a), a square tooth trowel was used to produce a series of 0.0625 in
(1.6 mm) high by 0.0625 in (1.6 mm) wide adhesive beads along the length of the beam, as
shown in Figure 5(b). This also served to remove any excess Hysol from the beam's surface,
and produced a uniform volume of adhesive on the beam as well.
Next, the CFRP panel was placed on top of the Hysol beads, shown in Figure 6, and
pressed into the adhesive while being concurrently smoothed with a paper towel. This served to
seat the panel on the beam as well as eliminating any excess air bubbles between the CFRP panel
and the beam.
After all six of the beams were prepared in this manner and keeping mind of the time so
as not to exceed the fifty minute pot life of the adhesive, the beams were then placed in a large
vacuum bag with 0.25 in (6.35 mm) foam spacers in between each beam. The perimeter of the
bag was sealed with a caulk compound. The vacuum bag itself was made of ordinary 0.006 in
(0.152 mm) thick polyethylene film. A vacuum pump was connected to the polyethylene bag via
a vacuum hose and a short length of 0.5 in (12.7 mm) galvanized pipe, which was then inserted
into the bag and also sealed with a caulk compound. A gage had been placed in line with the
length of galvanized pipe to measure the amount of vacuum pressure. Once the entire bag with
the six concrete beams inside was sealed, the pump was turned on. The pump was a Welch
Duoseal (Welch Vacuum Technology, Incorporated, Skokie, Illinois) two stage vacuum pump
which was driven by a 0.5 horsepower electric motor, with a clutch, pulley, and V-belt attached.
The performance specifications of the vacuum pump are shown in Table 5. Once any minor leaks
in the bag had been found and sealed, the vacuum pump quickly pulled the bag down to 28.5 in


20
(b)
Figure 5. Epoxy Adhesive Being Placed on Surface of Beams


21
Figure 6. CFRP Panel Being Placed on Beams
Table 5. Vacuum Pump Specifications (New Pump)
Free Air Displacement
5.6 ft3/min
Pump Rotational Speed
525 RPM
Guaranteed Partial Pressure
29.5 in Hg (14.5 psi)
Hg (14.0 psi), since the initial volume of air trapped inside the bag was relatively small. The
pressure differential created by the vacuum placed the beams into uniform hydrostatic


22
compression, placing a uniformly distributed force of 1260 lbs (5.62 kN) on each of the six 90.0
in (580.6 cm ) surfaces of each of the six beams. This fixturing force held the six CFRP panels
firmly in place on top of the beams until the Hysol reached its handling strength of 750 psi
(5.17 MPa) tensile lap shear strength in 6 8 hours at 77 F (28.9 C). The vacuum pump
typically ran for seven hours at a time for each batch of six beams to meet this curing schedule.
Afterward, the beams were then removed from the bag and allowed to cure for the remainder of
the three day cure schedule.
This process was used to bond 12 one ply panels, 12 two ply panels, and 24 three ply
panels to the bottom or tension side of these 48 concrete beam samples. The remaining 12
beams had three ply panels bonded to both the sides as well as the bottoms. For these beams,
each of the three sides were bonded separately following the procedures above. Chajes et al.
[18] also used a vacuum fixturing process to bond some composite strengthening panels to some
beams, but their process was unknown to the author at the time of this work. A patent is
currently under review for the vacuum fixturing process used in this investigation.
Characterization of the Concrete
In order to determine the material properties of the concrete used to fabricate the beam
test specimens, a series of quasistatic and dynamic loading experiments were conducted on the
samples of concrete which were cast at the same time as the beam samples were cast.
Static Compression Tests
The first tests conducted were static compression tests on 4.0 in (10.16 cm) diameter by
8.0 in ( 20.32 cm) long cylinders. Four cylinders from each batch of concrete were tested in
compression in a Forney System 2000 (Forney Incorporated, Wampum, Pennsylvania)
loadframe which has a 400,000 lb (1,779 kN) capacity. The four failure loads were averaged,
2 2
and then divided by the area of the test cylinder's cross section, which was 12.57 in (81.07 cm )


23
to yield the so-called unconfined compressive strength of the concrete, f c. Appendix A
summarizes the results of these static compression tests.
The unconfmed compressive strength may be associated with the hypothetical,
statistically derived compressive stress-strain curve in Figure 7 shown below, where the peak
compressive stress is the value of f c found in the tests.
Figure 7. Flypothetical Compressive Stress-Strain Curve
Certain properties of the stress-strain curve are necessarily described. The tangent to the
curve at the origin is called the initial tangent modulus of elasticity, Eci. A line drawn from the
origin to a point on the curve at which fc = 0.45 fc is called the secant modulus of elasticity, Ecs.
For low strength concrete, Eci and Ecs differ widely. For high strength concrete, there is
practically no difference between the two values. For lightweight concrete the initial slope is


24
somewhat less than for normal weight concrete, and the maximum stress f c occurs for larger
strain values. sc is the compressive strain associated with f c. It is commonly accepted that the
post-peak or strain softening portion of the stress-strain curve is terminated at a stress level of
0.85f c which is referred to as the concrete's ultimate strength. The strain associated with 0.85fc
is therefore the ultimate strain, denoted by eu.
Figure 8 is an actual stress-strain curve for lightweight concrete, generated at the Tyndall
Air Force Base structures laboratory in 1994. Unfortunately, the curve stops at fc which is about
6 ksi (41.4 MPa), and doesn't show the strain softening part of the curve due to strain gage
failure. However, an important feature is present in this curve which differentiates lightweight
concrete from its normal weight counterpart; the strain at f c. Typical values for s c, the strain at
fc, for normal weight concrete, are 0.002 in/in (cm/cm). Here we note that for the lightweight
concrete, the strain value ec at f c is larger, about 0.003 in/in (cm/cm). If we assume that the
strain softening part of the stress-strain curve is symmetric, we may extrapolate the value at eu,
the ultimate strain at rupture, which is associated with a stress of 0.85fc in the strain softening
region. This value is found to be about 0.004 in/in (cm/cm), which is also larger than that of the
typical value of 0.003 in/in (cm/cm) for su for normal weight concrete. Not only does the
maximum stress occur at a larger strain, but also the rupture stress has a larger strain value
associated with it as well.
Knowing the values of f c from quasistatic testing allows us to directly calculate the
tangent and secant moduli from standard empirical equations developed by the American
Concrete Institute (ACI). In general practice, the secant modulus of elasticity is used whenever
an elastic modulus value is needed, and is simply referred to as Ec. The ACI equation for the
secant modulus is defined as


25
O 0.0005 0 001 0.0015 0.002 0.0025 0.003 0.0035
Strain
Figure 8. Stress Strain Curve for Lightweight Concrete (Courtesy of L.C. Muszynski, 1994)
Ec = 33 w
3/2
(fc)
1/2
(1)
where
Ec Secant Modulus (psi)
w Unit Weight (lbs/ft3)
fc Unconfined Compressive Strength (psi)


26
In addition to the 4.0 in (10.16 cm) diameter by 8.0 in (20.32 cm) long samples, six
2.0 in (5.08 cm) diameter by approximately 2.0 in (5.08 cm) long specimens were also tested in
direct compression, in order to provide a direct sample size comparison to the Hopkinson bar
samples. A load rate of approximately 2000 Ibs/min (8.9 kN/min) was used for the first three
samples, and then a load rate of approximately 20,000 lbs/min (89.0 kN/min) was used for the
last three samples. It should be noted that the strain rate was calculated by dividing the stress
rate by a secant modulus which was calculated from an average unconfined compressive failure
stress of 6,708 psi (46.3 MPa). In addition, the dynamic increase factor is the ratio of the
dynamic failure stress to the average static failure stress. The dynamic increase factor is
subsequently plotted as a function of the log of the strain rate. For these quasistatic strain rates,
there is clearly no increase in the compressive strength of the concrete, as expected. Table 6
summarizes the results of these direct compression tests on the 2.0 in (5.08 cm) diameter
specimens.
Static Splitting Tensile Tests
It has been well established that concrete is about an order of magnitude weaker in
tension than in compression. See for example, Neville [23] or Winter and Nilson [24], The
tensile or fracture strength of concrete is an important material property, since tensile failure is
the dominant failure mode in concrete. In order to determine the tensile strength of the concrete
test specimens used in this investigation, a series of splitting tensile or Brazilian tests were
conducted in the Forney System 2000 compression load frame. The method of conducting
splitting tensile tests has also been well established. See, for example, Neville [23], This
indirect method of applying tension in the form of splitting was first suggested by a Brazilian
named Fernando Carneiro, hence the name Brazilian test, although the method was also
developed independently in Japan. In this test, the cylindrical sample is placed with its


27
Table 6. Quasistatic Direct Compression Tests on 2.0 in (5.08 cm) Diameter
by 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Length
in (cm)
Load at
Failure
lbs (kN)
Time to
Failure
min (sec)
Load
lbs/min
(kN/min)
Stress
psi
(MPa)
Stress
psi/sec
(kPa/sec)
Strain*
Rate
1/sec
Dynamic
Increase
Factor
1
2.006
(5.095)
21,760
(96.8)
11
(660)
1979
(8.8)
6926
(43.4)
10.5
(72.4)
3.03 x
10'6
1.03
2
2.01
(5.105)
20,050
(89.2)
9.98
(599)
2055
(9.1)
6525
(45.0)
10.9
(75.2)
3.15 x
10-6
0.97
3
2.012
(5.11)
20,960
(93.2)
10
(600)
2092
(9.3)
6672
(46.0)
11.1
(76.6)
3.20 x
lO'6
0.99
4
2.012
(5.11)
22,140
(98.5)
1.15
(69)
19,246
(85.6)
7047
(48.6)
102.1
(704.1)
2.95 x
10'5
1.06
5
2.013
(5.113)
17,760
(79.)
1
(60)
17,810
(79.2)
5653
(39.0)
94.2
(649.7)
2.72 x
10'5
0.85
6
2.012
(5.11)
22,760
(101.2)
1.1
(66)
20,696
(92.1)
7245
(50.0)
109.8
(757.2)
3.17 x
10'5
1.09
* Based on a Secant Modulus Calculated From an Average Compressive Failure
Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)
longitudinal axis between and parallel the platens of the load frame as shown in Figure 9. The
load is increased at a constant load rate until failure by splitting along the vertical diameter takes
place. These so-called splitting tensile tests were conducted in lieu of the more difficult direct
tensile test so as not in incur the difficulties of holding the specimens to achieve axial tension,
and the uncertainties of secondary stresses induced by the holding devices. Again, the samples
tested were from the same batches of concrete from which the beams were cast. Sample sizes
were approximately 2.0 in (5.08 cm) in diameter and 2.0 in ( 5.08 cm) long.
Calculation of the horizontal tensile strength follows the derivation of Boussinesq (1892)
as given by Malvern [25] and Timoshenko [26], An element on the vertical diameter of the
cylinder is subjected to a compression stress of


28
P
P
H
m
(5.08 cm)
D=2 in
(5.08 cm)
.
Figure 9. Diagram of the Splitting Tensile or Brazilian Test
and a horizontal tensile stress of
2P
JtLD
D2
r(D-r)
where
f,=
2P
7tLD
ac Compressive Stress, psi
P Compressive Load, lbs
L Length of Specimen, in
D Diameter of Specimen, in
r Distance of the Element from the Upper Load, in
D-r Distance of the Element from the Lower Load, in
(2)
(3)


29
ft Tensile Stress, psi
Similar to the direct compression tests, a series of six quasistatic splitting tensile tests
were also conducted at two different load rates, one load rate being approximately 2,000 Ibs/min
(8.9 kN/min) and the other about 10,000 lbs/min (44.5 kN/min). Knowing the load and time to
failure allows calculation of the stress and stress rate, respectively, using Equation (3). Once the
stress rate is known, the strain rate may be calculated by dividing by the secant modulus,
determined previously as 3.46 x 106 psi (23.89 GPa). Table 7 summarizes the results of these
splitting tensile tests on the 2.0 in (5.08 cm) diameter by approximately 2.0 (5.08 cm) long
specimens. Similar to the quasistatic direct compression tests, the dynamic increase factor is
calculated from the ratio of the splitting tensile stress for the three tests at a given load rate.
Obviously, there is no increase in strength at these low strain rates; hence the dynamic increase
factors are all ~ 1.0.
Dynamic Compression Tests
Strain rate effects on concrete strength have been examined experimentally by many
investigators [27 38]. However, in order to quantify the strain rate sensitivity of this particular
lightweight concrete in both tension and compression, a series of Split-Hopkinson Pressure Bar
(SHPB) tests were conducted on a 2.0 inch (5.08 cm) diameter SHPB located at Tyndall Air
Force Base, Florida. This particular SHPB has been described in full detail by Ross [38]. In
particular, compression tests on concrete from strain rates of 1/sec to 500/sec have been
conducted using this system. A schematic of the SHPB system is shown in Figure 10.
The striker bar impacts the incident bar which induces an elastic wave whose magnitude
and pulse length are proportional to the striker bar velocity and length, respectively. This elastic
wave is propagated down the length of the incident bar and impinges upon the cylindrical
concrete specimen, where part of the elastic wave is transmitted into the transmitter bar, and part
of the wave is reflected back into the incident bar. Ross [38] shows that for a specimen whose


Table 7. Quasistatic Indirect Splitting Tensile Tests on 2.0 in ( 5.08 cm) Diameter
by 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Diameter
in (cm)
Specimen
Length
in (cm)
Load at
Failure
lbs (kN)
Time to
Failure
min (sec)
Load Rate
lbs/min
(kN/min)
Stress
psi (MPa)
Stress Rate,
psi/sec
(kPa/sec)
Strain* Rate,
1/sec
Dynamic
Increase
Factor
1
1.99
(5.06)
2.012
(5.111)
4888
(21.7)
2.44
(146.4)
2003
(8.9)
776
(5.4)
5.3
(36.6)
1.53 x 10
1.23
2
1.99
(5.06)
2.012
(5.111)
3840
(17.1)
1.92
(115.2)
2000
(8.9)
611
(4.2)
5.3
(36.6)
1.53 x 10'
0.97
3
1.99
(5.06)
2.012
(5.111)
3160
(14.1)
1.58
(94.8)
2000
(8.9)
502
(3.5)
5.3
(36.6)
1.53 x 10'
0.8
4
2.00
(5.08)
2.008
(5.28)
4520
(20.1)
0.67
(40.0)
6780
(30.2)
717
(4.9)
17.9
(123.5)
5.17 x 10
1.01
5
2.00
(5.08)
2.008
(5.28)
4180
(18.6)
0.44
(26.5)
9464
(42.1)
663
(4.6)
25.00
(172.4)
7.22 x 10'
0.94
6
2.00
(5.08)
2.008
(5.28)
4680
(20.8)
0.43
(26.0)
10,800
(48.0)
740
(5.1)
28.5
(196.6)
8.22 x 1 O'6
1.05
* Based on a Secant Modulus Calculated From an Average Compressive
Failure Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)


31
Striker Bar
V
Sample
Incident Bar
Strain i
Gage 1
1
Transmitter Bar
Bridge
Strain i
Gage 1
Bridge
Strain Gage
Strain Gage
Conditioner
Conditioner
Recorder
Digital
Oscilloscope
Figure 10. Schematic of the Compressive Split-Hopkinson Pressure Bar
length is small compared to the length of the elastic wave, the integral of the reflected pulse is
proportional to the strain in the specimen. The incident and reflected pulses are recorded by a
strain gage located on the incident bar, and the transmitted pulse is recorded by a strain gage
located on the transmitter bar. The strains associated with each of these three pulses are
monitored by a full bridge of strain gages, which is then amplified and conditioned using a strain
gage conditioning amplifier, Model 2311 (Measurements Group, Raleigh, North Carolina). The
signals are then recorded on a Nicolet 4094 digital oscilloscope (Nicolet Instrument
Corporation, Madison, Wisconsin). The traces are subsequently stored for further processing
and analysis on a disk recorder. Following these procedures, a series of seventeen direct
compression tests were run on the SHPB, using a 26.0 in (66.04 cm) striker bar, and gun
chamber pressures of 20 200 psi (0.138 1.379 MPa). For these direct compression tests, the
specimens were lightly lubricated using a molybdenum disulfide grease, and pressed between the


32
incident and transmitter bars. They were held in place by friction between the two bars. Ross
[38] developed the following conversion factors, specific to this SHPB, to determine the incident
and transmitted stresses, and the strain rate from the strain gage data.
ctt = volts/gain x (2.058 x 10 ) (4)
6
6 = volts/gain x (1.4 x 10 ) (5)
where
ctt Transmitted Compressive Stress, psi
Strain Rate, 1/sec
Once the strain rate and compressive failure stress has been calculated, the dynamic increase
factor may be calculated by dividing the failure stress by the average of the three quasistatic
failure stresses, which was previously found to be 6708 psi (46.3 MPa). Table 8 shows the data
for the seventeen SHPB direct compression tests using the 26.0 in (66.04 cm) striker bar.
Dynamic Splitting Tensile Tests
To understand the potential dynamic strength enhancement due to strain rate effects of
lightweight concrete in tension, a series of thirteen splitting tensile SHPB experiments were also
conducted. The splitting tensile test was chosen over the more difficult to conduct direct tensile
test so as not to incur the difficulty of clamping or holding the specimen to achieve axial tension,
and the uncertainty of developing secondary stresses in the specimen induced by the holding
fixtures.
With regard to the specimen, it is rotated ninety degrees with respect to the incident and
transmitter bars. Two small pieces of steel stock, 0.25 x 0.25 x 2.0 in (6.35 x 6.35 x 50.8 mm) in
size, each with one side radiused to fit the contour of the concrete specimen, are affixed to the
outside of the specimen along a diameter line, parallel to each other, to help distribute the load as
shown in Figure 11. Figure 12 shows the orientation of the specimen in the SHPB. For the


33
Table 8. Direct Compression Tests on 2.0 in (5.08 cm) Diameter By
2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Gun
Pressure
psi (kPa)
Amplifier
Gain
1A/1B
Transmitted
Stress,
v/psi/MPa
Strain
Rate
v/l/sec
Dynamic
Increase
Factor (DIF)
DCLW 1
20
(137.9)
100/1000
4.22/8799/62.7
0.448/62.7
1.31
DCLW 2
30
(206.9)
100/1000
4.83/9940/68.6
0.770/107.8
1.48
DCLW 3
50
(344.8)
100/100
0.535/11,010/75.9
1.098/153.7
1.64
DCLW 4
100
(689.7)
100/100
0.705/14,509/100.1
1.678/234.9
2.16
DCLW 5
20
(137.9)
100/1000
4.365/8986/62.0
0.474/66.4
1.34
DCLW 6
30
(206.9)
100/1000
4.81/9899/68.3
0.720/100.8
1.48
DCLW 7
50
(344.8)
100/100
0.494/10,167/70.1
1.026/143.6
1.52
DCLW 8
100
(689.7)
100/1000
6.77/13,936/96.1
1.524/213.4
2.08
DCLW 9
100
(689.7)
100/1000
6.705/13,799/95.2
1.540/215.6
2.06
DCLW 10
50
(344.8)
100/1000
5.23/10,763/74.2
0.984/137.8
1.60
DCLW 11
30
(206.9)
100/1000
4.39/9035/62.3
0.720/100.8
1.35
DCLW 12
20
(137.9)
100/1000
4.57/9405/64.9
0.536/75.0
1.40
DCLW 13
150
(1034.5)
100/1000
7.48/15,394/106.2
1.940/271.6
2.30
DCLW 14
150
(1034.5)
100/100
0.67/13,789/95.1
1.870/261.8
2.06
DCLW 15
150
(1034.5)
100/100
0.734/15,106/104.2
1.920/269.0
2.25
DCLW 16
200
(1379.3)
100/100
0.754/15,517/107.0
2.250/315.0
2.31
DCLW 17
200
(1379.3)
100/100
0.66/13,583/93.7
2.340/328.0
2.03


34
P
2.0 in
(5.08 cm)
Figure 11. Schematic of Test Specimen Preparation for SHPB Splitting Tensile Test
TOP VIEW
Figure 12. Side and Top Views of Test Specimen Orientation for SHPB Splitting Tensile Test


35
splitting tensile test, the incident and transmitted signals are both compressive, and the reflected
signal is tensile. For analysis purposes, the peak of the transmitted compressive signal is
converted to a load which is then assumed to be the peak load applied to the specimen, similar to
the quasistatic loading case shown in Figure 9. The static tensile stress ft normal to the load
direction is given by Equation (3). Equation (4) is the conversion from volts to transmitted
compressive stress, and must be modified in order to calculate the splitting tensile stress. It is
easily shown that all that is necessary to modify Equation (4) to account for splitting tensile
stress is to divide by the specimen length, or simply
crT = volts/gain x (1/L) x (2.058 x 10 ) (6)
where
L Specimen Length, in
To calculate the strain rate, pick two points on the ascending or pre-peak portion of the
transmitted compressive signal, and calculate the slope. Then, using Equation (6), we can
calculate the splitting tensile stress rate. Finally, dividing by the quasistatic secant modulus,
which was calculated from an average compressive failure stress of 6,708 psi (46.3 MPa), allows
us to compute the strain rate. One may question whether a statically determined value of the
secant modulus is a valid parameter to use to calculate strain rate from stress rate. However,
Ross [38] states that the dynamic modulus data obtained from SHPB tests are not valid since
elastic deformation occurs in the rise time of the load pulse, and during that time the specimen is
not uniformly loaded along its length. Nonuniformly loaded specimens have nonuniform strain
distributions which may yield false modulus data. Additionally, John and Shah [39] assume the
modulus to be rate independent, which has also been observed by Gopalaratnam and Shah [40],
and Tinic and Bruhwiler [41].


36
Once the strain rate and splitting tensile failure stresses have been calculated, the
dynamic increase factor may again be calculated, by dividing the failure stress by the average of
the three quasistatic splitting tensile failure stresses, found previously to be 630 psi (4.3 MPa).
Table 9 shows the data for the thirteen SHPB indirect splitting tensile tests. The first nine tests
Table 9. SHPB Indirect Splitting Tensile Tests on 2.0 in (5.08 cm) Diameter
By 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Length
in (cm)
Gun
Pressure
psi (kPa)
Amplifier
Gain
1A/1B
Transmitted
Stress, a,
v/psi/MPa
Strain *
Rate,
mV 1
gs sec
Dynamic
Increase Factor
(DIF)
SCLW 1
2.008
(5.100)
8
(55.2)
1000/1000
0.968/992/6.8
20.2/5.98
1.58
SCLW 2
2.012
(5.111)
8
(55.2)
1000/1000
0.950/972/6.7
16.2/4.77
1.54
SCLW 3
2.013
(5.113)
8
(55.2)
1000/1000
1.164/1190/8.2
25.1/7.45
1.89
SCLW 4
2.012
(5.111)
30
(206.9)
100/1000
1.650/1688/11.6
120.2/17.25
2.68
SCLW 5
2.009
(5.103)
30
(206.9)
100/1000
1.434/1469/10.1
93.5/13.44
2.33
SCLW 6
2.015
(5.118)
30
(206.9)
100/1000
1.732/1769/12.2
133.6/19.16
2.81
SCLW 7
2.013
(5.113)
50
(344.8)
100/1000
1.288/1317/9.1
50.0/14.74
2.09
SCLW 8
2.012
(5.111)
50
(344.8)
100/1000
0.808/2849/12.8
66.5/19.60
2.94
SCLW 9
2.013
(5.113)
50
(344.8)
100/1000
1.728/1767/12.2
66.8/19.70
2.80
SCLW 10
2.011
(5.108)
15
(103.5)
1000/1000
1.012/1036/7.1
3.7/1.03
1.64
SCLW 11
2.015
(5.118)
15
(103.5)
1000/1000
0.878/897/6.2
5.1/1.50
1.42
SCLW 12
2.011
(5.108)
20
(137.9)
1000/1000
0.690/706/4.9
5.0/1.48
1.12
SCLW 13
2.013
(5.113)
20
(137.9)
1000/1000
1.050/1073/7.4
6.9/2.04
1.70
* Based on a Secant Modulus Calculated From an Average Compressive Failure
Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)


37
were conducted with a 26.0 in (66.04 cm) striker bar; the last four tests were conducted with a
50.0 in (127 cm) striker bar. The longer striker bar allowed the samples to be loaded at a lower
strain rate while maintaining gun chamber pressures in the 15 20 psi (10.35 137.9 kPa) range.
It was difficult to lower the chamber pressure below 8 psi (55.2 kPa) with the 26.0 in (66.04 cm)
striker bar.
Test Results
Failure of the static direct compression samples was by crushing, and the six samples are
almost identical in appearance. These tests were used as a basis for comparison with the other
data, since the unconfined compressive stress is the major property used in almost all discussion
of concrete data. The SHPB direct compression results show that as the load rate, hence the
strain rate increases, the number of fracture surfaces increases, hence the amount of
pulverization of the sample increases.
Failure of both the static and dynamic splitting tensile cylinders are almost identical,
breaking along a diameter plane. For the SHPB splitting tensile tests, with increasing load rate,
the split cylinder halves have increasing velocity and additional fractures occur during impacts
with the side walls of the debris catcher on the SHPB. However, Ross [38] states that during
splitting tensile tests in which high speed photography was used, the halves appear to be intact
after splitting. In fact, the films show that the fracture begins to occur first near the center of the
test specimen, which is also borne out by numerical simulations, which show that the tensile
stresses are larger in the center of the specimen than near the edges.
Figure 13 is a graph of the results for all of the quasistatic and SHPB direct compression
tests. The dynamic increase factor (DIF) is plotted as a function of the logarithm of the strain
rate for five different strength normal weight concrete mixes, as well as for the lightweight
concrete under investigation. One may conclude from this data that in compression the


38
CONCRETE STRAIN RATE EFFECTS
COMPRESSIVE STRENGTH
X
H
O
z
H
cn
U
P

o
?-
7
6
5
O 4
P
2 3
2
1
0
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
LOG STRAIN RATE, 1 / SECONDS
|a
9 mifi
IF u #
B E Mix, fc = 7900 psi 4 F Mix, fc = 8250 psi A G Mix, fc = 5700 psi a H Mix, fc = 5600 psi
0 J Mix, fc = 4060 psi ^ LW Mix, fc = 5775 psi
Figure 13. Concrete Strain Rate Effects Compressive Strength
lightweight concrete behaves identically, at least within the scatter of the data, to its normal
weight counterpart, whether quasistatically or dynamically. The lightweight concrete shows
moderate strain rate sensitivity, similar to its normal weight counterpart, with increases in
strength up to 2.3 times the static value at strain rates from 100 to 300/sec.
Figure 14 shows the analogous splitting tensile data. In tension, the lightweight concrete
does not appear to fall within the data scatter for the five different strength normal weight
concrete mixes. In fact, the data show that the lightweight concrete is less strain rate sensitive
than its normal weight counterparts, being shifted about a half-decade in strain rate for a constant
dynamic increase factor. Conversely, this means that the lightweight concrete must be loaded
approximately three times faster in order to have the same dynamic strength as normal weight
concrete. In either case, the normal and lightweight concrete have a higher strain rate sensitivity


39
x 7
P
§ 6
3 5
H 0
C/3
Ho 4
H r
| 3
m a
U
§
<
-7
CONCRETE STRAIN RATE EFFECTS
TENSILE STRENGTH


\

i
.n^ ...
* &
tVi *
-4 -3 -2 -1 0
LOG STRAIN RATE, 1 / SECONDS
B E Mix, fc = 7900 psi F Mix, f c = 8250 psi A G Mix, f c = 5700 psi D H Mix, fc = 5600 psi
0 J Mix, f c = 4060 psi ^ LW Mix, f c = 5775 psi
Figure 14. Concrete Strain Rate Effects Tensile Strength
in tension than in compression, which is consistent with the results of other researchers [31, 35,
36, 38, 39, 41] work on normal weight concrete. The author is unaware of any previous strain
rate sensitivity studies on lightweight concrete.
Static Beam Bending Experiments
The next step towards understanding the potential benefits of external application of
CFRP panels to concrete beams is a careful study of the static bending behavior of the beams
when subject to a quasistatic center point loading condition, simply supported, as shown in
Figure 15. Plain concrete beams, as well as beams with one, two, and three ply CFRP panels
bonded to the bottom or tension side of the beams were tested. Additionally, two more beams in
which three ply CFRP was bonded to the sides as well as the bottom, were tested.
Description of the MTS~ Load Frame
The three point bending tests were conducted on the MTS 880 load frame, (MTS
Systems Corporation, Minneapolis, Minnesota). The load frame may be run in either a load


40
P
Figure 15. Schematic of Quasistatic 3 Point Bending Tests
control or a displacement control mode, and is capable of loading material specimens up to
50,000 lbs (222.95 kN) in both uniaxial tension or compression. However, special tension
platens are required for cementing the concrete tensile specimens when conducting tension
experiments. Figure 16 shows a photograph of the MTS load frame with a concrete beam test
specimen.
The supporting platform on the machine was modified to accommodate the supported
span length of 27.0 in (68.58 cm) by placing roller supports in a notched plate which was then
bolted to the support platform by a series of Allen head bolts as shown in Figure 17. Since the
beams were 30.0 in (76.2 cm) long, and the supported span length was 27.0 in (68.68 cm) there
was a 1.5 in (3.81 cm) overhang on each end of the beam. The load cell was fitted with a special
half-cylinder platen to load the entire width of the beam uniformly at its center.


41
Figure 16. MTS Load Frame located at Wright Laboratory, Tyndall Air Force Base, Florida
Figure 17. MTS Support Platform Arrangement


42
Instrumentation Used/Measurements Made
(R)
The beams were loaded using the load control mode of operation on the MTS at a load
rate of 2 Ibs/sec. The crosshead load, time, and deflection were recorded every two seconds by
the data acquisition software resident in the MTS controller. Beam midpoint displacement was
also recorded using a small linear voltage displacement transducer (LVDT), which was mounted
separately on the MTS supporting platform. A small, flat circular tab was glued to the bottom
center of each beam to facilitate attachment of the LVDT to the beam's midpoint. Midspan
deflection was also recorded by the MTS data acquisition system every two seconds. Load to
failure versus deflection curves could then be generated for each beam tested; therefore, the
static fracture energy could be computed by calculating the area under the load to failure versus
deflection curve.
Test Results
A total of 16 beams were tested to failure on the MTS load frame in three point
bending. Since plain concrete is a brittle material, failure occurs abruptly and catastrophically.
Beams with some tension steel reinforcement in them fail somewhat more gracefully, due to
the ductile nature of steel. Reinforcing steel used in concrete structural elements generally follow
an elastic plastic with strain hardening stress strain curve, as shown in Figure 18, with yield
points typically in the 40 60 ksi (276 414 MPa) range. Typical elastic moduli are in the 29
Msi (200 GPa) range, with strain hardening moduli in the 1.2 Msi (8.3 GPa) range. As
mentioned previously, due to the small cross section of these beams, no tension steel was used
in them.
A typical load displacement curve for the plain concrete beam is shown in Figure 19.
Since these beams have neither steel nor CFRP tension reinforcement in them, they remain
elastic until failure, at which time the lower "fibers" of the beam reach a stress level which


43
Figure 18. Typical Stress-Strain Curve For Reinforcing Steel Used In Concrete Structures
LOAD vs DISPLACEMENT
BEAM LW5-38, PLAIN CONCRETE
2
§
o
_i
700
600 I
500 J
400
300 j
200
100 J
ol
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
DISPLACEMENT, INCHES
Figure 19. Load Displacement Curve for Beam LW5-38, Plain Concrete Beam


44
exceeds the modulus of rupture value for lightweight concrete. A crack merely forms at the
bottom of the beam, and runs to the upper surface, simply breaking the beam into two pieces.
Since the modulus of rupture for concrete is on the order of 10 12 percent of the
unconfined compressive strength f c, the loads at failure for these beams are quite low, typically
about 640 lbs (2898 N). The corresponding displacement is also quite small; therefore the
fracture strengths are also quite small for the plain concrete beams. Appendix B contains the
load displacement curves for the three plain concrete beams tested.
Next, beams reinforced with one ply CFRP on the bottom were tested. Figure 20 shows
the load displacement curve from beam LW2-16. Immediately, one notices a significant
difference in the appearance of the load displacement curve in that there are two distinct
regions; an initial all elastic region with a much higher slope or stiffness than the second
region, which continues until failure occurs at approximately 1150 lbs (5116 N). The
displacement at failure is 0.160 in (4.07 mm). The failure load for the beam with one ply CFRP
is 1.8 times higher than for the plain concrete beam. The stiffness in Region 1 is 55,700 lbs/in
(97.5 kN/cm), and the stiffness in region 2 is 2960 lbs/in (5.18 kN/cm). The beam also fails by a
different mechanism. No longer does a flexural crack merely split the beam in two. With the
addition of the CFRP, the flexural crack(s) still occur in the center cross section of the beam, but
the beam now fails by a combination of flexure and shear at a distance of about L/4 on either
side of the beam centerline. As soon as the combination of bending and shear stress combine to
produce a maximum principal stress higher than the tensile fracture stress, the beam fails. This
O
crack starts on the bottom of the beam and runs upward at a 45 angle until it reaches the neutral
axis of the beam, which itself has shifted upward when the initial flexural crack(s) first appeared,
as the concrete reached its modulus of rupture. Figure 21 is a photograph of beam LW2 16


45
LOAD vs DISPLACEMENT
BEAM LW2-I6,1 PLY CFRP BOTTOM ONLY
3000 -j
2500 -
VO
0 0.05 0.1 0.15 0.2 0.25
DISPLACEMENT, INCHES
Figure 20. Load Displacement Curve for Beam LW2-16, Plain Concrete Beam with One Ply
CFRP
Figure 21. Typical Failure of Concrete Beam Reinforced with One Ply CFRP


46
with one ply CFRP, loaded to failure. This shift in the neutral axis has been previously studied
by Sierakowski et al. [21], and Ross et al. [22], and is the basis of the section analyses which are
derived and discussed in detail in Chapter 3. Furthermore, as the outer portion of the now failed
beam rotates upward, the peel strength of the Hysol epoxy adhesive, 30 lbs/linear inch (52.5
N/linear cm) is exceeded, and the CFRP delaminates from the bottom of the beam. This
delamination consistently originated at the point where the crack started at the bottom of the
beam and ran to the end of the beam, as shown in Figure 21 above.
Next, three beams were tested with two ply CFRP on the bottom or tension side of the
beams. The failure loads increased to about 2300 lbs (10.22 kN), with corresponding
displacements of about 0.241 in (6.13 mm). The shape of the load-displacement curve was very
similar to that of the beams reinforced with one ply CFRP, with two distinct regions; one all
elastic region with a high stiffness 60,000 lbs/in (105.26 kN/cm), and then an abrupt change in
slope as the flexural cracking occurs, with a concomitant reduction in stiffness to about 6570
lbs/in (11.51 kN/cm). The failure is quite abrupt, and occurs when the combination of moment
and shear produce stresses which exceed the tensile strength of the concrete, usually
symmetrically distributed about the beams' centerline, at about the quarter span location. Figure
22 shows a photograph of beam LW4-27, a plain concrete beam with two ply CFRP, loaded to
failure. As the failure occurred and the outer portion of the failed beam rotated upward, the
CFRP was peeled off the bottom of the beam, starting at 7.5 in (19.05 cm) from the beam's
center, and continuing outward to the end of the beam, as shown in Figure 23.
Two concrete beams with three ply CFRP were tested next, and the load displacement
curve for beam LW9-67 is shown in Figure 24. The load-displacement curve is similar to those
shown for the one and two ply beams, with two distinct regions; the first all elastic region has a
stiffness of about 59,100 lbs/in (103.50 kN/cm) and the second region has a stiffness of


47
Figure 22. Typical Failure of Concrete Beam Reinforced with Two Ply CFRP
n ii i
1
DELAMINATED
5m ;\
(12.7 cm)
SMALL FLEXURAL
CRACKS
CFRP
7.5 in
(19.05 cm)
Figure 23. Damage Assessment of Beam LW4-27 with Two Ply CFRP


48
Figure 24. Load Displacement Curve for Beam LW9-67, Plain Concrete Beam with Three Ply
CFRP
9797 lbs/in (17.16 kN/cm). The initial elastic stiffness for the two ply beam is slightly greater
than the three ply beam because its unconfined compressive strength f c is larger, hence, its
modulus of rupture value is larger, which translates to a slightly higher elastic stiffness. After
failure occurs, one notices that the CFRP does not delaminate from the initial crack location on
the bottom of the beam and continue along the bottom of the beam to the end, as in the one and
two ply beams. Instead, the CFRP appears to have been pulled out of the concrete for only a
small portion along the bottom, and then is still securely bonded to the remainder of the beam on
out to end, as shown in the Figure 25.
Interestingly enough, the 2448 pounds (10.89 kN) average total load and the associated
displacements at failure are both not significantly different for these three ply beams than the
values for the two ply beams. In fact, the total displacements at failure are less. This may be due
to the fact that the stiffness of the three ply beam at failure is about 3500 lbs/in (6.13 kN/cm),


49
Figure 25. Damage Assessment of Beam LW9-67 with Three Ply CFRP
which is 49 percent greater than the stiffness of the two ply beam, which translates to higher
bending and shear stresses for a given amount of displacement.
Three more beams were tested in the MTS machine with three ply CFRP on the bottom
or tension side of the beams, except that for these beams the concrete mix was modified with
0.75 in (1.905 cm) long nylon fibers, at three pounds (1.36 kg) per cubic yard. Figure 26 shows
the load displacement curve for beam LWF10-76, which shows the two characteristic regions.
The first all elastic region has an initial stiffness of 54,900 lbs/in (96.15 kN/cm), and the second
region has a slope or stiffness of about 10,390 lbs/in (18.20 kN/cm). The failure loads averaged
2349 lbs (10.45 kN), and the failure displacements averaged 0.175 in (4.45 mm). Figure 27
shows a diagram of the post-test damage on beam LWF 10-76, which shows damage quite similar
to the plain concrete beams with three ply CFRP. In fact, it is impossible to discern any
qualitative or quantitative difference, within the scatter of the data, between the beams with
nylon fibers and three ply CFRP, and beams without nylon fibers and three ply CFRP. Figure 28


50
3500
LOAD vs DISPLACEMENT
BEAM LWF10-76, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY
0.1 0.15
DISPLACEMENT, INCHES
0.2
0.25
Figure 26. Load-Displacement Curve for Beam LWF 10-76, Nylon Fiber
Concrete Beam with Three Ply CFRP
Figure 27. Damage Assessment of Beam LWF 10-76 with Three Ply CFRP


51
shows a photograph of the post-test damage inflicted on beam LWF10-76. The nylon fibers did
not appear to influence the failure mechanism either.
Figure 28. Typical Failure of Nylon Fiber Concrete Beam Reinforced with Three Ply CFRP
The last type of beams tested were plain concrete beams externally reinforced with three
ply CFRP on the bottom as well as both sides of the beam, shown in Figure 29. Two beams were
tested in this configuration, and the load displacement curve for beam LW7-53 is shown in
Figure 30. Unfortunately, the two beams tested yielded a large difference in experimental
results; beam LW9-70 failed at 4060 lbs (18.06 kN), and beam LW7-53 failed at 5297 lbs (23.56
kN), with displacements at failure of 0.205 in (5.21 mm) and 0.284 (7.21 mm) respectively. The
displacements were not significantly greater than those of the two ply beams. However, the
loads at failure were a factor of two higher than for those beams with either two or three ply
CFRP on the bottom only.


52
Figure 29. Concrete Beam with Three Ply CFRP on Bottom and Sides
Figure 30. Load Displacement Curve for Beam LW7-53, Plain Concrete Beam
Reinforced with Three Ply CFRP on Bottom and Sides


53
The load displacement curves are of the same two region form, with an initial elastic
stiffness of about 63,200 lbs/in (110.68 kN/cm), an abrupt change of slope at the concrete's
modulus of rupture point, with a resulting reduced stiffness of about 16,709 lbs/in (29.26 kN/cm)
until failure occurs. The initial elastic stiffness is roughly the same as that of the other beams,
with a slight increase which may be attributed to the side CFRP. However, after the modulus of
rupture is reached within the concrete and the slope abruptly changes, the beams remain
substantially stiffer than the other four types of beams reinforced with bottom CFRP. Table 10
below is a summary of the average initial elastic stiffnesses, and the average post-flexural
Table 10. Summary of Beam Stiffnesses from MTS Static Three-Point Bending Experiments
Number of Plies of CFRP
Elastic Stiffness
lbs/in (kN/cm)
Post-MOR Stiffness
lbs/in (kN/cm)
0
45,714
(80.06)
55,700
2,960
1
(97.55)
(5.18)
60,100
6,570
2
(105.26)
(11.51)
59,100
9,797
3
(103.50)
(17.16)
54,900
10,390
3*
(96.15)
(18.20)
63,200
16,709
3**
(110.68)
(29.26)
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard
** Three ply CFRP Both Sides and Bottom
cracking or post-modulus of rupture (MOR) stiffnesses for all six types of beams tested. Note
that the beams with three ply CFRP on all three sides are approximately 60 70 percent stiffer
than the other two types of beams reinforced with three ply CFRP on the bottom only. Figure 31
below shows the post test damage of beams LW9-70 and LW7-53. Note that the side CFRP is
split and buckled along the major shear failure crack in the concrete, which is at 45 degrees from


54
Figure 31. Typical Failure of Concrete Beam Reinforced on Bottom
and Both Sides with Three Ply CFRP
the vertical as shown in Figure 32. It is theorized that the side CFRP prevents the concrete from
rotating or lifting upward upon failure, thereby preventing the peeling or delamination failures
seen in the one and two ply test specimens earlier. The side CFRP also holds all the pieces of the
failed beam together as one unit. In fact, the beams still maintain a limited capacity to carry a
load, although this fact was not quantified during the course of this investigation.
Calculation of Fracture Energies
Failure may be defined in many ways for a structure or structural element. In this case,
it is the abrupt termination in the ability of the beam to take any more load increments, which for
these beam bending experiments run in load control, was two lbs/sec. Once the maximum
principal stress at some location in the beam exceeded the concrete's tensile strength, the
concrete fails, hence the beam fails.


55
Figure 32. Damage Assessment of Beam LW7-53 with Three Ply CFRP Bottom and Sides
Fracture energies were calculated for all of the beams in an attempt to quantify the
capability of the beams to absorb energy. Fracture energy was then defined as the area under the
load displacement curve until failure occurred; the area under the load displacement curve
was found by numerical integration using the Trapezoid Rule.
Assuming that a straight line approximation between each set of load displacement
points is reasonable, the area of each trapezoidal segment is calculated and summed, and the
areas or fracture energies of all sixteen experimental load displacement curves were so
determined. Appendix C is a tabular summary of all of the relevant data from the static MTS
three point bending experiments. Also displayed in the table are load increase, displacement
increase, and fracture energy increase ratios. The load increase ratio is defined as the peak load
divided by the average peak load of the three plain concrete beams, which was 637.3 lbs (2.83
kN). The displacement increase ratio is defined as the maximum displacement divided by the


56
maximum displacement recorded on plain concrete beam LW5-38. The reason all three
displacements were not averaged and used as the baseline as in the case of the load increase ratio
was because the LVDT did not work very well on beams LW5-33 and LW5-36, as can be seen
by their respective load displacement curves in Appendix B, where the LVDT did not begin
recording displacement immediately after loading began but instead abruptly jumped up to some
value before it worked properly. Finally, the fracture energy increase ratio is defined as the
fracture energy divided by the fracture energy calculated from plain concrete beam LW5-38.
Table 11 shows the average load, displacement, and fracture energy increases computed
from the results of the table presented in Appendix C. Several conclusions may be drawn after
Table 11. Average Load, Displacement, and Fracture Energy Increase
Ratios Compared to Plain Concrete
No. of
Plies of
CFRP
Average
Load
Increase
Ratio
Average
Displacement
Increase
Ratio
Average
Fracture
Energy
Increase
Ratio
1
1.80
11.45
29.83
2
3.60
17.24
77.76
3
3.85
12.86
59.94
3*
3.68
12.53
58.38
3**
7.34
17.47
145.84
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard
** Three Ply CFRP Both Sides and Bottom
careful review of Table 11. Beams with three ply CFRP on the bottom and sides are clearly able
to take the most load before failure, with an average load increase ratio of 7.34. Somewhat
surprising is that the beams with nylon fibers in them and reinforced with three ply CFRP appear
to offer little advantage in either load or displacement to failure over the plain lightweight


57
concrete beams with three ply CFRP on the bottom only. One might theorize that the fibers
should provide some additional energy absorption capability in the form of crack attenuation
capability, but the average fracture energy increase ratio is 58.38, which is slightly less than the
plain lightweight concrete mix with three ply CFRP with a 59.94 fracture energy increase ratio.
In fact, the three ply fibrous concrete, three ply regular lightweight concrete, and two ply regular
lightweight concrete all have similar total load to failure capabilities, with the one ply CFRP
beams having a factor of two less load carrying capacity. Most surprising are the relative
ductilities of the beams; the two ply beams are the most ductile of the bottom or tension side
only reinforced beams. The three sided beams are roughly equivalent in ductility. Therefore, the
fracture energy increase ratio is the highest for the two ply beams when compared to its tension
reinforced only counterparts, which is solely attributed to this increased ductility. Since the two
ply and three sided three ply beams have roughly the same ductility, the factor of two increase in
fracture energy of the three sided three ply beams is directly proportional to its load increase
ratio.
In summary, it has been experimentally demonstrated that beams with three ply CFRP
on the bottom and sides have the highest load, displacement, and fracture energy increase ratios.
Beams with two ply CFRP on the bottom only have the next highest fracture energy increase
ratio, due to their high ductility and relatively high load increase ratio. Conversely, the
experimental evidence indicates little to no benefit is realized using nylon fibers in the concrete
mix as a potential technique to increase the static load, displacement, or fracture energy capacity,
when compared to similarly reinforced beams without the nylon fibers added to the mix. There
are still tremendous gains over the plain concrete beams in load and displacement capacity even
with the bottom only one, two, and three ply CFRP reinforced beams. Increases of two to four in
load and 11 to 17 in displacement are quite easily achieved with the addition of the CFRP.


58
Consequently, increases in fracture energy from 30 to 78 are also achieved, when compared to
the baseline plain concrete beams.
Dynamic Beam Bending Experiments
In order to determine the dynamic response and behavior of beams externally reinforced
with CFRP panels, a series of 54 drop weight tests were conducted in which the six beam types
were subjected to impulsive center point loading, shown in Figure 33. Mindess and Banthia
Figure 33. Simply Supported Beam With Dynamic Center Point Load
[27 33] have conducted drop weight impact tests on plain and conventional steel reinforced
concrete beam specimens with dimensions (length x width x height) 55.12 x 3.94 x 4.92 inches
(140.0 x 10.0 x 12.5 cm) and a simply supported span length of 37.80 in (96.0 cm). However,
their applied loads were not impulsive. Here, we define the term impulsive load as a load pulse
with a temporal duration less than 25 percent of the fundamental period of the structure


59
undergoing loading; in this case the structural element is a simply supported beam. None of the
previously cited CFRP reinforced concrete beam work [References 1 22] involved dynamic
loading, and for those conventionally reinforced beams dynamically tested by Mindess and
Banthia, the loading was not impulsive. Unfortunately, short duration impulsive loads greatly
complicate the process of analyzing the test results, and will be discussed in detail later.
Description Of The Wvle Laboratories Drop Weight Impact Machine
The test apparatus used in this investigation was a Wyle Laboratories SKM01 drop
weight impact machine (Wyle Laboratories, El Segundo, California). Figure 34 is an overall
view of the test apparatus.
The overall height of the apparatus is about 15 feet (4.6 m) above the floor. To provide
as rigid a platform as possible, the bottom frame of the impact machine is bolted to an 8 in
(20.32 cm) thick reinforced concrete floor slab, with four 0.75 in (1.905 cm) diameter by 10
threads per inch bolts with corresponding nuts and lock washers. The impact machine uses a tup
carriage to deliver the weight and instrumented tup to the test sample. The tup carriage relies on
two bearings to guide it along two Rockwell C60 hardened steel shafts to the impact location on
the test sample.
The tup carriage is machined out of aircraft quality 7075-T6 aluminum, and various
weights (steel plates) may be added in equal increments to each side of the tup carriage up to 100
lbs (45.5 kg) total. The additional weight(s) also structurally reinforces the tup carriage. The
weights are attached to the tup carriage via two load pins and a bottom weight plate. Four 0.625
in (1.59 cm) diameter by eighteen threads per inch socket head cap screws attach the additional
weights to the bottom plate, and two 1.0 in (2.54 cm) diameter by eight threads per inch nuts
fasten the weights to the load pins. The tup is attached to the tup carriage by four 0.125 in
(0.3175 cm) diameter by eighteen threads per inch socket head cap screws. The tup is machined


60
Figure 34. Overall View Of The Instrumented Drop Weight Impact Machine
from a Rockwell C40 hardened piece of 4340 steel, and the 4.0 in (10.16 cm) wide impacting
edge conforms to the Charpy impact requirements of ASTM D256-90b, paragraph 11.3
(formerly D256-56 10(d)). The tup itself weighs 1.5 lbs (0.68 kg). The total mass of the tup
assembly and weights used in this investigation was carefully weighed and found to be 96.14 lbs
(43.7 kg). The Wyle drop weight impact machine has the capability to provide a 10 ft (3.05 m)
free fall height; however, the highest drop height in this study was 2 ft (0.61 m).


61
Instrumentation Used
Some of the instrumentation used in the dynamic testing was attached to the drop weight
machine itself, some was attached to the beam test specimens, and yet another was external to
both the test machine and specimens. In order to measure the average acceleration and velocity
of the impact hammer as it drops and impacts the beam specimens, a photocell was mounted on
one of the two steel guide rails of the machine. The photocell's position could be adjusted up or
down so that the velocity of the tup could be measured just prior to impact. The photocell had a
slot in which a strip of aluminum that was attached to the impact hammer could pass through,
and break the continuity of the light source in the photocell. This strip of aluminum was
measured to be 1.2355 in (3.1382 cm) in width. An electronic counter was connected to the
photocell, and it recorded the amount of time that the light source was broken or off during the
passage of the aluminum strip through the slot in the photocell. The width of the aluminum strip
divided by the time recorded by the counter gave the impact velocity of the tup.
From conservation of energy considerations, a weight will fall under the influence of
gravity g and impact with a velocity of
v = V2gh (7)
when dropped from a height h above some datum, as shown in Figure 35. Unfortunately, the
acceleration of the impact hammer and tup assembly is not the acceleration of gravity g, due to
unavoidable friction between the hammer bearings and the steel guide rails. However, by using
the photocell to measure the impact velocity on a series of drop tests, and from Equation (7), the
acceleration of the hammer and tup assembly was consistently found to be about 0.9g. As a
result, careful attention was paid to the guide rails to minimize this friction effect; frequent


62
m
ii
o
t
T
.1
1
h
m
i
T
i
m
\
\
\
\
\
///////
Figure 35. Sketch Of Mass m Dropped At Rest From Height h
cleaning of the steel guide rails with acetone between tests seemed to reduce the variability in the
machine's acceleration constant.
Calibration Of The Tup
The total load or resistance that the tup develops as it impacts the specimen was
measured by using the average of two sets of four electrical resistance strain gages, each set of
four strain gages being mounted on either side of the tup, as shown in Figure 36. The strain
gages were from Micro Measurements (Micro Measurements Division, Measurements
Group Incorporated, Raleigh, North Carolina). Two of the gages in each circuit measure
compressive strain directly, and the other two measure the transverse strain. Each circuit forms a
full four-gage bridge which is inherently temperature compensated, and gives increased
sensitivity from the Poisson effect in the transverse gages. An additional benefit to providing a


63
Figure 36. The Tup And Its Circuit
full bridge of strain gages on either side of the tup is to eliminate any bending effects in the tup
which may occur during a test; when the two output voltage signals are averaged, the effects of
bending are neutralized. The gages are connected to an Ectron Model 563H transducer
conditioning amplifier (Ectron Corporation, San Diego, California). Excitation voltage, output
gain, and bridge balancing are all manually set on the transducer conditioning amplifier.
Excitation voltage was nominally 5 Vdc, and the output gain was set at 100. Connections from
the transducer conditioning amplifier to the digital oscilloscope are direct and straightforward as
shown in Figure 37. The oscilloscope used was a Nicolet Model 4094B (Nicolet Instrument
Corporation, Madison, Wisconsin) four channel digital device with dual disk recording
capability. Normal operation used one disk for recording up to four channels of data in
conjunction with the second math pack disk being used for titles, delays, etc.


64
Figure 37. Schematic of the Instrumentation Used In The Dynamic Beam Bending Experiments


65
Calibration of the tup was done statically, by loading it in compression in the Forney*
load frame. The method is similar with one used by Bentur et al. [27] to calibrate the tup in a
drop weight machine at the University of British Columbia. In the present work, the tup was
disconnected from the drop weight machine and bolted to a steel plate, such that the blade of the
tup was perpendicular to the steel plate. The steel plate was then placed on the bottom platen of
the Forney load frame, with the tup facing upward. The crosshead was then lowered until it just
touched the edge of the tup. The two full strain gage bridges were balanced on the transducer
conditioning amplifier, and their output was connected to the Forney's data acquisition system.
The tup was then loaded to 50 kips (222.4 kN) at a load rate of 75 lbs/sec. A load of 50 kips
(222.4 kN) is about 45 percent of the compressive yield load for the 4340 steel from which the
tup is machined. It was not anticipated that the tup would be loaded much beyond 10 kips (44.5
kN) during actual dynamic testing, so 50 kips (222.4 kN) was deemed sufficient for calibration
purposes.
Results for the first tup calibration test are shown in Figure 38. Plotted is the output
voltage (amplified 100 times) versus the tup load. Each strain gage bridge is plotted separately
STATIC TUP CALIBRATION #1
FORNEY LOAD FRAME
Figure 38. Strain Gage Output Voltage Versus Load


66
versus the tup load, so there are two lines on the plot. Since the tup is responding elastically, the
strains and therefore the voltage are linear, as one would expect. After the first calibration test
was completed, a second test was conducted to ensure repeatability. The same linear response
was almost identically duplicated.
To establish an overall calibration factor for the tup, the slopes of all four voltage versus
load curves (lines) from both tup calibration tests were calculated and averaged together. This
value was found to be 30,277.3 lbs/volt (134.7 kN/volt). Assuming the frequency response of
the tup is much greater than the frequency response of the specimen, one may then use this
calibration, which is quite specific to tup geometry, strain gage excitation voltage, and transducer
conditioning amplifier gain, to convert a voltage versus time signal to a load versus time signal
in the dynamic tests.
After the tup was calibrated, it was installed back onto the impact hammer to begin
dynamic testing. Both bridge outputs from the Ectron signal conditioning amplifier were re
connected to the oscilloscopes, as shown in Figure 37. Since the tup load signal was used as the
internal trigger for all of the instrumentation, each bridge output was connected to a separate
oscilloscope. An internal trigger uses a rise in the load signal above a preset threshold value to
trigger data collection. A general rule of thumb when using an internal or load trigger for the
rest of the instrumentation is that the trigger level be set at roughly ten percent of the expected
maximum load to ensure that the trigger level is well above any background noise. A typical
value used for the trigger voltage was 20 mV, which equates to 605.5 lbs (2.69 kN) of load as
sensed by the tup. Additionally, an adequate number of data points collected prior to the trigger
signal must also be saved, in order to capture the initial rise in the tup load. Typically, 500 psec
of pre trigger data points was sufficient to record the initial portion of the tups load time
signal.


67
Instrumentation attached to the beam test specimens consisted of a series of electrical
resistance strain gages, which were applied to one side on the center cross-section of each beam,
as shown in Figure 39. These strain gages were also from Micro-Measurements.
I
Figure 39. Sketch of Strain Gage Locations on the Beam
The strain gage located on the top of the beam was chosen as gage number one. It was
located at a distance of 2.75 in (6.99 cm) from the bottom of each beam. Unfortunately, the gage
could not be located on the top of the beam, due to the fact that it would be in the same location
that the tup would be striking and loading the beam. Gage number one was usually 1.9 in (4.8
cm) in length, in order to measure the representative strain over at least five aggregate diameters
in the concrete. However, when the sides of the beam were reinforced with CFRP, gage number
one was reduced in length to 0.25 in (0.64 cm), since strain in the CFRP is more uniform.


68
Strain gage number two was located on the geometric center of the beam and the initial
neutral axis location, as shown in Figure 39. Similar to the case with strain gage number one,
when the gage was mounted on concrete, it was 1.9 in (4.8 cm) in length, but when the beams
had side CFRP, the smaller 0.25 in (0.64 cm) gage length was used.
The bottom strain gage was chosen as gage number three. Usually, the shorter 0.25 in
(0.64 cm) gage length was used, except for the case of the unreinforced concrete beams, when
the 1.9 in (4.8 cm) gage length was used. Not all of the beams had three gages at the center cross
section; this three gage configuration was only used on every third beam tested. The remainder
of the beams had gages on the top or position one, and the bottom of the beam or position three.
The gages were mounted on the concrete beams in accordance with the manufacturer's
recommendations for surface cleaning and preparation, which were very similar to the
procedures followed for application of the Hysol adhesive and CFRP. In addition to surface
cleaning of the concrete, a bed of epoxy was initially applied to the area in which the 1.9 in (4.8
cm) concrete gage was to be affixed, in order to fill in any voids in the surface. After the epoxy
cured, it was block sanded back to the original surface height. The gages are usually fixtured in
place while the M-Bond AE-10 (Micro-Measurements Division, Measurements Group
Incorporated, Raleigh, North Carolina) two part epoxy cures, with a clamping device which
applies 5-20 psi (34.5 137.9 kPa) of pressure. However, since the vacuum bag technique
worked so well for fixturing the CFRP while the Hysol epoxy cured and produced 14 psi (96.55
kPa) of uniform hydrostatic pressure, this technique was also used to fixture all of the strain
gages while the M-Bond AE-10 epoxy cured as well. Typically, it took six hours for the epoxy
to cure at 75F (23.9C); all 54 beams were each left in the vacuum bag for this period of time.
Three completion bridges consisting of three 120 ohm resistors were made, and bridge
output was connected directly to the signal conditioning amplifier, where the gages could be


69
manually balanced. Excitation voltage for the bridges was nominally 5 Vdc, and the
amplification or gain was set at 10. In order to convert the strain gage output voltage versus time
to strain versus time requires elementary analysis of the Wheatstone Bridge circuit shown in
Figure 40, where one of the "resistors" in the circuit is the strain gage. When the gage is applied
Figure 40. Strain Gage Bridge Circuit
to the beam and subjected to a strain, say tensile, the magnitude of the resistance in the gage
would be increased, causing the bridge to become unbalanced. The magnitude of this imbalance
is measured as output of the bridge, and is proportional to the strain at that location in the beam.
The output of the bridge is measured as a change in voltage which is given by
QVe ARg AR3 AR2 AR4
(l + Q)2 Rg + R3 R2 R4
AV =
(8)


70
where
AV Change in Output Voltage, volts
Ve Excitation Voltage, volts
Rg Nominal Gage Resistance, ohms
ARg Change in Gage Resistance, ohms
Rt, R2, R3 Bridge Completion Resistor Resistances, ohms
AR), AR2, AR3 Bridge Completion Resistor Change in Resistance, ohms
The strain gage and the bridge completion resistors all have the same resistance so that
and Equation (8) becomes
a- R
R
gage
AV = ^
4 R
(9)
Each strain gage has a gage factor commonly denoted by GF. The gage factor is defined as the
ratio of the unit change in resistance to the unit change in length, which is given by
GF =
AR/R
AL/L
(10)
where
A R Total Change in Gage Resistance, ohms
AL Total Change in Gage Length, in
R Gage Resistance, ohms
L Gage Length, in
The gage factor is also an index of the strain sensitivity of a gage and is a constant for the small
range of resistance changes and strains normally encountered. Equation (10) may be rewritten as


71
= e(GF)
R
which when substituted into Equation (9) yields
(11)
e
f)
v vgf;
AV
(12)
Using the appropriate gage factors, excitation voltage, and amplifier gain in Equation (12) yields
the following calibration factors for the strain gages
s = 0.0381AV (13)
for the 0.25 in (0.635 cm) strain gages which were used on the CFRP and
E = 0.03 77AV (14)
for the 1.9 in (4.826 cm) strain gages, which were used directly on the concrete. These factors
may then be used to convert voltage versus time to strain versus time.
Piezoresistive accelerometers were also mounted on the beams in an attempt to measure
the distribution of acceleration along the length of the beam. Endevco model 7270A (Endevco
Corporation, San Juan Capistrano, California) piezoresistive accelerometers with a range of
20,000 g's were mounted side by side 4.5 in (11.43 cm) from beam midpoint. The
accelerometers have a frequency response of 50 kHz, and are very low mass, being etched from a
single piece of silicon, which includes the inertial mass and an active full Wheatstone Bridge
circuit, complete with an on chip zero balance network. The accelerometers were then
connected to the Ectron signal conditioning amplifier, and subsequently to the Nicolet
recording oscilloscopes, as shown in Figure 37. Nominal excitation voltage was 5 Vdc, and the
gain was set at 5. Each accelerometer had its own calibration factor, given by the manufacturer,
which converted voltage versus time to gs versus time.
Displacement versus time at beam midpoint was measured directly using a noncontact,
linear proximity measuring system. A Kaman Instruments Model KD2300 multipurpose,


72
variable impedance transducer (Kaman Instrumentation Corporation, Colorado Springs,
Colorado) with a 0.5 in (1.27 cm) measuring range was used. Since the beam static failure
displacements were on the order of 0.3 in (0.762 cm), this measuring range was deemed adequate
for dynamic testing. The sensor head was mounted underneath the beams, flush with the support
platform of the drop weight machine. A piece of non-magnetic aluminum tape was affixed to the
bottom midpoint of each beam, which was used as a target for the sensor head. The system had
its own signal conditioning electronics package, where zero, gain and linearity adjustments could
be made, so its output was hooked directly to the recording oscilloscope. The output voltage of
the system is proportional to the distance between the face of the sensor and the metallic target
located on the beam, so merely inverting the signal gave displacement versus time directly. The
frequency response of the system was 50 kHz.
The last piece of external instrumentation used in the dynamic beam bending
experiments was a 0.63 in (16 mm) high speed framing camera. The type of camera used was a
Photec Model 0061-0132A high speed rotating prism camera system (Photographic Analysis
Incorporated, Wayne, New Jersey), and was operated at a framing rate of 10,000 frames per
second. The camera was mounted to the side of the test specimens such that the lens of the
camera viewed the side of the beams, and was at the same height as the bottom of the beam. At
10,000 frames per second, additional lighting was required to illuminate the beam test specimens
during the impact event. The aperture setting on the camera was such that approximately 8 in
(20.32 cm) on either side of the beams midpoint was visible during the impact event. Due to the
high costs associated with developing high speed films, the framing camera was only used on
every third drop weight impact test. This coincided with those beams which had the full
complement of three strain gages mounted on the center cross section; these tests were denoted
the highly instrumented tests. Eighteen of these so-called highly instrumented impact tests were


73
conducted out of a total of fifty-four dynamic beam bending experiments. The high speed film
records also helped to determine the timing of the beam failure, the crack patterns, and verified
the midpoint deflection versus time data from the displacement gage, since each frame yielded
information in 100 psec increments. The eighteen high speed film records were also recorded on
0.5 in (1.27 cm) videotape for further study. However, detailed analysis of the high speed films
required the use of a Photo-Optical Data Analyzer, Model 224A (L-W International, Woodland
Hills, California) to view the individual events recorded on the films. The data analyzer allowed
stop motion, frame by frame, and 24 frame per second operation.
Method of Test
Prior to any actual impact experiments in which data was recorded, several dozen
preliminary drop tests were conducted in order to verify that all of the instrumentation was
working properly. Since the beams dynamic failure or fracture energies were initially unknown,
and the impact hammer weight was held constant at several iterations on the correct starting drop
height were required for each one of the six beam types in order to find incipient failure. Once
incipient failure was found the drop height could then be increased which not only increased the
incident energy, but also increased the load rate and hence the strain rate in the beam test
specimens. Additionally, once the tup load pulse amplitudes and durations were known, trigger
levels and delay times could be set on the recording oscilloscopes; typically 20 mV for the
trigger level and 500 psec for the delay or amount of pre trigger information saved.
It was originally desired that all six beam types be tested at the same three drop heights.
However, due to the large difference in beam stiffnesses and fracture energies, this became
impractical if not essentially impossible to do. In addition, the inertial forces turned out to be the
dominant forces during the loading process, since the beams were so massive and brittle. The
drop height for incipient failure on the stiffest, most energy resistant beam, which were those


74
beams reinforced on all three sides with three ply CFRP, would have produced an overwhelming
inertial force on the least stiff beams, which were the plain concrete beams, and preclude
recovery of the dynamic bending load for these types of beams. However, as much overlap in
drop height between the six different stiffness categories of beams as practical was kept, in order
to allow direct comparison between consecutive beam stiffness classes.
In order to conduct a drop weight test, several preliminary procedures needed to be
accomplished. First, the wire leads for the strain gages had to be attached, the accelerometers
had to be glued to the beams using a special adhesive and adhesive accelerator, and the
aluminum tape was affixed to the bottom of the beam for use as a target for the displacement
gage. The beam was then placed on its supports in the drop weight machine, the leads for the
strain gages were attached to the completion bridges, and the accelerometer leads were
connected to the Ectron signal conditioning amplifier. The excitation voltages for all gages and
accelerometers were checked prior to each test, and the strain gage Wheatstone bridges on the
beam and on the tup were balanced. The tup was then raised to the selected drop height and
measured with a scale, making any fine adjustments in height to within 0.031 in (0.8 mm). The
impact hammer guide rails were wiped with acetone, to minimize frictional effects. The
displacement gage was zeroed at 0.5 volts, and the photocell timer was reset to zero. If the high
speed camera was being used, it was loaded with 450 ft (137.2 m) of film, and the floodlights
were turned on and adjusted to properly illuminate the test specimen. One final check of all the
channels on the Nicolet recording oscilloscope was made, and an intentional trigger was
introduced into the tup strain gage bridges to make sure all the channels would record data.
When all the instrumentation was ready, a countdown would be given allowing enough time for
the high speed framing camera to spin up, when finally the impact hammer would be released,
causing the tup to strike the beam at its midpoint, and subsequently induce failure. The eight


75
channels of data are then transferred from the oscilloscopes to the disk recorders where the data
were written on 5.25 in (13.34 cm) floppy disks. Twenty records, four channels on each record,
were written on each floppy disk. In total, ten floppy disks worth of preliminary and final data
were collected during the course of this investigation. The data was then manipulated using a
scientific and engineering data processing program called VU-POINT Version 2.0, written by
(R)
Maxwell Laboratories, Incorporated, La Jolla, California. The VU-POINT software analyzes,
modifies and plots time series data recorded by waveform digitizes such as the Nicolet 4094
digital oscilloscope.
Fifty-four drop weight tests were conducted on the six different beam types in this
manner, and data were recorded on 51 of the 54 tests conducted. Data was lost on one plain
concrete beam, one two ply CFRP beam, and one beam with three ply CFRP on all three sides,
due to either to spurious or late triggers in the instrumentation system.
Interpretation of Test Results
To illustrate the procedure which was developed to analyze and interpret the data for all
of the drop weight tests, the data from one specific beam, LW3-20, is analyzed in detail. This
procedure was followed on all subsequent tests. This particular beam was reinforced with one
ply CFRP, the hammer weight was 96.14 lbs (43.7 kg), and the drop height was 8.0 in (20.32
cm). Figure 41(a) shows the beam in the drop weight machine ready to be tested, and Figure
41(b) shows the post-test results. Figure 42(a) shows a side view of beams LW3-20, LW2-12,
and LW2-13, all tested at the same condition. The post test damage from the dynamic tests
shown in Figure 42(a) is remarkably similar to the same beam type tested under quasistatic
loading conditions shown in Figure 42(b). This implies that the failure mechanism(s) is the same
for both static and dynamic loading conditions. Results and discussion of all the dynamic tests
are presented later in this chapter.


76
(b) Post Test
Figure 41. Pre and Post Test Results for Drop Weight Test on Beam LW3-20


77
(b)
Figure 42. Post Test Results for (a) Drop Weight and (b) Quasistatic
Bending Tests on One Ply CFRP Beams


78
As the tup strikes the beam test specimen, it records the stressing or bending load in the
beam, the beams inertia, as well as any damping forces which may also be present. However, it
will be shown later that due to the nature of the load pulses, the damping forces can be assumed
negligible in the analysis of the tup load constituents. Figures 43 and 44 show a typical set of
tup loads which were recorded on beam LW3-20. This load pulse is quite typical in shape, but
not in amplitude nor duration, to the other tup loads recorded on other beam types. Generally
speaking, larger drop heights produced larger amplitude and shorter duration tup loads. Figure
43(a) shows the tup load as recorded by strain gage bridge A, and 43(b) shows the tup load
recorded by strain gage bridge B. Figure 44(a) shows the average of the two signals. It should
be pointed out here that the actual load pulse is the first pulse, with an amplitude of 6160 lbs
(27.40 kN). The later two smaller amplitude pulses are merely rebounds of the tup, recorded
after the beam has already failed, shown here for completeness. The reason the load pulse
actually starts at about -150 psec and not zero is because the tup signal has not been time shifted
to account for the 50 mV trigger voltage (t = 0) and its corresponding 500 psec of saved pre -
trigger information. In fact, none of the signals in this study have been time shifted in an attempt
to keep the temporal nature of the different measurements as unbiased as possible. Quite simply,
it is the relative times in which the different events occur, as measured by the instrumentation,
which is of greatest interest and relevance. Looking closely at the end of the load pulse in
Figures 43(a) and (b), one notices that the tup has undergone a small amount of bending, as
evidenced by the small positive load measured by bridge A at the end of the load pulse, Figure
43(a), and the negative load sensed by bridge B in Figure 43(b). But when the two signals are
averaged together in Figure 44(a), the bending effect is nullified.
Figure 44(b) shows the average load pulse which has been digitally filtered using the 5
kHz low pass filter shown in Figure 45. The low pass filter resident in the VU-POINT software


79
(a)
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
Figure 43. Tup A and Tup B Load Versus Time Curves for Beam LW3-20


80
(a)
6000.0
5000.0
1 1 1 1 1 1 r~
TUP A/B AUG LOAD vs TIME
BEAM LW3-20
96.14 LBS 9 8 IN
1 PLY CFRP BOTTOM ONLY
20 JUNE 1995
5KHz LOU PASS FILTER
o 4000.0

o
j
u co 3000.0
i £
3
£ 2000.0
Ou
H 1000.0
0.0
0.0s
J I I I I I I I I L
1.0ms 2.0ms
TIME
3.0ms
(b)
Figure 44. Tup A and Tup B Average Load Versus Time Curves for Beam LW3-20


81
allows specification of the cutoff frequency (50 percent transmission), in this case 5 kHz, the
sharpness or width of the cutoff, in this case 1.25 kHz, and the gain beyond the cutoff, in this
case 0.01. The number of filter terms in Figure 45 is the number of data points that are
smoothed.
Sunnary of LOW-PASS Filter Response
Trans ition-Freq(s)=5.0khz S Trans ition-Width=l.25khz
Max. Response outside of pass-band=0.01
1787 Fi Iter Terns
fiPPROX. FILTER TRANSMISSION vs. FREQ.
0 hz 5.0khz
Figure 45. 5 kHz Low Pass Filter
A 5 kHz cutoff frequency was consistently used throughout this study on all signals
which required low pass filtering. This is reasonable, since the fundamental frequencies for
these beam types varies from 170 to 300 Hz, an order of magnitude below the chosen cutoff


82
frequency. Therefore, there is little danger of filtering out any high frequency components
which are important to the dynamic mechanical behavior of this system.
The raw displacement versus time data from the noncontact, linear proximity measuring
system is shown in Figure 46(a). Note that the load pulse actually starts at -205 psec and ends at
540 psec, but that the beam does not begin to displace until about 100 psec. After beam failure
occurs there is a decrease in the slope of the displacement versus time curve, at which point the
slope becomes constant as the beam merely falls down and comes to rest on the support platform
of the drop weight machine. Figure 46(b) is the smoothed displacement versus time curve after
passage through the 5 kHz low pass filter. The smoothed displacement versus time curve is
differentiated with respect to time, and the resulting velocity versus time is shown in Figure
47(a). Notice that the peak velocity of 102 in/sec (259.08 cm/sec) occurs at about 500 psec, then
rapidly drops off after the failure occurs, and then continues to oscillate at about 60 in/sec (152.4
cm/sec). At a drop height of 8 in (20.32 cm) the tup strikes the beam at 75 in/sec (189.5 cm/sec),
and transfers its momentum to the beam. Since the mass of the tup is greater than the mass of
the beam, the beam achieves a higher velocity than the tup striking velocity. The velocity versus
time curve is then differentiated with respect to time to yield acceleration versus time, the results
of which are shown in Figure 47 (b). The acceleration of the beam peaks at about 200 psec, with
an amplitude of 750 gs. The time of peak acceleration is consistent with an abrupt change of
slope in the strain versus time data, as the failure process in the beam initiates. Peak strain rates
occur at this time as well, and will be discussed in detail later in this chapter.
Consider, for a moment, the beam as a single degree of freedom system, with the
acceleration versus time known at beam midpoint. In order to estimate the inertial force or
resistance to motion that the beam offers when it is set into motion, the beams equivalent mass


83
lit i i i i i iiiiiiii
0.0s 1.0ms 2.0ms 3.0ms
TIME
(a)
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
Figure 46. Displacement versus Time Curves for Beam LW3-20


84
(a)
H
cc
w
ij
1.0ms
2.0ms
(b)
Figure 47. Velocity and Acceleration versus Time Curves for Beam LW3-20


85
must be calculated. Once the equivalent mass is known, the acceleration versus time curve may
be multiplied by the equivalent mass to calculate the beams inertial load versus time curve. At
the outset of this investigation, it was thought that the beam would initially assume a deflected
shape consistent with the fundamental mode of vibration as shown in Figure 48(a), but this was
P(t)
1
£
P(t)
JL \/
(b)
Figure 48. General versus Localized Bending
not the case. Since the loading may be considered impulsive, the beam as a whole does not have
time to react to the sudden blow imparted by the tup to the center of the beam. The initial
deflection is therefore localized about the beams midpoint, as shown in Figure 48(b), and the
original span length of the beam has little significance in calculating the beams equivalent mass.
In time, the deflection assumes a fully developed plastic state. A method for determining the
beams average equivalent mass during the time of the load pulse was developed based on the
assumption of a rigid, perfectly plastic beam, utilizing the concept of a traveling plastic hinge


86
and is discussed in detail in Chapter 3. Once the equivalent mass is known, the beams inertial
load versus time behavior is therefore determined. Figure 49(a) shows the tup or total load and
inertial load versus time curves plotted together. Note that the majority of the load measured by
the tup is represented by the inertial load. At the end of the load pulse, they are coincident.
Taking liberty at the present time to assume no damping forces present, the difference in
amplitude between the tup or total load and the inertial load represents the bending load. Figure
49(b) show the bending load versus time curve. The bending load has a peak amplitude of 2600
lbs (11.57 kN) at 275 psec, and returns to zero at 420 psec. Since the peak load of the tup is
4300 lbs (19.13 kN), failure to account for the inertial load will lead to misleading conclusions
being drawn about the bending load, and hence, the fracture energy. This observation has also
been noted by other researchers [32, 35, 42],
In order to calculate the amount of energy consumed by the beam up until failure occurs,
bending load versus time was plotted against displacement versus time, creating a dynamic
bending load versus displacement curve. This curve is shown in Figure 50(a). Recall that the
bending load versus time curve showed that the bending load dropped to zero at about 420 psec,
which in load displacement space, translates to a displacement of 0.022 in (0.564 mm) at
failure. At the point of failure, the beam stops receiving energy from the impact hammer, and
the tup load falls to zero. The energy consumed by the beam up until failure occurs is sometimes
called the toughness or simply the fracture energy. The area under the dynamic bending load
versus displacement curve therefore represents the fracture energy or the amount of energy
required to create two new fracture surfaces. In the case of plain concrete beams, the beams
break into two halves and the two broken halves swing about their supports away from the tup.
Although these beam halves may retain considerable kinetic energy, they have no bending or
strain energy left in them.


87
0.0s 1.0ms 2.0ms
TIME
(a)
(b)
Figure 49. Tup, Inertial, and Bending Load versus Time Curves for Beam LW3-20


88
(a)
250.0us 0.0s 250.0us 500.0us
TIME
(b)
Figure 50. Bending Load versus Displacement and Fracture Energy versus Time Curves for
Beam LW3-20


89
The area under the bending load versus displacement curve was found by numerical
integration in VU-POINT using Simpsons Rule. The resulting dynamic fracture energy versus
time curve is shown plotted in Figure 50(b). The fracture energy at the beam failure time of 420
psec was 3.6 ftlbs (4.9 N-m).
Data from the strain gages mounted on the center cross section was useful not only for
determining the strains, strain rates, and stresses, but also for verifying the initial onset of failure
in the beams as well. Figure 51(a) shows the strain versus time traces for strain gages number
one and three. Recall that Figure 39 showed the location of all of the strain gages. Strain gage
number three indicating tensile strain, shows an abrupt change of slope at 200 psec indicating
that the concrete has begun to fail in tension, and a crack is in the process of propagating from
the bottom of the beam to the top surface. Meanwhile, strain gage number one, located on the
concrete near the top surface, goes into compression and remains in that state until failure
occurs. The third strain gage, strain gage number two, located midway on the beams center
cross section, was not used on this particular test.
Differentiating the strains with respect to time yielded the strain rates. Its interesting to
note that the peak tensile strain rate recorded in gage number three shown in Figure 51(b) occurs
at almost the same time as the peak inertial load; about 200 psec, with a magnitude of 18.5/sec.
The shape of gage number threes strain rate versus time pulse is also curiously similar to that of
the inertial load versus time pulse as well. Strain gage number ones compressive strain rate
only reaches about 2/sec during the loading event. Both gages indicate that the strain rates return
to and oscillate about zero after 800 psec.
Figure 51(c) shows the stress versus time curves for gages one and three. The curves
were obtained by multiplying the strain versus time curves by their respective elastic moduli;


90
0.0s 1.0ms 2.0ms 3.0ms
TIME
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
0.0s 1.6ms 2.0ms 3.0ms
TIME
Figure 51. Strain, Strain Rate, and Stress versus Time for Beam LW3-20


91
20 x 106 psi (137.93 GPa) for the CFRP, gage number three, and 3.53 x 106 psi (24.35 GPa) for
the concrete, gage number one. The tensile stress in the CFRP is about 60 times higher than the
concrete compressive stress.
Damping Loads
All dynamic structural systems contain damping to some degree. However, the effect is
not significant if the loading is impulsive and only the maximum response is being investigated.
Damping in structures may arise due to different physical phenomena. In some cases, it may be
due to resistance provided by the medium surrounding the structure such as water, air, or soil. It
is also due to the loss of energy associated with slippage of structural connections either between
members or between the structure and the supports. It may also involve internal molecular
friction of the material itself. In any case, the effect is one of a force opposing the motion.
The analyses conducted during the course of this investigation assumed negligible
damping forces, primarily due to the impulsive nature of the loading. Furthermore, the beams
fail prior to any continuing state of vibration; a state in which damping generally is included. To
further illustrate the validity of the assumption, a calculation was carried out to determine the
relative magnitude of the damping force, and compare it to the other forces present. For the
purposes of this analysis, the damping was assumed to be of the viscous type (as opposed to
Coulomb, or constant damping) where the damping force is proportional to but opposite in
direction of the velocity. Viscous damping is the most commonly assumed for structural
analysis. The amount of damping which removes all vibration is referred to as critical damping,
which is a convenient reference point, since most structures typically have between five and ten
percent of critical damping.
Figure 52(a) shows beam midpoint velocity versus time for beam LW2-11. This is a
typical velocity profile for this beam type (one ply CFRP) and tup drop height (12 in/30.48 cm).


Full Text
LD
1780
199ip
; j55
UNIVERSITY OF FLORIDA


DYNAMIC RESPONSE OF CONCRETE BEAMS
EXTERNALLY REINFORCED WITH
CARBON FIBER REINFORCED PLASTIC
By
DAVID MARK JEROME
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1996

This work is dedicated to my wife Elisabetta Lidia, and our twin sons Matthew Allen and
Nathan Kelley, who were born on November 1, 1993, during the course of this investigation.

ACKNOWLEDGMENTS
The author would like to acknowledge the advice and counsel given to him by his faculty
supervisory committee: Professors E. K. Walsh, D. M. Belk, J. E. Milton and C. S. Anderson.
Special recognition is reserved for Professor C. A. Ross, supervisory committee chair, without
whose advice, mentorship, encouragement, and friendship, this research would not have been
possible.
The author would like to acknowledge the following individuals at the Wright
Laboratory Air Base Technology Branch at Tyndall Air Force Base, Florida, for their
contributions listed below: Mr. Dale W. Wahlstrom for fabrication of the concrete beams and
Split Hopkinson Pressure Bar (SHPB) samples, and for assistance with the compressive and
splitting tension tests on the Forney load frame; Mr. William C. Naylor for strain gage and
instrumentation assistance; Mr. Dean W. Flitzelberger for assistance with the Material Test
System (MTS ) load frame; Mr. Francis W. Barrett III for high speed camera assistance on the
drop weight impact machine; and Mr. Carl R. Hollopeter for maintenance of the drop weight
impact machine and instrumentation assistance. Use of the Forney® and MTS® load frames and
drop weight impact machine, all of which are owned and supported by the Wright Laboratory
Air Base Technology Branch, Tyndall Air Force Base, Florida, is also gratefully acknowledged.
The author also wishes to acknowledge the assistance of Ms. Cathy A. Rickard for ably
typing a large portion of the manuscript, and Mr. Danny R. Brubaker and Mr. Bruce C. Patterson
for help in preparing several of the figures.
Finally, the author wishes to acknowledge the help and support of the Wright Laboratory
Armament Directorate at Eglin Air Force Base, Florida, during his educational endeavors.

TABLE OF CONTENTS
page
ACKNOWLEDGMENTS iii
KEY TO ABBREVIATIONS vi
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
Objective 1
Background 1
Approach 10
2 EXPERIMENTAL WORK 13
Fabrication of Concrete Beam Specimens 13
Application of the CFRP 15
Surface Preparation 16
Characterization of the Concrete 22
Static Compression Tests 22
Static Splitting Tensile Tests 26
Dynamic Compression Tests 29
Dynamic Splitting Tensile Tests 32
Test Results 37
Static Beam Bending Experiments 39
Description of the MTS® Load Frame 39
Instrumentation Used/Measurements Made 42
Test Results 42
Calculation of Fracture Energies 54
Dynamic Beam Bending Experiments 58
Description of the Wyle Laboratories Drop Weight Impact Machine 59
Instrumentation Used 61
Calibration of the Tup 62
Method of Test 73
Interpretation of Test Results 75
Damping Loads 91
Results and Discussion 93
Comparison of Dynamic and Static Test Results 97
IV

3ANALYTICAL MODEL.
103
Section Analysis 103
Region 1 - All Elastic 105
Region 2 - Cracked Tension Concrete, All Other Elastic 109
Region 3 - Cracked Tension Concrete, Inelastic Compression
Concrete, All Other Elastic 112
Comparison to MTS® Test Data 115
Determination of Beam Equivalent Mass 121
Single Degree of Freedom Representation of a Beam Subject to a
Half - Sine Pulse Impulsive Load 125
Comparison to Drop Weight Impact Test Data 130
4 FINITE ELEMENT METHOD CALCULATIONS 134
Description of the ADINA FEM Computer Code 134
The Concrete Material Model 137
The CFRP Material Model 141
Concrete Beam with CFRP - Finite Element Model 141
Dynamic Loading Calculations 142
Results and Comparison to Test Data 144
5 SUMMARY AND DISCUSSION 157
Characterization of the Concrete 158
Static Beam Bending Experiments 159
Dynamic Beam Bending Experiments 160
Static versus Dynamic Beam Bending Experiments 161
Analytical Model 162
Finite Element Method (FEM) Calculations 163
Future Research 164
6 CONCLUSIONS 167
REFERENCES 170
APPENDICES
A SUMMARY OF CONCRETE UNCONFINED COMPRESSIVE
STRENGTHS 174
B MTS® LOAD - DISPLACEMENT CURVES 177
C SUMMARY OF STATIC THREE POINT BENDING EXPERIMENTS 194
D DYNAMIC BEAM BENDING EXPERIMENTAL DATA - TABULAR
SUMMARY 197
E ANALYTICAL MODEL COMPUTER PROGRAM 202
F ADINA INPUT AND PLOT FILES FOR BEAM LW6-43 222
G ADINA INPUT AND PLOT FILES FOR BEAM LW9-66 229
BIOGRAPHICAL SKETCH 236
v

KEY TO ABBREVIATIONS
American Concrete Institute (ACI)
Automatic, Dynamic, Incremental, Nonlinear Analysis (ADINA)
Carbon Fiber Reinforced Plastic (CFRP)
Fiber Reinforced Plastic (FRP)
Finite Element Method (FEM)
Glass Fiber Reinforced Plastic (GFRP)
Linear Voltage Displacement Transducer (LVDT)
Material Test System (MTS)
Methyl Ethyl Ketone (MEK)
Modulus of Rupture (MOR)
Single Degree of Freedom (SDOF)
Split Hopkinson Pressure Bar (SHPB)
Swiss Federal Laboratories for Testing and Research (EMPA)
vi

Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy
DYNAMIC RESPONSE OF CONCRETE BEAMS
EXTERNALLY REINFORCED WITH
CARBON FIBER REINFORCED PLASTIC
By
DAVID MARK JEROME
May 1996
Chair: C. Allen Ross
Cochair: James E. Milton
Major Department: Aerospace Engineering, Mechanics, and Engineering Science
A series of 54 laboratory scale concrete beams 3 x 3 x 30 in (7.62 x 7.62 x 76.2 cm) in
size were impulsively loaded to failure in a drop weight impact machine. An additional 16
beams were quasistatically loaded to failure in a load frame. The beams had no internal steel
reinforcement, but instead were externally reinforced on the bottom or tension side of the beams
with one, two, and three ply unidirectional carbon fiber reinforced plastic (CFRP) panels. In
addition, several of the beams were also reinforced on the sides as well as the bottom with three
ply CFRP. The beams were simply supported and loaded at beam midspan.
The lightweight concrete used in the test specimens was characterized via quasistatic and
dynamic compression and splitting tensile tests. In compression, the concrete behaved the same
as normal weight concrete. In tension, however, its behavior was quite different.
The beams sustained dynamic loads with amplitudes up to 10 kips (44.5 kN) and
durations less than 1 millisecond. Measurements including total load, midspan displacement and
strains, and a high speed framing camera (10,000 frames/sec) gave insight into failure
vii

mechanisms. Failure to account for inertia in the loads will result in incorrect calculation of the
beam’s peak bending load.
The quasistatic bending tests were conducted in load control at 2 lbs/sec (8.9 N/sec).
Both crosshead load and midspan displacement were recorded.
Fracture energies were determined by calculating the areas under the bending load
versus displacement curves. Dynamic fracture energies and peak displacements for the CFRP
reinforced beams were always less than the static values. However, dynamic peak bending loads
were 2-3 times larger than the corresponding static values. This implies that for a given load
rate, a beam has a fixed capacity to absorb energy, dictated by the peak bending load and limited
by displacement, emphasizing the brittle nature of concrete when loaded dynamically.
The beams were analytically idealized as a single degree of freedom (SDOF) system,
subjected to a half sine pulse impulsive load. The SDOF analysis generally overpredicted the
displacement-time behavior of the beams, due to modeling limitations.
Dynamic beam behavior was also studied numerically using the finite element method.
Excellent agreement with experimental evidence was obtained.
viii

CHAPTER 1
INTRODUCTION
Objective
The objective of this investigation was to understand the dynamic behavior of plain
lightweight concrete beams and plain lightweight concrete beams embedded with nylon fibers,
both of which were externally reinforced with carbon fiber reinforced plastic (CFRP) panels.
One, two, and three ply external CFRP strengthening panels were bonded to the bottom or
tension side of laboratory size (3.0 x 3.0 x 30.0 in/7.6 x 7.6 x 76.2 cm) concrete beams. In
addition, three ply CFRP was also applied to the sides as well as the bottom of some of the
beams. A schematic of the externally reinforced beam cross-section is shown in Figure 1. A
total of seventy-two beams were tested statically and dynamically. The experimental and
analytical studies of the static and dynamic response of these beams is the basis for this
dissertation.
Background
Since the mid sixties, existing concrete structures in Europe, South Africa, the US, and
Japan had all been externally post-strengthened by bonding steel plates to the bottom or tension
sides of the beams or slabs. The first actual use of externally applied fibrous composite
materials to enhance the load carrying capability of a concrete structure is attributed to a
researcher in Switzerland in 1989. Kaiser [1] is credited as the first worker to employ carbon
fiber reinforced plastic (CFRP) to post-strengthen concrete beams. Motivation for Kaiser's work
was driven by several factors: difficulties in manipulating the unwieldy steel plates into position
1

2
T
3 in
(7.62 cm)
I
Figure 1. Cross Section of Externally Strengthened Concrete Beam
at the construction site, corrosion at the steel/concrete interface, improper formation of
construction joints, and a limited selection of steel plate sizes. The CFRP material system, with
such desirable material attributes as being lightweight, being very strong (150-320 Ksi/1.034 -
2.206 GPa tensile strength), having outstanding corrosion resistance, and having excellent
fatigue properties, seemed a logical solution to the post-strengthening problem. Kaiser tested
twenty-six beams with a span of 6.56 ft (2 m), and one with a span of 22.96 ft (7 m), both
employing unidirectional CFRP on the bottom or tension side of the beams. He concluded that
the calculation of flexure in reinforced concrete elements post-strengthened with CFRP could be
performed similar to conventionally reinforced concrete structural elements, but that shear
cracking in the concrete could lead to delamination of the CFRP. He also concluded that flexural
cracking did not seem to influence the loading capacity of the beam, and that the addition of the
CFRP on the bottom of the beam led to a finer crack distribution in the beam when compared
with the plain concrete beam. Additionally, he stated that the difference in the thermal

3
expansion coefficients between concrete and the CFRP resulted in stresses at the CFRP/concrete
interface due to changes in temperature, but that after one hundred frost cycles ranging from -13
O O .
to +68 F (-25 to +20 C), no deleterious effects were noted on the load capacity of the beams.
In 1987, Meier and colleagues at the Swiss Federal Laboratories for Materials Testing
and Research (EMPA) conducted feasibility studies on the use of CFRP to rehabilitate the beams
in a bridge; in particular, as cables for cable-stayed and suspension bridges, prestressing tendons,
and other applications where steel repair plates could be replaced with advanced composite
panels [2, 3]. This may have been the genesis of Kaiser's pioneering work. Their more recent
work [4, 5, 6, 7] continues to focus on the large variety of modem materials recently employed
or soon to find application in modern bridge construction, focusing primarily on the use of
fibrous composites. Meier [7] also describes the first use of CFRP to rehabilitate a damaged
bridge in Switzerland in 1991. The Ibach bridge, in the county of Lucerne, was damaged when
construction workers were coring the bottom of a 127.92 ft (39 m) span for a traffic light and
accidentally severed several of the wires in a prestressing tendon in the outer web. The bridge
was successfully repaired with only 13.6 lbs (6.2 kg) of CFRP, compared with the 385 lbs (175
kg) of steel that normally would have been used to repair the bridge.
Triantafillou and Plevris [8] have studied the failure mechanisms of fiber reinforced
plastic (FRP) strengthened concrete beams subjected to a bending load which included: steel
yield-CFRP rupture, steel yield-concrete crushing, compressive failure, and crushing. In
addition to discussing the concept of concrete strengthening with prestressed FRP composite
sheets, they obtained equations describing each failure mechanism and produced diagrams
showing the beam designs for which each failure mechanism is dominant. In collaboration with
colleagues at EMPA in Switzerland, they have also studied hybrid box beams composed of
aluminum and CFRP to meet the requirements of low weight, high strength and stiffness, relative

4
ease of design and fabrication, and high reliability [9], They analyzed the problem of a
minimum weight design, CFRP-reinforced, thin-walled rectangular aluminum section subjected
to given strength and stiffness constraints, and showed that at the optimum design, various local
failure mechanisms occur almost simultaneously, while beam stiffness rarely controls.
Triantafillou and Meier [10] studied the basic mechanics of glass fiber-reinforced plastic (GFRP)
box beams, which combine a layer of concrete in the compression zone and a unidirectional
CFRP laminate in the tension zone. A design methodology for the hybrid sections is also
presented, based on a complete set of stiffness, strength, and ductility design requirements. The
method of strengthening and reinforcing concrete beams using thin FRP sheets as externally
bonded reinforcement, described by Triantafillou and Plevris [8], can be extended if the sheets
are prestressed before they are applied to the concrete surface [11], This prestressing technique
is being considered in the building of new structures, particularly bridges, and the strengthening
of existing ones. An analytical model was developed which predicts the maximum achievable
prestress level, so that the prestressed FRP system does not crack and fail near the two ends upon
release of the prestress force. The flexural behavior of concrete beams prestressed with
externally bonded FRP sheets is discussed, based upon four experiments and their subsequent
analysis. Test results indicate a modest 25 percent improvement in the flexural strength of the
four beams when prestressed FRP panels were used, compared with when the non-prestressed
FRP panels were used.
In China, Shijie and Ruixian [12] have analyzed the adhesive bond characteristics of
GFRPs used to externally reinforce concrete beams. Static loading and fatigue tests were carried
out on GFRP reinforced beams with spans of 13.12 ft (4 m). They concluded that the fatigue life
of a GFRP strengthened beam is approximately three times that of the unreinforced beam.
Applications for repairing two highway bridges were also discussed.

5
At the University of Arizona, Saadatmanesh and Ehsani [13] conducted four tests on 5 ft
(1.52 m) long concrete beams that had been externally reinforced with GFRP plates which were
bonded to the tension side of the beams. One beam was unreinforced, and designated as the
control beam. One #3 steel tensile reinforcement rod was also cast into the beams. The beams
were simply supported and subjected to two concentrated loads symmetrically placed 3 in (7.62
cm) on either side of the midspan. Deflection at the midspan, as well as strains on the surface of
the GFRP plates, were measured as the beams were loaded to failure. Load versus strain and
load versus deflection plots were obtained on all five tests. They concluded that strengthening
concrete beams with epoxy bonded GFRP plates is a feasible way of increasing the load carrying
capacity of existing beams and bridges, and that selection of a suitable epoxy for bonding the
GFRP panels onto the beams is necessary to improve the ultimate capacity of the beam. They
noted that the epoxy adhesive should have sufficient stiffness and strength to transfer the shear
force between the composite plate and the concrete.
Sharif et al. [14] and Ziraba et al. [15], in the Department of Civil Engineering at King
Fahd University of Petroleum and Minerals in Saudi Arabia, experimentally investigated the
repair of initially-loaded reinforced concrete beams with epoxy-bonded fiberglass reinforced
plastic (FRP) plates. The reinforced concrete beams were initially loaded to 85 percent of their
ultimate flexural capacity and subsequently repaired with the epoxy-bonded FRP plates and then
re-loaded to failure. Different repair and plate anchoring schemes were investigated to try and
eliminate premature failure of the FRP plate/bond due to the high concentration of shear stress,
and to promote ductile behavior of the beam. Load-deflection curves for the FRP repaired beams
are presented, and the different failure modes for each beam are discussed. Their results,
consistent with those obtained by other researchers, indicated that the flexural strength of the

6
FRP repaired beams is greater than that of the control beams, and that the ductile behavior of the
FRP repaired beams is inversely proportional to FRP plate thickness.
University of Nevada researchers Gordaninejad et al. [16] focused on the behavior of
composite bridge girders constructed from CFRP sections and concrete slabs. Their study
examined four-point bending of three beams; one plain graphite/epoxy I-beam, and two beams
constructed from concrete slab and graphite/epoxy sections which were adhesively bonded
together. All three beams were one-eighth scale models of bridge girders. Theoretical and
experimental studies were performed on the plain I-beam sections to develop a basic
understanding of the beams flexural behavior. The tests and analyses were then extended to the
composite girders. They concluded that slip at the interface between the concrete slab and the
graphite/epoxy beam had a minor effect on the failure load, but produced a significant reduction
in stiffness of the composite section.
Chajes et al. [17,18] at the University of Delaware tested a series of reinforced concrete
beams in four-point bending to determine the ability of several different types of externally
bonded composite fabric materials to improve the beam's flexural capacity. The different fabric
materials were chosen to allow a variety of fabric stiffnesses and strengths to be studied. The
fabrics used were made of aramid, E-glass, and graphite fibers, and were bonded to the beams
using Sikadur 32, a two-component, high modulus, high strength, construction epoxy. A series
of pull-off tests were run to investigate the bond strength of the adhesive, and it was determined
that a single layer of aramid, a triple layer of E-glass, and a double layer of graphite fabric can be
expected to develop full tensile capacity in approximately 2.0 in (50.8 mm) for both the E-glass
and graphite fabric, and in approximately 3.0 in (76.2 mm) for the aramid fabric. Test results
indicated that the external fabric reinforced beams yielded an increase of approximately 50
percent in both flexural capacity and stiffness. For the beams reinforced with graphite fiber and

7
E-glass fabrics, failures were attributed to fabric tensile failure in the beam's constant moment
section, whereas the beams reinforced with aramid fabric failed due to crushing of the concrete
in compression. An analytical model based on the stress-strain relationships of the concrete,
steel, and composite fabrics was also developed, and comparisons were made with the
experimental results.
Recently, the Japanese have also investigated the use of carbon fiber reinforced plastic to
retrofit several large scale structures in the field [19]. The Japanese work is different from the
European, Saudi, and U.S. approaches in that it makes use of dry fiber CFRP sheets rather than
the pre-preg plates or strips. Two petrochemical industries, the Tonen Corporation and
Mitsubishi Chemical, have aggressively pursued this technology, and have developed proprietary
processes to impregnate the dry CFRP sheets with various epoxy resins. This technique
effectively eliminates the requirement for mechanical or vacuum fixturing when applying the
CFRP sheets to the structure. Unfortunately, the available literature describing their processes
are proprietary and are unavailable through normal journal articles or library access. The only
other work cited were Japanese symposia proceedings, which were also unavailable through
normal library channels.
The U.S. military has also recognized the potential of advanced composite materials to
strengthen and rehabilitate existing concrete structures. In the past, massive concrete protective
structures were constructed to shelter military personnel and equipment from conventional
weapon attacks. Now, as the US military moves into an era in which "mobility" is becoming
more and more important, new materials must be sought out that can be easily transported and
used for rapid construction while still maintaining adequate levels of protection for personnel
and equipment. Composite materials, typically used by the military solely in aerospace
applications, are now being investigated to potentially solve the burgeoning demand for

8
lightweight, durable construction materials. To investigate CFRP's potential for Air Force
applications, Hughes and Strickland [20] at Tyndall Air Force Base in Florida, in conjunction
with faculty members from the University of Florida, Ohio State, and Auburn, have conducted a
series of tests on small concrete beams reinforced with uniaxial CFRP strips on the tension side
of the beams. Initial tests were performed on 1.0 x 1.0 x 12.0 in (2.54 x 2.54 x 30.48 cm) and 2.0
x 2.0 x 12.0 in (5.08 x 5.08 x 30.48 cm) concrete beams in both three and four-point bending
tests. The number of plies in the CFRP reinforcing strip was varied from one to three, and three
different cementitious mixes were used for the beam material. Additionally, CFRP strips were
also applied along the sides of some of the beams. Test results indicated that the failure load was
increased by 4.4 to 9.0 times the baseline failure load when the CFRP strips were applied to the
beams, when compared to plain concrete beams without such strips.
The most recent work to date on concrete beams externally reinforced with CFRP has
been conducted by Chajes et al. [18], Sierakowski et al. [21], and Ross et al. [22], Sierakowski
et al. [21] used a strength of materials approach to develop an analytical model which evaluates
the shift in the neutral axis that occurs in statically loaded concrete beams that have been
externally reinforced with CFRP strips. The shift in the neutral axis of the beam occurs when the
tensile stress in the concrete at the bottom of the beam exceeds the fracture stress of the concrete,
which is roughly 10 percent of the concrete's compressive strength. It is this cracking that causes
a shift in the neutral axis of the beam, and a subsequent change in the beam's planar moment of
inertia. In this analysis, two particular cases were studied. In the first case, the externally
bonded CFRP strips were placed on the bottom or tension side of the beams only, and in the
second case, the strips were placed on the bottom as well as the sides of the beams. The
analytical model is compared to experimental test data gathered on some small scale beams, 3.0
x 3.0 x 30.0 in (7.62 x 7.62 x 76.2 cm ) in size, and some large scale beams, 0.67 x 0.67 x 9.0 ft

9
(0.20 x 0.20 x 2.74 m) in size, both of which were tested in third-point loading in the structures
laboratory owned by the Wright Laboratory Air Base Technology Branch at Tyndall Air Force
Base, Florida. Reasonable agreement between experiment and analysis was obtained. Ross et al.
[22] performed a complete review and analysis of twenty-two experiments that were conducted
in the summer of 1994 also at the Tyndall Air Force Base, Florida, structures laboratory. The
concrete beams were again 0.67 x 0.67 x 9.0 ft (0.20 x 0.20 x 2.74 m), had three ply
unidirectional CFRP on the bottom or tension side of the beams, and were all tested to failure
using a third point loading test frame. The tension steel varied in size from number 3s to number
8s in the beams. Complete section analysis was conducted on the beam cross sections and load-
displacement curves were calculated and compared with the test beams both with and without
CFRP. In an effort to quantify the strengthening effects of the CFRP, an enhancement ratio,
defined as the peak load on a beam with a given steel percentage and CFRP, divided by the peak
load of the same beam with steel only, was obtained and is shown in Figure 2. Results show
considerable enhancement for the beams with the lower tensile steel ratios, and little to no
enhancement for beams with high tensile steel ratios. Ross et al. [22] also concluded that for the
experiments in which concrete beams reinforced with CFRP were tested, the weakest link
appears to be in the shear strength of the adhesive/CFRP interface. When delamination occurs
between the CFRP and adhesive, the tensile strength of the CFRP is not used to its fullest
potential, and the authors recommend further work on improving the bond between the CFRP
and the concrete. The principal conclusion of the study was that a quantifiable strength
enhancement was obtained by the addition of a very thin (0.0175 in/0.4445 mm) strip of CFRP
added to the bottom side of the beams, especially for those beams with tensile steel ratios of one
percent or less.

10
ER.ENHANCEMENT RATIO
PS, PERCENT TENSION STEEL
0 3 4 5 6 7 8
STEEL BAR SIZE
Figure 2. CFRP Enhancement Ratio [22]
Approach
After a thorough and careful review of this literature, it became evident that the approach
for this investigation should consider:
1. A systematic experimental study of the effect of varying the number of plies of
CFRP on the static load carrying capacity of concrete beams.
2. An experimental study of the dynamic behavior of concrete beams both with and
without external CFRP reinforcing panels.
3. Development of a comprehensive analytical model which attempts to predict not
only the static load-displacement behavior of the beams, but the dynamic
displacement-time behavior as well.

11
4. Finite element method (FEM) calculations of the dynamic response of beams
both with and without external CFRP reinforcing panels.
In order to carry out a systematic experimental study of the effect of varying the number
of plies of the CFRP on both the static and dynamic load carrying capacity of the beams, the
approach was to test laboratory scale concrete beams with none, one, two, and three plies of
unidirectional CFRP to failure both statically, using a MTS load frame, and dynamically, using
a drop weight test machine. All tests were conducted in center-point loading mode, as shown in
Figure 3. Eighteen static and fifty-four dynamic beam bending experiments were attempted,
based on the test matrix shown in Table 1 below.
Figure 3. Center or Three Point Bending Mode Used in Static and Dynamic Tests
It has already been well-established that concrete, although brittle, is a rate sensitive
material both in tension and compression. In order to thoroughly characterize the concrete used
in the beam test samples, the approach was to conduct a series of static and dynamic

12
Table 1. Test Matrix for Static and Dynamic Beam Bending Experiments
Dynamic Tests
Static
Number of
Number of
Tests
Iterations
Beams
Beam
* Drop Height # 1 Drop Height #2 Drop Height #3
BO
X X
X X
3
12
B1
X X
X X
3
12
B2
X X
X X
3
12
B3
X X
X X
3
12
B4
X X
X X
3
12
B5
X X
X X
3
12
Total Number Beams
72
Ksyl
BO
Plain Concrete Beam
B1
Concrete Beam with 1 ply
CFRP
on Bottom of Beam
B2
Concrete Beam with 2 ply
CFRP
on Bottom of Beam
B3
Concrete Beam with 3 ply
CFRP
on Bottom of Beam
B4
Concrete Beam with 3 ply
CFRP
on Bottom and Sides of Beam
B5
Fibrous Concrete Beam with 3 ply CFRP on Bottom of Beam
compression and splitting tension tests. Standard compression tests on concrete cylinders were
also required to ensure that the static unconfined compressive strength f c was known for each
beam. This attribute is very important, since other material parameters which were required for
subsequent analyses and modeling are typically derived from it, such as the secant modulus and
modulus of rupture. This was the approach taken for the research.

CHAPTER 2
EXPERIMENTAL WORK
Fabrication Of Concrete Beam Specimens
A total of 72 lightweight concrete beams were prepared for the experimental portion of
the research. The beam size chosen was 3.0 x 3.0 x 30.0 in ( 7.62 x 7.62 x 76.2 cm), and the
aggregate used was a lightweight aggregate with the trade name Solite , which reduced the
overall density of the concrete by about 20 percent. The aggregate used in these beams was
sieved to pass a 0.375 in ( 0.925 cm) sieve, but was retained on a 0.25 in (0.635 cm) sieve. No
tension steel reinforcement was used in any of the beams, due to their small size. Each batch of
concrete beams were cast in lots of eight, with four 4.0 in (10.16 cm) diameter by 8.0 in (20.32
cm) long cylinders cast at the same time as the beams for subsequent material property testing.
The mix proportions in Table 2 were used for each lot of eight beams and four cylinders cast.
Table 2. Lightweight Concrete Mix Proportions
Type 1 Portland Cement
35.25 lbs (16.02 kg)
Concrete Sand
85.63 lbs (38.92 kg)
Lightweight Aggregate
69.38 lbs (31.54 kg)
Water
19.00 lbs (8.64 kg)
13

14
A 5.6 percent moisture content in the aggregate was chosen as the standard. The
aggregate was soaked in water for one day and then allowed to drain off overnight prior to
combining it with the other mix constituents to facilitate the mixing process. If the moisture
content of the mix turned out to be too high, the amounts of aggregate and water were then
adjusted to account for the difference. This made a workable mix that flowed well on the
vibrator, with no measurable slump. The unit weight for all of the beams cast was 118 lbs/ft3
(1892.7 kg/m3). Table 3 lists the beam designators and the dates on which those lots of beams
were cast.
Table 3. Beam Designator and Cast Schedule
BEAM DESIGNATOR
DATE CAST
LW1-1 through LW1-8
16 November 1994
LW2-9 through LW2-16
29 November 1994
LW3-17 through LW3-24
6 December 1994
LW4-25 through LW4-32
14 December 1994
LW5-33 through LW5-40
17 January 1995
LW6-41 through LW6-48
18 January 1995
LW7-49 through LW7-56
23 February 1995
LW8-57 through LW8-64
28 February 1995
LW9-65 through LW9-72
2 March 1995
LW10-73 through LW10-80
7 March 1995
LW11-81 through LW 11-88
9 March 1995

15
Additionally, another half batch of concrete was cast into one 12.0 x 12.0 x 6.0 in (30.48
x 30.48 x 15.24 cm) box to make samples for Split Hopkinson Pressure Bar (SHPB) testing
along with two additional 4.0 in (10.16 cm) diameter by 8.0 in (20.32 cm) long cylinders, and
one additional 6.0 in (15.24 cm) diameter by 12.0 in (30.48 cm) long cylinder for subsequent
concrete characterization testing. These samples were cast on 29 November 1994. For the last
two sets of beams, labeled LWF10 and LWF11, three pounds of nylon fibers per cubic yard (1.78
kg/m3) were added to the mix to make two sets of fibrous concrete beams.
After casting, each beam was initially cured underwater for 28 days, and then cured in
air until it was subsequently tested. The minimum recommended time for concrete test
specimens between fabrication and conducting any type of dynamic testing is 60 days; for these
beams the minimum time turned out to be 90 days. This is to allow time for the concrete to
achieve sufficient strength, since its strength increases with time.
Application of the CFRP
Once the concrete beams had sufficiently dried after the 28 day water cure, the process
of bonding the CFRP to the beams could begin. The one, two, and three ply CFRP panels were
supplied by the Wright Laboratory Materials Directorate, Wright Patterson Air Force Base,
Ohio. The panels were all of the same material, an AS4C/1919 graphite epoxy. The C
designation denotes a commercial grade of AS4 graphite epoxy, and the 1919 denotes a 250° F
(121.1 °C) cure. Table 4 contains the relevant material properties of AS4C/1919 CFRP.
All of the CFRP panels came as pre-cut panels 3.0 x 30.0 in ( 7.62 x 76.2 cm) in size.
The cured ply thicknesses of the one, two, and three ply panels were individually measured with
a micrometer; the other material properties were provided by the Wright Laboratory Materials
Directorate.

16
Table 4. Pre-Preg CFRP Material Properties
0° Tensile Strength
320 ksi (2206.9 MPa)
O
0 Tensile Modulus
20 x 106 psi (137.9 GPa)
Fiber Volume
60 percent
Cured Ply Thickness
one ply
0.0085 in (0.2159 mm)
two ply
0.0140 in (0.3556 mm)
three ply
0.0195 in (0.4953 mm)
The adhesive used to bond the CFRP panels to the concrete beams was a thixotropic,
modified, two part epoxy engineering adhesive manufactured by the Dexter Corporation in
Seabrook, New Flampshire with the trade name Hysol . This adhesive was recommended for its
toughness, flexibility, and efficacy in bonding dissimilar materials. The adhesive has excellent
O O
peel and lap shear strengths, and has a pot-life of 40 - 50 minutes at 77 F (28.9 C) for 0.55 lbs
(250 gms) of mixed adhesive. The two part epoxy was mixed in equal parts (by either weight or
volume since the densities of the two parts were the same) and mixed thoroughly until the off-
white Part A and black Part B were a uniform gray in color. Some heat buildup or exotherm was
noticed during the mixing process, but since less than 1 cup of Part A and Part B were ever
mixed at one time, no excessive exotherm or heat buildup ever developed.
Surface Preparation
As with any bonding or adhesive process, proper surface preparation of the bonded
materials is a necessity. The bonding surfaces should be clean, dry, and properly prepared. The

17
bonded parts should be held in intimate contact until the adhesive is set. It was not necessary to
O
maintain fixturing for the entire adhesive cure schedule (3 days @ 77 F) but only until handling
strength is achieved. The manufacturer considers handling strength to be the same as tensile lap
0
shear strength. Handling strength of 750 psi (5.2 MPa) is achieved in 6 - 8 hours @11 V
O
(28.9 C). Figure 4 is the manufacturer's graph of tensile lap shear strength versus time at room
O
temperature 77 F.
TIME AT ROOM TEMPERATURE, DAYS
• • • (K)
Figure 4. Tensile Lap Shear Strength (psi) versus Time (days) for Dexter Hysol Adhesive
To properly prepare the concrete and CFRP surfaces to be bonded, all grease, oil, and
foreign particles had to be removed from each surface. As with most high performance
engineering adhesives, this step is critical, since for good wetting, the surface to be bonded
should have a higher surface tension than the adhesive. Therefore, the adhesive manufacturer's

18
recommendations were closely followed to ensure proper surface preparation of the concrete and
CFRP. For the CFRP, the following surface preparation steps were taken:
1. Use peel ply side of CFRP for bonding surface.
2. Degrease surface with methyl ethyl ketone (MEK).
3. Lightly abrade surface with medium grit emery paper. Take extreme care to
avoid exposing the carbon reinforcing fibers. (This is particularly important for
the one ply CFRP panels since they are only 0.0085 in (0.2159 mm) thick).
4. Repeat step 2; degrease surface with methyl ethyl ketone (MEK).
Since the concrete beams were new, recently cast test specimens, it was extremely
important to make sure that the de - molding release agent, which was either 10W-30 motor oil
or WD-40 lubricant, was completely removed from the bottoms and sides of the beams. The
following procedure was used to prepare the surfaces of the concrete beams:
1. Degrease the surface to be bonded with acetone.
2. Repeat step 1, if excessive de-molding oil is noted on the exterior surface.
3. Mechanically scarify (abrade) the surface with a grinding wheel.
4. Repeat step 1, degrease the surface with acetone.
5. Remove dust and concrete particulates with a stiff bristle whisk broom.
The cleaned concrete and CFRP surfaces should be bonded as soon as possible after the surface
preparation procedures have been accomplished, and the bonding procedure itself should be
performed in a room separate from the room or area in which the cleaning and surface
preparation procedures are accomplished.
After the above surface preparation steps were completed, the CFRP strengthening
panels were ready to be bonded. Six beams and six CFRP panels could be reasonably handled
and bonded at one time. One third of a cup of Part A and one third of a cup of Part B Hysol®

19
adhesive was sufficient to produce a 0.030 in (0.762 cm) bondline between the CFRP and the
beam for six specimens. After the Hysol was spread on the surface with a wide-bladed putty
knife as shown in Figure 5(a), a square tooth trowel was used to produce a series of 0.0625 in
(1.6 mm) high by 0.0625 in (1.6 mm) wide adhesive beads along the length of the beam, as
shown in Figure 5(b). This also served to remove any excess Hysol from the beam's surface,
and produced a uniform volume of adhesive on the beam as well.
Next, the CFRP panel was placed on top of the Hysol beads, shown in Figure 6, and
pressed into the adhesive while being concurrently smoothed with a paper towel. This served to
seat the panel on the beam as well as eliminating any excess air bubbles between the CFRP panel
and the beam.
After all six of the beams were prepared in this manner and keeping mind of the time so
as not to exceed the fifty minute pot life of the adhesive, the beams were then placed in a large
vacuum bag with 0.25 in (6.35 mm) foam spacers in between each beam. The perimeter of the
bag was sealed with a caulk compound. The vacuum bag itself was made of ordinary 0.006 in
(0.152 mm) thick polyethylene film. A vacuum pump was connected to the polyethylene bag via
a vacuum hose and a short length of 0.5 in (12.7 mm) galvanized pipe, which was then inserted
into the bag and also sealed with a caulk compound. A gage had been placed in line with the
length of galvanized pipe to measure the amount of vacuum pressure. Once the entire bag with
the six concrete beams inside was sealed, the pump was turned on. The pump was a Welch
Duoseal (Welch Vacuum Technology, Incorporated, Skokie, Illinois) two stage vacuum pump
which was driven by a 0.5 horsepower electric motor, with a clutch, pulley, and V-belt attached.
The performance specifications of the vacuum pump are shown in Table 5. Once any minor leaks
in the bag had been found and sealed, the vacuum pump quickly pulled the bag down to 28.5 in

20
Figure 5. Epoxy Adhesive Being Placed on Surface of Beams

21
Figure 6. CFRP Panel Being Placed on Beams
Table 5. Vacuum Pump Specifications (New Pump)
Free Air Displacement
5.6 ft3/min
Pump Rotational Speed
525 RPM
Guaranteed Partial Pressure
29.5 in Hg (14.5 psi)
Hg (14.0 psi), since the initial volume of air trapped inside the bag was relatively small. The
pressure differential created by the vacuum placed the beams into uniform hydrostatic

22
compression, placing a uniformly distributed force of 1260 lbs (5.62 kN) on each of the six 90.0
in (580.6 cm ) surfaces of each of the six beams. This fixturing force held the six CFRP panels
firmly in place on top of the beams until the Hysol reached its handling strength of 750 psi
(5.17 MPa) tensile lap shear strength in 6 - 8 hours at 77° F (28.9° C). The vacuum pump
typically ran for seven hours at a time for each batch of six beams to meet this curing schedule.
Afterward, the beams were then removed from the bag and allowed to cure for the remainder of
the three day cure schedule.
This process was used to bond 12 one ply panels, 12 two ply panels, and 24 three ply
panels to the bottom or tension side of these 48 concrete beam samples. The remaining 12
beams had three ply panels bonded to both the sides as well as the bottoms. For these beams,
each of the three sides were bonded separately following the procedures above. Chajes et al.
[18] also used a vacuum fixturing process to bond some composite strengthening panels to some
beams, but their process was unknown to the author at the time of this work. A patent is
currently under review for the vacuum fixturing process used in this investigation.
Characterization of the Concrete
In order to determine the material properties of the concrete used to fabricate the beam
test specimens, a series of quasistatic and dynamic loading experiments were conducted on the
samples of concrete which were cast at the same time as the beam samples were cast.
Static Compression Tests
The first tests conducted were static compression tests on 4.0 in (10.16 cm) diameter by
8.0 in ( 20.32 cm) long cylinders. Four cylinders from each batch of concrete were tested in
compression in a Forney System 2000 (Forney Incorporated, Wampum, Pennsylvania)
loadframe which has a 400,000 lb (1,779 kN) capacity. The four failure loads were averaged,
2 2
and then divided by the area of the test cylinder's cross section, which was 12.57 in (81.07 cm )

23
to yield the so-called unconfined compressive strength of the concrete, f c. Appendix A
summarizes the results of these static compression tests.
The unconfmed compressive strength may be associated with the hypothetical,
statistically derived compressive stress-strain curve in Figure 7 shown below, where the peak
compressive stress is the value of f c found in the tests.
Figure 7. Flypothetical Compressive Stress-Strain Curve
Certain properties of the stress-strain curve are necessarily described. The tangent to the
curve at the origin is called the initial tangent modulus of elasticity, Eci. A line drawn from the
origin to a point on the curve at which fc = 0.45 fc is called the secant modulus of elasticity, Ecs.
For low strength concrete, Eci and Ecs differ widely. For high strength concrete, there is
practically no difference between the two values. For lightweight concrete the initial slope is

24
somewhat less than for normal weight concrete, and the maximum stress f c occurs for larger
strain values. sc is the compressive strain associated with f c. It is commonly accepted that the
post-peak or strain softening portion of the stress-strain curve is terminated at a stress level of
0.85f c which is referred to as the concrete's ultimate strength. The strain associated with 0.85fc
is therefore the ultimate strain, denoted by eu.
Figure 8 is an actual stress-strain curve for lightweight concrete, generated at the Tyndall
Air Force Base structures laboratory in 1994. Unfortunately, the curve stops at fc which is about
6 ksi (41.4 MPa), and doesn't show the strain softening part of the curve due to strain gage
failure. However, an important feature is present in this curve which differentiates lightweight
concrete from its normal weight counterpart; the strain at f c. Typical values for s c, the strain at
fc, for normal weight concrete, are 0.002 in/in (cm/cm). Here we note that for the lightweight
concrete, the strain value ec at f c is larger, about 0.003 in/in (cm/cm). If we assume that the
strain softening part of the stress-strain curve is symmetric, we may extrapolate the value at eu,
the ultimate strain at rupture, which is associated with a stress of 0.85fc in the strain softening
region. This value is found to be about 0.004 in/in (cm/cm), which is also larger than that of the
typical value of 0.003 in/in (cm/cm) for su for normal weight concrete. Not only does the
maximum stress occur at a larger strain, but also the rupture stress has a larger strain value
associated with it as well.
Knowing the values of f c from quasistatic testing allows us to directly calculate the
tangent and secant moduli from standard empirical equations developed by the American
Concrete Institute (ACI). In general practice, the secant modulus of elasticity is used whenever
an elastic modulus value is needed, and is simply referred to as Ec. The ACI equation for the
secant modulus is defined as

25
Figure 8. Stress - Strain Curve for Lightweight Concrete (Courtesy of L.C. Muszynski, 1994)
Ec = 33 w3/2 (fc)I/2 (1)
where
Ec Secant Modulus (psi)
w Unit Weight (lbs/ft3)
fc Unconfined Compressive Strength (psi)

26
In addition to the 4.0 in (10.16 cm) diameter by 8.0 in (20.32 cm) long samples, six
2.0 in (5.08 cm) diameter by approximately 2.0 in (5.08 cm) long specimens were also tested in
direct compression, in order to provide a direct sample size comparison to the Hopkinson bar
samples. A load rate of approximately 2000 Ibs/min (8.9 kN/min) was used for the first three
samples, and then a load rate of approximately 20,000 lbs/min (89.0 kN/min) was used for the
last three samples. It should be noted that the strain rate was calculated by dividing the stress
rate by a secant modulus which was calculated from an average unconfmed compressive failure
stress of 6,708 psi (46.3 MPa). In addition, the dynamic increase factor is the ratio of the
dynamic failure stress to the average static failure stress. The dynamic increase factor is
subsequently plotted as a function of the log of the strain rate. For these quasistatic strain rates,
there is clearly no increase in the compressive strength of the concrete, as expected. Table 6
summarizes the results of these direct compression tests on the 2.0 in (5.08 cm) diameter
specimens.
Static Splitting Tensile Tests
It has been well established that concrete is about an order of magnitude weaker in
tension than in compression. See for example, Neville [23] or Winter and Nilson [24], The
tensile or fracture strength of concrete is an important material property, since tensile failure is
the dominant failure mode in concrete. In order to determine the tensile strength of the concrete
test specimens used in this investigation, a series of splitting tensile or Brazilian tests were
conducted in the Forney® System 2000 compression load frame. The method of conducting
splitting tensile tests has also been well established. See, for example, Neville [23], This
indirect method of applying tension in the form of splitting was first suggested by a Brazilian
named Fernando Carneiro, hence the name Brazilian test, although the method was also
developed independently in Japan. In this test, the cylindrical sample is placed with its

27
Table 6. Quasistatic Direct Compression Tests on 2.0 in (5.08 cm) Diameter
by 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Length
in (cm)
Load at
Failure
lbs (kN)
Time to
Failure
min (sec)
Load
lbs/min
(kN/min)
Stress
psi
(MPa)
Stress
ps i/sec
(kPa/sec)
Strain*
Rate
1/sec
Dynamic
Increase
Factor
1
2.006
(5.095)
21,760
(96.8)
11
(660)
1979
(8.8)
6926
(43.4)
10.5
(72.4)
3.03 x
10'6
1.03
2
2.01
(5.105)
20,050
(89.2)
9.98
(599)
2055
(9.1)
6525
(45.0)
10.9
(75.2)
3.15 x
10-6
0.97
3
2.012
(5.11)
20,960
(93.2)
10
(600)
2092
(9.3)
6672
(46.0)
11.1
(76.6)
3.20 x
lO'6
0.99
4
2.012
(5.11)
22,140
(98.5)
1.15
(69)
19,246
(85.6)
7047
(48.6)
102.1
(704.1)
2.95 x
10'5
1.06
5
2.013
(5.113)
17,760
(79.)
1
(60)
17,810
(79.2)
5653
(39.0)
94.2
(649.7)
2.72 x
10'5
0.85
6
2.012
(5.11)
22,760
(101.2)
1.1
(66)
20,696
(92.1)
7245
(50.0)
109.8
(757.2)
3.17 x
10'5
1.09
* Based on a Secant Modulus Calculated From an Average Compressive Failure
Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)
longitudinal axis between and parallel the platens of the load frame as shown in Figure 9. The
load is increased at a constant load rate until failure by splitting along the vertical diameter takes
place. These so-called splitting tensile tests were conducted in lieu of the more difficult direct
tensile test so as not in incur the difficulties of holding the specimens to achieve axial tension,
and the uncertainties of secondary stresses induced by the holding devices. Again, the samples
tested were from the same batches of concrete from which the beams were cast. Sample sizes
were approximately 2.0 in (5.08 cm) in diameter and 2.0 in ( 5.08 cm) long.
Calculation of the horizontal tensile strength follows the derivation of Boussinesq (1892)
as given by Malvern [25] and Timoshenko [26], An element on the vertical diameter of the
cylinder is subjected to a compression stress of

28
P
P
H
m
(5.08 cm)
D=2 in
(5.08 cm)
±.
Figure 9. Diagram of the Splitting Tensile or Brazilian Test
and a horizontal tensile stress of
°C =
2P
JtLD
D2
r(D-r)
where
f,=
2P
7tLD
ac Compressive Stress, psi
P Compressive Load, lbs
L Length of Specimen, in
D Diameter of Specimen, in
r Distance of the Element from the Upper Load, in
D-r Distance of the Element from the Lower Load, in
(2)
(3)

29
ft Tensile Stress, psi
Similar to the direct compression tests, a series of six quasistatic splitting tensile tests
were also conducted at two different load rates, one load rate being approximately 2,000 Ibs/min
(8.9 kN/min) and the other about 10,000 lbs/min (44.5 kN/min). Knowing the load and time to
failure allows calculation of the stress and stress rate, respectively, using Equation (3). Once the
stress rate is known, the strain rate may be calculated by dividing by the secant modulus,
determined previously as 3.46 x 106 psi (23.89 GPa). Table 7 summarizes the results of these
splitting tensile tests on the 2.0 in (5.08 cm) diameter by approximately 2.0 (5.08 cm) long
specimens. Similar to the quasistatic direct compression tests, the dynamic increase factor is
calculated from the ratio of the splitting tensile stress for the three tests at a given load rate.
Obviously, there is no increase in strength at these low strain rates; hence the dynamic increase
factors are all ~ 1.0.
Dynamic Compression Tests
Strain rate effects on concrete strength have been examined experimentally by many
investigators [27 - 38]. However, in order to quantify the strain rate sensitivity of this particular
lightweight concrete in both tension and compression, a series of Split-Hopkinson Pressure Bar
(SHPB) tests were conducted on a 2.0 inch (5.08 cm) diameter SHPB located at Tyndall Air
Force Base, Florida. This particular SHPB has been described in full detail by Ross [38]. In
particular, compression tests on concrete from strain rates of 1/sec to 500/sec have been
conducted using this system. A schematic of the SHPB system is shown in Figure 10.
The striker bar impacts the incident bar which induces an elastic wave whose magnitude
and pulse length are proportional to the striker bar velocity and length, respectively. This elastic
wave is propagated down the length of the incident bar and impinges upon the cylindrical
concrete specimen, where part of the elastic wave is transmitted into the transmitter bar, and part
of the wave is reflected back into the incident bar. Ross [38] shows that for a specimen whose

Table 7. Quasistatic Indirect Splitting Tensile Tests on 2.0 in ( 5.08 cm) Diameter
by 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Diameter
in (cm)
Specimen
Length
in (cm)
Load at
Failure
lbs (kN)
Time to
Failure
min (sec)
Load Rate
lbs/min
(kN/min)
Stress
psi (MPa)
Stress Rate,
psi/sec
(kPa/sec)
Strain* Rate,
1/sec
Dynamic
Increase
Factor
1
1.99
(5.06)
2.012
(5.111)
4888
(21.7)
2.44
(146.4)
2003
(8.9)
776
(5.4)
5.3
(36.6)
1.53 x 10 °
1.23
2
1.99
(5.06)
2.012
(5.111)
3840
(17.1)
1.92
(115.2)
2000
(8.9)
611
(4.2)
5.3
(36.6)
1.53 x 10'°
0.97
3
1.99
(5.06)
2.012
(5.111)
3160
(14.1)
1.58
(94.8)
2000
(8.9)
502
(3.5)
5.3
(36.6)
1.53 x 10'°
0.8
4
2.00
(5.08)
2.008
(5.28)
4520
(20.1)
0.67
(40.0)
6780
(30.2)
717
(4.9)
17.9
(123.5)
5.17 x 10 °
1.01
5
2.00
(5.08)
2.008
(5.28)
4180
(18.6)
0.44
(26.5)
9464
(42.1)
663
(4.6)
25.00
(172.4)
7.22 x 10'°
0.94
6
2.00
(5.08)
2.008
(5.28)
4680
(20.8)
0.43
(26.0)
10,800
(48.0)
740
(5.1)
28.5
(196.6)
8.22 x 1 O'6
1.05
* Based on a Secant Modulus Calculated From an Average Compressive
Failure Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)

31
Striker Bar
V
Sample
Incident Bar
Strain i
Gage 1
1
Transmitter Bar
Bridge
Strain i
Gage 1
Bridge
Strain Gage
Strain Gage
Conditioner
Conditioner
Recorder â– 
Digital
Oscilloscope
Figure 10. Schematic of the Compressive Split-Hopkinson Pressure Bar
length is small compared to the length of the elastic wave, the integral of the reflected pulse is
proportional to the strain in the specimen. The incident and reflected pulses are recorded by a
strain gage located on the incident bar, and the transmitted pulse is recorded by a strain gage
located on the transmitter bar. The strains associated with each of these three pulses are
monitored by a full bridge of strain gages, which is then amplified and conditioned using a strain
gage conditioning amplifier, Model 2311 (Measurements Group, Raleigh, North Carolina). The
signals are then recorded on a Nicolet 4094 digital oscilloscope (Nicolet Instrument
Corporation, Madison, Wisconsin). The traces are subsequently stored for further processing
and analysis on a disk recorder. Following these procedures, a series of seventeen direct
compression tests were run on the SHPB, using a 26.0 in (66.04 cm) striker bar, and gun
chamber pressures of 20 - 200 psi (0.138 - 1.379 MPa). For these direct compression tests, the
specimens were lightly lubricated using a molybdenum disulfide grease, and pressed between the

32
incident and transmitter bars. They were held in place by friction between the two bars. Ross
[38] developed the following conversion factors, specific to this SHPB, to determine the incident
and transmitted stresses, and the strain rate from the strain gage data.
ctt = volts/gain x (2.058 x 10 ) (4)
6
e = volts/gain x (1.4 x 10 ) (5)
where
ctt Transmitted Compressive Stress, psi
s Strain Rate, 1/sec
Once the strain rate and compressive failure stress has been calculated, the dynamic increase
factor may be calculated by dividing the failure stress by the average of the three quasistatic
failure stresses, which was previously found to be 6708 psi (46.3 MPa). Table 8 shows the data
for the seventeen SHPB direct compression tests using the 26.0 in (66.04 cm) striker bar.
Dynamic Splitting Tensile Tests
To understand the potential dynamic strength enhancement due to strain rate effects of
lightweight concrete in tension, a series of thirteen splitting tensile SHPB experiments were also
conducted. The splitting tensile test was chosen over the more difficult to conduct direct tensile
test so as not to incur the difficulty of clamping or holding the specimen to achieve axial tension,
and the uncertainty of developing secondary stresses in the specimen induced by the holding
fixtures.
With regard to the specimen, it is rotated ninety degrees with respect to the incident and
transmitter bars. Two small pieces of steel stock, 0.25 x 0.25 x 2.0 in (6.35 x 6.35 x 50.8 mm) in
size, each with one side radiused to fit the contour of the concrete specimen, are affixed to the
outside of the specimen along a diameter line, parallel to each other, to help distribute the load as
shown in Figure 11. Figure 12 shows the orientation of the specimen in the SHPB. For the

33
Table 8. Direct Compression Tests on 2.0 in (5.08 cm) Diameter By
2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Gun
Pressure
psi (kPa)
Amplifier
Gain
1A/1B
Transmitted
Stress,
v/psi/MPa
Strain
Rate
v/l/sec
Dynamic
Increase
Factor (DIF)
DCLW - 1
20
(137.9)
100/1000
4.22/8799/62.7
0.448/62.7
1.31
DCLW - 2
30
(206.9)
100/1000
4.83/9940/68.6
0.770/107.8
1.48
DCLW - 3
50
(344.8)
100/100
0.535/11,010/75.9
1.098/153.7
1.64
DCLW - 4
100
(689.7)
100/100
0.705/14,509/100.1
1.678/234.9
2.16
DCLW - 5
20
(137.9)
100/1000
4.365/8986/62.0
0.474/66.4
1.34
DCLW - 6
30
(206.9)
100/1000
4.81/9899/68.3
0.720/100.8
1.48
DCLW - 7
50
(344.8)
100/100
0.494/10,167/70.1
1.026/143.6
1.52
DCLW - 8
100
(689.7)
100/1000
6.77/13,936/96.1
1.524/213.4
2.08
DCLW - 9
100
(689.7)
100/1000
6.705/13,799/95.2
1.540/215.6
2.06
DCLW - 10
50
(344.8)
100/1000
5.23/10,763/74.2
0.984/137.8
1.60
DCLW - 11
30
(206.9)
100/1000
4.39/9035/62.3
0.720/100.8
1.35
DCLW - 12
20
(137.9)
100/1000
4.57/9405/64.9
0.536/75.0
1.40
DCLW - 13
150
(1034.5)
100/1000
7.48/15,394/106.2
1.940/271.6
2.30
DCLW - 14
150
(1034.5)
100/100
0.67/13,789/95.1
1.870/261.8
2.06
DCLW - 15
150
(1034.5)
100/100
0.734/15,106/104.2
1.920/269.0
2.25
DCLW - 16
200
(1379.3)
100/100
0.754/15,517/107.0
2.250/315.0
2.31
DCLW - 17
200
(1379.3)
100/100
0.66/13,583/93.7
2.340/328.0
2.03

34
P
2.0 in
(5.08 cm)
Figure 11. Schematic of Test Specimen Preparation for SHPB Splitting Tensile Test
TOP VIEW
Figure 12. Side and Top Views of Test Specimen Orientation for SHPB Splitting Tensile Test

35
splitting tensile test, the incident and transmitted signals are both compressive, and the reflected
signal is tensile. For analysis purposes, the peak of the transmitted compressive signal is
converted to a load which is then assumed to be the peak load applied to the specimen, similar to
the quasistatic loading case shown in Figure 9. The static tensile stress ft normal to the load
direction is given by Equation (3). Equation (4) is the conversion from volts to transmitted
compressive stress, and must be modified in order to calculate the splitting tensile stress. It is
easily shown that all that is necessary to modify Equation (4) to account for splitting tensile
stress is to divide by the specimen length, or simply
crT = volts/gain x (1/L) x (2.058 x 10 ) (6)
where
L Specimen Length, in
To calculate the strain rate, pick two points on the ascending or pre-peak portion of the
transmitted compressive signal, and calculate the slope. Then, using Equation (6), we can
calculate the splitting tensile stress rate. Finally, dividing by the quasistatic secant modulus,
which was calculated from an average compressive failure stress of 6,708 psi (46.3 MPa), allows
us to compute the strain rate. One may question whether a statically determined value of the
secant modulus is a valid parameter to use to calculate strain rate from stress rate. However,
Ross [38] states that the dynamic modulus data obtained from SHPB tests are not valid since
elastic deformation occurs in the rise time of the load pulse, and during that time the specimen is
not uniformly loaded along its length. Nonuniformly loaded specimens have nonuniform strain
distributions which may yield false modulus data. Additionally, John and Shah [39] assume the
modulus to be rate independent, which has also been observed by Gopalaratnam and Shah [40],
and Tinic and Bruhwiler [41].

36
Once the strain rate and splitting tensile failure stresses have been calculated, the
dynamic increase factor may again be calculated, by dividing the failure stress by the average of
the three quasistatic splitting tensile failure stresses, found previously to be 630 psi (4.3 MPa).
Table 9 shows the data for the thirteen SHPB indirect splitting tensile tests. The first nine tests
Table 9. SHPB Indirect Splitting Tensile Tests on 2.0 in (5.08 cm) Diameter
By 2.0 in (5.08 cm) Long Lightweight Concrete Specimens
Specimen
Number
Specimen
Length
in (cm)
Gun
Pressure
psi (kPa)
Amplifier
Gain
1A/1B
Transmitted
Stress, a,
v/psi/MPa
Strain *
Rate, é
mV 1
gs sec
Dynamic
Increase Factor
(DIF)
SCLW - 1
2.008
(5.100)
8
(55.2)
1000/1000
0.968/992/6.8
20.2/5.98
1.58
SCLW - 2
2.012
(5.111)
8
(55.2)
1000/1000
0.950/972/6.7
16.2/4.77
1.54
SCLW - 3
2.013
(5.113)
8
(55.2)
1000/1000
1.164/1190/8.2
25.1/7.45
1.89
SCLW - 4
2.012
(5.111)
30
(206.9)
100/1000
1.650/1688/11.6
120.2/17.25
2.68
SCLW - 5
2.009
(5.103)
30
(206.9)
100/1000
1.434/1469/10.1
93.5/13.44
2.33
SCLW - 6
2.015
(5.118)
30
(206.9)
100/1000
1.732/1769/12.2
133.6/19.16
2.81
SCLW - 7
2.013
(5.113)
50
(344.8)
100/1000
1.288/1317/9.1
50.0/14.74
2.09
SCLW - 8
2.012
(5.111)
50
(344.8)
100/1000
0.808/2849/12.8
66.5/19.60
2.94
SCLW - 9
2.013
(5.113)
50
(344.8)
100/1000
1.728/1767/12.2
66.8/19.70
2.80
SCLW - 10
2.011
(5.108)
15
(103.5)
1000/1000
1.012/1036/7.1
3.7/1.03
1.64
SCLW - 11
2.015
(5.118)
15
(103.5)
1000/1000
0.878/897/6.2
5.1/1.50
1.42
SCLW - 12
2.011
(5.108)
20
(137.9)
1000/1000
0.690/706/4.9
5.0/1.48
1.12
SCLW - 13
2.013
(5.113)
20
(137.9)
1000/1000
1.050/1073/7.4
6.9/2.04
1.70
* Based on a Secant Modulus Calculated From an Average Compressive Failure
Stress of 6708 psi (46.3 MPa); Ec = 3.46 x 106 psi (23.89 GPa)

37
were conducted with a 26.0 in (66.04 cm) striker bar; the last four tests were conducted with a
50.0 in (127 cm) striker bar. The longer striker bar allowed the samples to be loaded at a lower
strain rate while maintaining gun chamber pressures in the 15 - 20 psi (10.35 - 137.9 kPa) range.
It was difficult to lower the chamber pressure below 8 psi (55.2 kPa) with the 26.0 in (66.04 cm)
striker bar.
Test Results
Failure of the static direct compression samples was by crushing, and the six samples are
almost identical in appearance. These tests were used as a basis for comparison with the other
data, since the unconfined compressive stress is the major property used in almost all discussion
of concrete data. The SHPB direct compression results show that as the load rate, hence the
strain rate increases, the number of fracture surfaces increases, hence the amount of
pulverization of the sample increases.
Failure of both the static and dynamic splitting tensile cylinders are almost identical,
breaking along a diameter plane. For the SHPB splitting tensile tests, with increasing load rate,
the split cylinder halves have increasing velocity and additional fractures occur during impacts
with the side walls of the debris catcher on the SHPB. However, Ross [38] states that during
splitting tensile tests in which high speed photography was used, the halves appear to be intact
after splitting. In fact, the films show that the fracture begins to occur first near the center of the
test specimen, which is also borne out by numerical simulations, which show that the tensile
stresses are larger in the center of the specimen than near the edges.
Figure 13 is a graph of the results for all of the quasistatic and SHPB direct compression
tests. The dynamic increase factor (DIF) is plotted as a function of the logarithm of the strain
rate for five different strength normal weight concrete mixes, as well as for the lightweight
concrete under investigation. One may conclude from this data that in compression the

38
CONCRETE STRAIN RATE EFFECTS
COMPRESSIVE STRENGTH
7
6
5
O 4
P
2 3
2
1
0
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
LOG STRAIN RATE, 1 / SECONDS
|a
® m¡¡&,
**
IF u *
a E Mix, fc = 7900 psi * F Mix, fc = 8250 psi A G Mix, fc = 5700 psi a H Mix, fc = 5600 psi
0 J Mix, fc = 4060 psi ^ LW Mix, fc = 5775 psi
Figure 13. Concrete Strain Rate Effects - Compressive Strength
lightweight concrete behaves identically, at least within the scatter of the data, to its normal
weight counterpart, whether quasistatically or dynamically. The lightweight concrete shows
moderate strain rate sensitivity, similar to its normal weight counterpart, with increases in
strength up to 2.3 times the static value at strain rates from 100 to 300/sec.
Figure 14 shows the analogous splitting tensile data. In tension, the lightweight concrete
does not appear to fall within the data scatter for the five different strength normal weight
concrete mixes. In fact, the data show that the lightweight concrete is less strain rate sensitive
than its normal weight counterparts, being shifted about a half-decade in strain rate for a constant
dynamic increase factor. Conversely, this means that the lightweight concrete must be loaded
approximately three times faster in order to have the same dynamic strength as normal weight
concrete. In either case, the normal and lightweight concrete have a higher strain rate sensitivity

39
CONCRETE STRAIN RATE EFFECTS
TENSILE STRENGTH
â– 
â– 
\
♦
t
♦
»&
1
éVi. ♦
♦°
â–  LJo
X
P
o
z
I
(- C
Í5
u
2
<
-7
-4 -3 -2 -1 0
LOG STRAIN RATE, 1 / SECONDS
B E Mix, fc = 7900 psi * F Mix, f c = 8250 psi 4 G Mix, f c = 5700 psi D H Mix, fc = 5600 psi
0 J Mix, fc = 4060 psi ^ LW Mix, fc = 5775 psi
Figure 14. Concrete Strain Rate Effects - Tensile Strength
in tension than in compression, which is consistent with the results of other researchers [31,35,
36, 38, 39, 41] work on normal weight concrete. The author is unaware of any previous strain
rate sensitivity studies on lightweight concrete.
Static Beam Bending Experiments
The next step towards understanding the potential benefits of external application of
CFRP panels to concrete beams is a careful study of the static bending behavior of the beams
when subject to a quasistatic center point loading condition, simply supported, as shown in
Figure 15. Plain concrete beams, as well as beams with one, two, and three ply CFRP panels
bonded to the bottom or tension side of the beams were tested. Additionally, two more beams in
which three ply CFRP was bonded to the sides as well as the bottom, were tested.
Description of the MTS~ Load Frame
The three point bending tests were conducted on the MTS® 880 load frame, (MTS
Systems Corporation, Minneapolis, Minnesota). The load frame may be run in either a load

40
P
Figure 15. Schematic of Quasistatic 3 Point Bending Tests
control or a displacement control mode, and is capable of loading material specimens up to
50,000 lbs (222.95 kN) in both uniaxial tension or compression. However, special tension
platens are required for cementing the concrete tensile specimens when conducting tension
experiments. Figure 16 shows a photograph of the MTS load frame with a concrete beam test
specimen.
The supporting platform on the machine was modified to accommodate the supported
span length of 27.0 in (68.58 cm) by placing roller supports in a notched plate which was then
bolted to the support platform by a series of Allen head bolts as shown in Figure 17. Since the
beams were 30.0 in (76.2 cm) long, and the supported span length was 27.0 in (68.68 cm) there
was a 1.5 in (3.81 cm) overhang on each end of the beam. The load cell was fitted with a special
half-cylinder platen to load the entire width of the beam uniformly at its center.

41
Figure 16. MTS Load Frame located at Wright Laboratory, Tyndall Air Force Base, Florida
Figure 17. MTS Support Platform Arrangement

42
Instrumentation Used/Measurements Made
(R)
The beams were loaded using the load control mode of operation on the MTS , at a load
rate of 2 Ibs/sec. The crosshead load, time, and deflection were recorded every two seconds by
the data acquisition software resident in the MTS controller. Beam midpoint displacement was
also recorded using a small linear voltage displacement transducer (LVDT), which was mounted
separately on the MTS supporting platform. A small, flat circular tab was glued to the bottom
center of each beam to facilitate attachment of the LVDT to the beam's midpoint. Midspan
deflection was also recorded by the MTS data acquisition system every two seconds. Load to
failure versus deflection curves could then be generated for each beam tested; therefore, the
static fracture energy could be computed by calculating the area under the load to failure versus
deflection curve.
Test Results
A total of 16 beams were tested to failure on the MTS load frame in three point
bending. Since plain concrete is a brittle material, failure occurs abruptly and catastrophically.
Beams with some tension steel reinforcement in them fail somewhat more “gracefully”, due to
the ductile nature of steel. Reinforcing steel used in concrete structural elements generally follow
an elastic - plastic with strain hardening stress - strain curve, as shown in Figure 18, with yield
points typically in the 40 - 60 ksi (276 - 414 MPa) range. Typical elastic moduli are in the 29
Msi (200 GPa) range, with strain hardening moduli in the 1.2 Msi (8.3 GPa) range. As
mentioned previously, due to the small cross - section of these beams, no tension steel was used
in them.
A typical load - displacement curve for the plain concrete beam is shown in Figure 19.
Since these beams have neither steel nor CFRP tension reinforcement in them, they remain
elastic until failure, at which time the lower "fibers" of the beam reach a stress level which

43
Figure 18. Typical Stress-Strain Curve For Reinforcing Steel Used In Concrete Structures
LOAD vs DISPLACEMENT
BEAM LW5-38, PLAIN CONCRETE
700
600
_/ '
500
C/3
| 400
2
§ 300
o
J 200
100
0
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014
DISPLACEMENT, INCHES
Figure 19. Load - Displacement Curve for Beam LW5-38, Plain Concrete Beam

44
exceeds the modulus of rupture value for lightweight concrete. A crack merely forms at the
bottom of the beam, and runs to the upper surface, simply breaking the beam into two pieces.
Since the modulus of rupture for concrete is on the order of 10 - 12 percent of the
unconfined compressive strength f c, the loads at failure for these beams are quite low, typically
about 640 lbs (2898 N). The corresponding displacement is also quite small; therefore the
fracture strengths are also quite small for the plain concrete beams. Appendix B contains the
load displacement curves for the three plain concrete beams tested.
Next, beams reinforced with one ply CFRP on the bottom were tested. Figure 20 shows
the load - displacement curve from beam LW2-16. Immediately, one notices a significant
difference in the appearance of the load - displacement curve in that there are two distinct
regions; an initial all - elastic region with a much higher slope or stiffness than the second
region, which continues until failure occurs at approximately 1150 lbs (5116 N). The
displacement at failure is 0.160 in (4.07 mm). The failure load for the beam with one ply CFRP
is 1.8 times higher than for the plain concrete beam. The stiffness in Region 1 is 55,700 lbs/in
(97.5 kN/cm), and the stiffness in region 2 is 2960 lbs/in (5.18 kN/cm). The beam also fails by a
different mechanism. No longer does a flexural crack merely split the beam in two. With the
addition of the CFRP, the flexural crack(s) still occur in the center cross section of the beam, but
the beam now fails by a combination of flexure and shear at a distance of about L/4 on either
side of the beam centerline. As soon as the combination of bending and shear stress combine to
produce a maximum principal stress higher than the tensile fracture stress, the beam fails. This
O
crack starts on the bottom of the beam and runs upward at a 45 angle until it reaches the neutral
axis of the beam, which itself has shifted upward when the initial flexural crack(s) first appeared,
as the concrete reached its modulus of rupture. Figure 21 is a photograph of beam LW2 - 16

45
LOAD vs DISPLACEMENT
BEAM LW2-I6,1 PLY CFRP BOTTOM ONLY
3000 -j
2500 -
VO
0 0.05 0.1 0.15 0.2 0.25
DISPLACEMENT, INCHES
Figure 20. Load - Displacement Curve for Beam LW2-16, Plain Concrete Beam with One Ply
CFRP
Figure 21. Typical Failure of Concrete Beam Reinforced with One Ply CFRP

46
with one ply CFRP, loaded to failure. This shift in the neutral axis has been previously studied
by Sierakowski et al. [21], and Ross et al. [22], and is the basis of the section analyses which are
derived and discussed in detail in Chapter 3. Furthermore, as the outer portion of the now failed
beam rotates upward, the peel strength of the Flysol epoxy adhesive, 30 lbs/linear inch (52.5
N/linear cm) is exceeded, and the CFRP delaminates from the bottom of the beam. This
delamination consistently originated at the point where the crack started at the bottom of the
beam and ran to the end of the beam, as shown in Figure 21 above.
Next, three beams were tested with two ply CFRP on the bottom or tension side of the
beams. The failure loads increased to about 2300 lbs (10.22 kN), with corresponding
displacements of about 0.241 in (6.13 mm). The shape of the load-displacement curve was very
similar to that of the beams reinforced with one ply CFRP, with two distinct regions; one all¬
elastic region with a high stiffness 60,000 lbs/in (105.26 kN/cm), and then an abrupt change in
slope as the flexural cracking occurs, with a concomitant reduction in stiffness to about 6570
lbs/in (11.51 kN/cm). The failure is quite abrupt, and occurs when the combination of moment
and shear produce stresses which exceed the tensile strength of the concrete, usually
symmetrically distributed about the beams' centerline, at about the quarter span location. Figure
22 shows a photograph of beam LW4-27, a plain concrete beam with two ply CFRP, loaded to
failure. As the failure occurred and the outer portion of the failed beam rotated upward, the
CFRP was peeled off the bottom of the beam, starting at 7.5 in (19.05 cm) from the beam's
center, and continuing outward to the end of the beam, as shown in Figure 23.
Two concrete beams with three ply CFRP were tested next, and the load - displacement
curve for beam LW9-67 is shown in Figure 24. The load-displacement curve is similar to those
shown for the one and two ply beams, with two distinct regions; the first all - elastic region has a
stiffness of about 59,100 lbs/in (103.50 kN/cm) and the second region has a stiffness of

47
Figure 22. Typical Failure of Concrete Beam Reinforced with Two Ply CFRP
Figure 23. Damage Assessment of Beam LW4-27 with Two Ply CFRP

48
Figure 24. Load - Displacement Curve for Beam LW9-67, Plain Concrete Beam with Three Ply
CFRP
9797 lbs/in (17.16 kN/cm). The initial elastic stiffness for the two ply beam is slightly greater
than the three ply beam because its unconfined compressive strength f c is larger, hence, its
modulus of rupture value is larger, which translates to a slightly higher elastic stiffness. After
failure occurs, one notices that the CFRP does not delaminate from the initial crack location on
the bottom of the beam and continue along the bottom of the beam to the end, as in the one and
two ply beams. Instead, the CFRP appears to have been pulled out of the concrete for only a
small portion along the bottom, and then is still securely bonded to the remainder of the beam on
out to end, as shown in the Figure 25.
Interestingly enough, the 2448 pounds (10.89 kN) average total load and the associated
displacements at failure are both not significantly different for these three ply beams than the
values for the two ply beams. In fact, the total displacements at failure are less. This may be due
to the fact that the stiffness of the three ply beam at failure is about 3500 lbs/in (6.13 kN/cm),

49
Figure 25. Damage Assessment of Beam LW9-67 with Three Ply CFRP
which is 49 percent greater than the stiffness of the two ply beam, which translates to higher
bending and shear stresses for a given amount of displacement.
Three more beams were tested in the MTS machine with three ply CFRP on the bottom
or tension side of the beams, except that for these beams the concrete mix was modified with
0.75 in (1.905 cm) long nylon fibers, at three pounds (1.36 kg) per cubic yard. Figure 26 shows
the load - displacement curve for beam LWF10-76, which shows the two characteristic regions.
The first all - elastic region has an initial stiffness of 54,900 lbs/in (96.15 kN/cm), and the second
region has a slope or stiffness of about 10,390 lbs/in (18.20 kN/cm). The failure loads averaged
2349 lbs (10.45 kN), and the failure displacements averaged 0.175 in (4.45 mm). Figure 27
shows a diagram of the post-test damage on beam LWF 10-76, which shows damage quite similar
to the plain concrete beams with three ply CFRP. In fact, it is impossible to discern any
qualitative or quantitative difference, within the scatter of the data, between the beams with
nylon fibers and three ply CFRP, and beams without nylon fibers and three ply CFRP. Figure 28

50
3500
LOAD vs DISPLACEMENT
BEAM LWF10-76, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY
0.1 0.15
DISPLACEMENT, INCHES
0.2
0.25
Figure 26. Load-Displacement Curve for Beam LWF 10-76, Nylon Fiber
Concrete Beam with Three Ply CFRP
Figure 27. Damage Assessment of Beam LWF 10-76 with Three Ply CFRP

51
shows a photograph of the post-test damage inflicted on beam LWF10-76. The nylon fibers did
not appear to influence the failure mechanism either.
Figure 28. Typical Failure of Nylon Fiber Concrete Beam Reinforced with Three Ply CFRP
The last type of beams tested were plain concrete beams externally reinforced with three
ply CFRP on the bottom as well as both sides of the beam, shown in Figure 29. Two beams were
tested in this configuration, and the load - displacement curve for beam LW7-53 is shown in
Figure 30. Unfortunately, the two beams tested yielded a large difference in experimental
results; beam LW9-70 failed at 4060 lbs (18.06 kN), and beam LW7-53 failed at 5297 lbs (23.56
kN), with displacements at failure of 0.205 in (5.21 mm) and 0.284 (7.21 mm) respectively. The
displacements were not significantly greater than those of the two ply beams. However, the
loads at failure were a factor of two higher than for those beams with either two or three ply
CFRP on the bottom only.

52
Figure 29. Concrete Beam with Three Ply CFRP on Bottom and Sides
Figure 30. Load - Displacement Curve for Beam LW7-53, Plain Concrete Beam
Reinforced with Three Ply CFRP on Bottom and Sides

53
The load - displacement curves are of the same two region form, with an initial elastic
stiffness of about 63,200 lbs/in (110.68 kN/cm), an abrupt change of slope at the concrete's
modulus of rupture point, with a resulting reduced stiffness of about 16,709 lbs/in (29.26 kN/cm)
until failure occurs. The initial elastic stiffness is roughly the same as that of the other beams,
with a slight increase which may be attributed to the side CFRP. However, after the modulus of
rupture is reached within the concrete and the slope abruptly changes, the beams remain
substantially stiffer than the other four types of beams reinforced with bottom CFRP. Table 10
below is a summary of the average initial elastic stiffnesses, and the average post-flexural
Table 10. Summary of Beam Stiffnesses from MTS® Static Three-Point Bending Experiments
Number of Plies of CFRP
Elastic Stiffness
lbs/in (kN/cm)
Post-MOR Stiffness
lbs/in (kN/cm)
0
45,714
(80.06)
55,700
2,960
1
(97.55)
(5.18)
60,100
6,570
2
(105.26)
(11.51)
59,100
9,797
3
(103.50)
(17.16)
54,900
10,390
3*
(96.15)
(18.20)
63,200
16,709
3**
(110.68)
(29.26)
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard
** Three ply CFRP Both Sides and Bottom
cracking or post-modulus of rupture (MOR) stiffnesses for all six types of beams tested. Note
that the beams with three ply CFRP on all three sides are approximately 60 - 70 percent stiffer
than the other two types of beams reinforced with three ply CFRP on the bottom only. Figure 31
below shows the post - test damage of beams LW9-70 and LW7-53. Note that the side CFRP is
split and buckled along the major shear failure crack in the concrete, which is at 45 degrees from

54
Figure 31. Typical Failure of Concrete Beam Reinforced on Bottom
and Both Sides with Three Ply CFRP
the vertical as shown in Figure 32. It is theorized that the side CFRP prevents the concrete from
rotating or lifting upward upon failure, thereby preventing the peeling or delamination failures
seen in the one and two ply test specimens earlier. The side CFRP also holds all the pieces of the
failed beam together as one unit. In fact, the beams still maintain a limited capacity to carry a
load, although this fact was not quantified during the course of this investigation.
Calculation of Fracture Energies
Failure may be defined in many ways for a structure or structural element. In this case,
it is the abrupt termination in the ability of the beam to take any more load increments, which for
these beam bending experiments run in load control, was two lbs/sec. Once the maximum
principal stress at some location in the beam exceeded the concrete's tensile strength, the
concrete fails, hence the beam fails.

55
SIDE CFRP
BUCKLED
GOOD BOND
CFRP PULLED
OUT OF CONCRETE
4 in
(10.16 cm)
— 8 in —►
(20.32 cm) i
12 in
(30.48 cm)
Figure 32. Damage Assessment of Beam LW7-53 with Three Ply CFRP Bottom and Sides
Fracture energies were calculated for all of the beams in an attempt to quantify the
capability of the beams to absorb energy. Fracture energy was then defined as the area under the
load - displacement curve until failure occurred; the area under the load - displacement curve
was found by numerical integration using the Trapezoid Rule.
Assuming that a straight line approximation between each set of load - displacement
points is reasonable, the area of each trapezoidal segment is calculated and summed, and the
areas or fracture energies of all sixteen experimental load - displacement curves were so
determined. Appendix C is a tabular summary of all of the relevant data from the static MTS®
three point bending experiments. Also displayed in the table are load increase, displacement
increase, and fracture energy increase ratios. The load increase ratio is defined as the peak load
divided by the average peak load of the three plain concrete beams, which was 637.3 lbs (2.83
kN). The displacement increase ratio is defined as the maximum displacement divided by the

56
maximum displacement recorded on plain concrete beam LW5-38. The reason all three
displacements were not averaged and used as the baseline as in the case of the load increase ratio
was because the LVDT did not work very well on beams LW5-33 and LW5-36, as can be seen
by their respective load - displacement curves in Appendix B, where the LVDT did not begin
recording displacement immediately after loading began but instead abruptly jumped up to some
value before it worked properly. Finally, the fracture energy increase ratio is defined as the
fracture energy divided by the fracture energy calculated from plain concrete beam LW5-38.
Table 11 shows the average load, displacement, and fracture energy increases computed
from the results of the table presented in Appendix C. Several conclusions may be drawn after
Table 11. Average Load, Displacement, and Fracture Energy Increase
Ratios Compared to Plain Concrete
No. of
Plies of
CFRP
Average
Load
Increase
Ratio
Average
Displacement
Increase
Ratio
Average
Fracture
Energy
Increase
Ratio
1
1.80
11.45
29.83
2
3.60
17.24
77.76
3
3.85
12.86
59.94
3*
3.68
12.53
58.38
3**
7.34
17.47
145.84
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard
** Three Ply CFRP Both Sides and Bottom
careful review of Table 11. Beams with three ply CFRP on the bottom and sides are clearly able
to take the most load before failure, with an average load increase ratio of 7.34. Somewhat
surprising is that the beams with nylon fibers in them and reinforced with three ply CFRP appear
to offer little advantage in either load or displacement to failure over the plain lightweight

57
concrete beams with three ply CFRP on the bottom only. One might theorize that the fibers
should provide some additional energy absorption capability in the form of crack attenuation
capability, but the average fracture energy increase ratio is 58.38, which is slightly less than the
plain lightweight concrete mix with three ply CFRP with a 59.94 fracture energy increase ratio.
In fact, the three ply fibrous concrete, three ply regular lightweight concrete, and two ply regular
lightweight concrete all have similar total load to failure capabilities, with the one ply CFRP
beams having a factor of two less load carrying capacity. Most surprising are the relative
ductilities of the beams; the two ply beams are the most ductile of the bottom or tension side
only reinforced beams. The three sided beams are roughly equivalent in ductility. Therefore, the
fracture energy increase ratio is the highest for the two ply beams when compared to its tension
reinforced only counterparts, which is solely attributed to this increased ductility. Since the two
ply and three sided three ply beams have roughly the same ductility, the factor of two increase in
fracture energy of the three sided three ply beams is directly proportional to its load increase
ratio.
In summary, it has been experimentally demonstrated that beams with three ply CFRP
on the bottom and sides have the highest load, displacement, and fracture energy increase ratios.
Beams with two ply CFRP on the bottom only have the next highest fracture energy increase
ratio, due to their high ductility and relatively high load increase ratio. Conversely, the
experimental evidence indicates little to no benefit is realized using nylon fibers in the concrete
mix as a potential technique to increase the static load, displacement, or fracture energy capacity,
when compared to similarly reinforced beams without the nylon fibers added to the mix. There
are still tremendous gains over the plain concrete beams in load and displacement capacity even
with the bottom only one, two, and three ply CFRP reinforced beams. Increases of two to four in
load and 11 to 17 in displacement are quite easily achieved with the addition of the CFRP.

58
Consequently, increases in fracture energy from 30 to 78 are also achieved, when compared to
the baseline plain concrete beams.
Dynamic Beam Bending Experiments
In order to determine the dynamic response and behavior of beams externally reinforced
with CFRP panels, a series of 54 drop weight tests were conducted in which the six beam types
were subjected to impulsive center point loading, shown in Figure 33. Mindess and Banthia
Figure 33. Simply Supported Beam With Dynamic Center Point Load
[27 - 33] have conducted drop weight impact tests on plain and conventional steel reinforced
concrete beam specimens with dimensions (length x width x height) 55.12 x 3.94 x 4.92 inches
(140.0 x 10.0 x 12.5 cm) and a simply supported span length of 37.80 in (96.0 cm). However,
their applied loads were not impulsive. Here, we define the term impulsive load as a load pulse
with a temporal duration less than 25 percent of the fundamental period of the structure

59
undergoing loading; in this case the structural element is a simply supported beam. None of the
previously cited CFRP reinforced concrete beam work [References 1 - 22] involved dynamic
loading, and for those conventionally reinforced beams dynamically tested by Mindess and
Banthia, the loading was not impulsive. Unfortunately, short duration impulsive loads greatly
complicate the process of analyzing the test results, and will be discussed in detail later.
Description Of The Wvle Laboratories Drop Weight Impact Machine
The test apparatus used in this investigation was a Wyle Laboratories SKM01 drop
weight impact machine (Wyle Laboratories, El Segundo, California). Figure 34 is an overall
view of the test apparatus.
The overall height of the apparatus is about 15 feet (4.6 m) above the floor. To provide
as rigid a platform as possible, the bottom frame of the impact machine is bolted to an 8 in
(20.32 cm) thick reinforced concrete floor slab, with four 0.75 in (1.905 cm) diameter by 10
threads per inch bolts with corresponding nuts and lock washers. The impact machine uses a tup
carriage to deliver the weight and instrumented tup to the test sample. The tup carriage relies on
two bearings to guide it along two Rockwell C60 hardened steel shafts to the impact location on
the test sample.
The tup carriage is machined out of aircraft quality 7075-T6 aluminum, and various
weights (steel plates) may be added in equal increments to each side of the tup carriage up to 100
lbs (45.5 kg) total. The additional weight(s) also structurally reinforces the tup carriage. The
weights are attached to the tup carriage via two load pins and a bottom weight plate. Four 0.625
in (1.59 cm) diameter by eighteen threads per inch socket head cap screws attach the additional
weights to the bottom plate, and two 1.0 in (2.54 cm) diameter by eight threads per inch nuts
fasten the weights to the load pins. The tup is attached to the tup carriage by four 0.125 in
(0.3175 cm) diameter by eighteen threads per inch socket head cap screws. The tup is machined

60
Figure 34. Overall View Of The Instrumented Drop Weight Impact Machine
from a Rockwell C40 hardened piece of 4340 steel, and the 4.0 in (10.16 cm) wide impacting
edge conforms to the Charpy impact requirements of ASTM D256-90b, paragraph 11.3
(formerly D256-56 10(d)). The tup itself weighs 1.5 lbs (0.68 kg). The total mass of the tup
assembly and weights used in this investigation was carefully weighed and found to be 96.14 lbs
(43.7 kg). The Wyle drop weight impact machine has the capability to provide a 10 ft (3.05 m)
free fall height; however, the highest drop height in this study was 2 ft (0.61 m).

61
Instrumentation Used
Some of the instrumentation used in the dynamic testing was attached to the drop weight
machine itself, some was attached to the beam test specimens, and yet another was external to
both the test machine and specimens. In order to measure the average acceleration and velocity
of the impact hammer as it drops and impacts the beam specimens, a photocell was mounted on
one of the two steel guide rails of the machine. The photocell's position could be adjusted up or
down so that the velocity of the tup could be measured just prior to impact. The photocell had a
slot in which a strip of aluminum that was attached to the impact hammer could pass through,
and break the continuity of the light source in the photocell. This strip of aluminum was
measured to be 1.2355 in (3.1382 cm) in width. An electronic counter was connected to the
photocell, and it recorded the amount of time that the light source was broken or off during the
passage of the aluminum strip through the slot in the photocell. The width of the aluminum strip
divided by the time recorded by the counter gave the impact velocity of the tup.
From conservation of energy considerations, a weight will fall under the influence of
gravity g and impact with a velocity of
v = V2gh (7)
when dropped from a height h above some datum, as shown in Figure 35. Unfortunately, the
acceleration of the impact hammer and tup assembly is not the acceleration of gravity g, due to
unavoidable friction between the hammer bearings and the steel guide rails. However, by using
the photocell to measure the impact velocity on a series of drop tests, and from Equation (7), the
acceleration of the hammer and tup assembly was consistently found to be about 0.9g. As a
result, careful attention was paid to the guide rails to minimize this friction effect; frequent

62
m
ii
o
t
T
«1
1
m
T
h
1
*
1
m
gh
\
\
\
\
\
///////
Figure 35. Sketch Of Mass m Dropped At Rest From Height h
cleaning of the steel guide rails with acetone between tests seemed to reduce the variability in the
machine's acceleration constant.
Calibration Of The Tup
The total load or resistance that the tup develops as it impacts the specimen was
measured by using the average of two sets of four electrical resistance strain gages, each set of
four strain gages being mounted on either side of the tup, as shown in Figure 36. The strain
gages were from Micro - Measurements (Micro - Measurements Division, Measurements
Group Incorporated, Raleigh, North Carolina). Two of the gages in each circuit measure
compressive strain directly, and the other two measure the transverse strain. Each circuit forms a
full four-gage bridge which is inherently temperature compensated, and gives increased
sensitivity from the Poisson effect in the transverse gages. An additional benefit to providing a

63
Figure 36. The Tup And Its Circuit
full bridge of strain gages on either side of the tup is to eliminate any bending effects in the tup
which may occur during a test; when the two output voltage signals are averaged, the effects of
bending are neutralized. The gages are connected to an Ectron Model 563H transducer
conditioning amplifier (Ectron Corporation, San Diego, California). Excitation voltage, output
gain, and bridge balancing are all manually set on the transducer conditioning amplifier.
Excitation voltage was nominally 5 Vdc, and the output gain was set at 100. Connections from
the transducer conditioning amplifier to the digital oscilloscope are direct and straightforward as
shown in Figure 37. The oscilloscope used was a Nicolet Model 4094B (Nicolet Instrument
Corporation, Madison, Wisconsin) four channel digital device with dual disk recording
capability. Normal operation used one disk for recording up to four channels of data in
conjunction with the second math pack disk being used for titles, delays, etc.

64
Nicolet® 4094
Recorder
Nicolet® 4094
Recorder
MATH PACK
DISK 2
DISK 1
â– 
â– 
Ectron®
Model 563H
Signal
Conditioning
Amplifier
Completion
Bridge
Completion
Bridge
Completion
Bridge
i
Supply/Conditioning
' Nicolet
-^cill
® 4094 \
OSCODC B
<
0.
3
H
m
CL
3
CN
T—
CNI
CO
o
o
%
o
a
CD
CT
n
<â– 
<
W
CO
CO
P(t), V
Photocell
TUP
SG
Bridge B
0.012
Photocell
Timer
fe
- SG
Bridge A
W
SG #1 ACC1 ACC2 P
SG #2
1 - SG #3 ~
Displ Gage
/
CONCRETE
BEAM
TEST
SPECIMEN
SJL
Hi-Speed
Camera
Figure 37. Schematic of the Instrumentation Used In The Dynamic Beam Bending Experiments

65
Calibration of the tup was done statically, by loading it in compression in the Forney*
load frame. The method is similar with one used by Bentur et al. [27] to calibrate the tup in a
drop weight machine at the University of British Columbia. In the present work, the tup was
disconnected from the drop weight machine and bolted to a steel plate, such that the blade of the
tup was perpendicular to the steel plate. The steel plate was then placed on the bottom platen of
the Forney load frame, with the tup facing upward. The crosshead was then lowered until it just
touched the edge of the tup. The two full strain gage bridges were balanced on the transducer
conditioning amplifier, and their output was connected to the Forney's data acquisition system.
The tup was then loaded to 50 kips (222.4 kN) at a load rate of 75 lbs/sec. A load of 50 kips
(222.4 kN) is about 45 percent of the compressive yield load for the 4340 steel from which the
tup is machined. It was not anticipated that the tup would be loaded much beyond 10 kips (44.5
kN) during actual dynamic testing, so 50 kips (222.4 kN) was deemed sufficient for calibration
purposes.
Results for the first tup calibration test are shown in Figure 38. Plotted is the output
voltage (amplified 100 times) versus the tup load. Each strain gage bridge is plotted separately
STATIC TUP CALIBRATION #1
FORNEY LOAD FRAME
Figure 38. Strain Gage Output Voltage Versus Load

66
versus the tup load, so there are two lines on the plot. Since the tup is responding elastically, the
strains and therefore the voltage are linear, as one would expect. After the first calibration test
was completed, a second test was conducted to ensure repeatability. The same linear response
was almost identically duplicated.
To establish an overall calibration factor for the tup, the slopes of all four voltage versus
load curves (lines) from both tup calibration tests were calculated and averaged together. This
value was found to be 30,277.3 lbs/volt (134.7 kN/volt). Assuming the frequency response of
the tup is much greater than the frequency response of the specimen, one may then use this
calibration, which is quite specific to tup geometry, strain gage excitation voltage, and transducer
conditioning amplifier gain, to convert a voltage versus time signal to a load versus time signal
in the dynamic tests.
After the tup was calibrated, it was installed back onto the impact hammer to begin
dynamic testing. Both bridge outputs from the Ectron signal conditioning amplifier were re¬
connected to the oscilloscopes, as shown in Figure 37. Since the tup load signal was used as the
internal trigger for all of the instrumentation, each bridge output was connected to a separate
oscilloscope. An internal trigger uses a rise in the load signal above a preset threshold value to
trigger data collection. A general rule of thumb when using an internal or load trigger for the
rest of the instrumentation is that the trigger level be set at roughly ten percent of the expected
maximum load to ensure that the trigger level is well above any background noise. A typical
value used for the trigger voltage was 20 mV, which equates to 605.5 lbs (2.69 kN) of load as
sensed by the tup. Additionally, an adequate number of data points collected prior to the trigger
signal must also be saved, in order to capture the initial rise in the tup load. Typically, 500 psec
of pre - trigger data points was sufficient to record the initial portion of the tup’s load - time
signal.

67
Instrumentation attached to the beam test specimens consisted of a series of electrical
resistance strain gages, which were applied to one side on the center cross-section of each beam,
as shown in Figure 39. These strain gages were also from Micro-Measurements®.
Figure 39. Sketch of Strain Gage Locations on the Beam
The strain gage located on the top of the beam was chosen as gage number one. It was
located at a distance of 2.75 in (6.99 cm) from the bottom of each beam. Unfortunately, the gage
could not be located on the top of the beam, due to the fact that it would be in the same location
that the tup would be striking and loading the beam. Gage number one was usually 1.9 in (4.8
cm) in length, in order to measure the representative strain over at least five aggregate diameters
in the concrete. However, when the sides of the beam were reinforced with CFRP, gage number
one was reduced in length to 0.25 in (0.64 cm), since strain in the CFRP is more uniform.

68
Strain gage number two was located on the geometric center of the beam and the initial
neutral axis location, as shown in Figure 39. Similar to the case with strain gage number one,
when the gage was mounted on concrete, it was 1.9 in (4.8 cm) in length, but when the beams
had side CFRP, the smaller 0.25 in (0.64 cm) gage length was used.
The bottom strain gage was chosen as gage number three. Usually, the shorter 0.25 in
(0.64 cm) gage length was used, except for the case of the unreinforced concrete beams, when
the 1.9 in (4.8 cm) gage length was used. Not all of the beams had three gages at the center cross
section; this three gage configuration was only used on every third beam tested. The remainder
of the beams had gages on the top or position one, and the bottom of the beam or position three.
The gages were mounted on the concrete beams in accordance with the manufacturer's
recommendations for surface cleaning and preparation, which were very similar to the
procedures followed for application of the Hysol adhesive and CFRP. In addition to surface
cleaning of the concrete, a bed of epoxy was initially applied to the area in which the 1.9 in (4.8
cm) concrete gage was to be affixed, in order to fill in any voids in the surface. After the epoxy
cured, it was block sanded back to the original surface height. The gages are usually fixtured in
place while the M-Bond AE-10 (Micro-Measurements Division, Measurements Group
Incorporated, Raleigh, North Carolina) two part epoxy cures, with a clamping device which
applies 5-20 psi (34.5 - 137.9 kPa) of pressure. However, since the vacuum bag technique
worked so well for fixturing the CFRP while the Hysol epoxy cured and produced 14 psi (96.55
kPa) of uniform hydrostatic pressure, this technique was also used to fixture all of the strain
gages while the M-Bond AE-10 epoxy cured as well. Typically, it took six hours for the epoxy
to cure at 75°F (23.9°C); all 54 beams were each left in the vacuum bag for this period of time.
Three completion bridges consisting of three 120 ohm resistors were made, and bridge
output was connected directly to the signal conditioning amplifier, where the gages could be

69
manually balanced. Excitation voltage for the bridges was nominally 5 Vdc, and the
amplification or gain was set at 10. In order to convert the strain gage output voltage versus time
to strain versus time requires elementary analysis of the Wheatstone Bridge circuit shown in
Figure 40, where one of the "resistors" in the circuit is the strain gage. When the gage is applied
Figure 40. Strain Gage Bridge Circuit
to the beam and subjected to a strain, say tensile, the magnitude of the resistance in the gage
would be increased, causing the bridge to become unbalanced. The magnitude of this imbalance
is measured as output of the bridge, and is proportional to the strain at that location in the beam.
The output of the bridge is measured as a change in voltage which is given by
QVe ARg AR3 AR2 AR4
(l + Q)2 Rg + R3 R2 R4
AV =
(8)

70
where
AV Change in Output Voltage, volts
Ve Excitation Voltage, volts
Rg Nominal Gage Resistance, ohms
ARg Change in Gage Resistance, ohms
Rt, R2, R3 Bridge Completion Resistor Resistances, ohms
AR), AR2, AR3 Bridge Completion Resistor Change in Resistance, ohms
The strain gage and the bridge completion resistors all have the same resistance so that
and Equation (8) becomes
R
gage
AV = ^
4 R
(9)
Each strain gage has a gage factor commonly denoted by GF. The gage factor is defined as the
ratio of the unit change in resistance to the unit change in length, which is given by
GF =
AR/R
AL/L
(10)
where
A R Total Change in Gage Resistance, ohms
AL Total Change in Gage Length, in
R Gage Resistance, ohms
L Gage Length, in
The gage factor is also an index of the strain sensitivity of a gage and is a constant for the small
range of resistance changes and strains normally encountered. Equation (10) may be rewritten as

71
— = e(GF)
R
which when substituted into Equation (9) yields
(11)
e
—Í—)
v vgf;
AV
(12)
Using the appropriate gage factors, excitation voltage, and amplifier gain in Equation (12) yields
the following calibration factors for the strain gages
s = 0.0381AV (13)
for the 0.25 in (0.635 cm) strain gages which were used on the CFRP and
E = 0.03 77AV (14)
for the 1.9 in (4.826 cm) strain gages, which were used directly on the concrete. These factors
may then be used to convert voltage versus time to strain versus time.
Piezoresistive accelerometers were also mounted on the beams in an attempt to measure
the distribution of acceleration along the length of the beam. Endevco model 7270A (Endevco
Corporation, San Juan Capistrano, California) piezoresistive accelerometers with a range of ±
20,000 g's were mounted side by side 4.5 in (11.43 cm) from beam midpoint. The
accelerometers have a frequency response of 50 kHz, and are very low mass, being etched from a
single piece of silicon, which includes the inertial mass and an active full Wheatstone Bridge
circuit, complete with an on - chip zero balance network. The accelerometers were then
connected to the Ectron signal conditioning amplifier, and subsequently to the Nicolet
recording oscilloscopes, as shown in Figure 37. Nominal excitation voltage was 5 Vdc, and the
gain was set at 5. Each accelerometer had its own calibration factor, given by the manufacturer,
which converted voltage versus time to g’s versus time.
Displacement versus time at beam midpoint was measured directly using a noncontact,
linear proximity measuring system. A Kaman Instruments Model KD2300® multipurpose,

72
variable impedance transducer (Kaman Instrumentation Corporation, Colorado Springs,
Colorado) with a 0.5 in (1.27 cm) measuring range was used. Since the beam static failure
displacements were on the order of 0.3 in (0.762 cm), this measuring range was deemed adequate
for dynamic testing. The sensor head was mounted underneath the beams, flush with the support
platform of the drop weight machine. A piece of non-magnetic aluminum tape was affixed to the
bottom midpoint of each beam, which was used as a target for the sensor head. The system had
its own signal conditioning electronics package, where zero, gain and linearity adjustments could
be made, so its output was hooked directly to the recording oscilloscope. The output voltage of
the system is proportional to the distance between the face of the sensor and the metallic target
located on the beam, so merely inverting the signal gave displacement versus time directly. The
frequency response of the system was 50 kHz.
The last piece of external instrumentation used in the dynamic beam bending
experiments was a 0.63 in (16 mm) high speed framing camera. The type of camera used was a
Photec Model 0061-0132A high speed rotating prism camera system (Photographic Analysis
Incorporated, Wayne, New Jersey), and was operated at a framing rate of 10,000 frames per
second. The camera was mounted to the side of the test specimens such that the lens of the
camera viewed the side of the beams, and was at the same height as the bottom of the beam. At
10,000 frames per second, additional lighting was required to illuminate the beam test specimens
during the impact event. The aperture setting on the camera was such that approximately 8 in
(20.32 cm) on either side of the beam’s midpoint was visible during the impact event. Due to the
high costs associated with developing high speed films, the framing camera was only used on
every third drop weight impact test. This coincided with those beams which had the full
complement of three strain gages mounted on the center cross section; these tests were denoted
the highly instrumented tests. Eighteen of these so-called highly instrumented impact tests were

73
conducted out of a total of fifty-four dynamic beam bending experiments. The high speed film
records also helped to determine the timing of the beam failure, the crack patterns, and verified
the midpoint deflection versus time data from the displacement gage, since each frame yielded
information in 100 psec increments. The eighteen high speed film records were also recorded on
0.5 in (1.27 cm) videotape for further study. However, detailed analysis of the high speed films
required the use of a Photo-Optical Data Analyzer, Model 224A (L-W International, Woodland
Hills, California) to view the individual events recorded on the films. The data analyzer allowed
stop motion, frame by frame, and 24 frame per second operation.
Method of Test
Prior to any actual impact experiments in which data was recorded, several dozen
preliminary drop tests were conducted in order to verify that all of the instrumentation was
working properly. Since the beam’s dynamic failure or fracture energies were initially unknown,
and the impact hammer weight was held constant at several iterations on the correct starting drop
height were required for each one of the six beam types in order to find incipient failure. Once
incipient failure was found the drop height could then be increased which not only increased the
incident energy, but also increased the load rate and hence the strain rate in the beam test
specimens. Additionally, once the tup load pulse amplitudes and durations were known, trigger
levels and delay times could be set on the recording oscilloscopes; typically 20 mV for the
trigger level and 500 psec for the delay or amount of pre - trigger information saved.
It was originally desired that all six beam types be tested at the same three drop heights.
However, due to the large difference in beam stiffnesses and fracture energies, this became
impractical if not essentially impossible to do. In addition, the inertial forces turned out to be the
dominant forces during the loading process, since the beams were so massive and brittle. The
drop height for incipient failure on the stiffest, most energy resistant beam, which were those

74
beams reinforced on all three sides with three ply CFRP, would have produced an overwhelming
inertial force on the least stiff beams, which were the plain concrete beams, and preclude
recovery of the dynamic bending load for these types of beams. However, as much overlap in
drop height between the six different stiffness categories of beams as practical was kept, in order
to allow direct comparison between consecutive beam stiffness classes.
In order to conduct a drop weight test, several preliminary procedures needed to be
accomplished. First, the wire leads for the strain gages had to be attached, the accelerometers
had to be glued to the beams using a special adhesive and adhesive accelerator, and the
aluminum tape was affixed to the bottom of the beam for use as a target for the displacement
gage. The beam was then placed on its supports in the drop weight machine, the leads for the
strain gages were attached to the completion bridges, and the accelerometer leads were
connected to the Ectron signal conditioning amplifier. The excitation voltages for all gages and
accelerometers were checked prior to each test, and the strain gage Wheatstone bridges on the
beam and on the tup were balanced. The tup was then raised to the selected drop height and
measured with a scale, making any fine adjustments in height to within 0.031 in (0.8 mm). The
impact hammer guide rails were wiped with acetone, to minimize frictional effects. The
displacement gage was zeroed at 0.5 volts, and the photocell timer was reset to zero. If the high
speed camera was being used, it was loaded with 450 ft (137.2 m) of film, and the floodlights
were turned on and adjusted to properly illuminate the test specimen. One final check of all the
channels on the Nicolet recording oscilloscope was made, and an intentional trigger was
introduced into the tup strain gage bridges to make sure all the channels would record data.
When all the instrumentation was ready, a countdown would be given allowing enough time for
the high speed framing camera to spin up, when finally the impact hammer would be released,
causing the tup to strike the beam at its midpoint, and subsequently induce failure. The eight

75
channels of data are then transferred from the oscilloscopes to the disk recorders where the data
were written on 5.25 in (13.34 cm) floppy disks. Twenty records, four channels on each record,
were written on each floppy disk. In total, ten floppy disks worth of preliminary and final data
were collected during the course of this investigation. The data was then manipulated using a
scientific and engineering data processing program called VU-POINT , Version 2.0, written by
Maxwell Laboratories, Incorporated, La Jolla, California. The VU-POINT software analyzes,
modifies and plots time series data recorded by waveform digitizes such as the Nicolet 4094
digital oscilloscope.
Fifty-four drop weight tests were conducted on the six different beam types in this
manner, and data were recorded on 51 of the 54 tests conducted. Data was lost on one plain
concrete beam, one two ply CFRP beam, and one beam with three ply CFRP on all three sides,
due to either to spurious or late triggers in the instrumentation system.
Interpretation of Test Results
To illustrate the procedure which was developed to analyze and interpret the data for all
of the drop weight tests, the data from one specific beam, LW3-20, is analyzed in detail. This
procedure was followed on all subsequent tests. This particular beam was reinforced with one
ply CFRP, the hammer weight was 96.14 lbs (43.7 kg), and the drop height was 8.0 in (20.32
cm). Figure 41(a) shows the beam in the drop weight machine ready to be tested, and Figure
41(b) shows the post-test results. Figure 42(a) shows a side view of beams LW3-20, LW2-12,
and LW2-13, all tested at the same condition. The post - test damage from the dynamic tests
shown in Figure 42(a) is remarkably similar to the same beam type tested under quasistatic
loading conditions shown in Figure 42(b). This implies that the failure mechanism(s) is the same
for both static and dynamic loading conditions. Results and discussion of all the dynamic tests
are presented later in this chapter.

76
(b) Post - Test
Figure 41. Pre and Post - Test Results for Drop Weight Test on Beam LW3-20

77
(b)
Figure 42. Post - Test Results for (a) Drop Weight and (b) Quasistatic
Bending Tests on One Ply CFRP Beams

78
As the tup strikes the beam test specimen, it records the stressing or bending load in the
beam, the beam’s inertia, as well as any damping forces which may also be present. However, it
will be shown later that due to the nature of the load pulses, the damping forces can be assumed
negligible in the analysis of the tup load constituents. Figures 43 and 44 show a typical set of
tup loads which were recorded on beam LW3-20. This load pulse is quite typical in shape, but
not in amplitude nor duration, to the other tup loads recorded on other beam types. Generally
speaking, larger drop heights produced larger amplitude and shorter duration tup loads. Figure
43(a) shows the tup load as recorded by strain gage bridge A, and 43(b) shows the tup load
recorded by strain gage bridge B. Figure 44(a) shows the average of the two signals. It should
be pointed out here that the actual load pulse is the first pulse, with an amplitude of 6160 lbs
(27.40 kN). The later two smaller amplitude pulses are merely rebounds of the tup, recorded
after the beam has already failed, shown here for completeness. The reason the load pulse
actually starts at about -150 psec and not zero is because the tup signal has not been time shifted
to account for the 50 mV trigger voltage (t = 0) and its corresponding 500 psec of saved pre -
trigger information. In fact, none of the signals in this study have been time shifted in an attempt
to keep the temporal nature of the different measurements as unbiased as possible. Quite simply,
it is the relative times in which the different events occur, as measured by the instrumentation,
which is of greatest interest and relevance. Looking closely at the end of the load pulse in
Figures 43(a) and (b), one notices that the tup has undergone a small amount of bending, as
evidenced by the small positive load measured by bridge A at the end of the load pulse, Figure
43(a), and the negative load sensed by bridge B in Figure 43(b). But when the two signals are
averaged together in Figure 44(a), the bending effect is nullified.
Figure 44(b) shows the average load pulse which has been digitally filtered using the 5
kHz low pass filter shown in Figure 45. The low pass filter resident in the VU-POINT® software

79
(a)
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
Figure 43. Tup A and Tup B Load Versus Time Curves for Beam LW3-20

80
(a)
6000.0
5000.0
1 1 1 1 1 1 1 r
TUP A/B AUG LOAD vs TIME
BEAM LW3-20
96.14 LBS @ 8 IN
1 PLY CFRP BOTTOM ONLY
20 JUNE 1995
5KHz LOU PASS FILTER
a 4000.0
«
o
j
u co 3000.0
i S
-
\ £ 2000.0
Du
H 1000.0
0.0
0.0s
J I I I I I I I I L
1.0ms 2.0ms
TIME
3.0ms
(b)
Figure 44. Tup A and Tup B Average Load Versus Time Curves for Beam LW3-20

81
allows specification of the cutoff frequency (50 percent transmission), in this case 5 kHz, the
sharpness or width of the cutoff, in this case 1.25 kHz, and the gain beyond the cutoff, in this
case 0.01. The number of “filter terms” in Figure 45 is the number of data points that are
smoothed.
Sunnary of LOW-PASS Filter Response
Trans ition-Freq(s)=5.Bkhz S Trans ition-Width=l.25khz
Max. Response outside of pass-band=0.01
1787 Fi Iter Terns
APPROX. FILTER TRANSMISSION vs. FREQ.
0 hz 5.0khz
Figure 45. 5 kHz Low Pass Filter
A 5 kHz cutoff frequency was consistently used throughout this study on all signals
which required low pass filtering. This is reasonable, since the fundamental frequencies for
these beam types varies from 170 to 300 Hz, an order of magnitude below the chosen cutoff

82
frequency. Therefore, there is little danger of filtering out any high frequency components
which are important to the dynamic mechanical behavior of this system.
The raw displacement versus time data from the noncontact, linear proximity measuring
system is shown in Figure 46(a). Note that the load pulse actually starts at -205 psec and ends at
540 psec, but that the beam does not begin to displace until about 100 psec. After beam failure
occurs there is a decrease in the slope of the displacement versus time curve, at which point the
slope becomes constant as the beam merely falls down and comes to rest on the support platform
of the drop weight machine. Figure 46(b) is the smoothed displacement versus time curve after
passage through the 5 kHz low pass filter. The smoothed displacement versus time curve is
differentiated with respect to time, and the resulting velocity versus time is shown in Figure
47(a). Notice that the peak velocity of 102 in/sec (259.08 cm/sec) occurs at about 500 psec, then
rapidly drops off after the failure occurs, and then continues to oscillate at about 60 in/sec (152.4
cm/sec). At a drop height of 8 in (20.32 cm) the tup strikes the beam at 75 in/sec (189.5 cm/sec),
and transfers its momentum to the beam. Since the mass of the tup is greater than the mass of
the beam, the beam achieves a higher velocity than the tup striking velocity. The velocity versus
time curve is then differentiated with respect to time to yield acceleration versus time, the results
of which are shown in Figure 47 (b). The acceleration of the beam peaks at about 200 psec, with
an amplitude of 750 g’s. The time of peak acceleration is consistent with an abrupt change of
slope in the strain versus time data, as the failure process in the beam initiates. Peak strain rates
occur at this time as well, and will be discussed in detail later in this chapter.
Consider, for a moment, the beam as a single degree of freedom system, with the
acceleration versus time known at beam midpoint. In order to estimate the inertial force or
resistance to motion that the beam offers when it is set into motion, the beam’s equivalent mass

83
■ lit i i i i i i—i—i—i—i—i—i—i—
0.0s 1.0ms 2.0ms 3.0ms
TIME
(a)
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
Figure 46. Displacement versus Time Curves for Beam LW3-20

84
(a)
H
cc
w
ij
1.0ms
2.0ms
(b)
Figure 47. Velocity and Acceleration versus Time Curves for Beam LW3-20

85
must be calculated. Once the equivalent mass is known, the acceleration versus time curve may
be multiplied by the equivalent mass to calculate the beam’s inertial load versus time curve. At
the outset of this investigation, it was thought that the beam would initially assume a deflected
shape consistent with the fundamental mode of vibration as shown in Figure 48(a), but this was
P(t)
1
p(t)
\y ^
(b)
Figure 48. General versus Localized Bending
not the case. Since the loading may be considered impulsive, the beam as a whole does not have
time to react to the sudden blow imparted by the tup to the center of the beam. The initial
deflection is therefore localized about the beam’s midpoint, as shown in Figure 48(b), and the
original span length of the beam has little significance in calculating the beam’s equivalent mass.
In time, the deflection assumes a fully developed plastic state. A method for determining the
beam’s average equivalent mass during the time of the load pulse was developed based on the
assumption of a rigid, perfectly plastic beam, utilizing the concept of a traveling plastic hinge

86
and is discussed in detail in Chapter 3. Once the equivalent mass is known, the beams inertial
load versus time behavior is therefore determined. Figure 49(a) shows the tup or total load and
inertial load versus time curves plotted together. Note that the majority of the load measured by
the tup is represented by the inertial load. At the end of the load pulse, they are coincident.
Taking liberty at the present time to assume no damping forces present, the difference in
amplitude between the tup or total load and the inertial load represents the bending load. Figure
49(b) show the bending load versus time curve. The bending load has a peak amplitude of 2600
lbs (11.57 kN) at 275 psec, and returns to zero at 420 psec. Since the peak load of the tup is
4300 lbs (19.13 kN), failure to account for the inertial load will lead to misleading conclusions
being drawn about the bending load, and hence, the fracture energy. This observation has also
been noted by other researchers [32, 35, 42],
In order to calculate the amount of energy consumed by the beam up until failure occurs,
bending load versus time was plotted against displacement versus time, creating a dynamic
bending load versus displacement curve. This curve is shown in Figure 50(a). Recall that the
bending load versus time curve showed that the bending load dropped to zero at about 420 psec,
which in load - displacement space, translates to a displacement of 0.022 in (0.564 mm) at
failure. At the point of failure, the beam stops receiving energy from the impact hammer, and
the tup load falls to zero. The energy consumed by the beam up until failure occurs is sometimes
called the toughness or simply the fracture energy. The area under the dynamic bending load
versus displacement curve therefore represents the fracture energy or the amount of energy
required to create two new fracture surfaces. In the case of plain concrete beams, the beams
break into two halves and the two broken halves swing about their supports away from the tup.
Although these beam halves may retain considerable kinetic energy, they have no bending or
strain energy left in them.

87
0.0s 1.0ms 2.0ms
TIME
(a)
(b)
Figure 49. Tup, Inertial, and Bending Load versus Time Curves for Beam LW3-20

88
O
o
u
03
DISPLACEMENT, INCHES
(a)
250.0us 0.0s 250.0us 500.0us
TIME
(b)
Figure 50. Bending Load versus Displacement and Fracture Energy versus Time Curves for
Beam LW3-20

89
The area under the bending load versus displacement curve was found by numerical
integration in VU-POINT using Simpson’s Rule. The resulting dynamic fracture energy versus
time curve is shown plotted in Figure 50(b). The fracture energy at the beam failure time of 420
psec was 3.6 fit-lbs (4.9 N-m).
Data from the strain gages mounted on the center cross section was useful not only for
determining the strains, strain rates, and stresses, but also for verifying the initial onset of failure
in the beams as well. Figure 51(a) shows the strain versus time traces for strain gages number
one and three. Recall that Figure 39 showed the location of all of the strain gages. Strain gage
number three indicating tensile strain, shows an abrupt change of slope at 200 psec indicating
that the concrete has begun to fail in tension, and a crack is in the process of propagating from
the bottom of the beam to the top surface. Meanwhile, strain gage number one, located on the
concrete near the top surface, goes into compression and remains in that state until failure
occurs. The third strain gage, strain gage number two, located midway on the beam’s center
cross - section, was not used on this particular test.
Differentiating the strains with respect to time yielded the strain rates. Its interesting to
note that the peak tensile strain rate recorded in gage number three shown in Figure 51(b) occurs
at almost the same time as the peak inertial load; about 200 psec, with a magnitude of 18.5/sec.
The shape of gage number three’s strain rate versus time pulse is also curiously similar to that of
the inertial load versus time pulse as well. Strain gage number one’s compressive strain rate
only reaches about 2/sec during the loading event. Both gages indicate that the strain rates return
to and oscillate about zero after 800 psec.
Figure 51(c) shows the stress versus time curves for gages one and three. The curves
were obtained by multiplying the strain versus time curves by their respective elastic moduli;

90
0.0s 1.0ms 2.0ms 3.0ms
TIME
0.0s 1.0ms 2.0ms 3.0ms
TIME
(b)
0.0s 1.6ms 2.0ms 3.0ms
TIME
Figure 51. Strain, Strain Rate, and Stress versus Time for Beam LW3-20

91
20 x 106 psi (137.93 GPa) for the CFRP, gage number three, and 3.53 x 106 psi (24.35 GPa) for
the concrete, gage number one. The tensile stress in the CFRP is about 60 times higher than the
concrete compressive stress.
Damping Loads
All dynamic structural systems contain damping to some degree. However, the effect is
not significant if the loading is impulsive and only the maximum response is being investigated.
Damping in structures may arise due to different physical phenomena. In some cases, it may be
due to resistance provided by the medium surrounding the structure such as water, air, or soil. It
is also due to the loss of energy associated with slippage of structural connections either between
members or between the structure and the supports. It may also involve internal molecular
friction of the material itself. In any case, the effect is one of a force opposing the motion.
The analyses conducted during the course of this investigation assumed negligible
damping forces, primarily due to the impulsive nature of the loading. Furthermore, the beams
fail prior to any continuing state of vibration; a state in which damping generally is included. To
further illustrate the validity of the assumption, a calculation was carried out to determine the
relative magnitude of the damping force, and compare it to the other forces present. For the
purposes of this analysis, the damping was assumed to be of the viscous type (as opposed to
Coulomb, or constant damping) where the damping force is proportional to but opposite in
direction of the velocity. Viscous damping is the most commonly assumed for structural
analysis. The amount of damping which removes all vibration is referred to as critical damping,
which is a convenient reference point, since most structures typically have between five and ten
percent of critical damping.
Figure 52(a) shows beam midpoint velocity versus time for beam LW2-11. This is a
typical velocity profile for this beam type (one ply CFRP) and tup drop height (12 in/30.48 cm).

92
0.0s 1.0ms 2.0ms 3.0ms
TIME
0.0s
1.0ms
2.0ms
(b)
0.0s 1.0ms 2.0ms
TIME
(c)
Figure 52. Velocity versus Time, and Damping, Tup, and Inertial Loads versus Time for
Beam LW2-11

93
Assuming a five percent damping factor, Figure 52(b) shows the calculated damping load versus
time, and Figure 52(c) shows the damping load plotted against the total or tup load and the
inertial load versus time curves. When compared to the magnitudes of the tup and inertial loads,
the damping load is indeed insignificant. Furthermore, the damping load peaks at 600 psec,
which is at the end of the load pulse when the bending or stressing load has dropped to zero,
indicating that failure has occurred. Prior to 600 psec, the damping load is quite insignificant, as
the midpoint velocity has not yet fully developed.
Results and Discussion
The data from the remaining 50 beams tested were all analyzed in a similar fashion to
beam LW3-20. In order to organize the massive body of data collected, key parameters from
each test were recorded in tabular form; beam designator, unconfined compressive strength,
number of plies CFRP, drop height, peak tup load and pulse length, peak inertial and peak
bending loads, peak displacement, and lastly, the calculated fracture energy. This tabularized
data for all 51 experiments is presented as Appendix D. Since each of the 18 separate test
conditions was repeated three times each, the tabularized data recorded in Appendix D was
averaged together for each of the 18 test conditions, and is presented as Table 12.
Table 12 indicates that the average peak amplitude of the tup load increases with an
increase in drop height, along with concomitant increases in the peak inertial load and peak
bending load. Since most of the load recorded by the tup is inertial in nature, it is not surprising
to see the tup load increase with drop height; as the drop height increases, the hammer impact
velocity increases. From momentum considerations, the beam's velocity increases as well;
therefore, the force resisting the motion increases. The bending load was previously shown to be
the difference between the tup load and the inertial load. Table 12 shows an increase in peak
bending load with drop height. Bentur et al. [27] refers to peak bending load as “impact

Table 12. Summary of Dynamic Three-Point Bending Experiments
No. of
Plies CFRP
Drop
Height
in (cm)
Avg. Peak
Tup Load
lbs (kN)
Avg. Tup Load
Pulse Length
psec
Avg. Peak
Inertial Load
lbs (kN)
Avg. Peak
Bending Load
lbs (kN)
Avg. Peak
Displacement
in (mm)
Avg. Fracture
Energy
ft lbs (kNm)
0
2
(5.08)
2235
(9.94)
1011
1500
(6.67)
1600
(7.12)
0.0095
(0.241)
1.2
(1.63)
0
4
(10.16)
3700
(16.46)
863
3033
(13.49)
2233
(9.93)
0.0075
(0.191)
1.5
(2.03)
0
8
(20.32)
4242
(18.87)
937
4000
(17.79)
2900
(12.90)
0.0063
(0.160)
2.5
(3.39)
1
6
(15.24)
4733
(21.05)
937
4200
(18.68)
3017
(13.42)
0.0145
(0.368)
1.8
(2.44)
1
8
(20.32)
5760
(25.62)
802
4217
(18.76)
2900
(12.90)
0.0241
(0.612)
4.2
(5.70)
1
12
(30.48)
6050
(26.91)
878
5467
(24.32)
3733
(16.61)
0.0227
(0.577)
5.1
(6.92)
2
7
(17.78)
4910
(21.84)
925
4300
(19.13)
3025
(13.46)
0.0106
(0.269)
2.8
(3.80)
2
8
(20.32)
6613
(29.42)
719
5533
(24.61)
4117
(18.31)
0.025
(0.635)
6.5
(8.82)
2
12
(30.48)
7647
(34.02)
739
5750
(25.58)
4517
(20.09)
0.038
(0.965)
9.7
(13.16)
3
8
(20.32)
5283
(23.50)
949
4317
(19.20)
3100
(13.79)
0.041
(1.041)
7.0
(9.49)
3
12
(30.48)
6330
(28.16)
945
5267
(23.43)
3300
(14.68)
0.043
(1.092)
8.5
(11.53)
3
18
(45.72)
8433
(37.51)
838
5833
(25.95)
5183
(23.06)
0.07
(1.778)
18.4
(24.95)

Table 12—continued
No. of
Plies CFRP
Drop
Height
in (cm)
Avg. Peak
Tup Load
lbs (kN)
Avg. Tup Load
Pulse Length
psec
Avg. Peak
Inertial Load
lbs (kN)
Avg. Peak
Bending Load
lbs (kN)
Avg. Peak
Displacement
in (mm)
Avg. Fracture
Energy
fit-lbs (kN-m)
3+
8
(20.32)
5440
(24.20)
928
3967
(17.65)
2550
(11.34)
0.035
(0.889)
6.4
(8.68)
3+
12
(30.48)
7127
(31.70)
865
5100
(22.69)
4033
(17.94)
0.052
(1.321)
9.9
(13.43)
3+
18
(45.72)
8923
(39.69)
670
6283
(27.95)
5800
(25.80)
0.067
(1.702)
16.5
(22.38)
3 - Bottom
3 - Sides
16
(40.64)
8775
(39.03)
761
6800
(30.25)
6625
(29.47)
0.058
(1.473)
16.4
(22.24)
3 - Bottom
3 - Sides
18
(45.72)
8987
(39.98)
858
5783
(25.73)
5450
(24.24)
0.076
(1.930)
20.4
(27.67)
3 - Bottom
3 - Sides
24
(60.96)
10,067
(44.78)
849
6200
(27.58)
5967
(26.54)
0.077
(1.956)
25.2
(34.18)
+ Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard

96
strength”. This implies an increase in impact strength with an increase in the hammer drop
height, i.e. with an increase in the stressing or strain rate. Data from strain gage number three,
located on the bottom or tension side of the beams, clearly indicate an increase in strain rate with
an increase in drop height. For example, beam LW5-39, which was an unreinforced plain
concrete beam tested at an 8.0 in (20.32 cm) drop height, showed a peak strain rate of 20/sec at
250 psec, whereas a similar unreinforced beam, beam LW6-43, tested at a 2.0 in (5.08 cm) drop
height, showed a peak strain rate of 16.3/sec at 350 psec. From the SHPB splitting tensile tests,
these strain rates would yield a 2.5 - 2.8 times increase in the static tensile strength value. At the
2.0 in (5.08 cm) drop height, the average dynamic peak bending load was 2.5 times the static
value; at the 8.0 in (20.32 cm) drop height, the average dynamic peak bending load was 4.5 times
the static value. This will be discussed further in the next section. The dynamic peak bending
load increase was highest for the “less reinforced” or less stiff beams, gradually declining in
value as beam stiffness increased. The increase in peak bending load with drop height has also
been observed in conventionally reinforced concrete beams by Bentur et al. [27] and Banthia et
al. [30], although both references report on the same experimental data.
For the plain concrete beams, the maximum displacement (beam failure) decreases with
an increase in drop height. Flowever, for those beams reinforced with CFRP, the displacement at
failure increases with an increase in drop height. The latter observation has also been reported in
conventionally reinforced concrete beams by Mindess et al. [33], who also found that for low
drop heights, the peak displacement was similar to that under static loading.
Table 12 also shows a consistent increase in fracture energy with increasing drop height,
or increasing stress rate. Nonlinear fracture mechanics theory states that immediately ahead of a
moving crack is a zone of microcracking called the “process zone”. Reinhardt [43] suggests that
the size of the microcracking zone depends upon crack velocity; a faster crack has a larger zone

97
of microcracking ahead of it. Higher stress rates propagate cracks more quickly; therefore the
process zone will be larger. This increased microcracking may explain the increasing fracture
energies recorded at increasing drop heights or increasing stress rates. This increase in fracture
energy with drop height has also been observed in conventionally reinforced plain concrete
beams by Mindess et al. [33].
Also evident, is the trend in the tup load pulse duration to shorten with increasing drop
height. This would seem to indicate that the failure is occurring more quickly as the drop height
is increased. Since the tup measures the total resistance the beam offers to include both bending
and inertial forces, a foreshortening of the bending load versus time trace would also indicate a
foreshortening in the time to failure. Strictly speaking, however, not all of the six beam types
followed this trend.
Comparison of Dynamic and Static Test Results
Table 13 is a comparison between the static three point bending tests conducted on the
MTS load frame, and the dynamic three point bending tests conducted on the drop weight
machine.
The average static peak bending load was always less than the dynamic peak bending
load, even at the lowest drop heights. The increase in peak bending load with drop height is
attributed to strain rate effects in the concrete. As the beam stiffness increases with increasing
number of plies of CFRP, the ratio of peak dynamic load to peak static load, for a given drop
height, decreases.
It has been observed by other researchers [29 -31,33], that high strength concrete
behaves in a more brittle fashion dynamically, that it does statically. Banthia et al. [30] tested
concrete beams of two different strengths, 6090 psi (42 MPa) and 11,890 psi (82 MPa), and
found that the peak bending load (impact strength) for both strength beams increased with an

Table 13. Comparison of Dynamic and Static Three Point Bending Experiments
Number
of Plies
CFRP
Drop
Height
in (cm)
Avg. Dyn.
Peak Bending
Load
lbs (kN)
Avg. Static
Peak Bending
Load
lbs (kN)
Ratio
Dyn./Static
Load
Unitless
Avg.
Dyn.
Peak Displ.
in (mm)
Avg.
Static
Peak Displ.
in (mm)
Ratio
Static/Dyn.
Displ.
Unitless
Avg. Dyn.
Fract. Energy
ft-lbs (N m)
Avg. Static
Fracture
Energy
ft-lbs (N m)
Ratio
Static/Dyn.
Fracture Energy
Unitless
0
2
(5.08)
1600
(7.12)
637
(2.83)
2.51
0.0095
(0.241)
0.014
(0.356)
1.47
1.2
(1.63)
0.392
(0.531)
0.33
0
4
(10.16)
2233
(9.93)
3.50
0.0075
(0.091)
—
1.87
1.5
(2.03)
—
0.26
0
8
(20.32)
2900
(12.90)
4.55
0.0063
(0.160)
—
2.22
2.5
(3.39)
—
0.16
1
6
(15.24)
3017
(13.42)
1149
(5.11)
2.63
(0.0145)
(0.368)
0.160
(4.072)
11.03
1.8
(2.44)
11.747
(15.932)
6.53
1
8
(20.32)
2900
(12.90)
—
2.52
(0.0241)
(0.612)
—
6.64
4.2
(5.70)
—
2.80
1
12
(30.48)
3733
(16.61)
—
3.25
(0.0227)
(0.577)
—
7.05
5.1
(6.92)
—
2.30
2
7
(17.78)
3025
(13.46)
2298
(10.22)
1.31
0.0106
(0.269)
0.241
(6.130)
22.74
2.8
(3.80)
30.483
(41.341)
10.89
2
8
(20.32)
4117
(18.31)
—
1.79
0.025
(0.635)
—
9.64
6.5
(8.82)
—
4.69
2
12
(30.48)
4517
(20.09)
—
1.97
0.038
(0.965)
—
6.34
9.7
(13.16)
—
3.14

Table 13—continued
Number
of Plies
CFRP
Drop
Height
in (cm)
Avg. Dyn.
Peak Bending
Load
lbs (kN)
Avg. Static
Peak Bending
Load
lbs (kN)
Ratio
Dyn ./Static
Load
Unitless
Avg.
Dyn.
Peak Displ.
in (mm)
Avg.
Static
Peak Displ.
in (mm)
Ratio
Static/Dyn.
Displ.
Unitless
Avg. Dyn.
Fract. Energy
ft lbs (N m)
Avg. Static
Fracture
Energy
ft lbs (N m)
Ratio
Static/Dyn.
Fracture Energy
Unitless
3
8
(20.32)
3100
(13.79)
2248
(10.00)
1.26
0.041
(1.041)
0.180
(4.572)
4.39
7.0
(9.49)
23.497
(31.867)
3.36
3
12
(30.48)
3300
(14.68)
—
1.35
0.043
(1.092)
—
4.18
8.5
(11.53)
—
2.76
3
18
(45.72)
5183
(23.06)
—
2.12
0.070
(1.178)
—
2.57
18.4
(24.95)
—
1.28
3+
8
(20.32)
2550
(11.34)
2349
(10.45)
1.08
0.035
(0.889)
0.175
(4.453)
5.00
6.4
(8.68)
22.884
(31.037)
3.58
3+
12
(30.48)
4033
(17.94)
—
1.72
0.052
(1.321)
—
3.37
9.9
(13.43)
—
2.31
3+
18
(45.72)
5800
(25.80)
—
2.47
0.067
(1.702)
—
2.61
16.5
(22.38)
—
1.39
3 - Bottom
3 - Sides
16
(40.64)
6625
(29.47)
4679
(20.81)
1.42
0.058
(1.473)
0.245
(6.210)
4.22
16.4
(22.24)
57.170
(77.534)
3.49
3 - Bottom
3 - Sides
18
(45.72)
5450
(24.24)
—
1.17
0.076
(1.930)
—
3.22
20.4
(27.67)
2.80
3 - Bottom
3 - Sides
24
(60.96)
5967
(26.54)
—
1.28
0.077
(1.956)
—
3.18
25.2
(34.18)
2.27
+ Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard

100
increase in drop height. Furthermore, the higher strength concrete beam showed a higher
bending load (impact strength) for the same drop height. Both types of beams required higher
fracture energies at increasing drop heights. However, Banthia et al. [30] also found that the
dynamic fracture energy was lower for the higher strength concrete beams when compared to its
static fracture energy value. Conversely, the 6090 psi (42 MPa) concrete beams always showed
a higher dynamic fracture energy when compared to the static value.
Table 13 indicates that the dynamic fracture energy is larger than the static fracture
energy for the plain concrete beams, which consistent with the findings of Banthia et al. [30] for
the 6090 psi (42 MPa) concrete beams. However, for the remainder of the beams all reinforced
with external CFRP, the dynamic fracture energies were all consistently less than the static
fracture energies, similar to the high strength beam results of Banthia et al. [30]. A possible
explanation for this is that the addition of external CFRP significantly stiffens the beam (see, for
example, Table 10), thereby enhancing the beam’s brittle behavior when loaded dynamically.
The flexural stiffness of a beam is the product of its modulus times its planar moment of inertia.
Since the elastic modulus of concrete is calculated directly from its unconfmed compressive
strength, the higher the compressive strength the higher the elastic modulus, hence the higher the
flexural stiffness. This explains why a beam made of high strength concrete would have a higher
flexural stiffness than a beam made from normal strength concrete. This may also explain why a
high strength plain concrete beam is dynamically equivalent to a normal strength concrete beam,
when it has been externally reinforced with CFRP.
The peak displacements were always less under dynamic loading when compared to the
static case. Except for the plain concrete beams as discussed above, the dynamic fracture
energies were always less than the static values, even though the peak dynamic bending loads
were typically 2-3 times higher than the peak static bending loads. This would seem to indicate

101
that displacement is the limiting parameter in determining the beam’s dynamic fracture
toughness. It also implies that for a given drop height, i.e. strain rate, that a beam has a fixed
capacity to absorb energy, dictated by the concrete’s impact strength and limited by
displacement, thereby emphasizing the brittle nature of concrete.
The mechanism by which the beams failed dynamically did not change appreciably from
the quasistatic loading case; failure of all beams with only tensile CFRP reinforcement was one
of shear failure of the concrete at approximately one quarter span from beam midspan, followed
by delamination and peeling of the CFRP. Failure of beams with CFRP on the sides as well as
the bottom was concrete shear failure followed by CFRP side panel splitting and buckling along
the major shear crack in the concrete. Similar to the SFIPB results, some additional cracking of
the concrete was noticed on the dynamic tests; as the load rate hence the strain rate increases
with increasing drop height, the number of fracture surfaces also increases.
Table 14 shows a comparison of the six beam types tested at three drop heights; 8, 12,
and 18 in (20.32, 30.48, and 45.72 cm). At the 8 in (20.32 cm) drop height, there is an increase
in fracture energy with increasing number of plies of CFRP, except for those beams reinforced
with nylon fibers. Those beams reinforced with two ply CFRP have almost the same capacity to
absorb energy as the three ply beams. Recall that under static loading conditions, the two ply
CFRP beams had the greatest fracture energy of all beams reinforced on the bottom only. At the
12 in (30.48 cm) drop height, this trend continues; an increase in fracture energy with increasing
number of plies, and remarkable performance from the two ply beams. Finally, at 18 in (45.72
cm), there are only three beams types available for comparison; three ply CFRP, three ply CFRP
with fibrous concrete, and three ply CFRP on both bottom and sides. Once again, those beams
reinforced on all three sides with three ply CFRP had the highest energy absorption capacity of

102
Table 14. Comparison of Six Beam Types at Three Different Drop Heights
No. of
Plies CFRP
Avg.
Tup Load
lbs (kN)
Avg. Tup
Load Pulse
(isec
Avg. Peak
Inertial Load
lbs (kN)
Avg. Peak
Bending Load
lbs (kN)
Avg. Peak
Displacement
in (mm)
Avg. Fracture
Energy
fit-lbs (N-m)
8 in (20.32 cm)
Drop Height
0
4242
(18.87)
937
4000
(17.79)
2900
(12.90)
0.0063
(0.160)
2.5
(3.39)
1
5760
(25.62)
802
4217
(18.76)
2900
(12.90)
0.0241
(0.612)
4.2
(5.70)
2
6613
(29.42)
719
5533
(24.61)
4117
(18.31)
0.025
(0.635)
6.5
(8.82)
3
5283
(23.50)
949
4317
(19.20)
3100
(13.79)
0.041
(1.041)
7.0
(9.49)
3+
5440
(24.20)
928
3967
(17.64)
2550
(11.34)
0.035
(0.889)
6.4
(8.68)
12 in (30.48 cm)
Drop Height
1
6050
(26.91)
878
5467
(24.32)
3733
(16.61)
0.0227
(0.577)
5.1
(6.92)
2
7647
(34.02)
739
5750
(25.58)
4517
(20.09)
0.038
(0.965)
9.7
(13.16)
3
6330
(28.16)
945
5267
(23.43)
3300
(14.68)
0.043
(1.092)
8.5
(11.53)
3+
7127
(31.70)
865
5100
(22.69)
4033
(17.94)
0.052
(1.321)
9.9
(13.43)
18 in (45.72 cm) Drop Height
3
8433
(37.51)
838
5833
(25.95)
5183
(23.06)
0.07
(1.178)
18.4
(24.95)
3+
8923
(39.69)
670
6283
(27.95)
5800
(25.8)
0.067
(1.702)
16.5
(22.38)
3 - Bottom
3 - Sides
8987
(39.98)
858
5783
(25.73)
5450
(24.24)
0.076
(1.930)
20.4
(27.67)
+ Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard
all beams tested at that drop height. Recall that under static loading conditions, the three-sided
beams also had the highest recorded fracture energies.

CHAPTER 3
ANALYTICAL MODEL
An analytical model has been developed which attempts to predict the static load-
displacement behavior, as well as the dynamic displacement-time behavior. The static load-
displacement response portion of the model is based on classical beam section analyses, which
have been modified to take into account the external reinforcement of the CFRP. The dynamic
displacement-time behavior is modeled by rigorous (closed form) analysis of single degree of
freedom representations of the CFRP reinforced beams, which have been subjected to a half-
sine pulse impulsive loads. The complete analytical model was programmed using the
MATHCAD software package, which runs on a personal computer.
Section Analysis
The analysis of the quasistatic load - displacement behavior of concrete beams externally
reinforced with CFRP is based on elastic - plastic section analysis. Ross et al. [22] used this
method to analyze some large concrete beams 8.0 x 8.0 x 120.0 in (20.32 x 20.32 x 304.8 cm) in
size, externally reinforced with three ply CFRP on the bottom, and tested to failure in third point
loading. The beams had a composite cross - section of concrete, steel, and CFRP.
The basic assumption used in the analysis is that concrete fractures in tension at a very
low stress, causing the beam’s neutral axis to shift upward a rather large amount. This shift in
the neutral axis is calculated and used to determine the beam’s residual resisting moment. It is
assumed that the distribution of strain in the beam is linear, and that plane cross-sections remain
planar. The assumption of a linear strain distribution has been verified experimentally [21].
103

104
Since these beams did not use tension steel, the composite cross-section only contains
lightweight concrete and CFRP. The addition of side CFRP to the beam cross-section is an
extension to the model previously developed by Ross et al. [22], The analyses were
accomplished using the various regions of the idealized load-displacement curve shown in
Figure 53. The curve is divided into three regions, each terminated by a similarly numbered
Region 3 - Cracked Tension Concrete,
Displacement
Figure 53. Idealized Load - Displacement Curve and Assumptions Used in Analytical
Beam Section Analysis
point. These three numbered points plus the origin are connected by straight lines to define the
calculated load-displacement curve. The stress - strain curve for the lightweight concrete was
previously given as Figure 8. The stress - strain curve for the CFRP is assumed linear elastic
until failure, with a modulus of 20 x 106 psi (137.93 GPa). Each of these three regions and their

105
respective terminating points shown in Figure 53 are each derived below, based on the beam
cross - section shown in Figure 54.
where
h Beam Depth = 3.0 in (7.62 cm)
t Bottom CFRP Thickness = 0.0085/0.014/0.0195 in (0.261/0.356/0.495 mm)
b Beam Width = 3.0 in (7.62 cm)
tj Side CFRP Thickness = 0.0195 in (0.495 cm)
c Neutral Axis Location From Top of Beam
Figure 54. Beam Cross - Section Used in Section Analysis
Region 1 - All Elastic
In this region, all materials in the cross section are assumed to behave elastically until
fracture of the concrete occurs in bending in the tensile region of the concrete, commonly
referred to as the modulus of rupture. Figure 55 shows the strain and stress/load distribution for

106
(b) Stress and Load Distribution
where
sc Compressive Strain, Concrete
ec sf Compressive Strain, Side CFRP
et Tensile Strain, Concrete
ef Tensile Strain, Bottom CFRP
£, sf Tensile Strain, Side CFRP
Cc Compressive Force, Concrete
Csf Compressive Force, Side CFRP
Tc Tensile Force, Concrete
Tsf Tensile Force, Side CFRP
Tf Tensile Force, Bottom CFRP
Figure 55. Strain, Stress, and Load Distribution for Region 1-All-Elastic

107
this region. All strains may be written in terms of the concrete strain sc at the top of the beam,
and they become
e, - Ef - e, sf - sc
8c,sf
h -c
(15)
Since all materials in this region are elastic, the stress may be expressed as the product of strain
times the modulus of elasticity, the forces acting on the cross - section may be written as
Compressive Concrete Force
C.-e,E.ir
(16)
Compressive Side CFRP Force
Csf -ecEf ~2 ^
Tensile Concrete Force
(h^ bO^c,
Tensile Side CFRP Force
T . _ (h-c) ts(h-c)
*sf -8c bf z
c 2
Tensile Bottom CFRP Force
T, = Mh-c)Efbt
where
(17)
(18)
(19)
(20)
Ec Concrete Elastic Modulus, psi
Ef CFRP Elastic Modulus, 20 x 106 psi (137.9 GPa)

108
Assuming flexural response, the sum of the normal forces acting on the cross - section of the
beam is zero. This may be expressed as
X F = Cc + Csf - Tc - Tsf - Tf = 0 (21)
Substituting the expressions for these forces from Equations (16) - (20) above into Equation (21)
yields a linear equation in c, which when solved yields the location of the neutral axis position in
Region 1.
bh2 +2*^h(tsh + bt)
c = (22)
2bh + 2 —- (2tsh + bt)
E„
The slope of Region 1 may be derived from the load-displacement relationship for three point
bending
where
K, (El),
K, Slope in Region 1, lbs/in
L Beam Span Length = 27 in (68.58 cm)
(23)
, 2
(EI)j Composite Flexural Stiffness in Region 1, lbs in , where I is the Planar Moment
of Inertia of the Elastic Portions of the Beam Cross - Section
Having determined the neutral axis location c, the flexural stiffness may be written as
(El), =E(
2E(
be
12
+ 2E,
ir+cH§
+ E,
¿(h-c)> + b(h-c(^)
ts(h-c)3
12
+ (h-c)t.
h-c
+ E,
— + bt(h-c + -
12 V 2
(24)
The planar moment of inertia for the composite section is determined from the relation

109
I.=
(El),
bh
bh + 2tsh + tb.
+ E4
f 2tsh + tb "
bh + 2tsh + tb>
(25)
The bending moment M, and corresponding load P,, are related through the relation for three
point bending as
PiL
M, =•
(26)
and are determined using the modulus of rupture relation
f =
M,(h -c) P,L
h - c
vT
(27)
which may then be used to solve for the displacement 8,
8,=
P,L3
48(EI),
(28)
With this, Point 1 is determined as P, ,8,.
Region 2 - Cracked Tension Concrete. All Other Elastic
For Region 2, the concrete is cracked below the neutral axis and is not active in bending.
At the end of this region, Point 2, the concrete will have reached its peak value of f and its
associated strain of e’ as shown in Figure 56. Again, all strains may be written in terms of the
concrete’s peak unconfined compressive stress value f, and its associated strain ec
ec,sf =ec =0.003
.(h-c^l (30)
£.,sf =£f =ec
Referring to Figure 56(b), the forces acting on the uncracked sections may be written as

110
at fc
(b) Stress and Load
Distribution at fc
Figure 56. Stress, Strain, and Load Distribution at fc
Concrete Compressive Force
Side CFRP Compressive Force
C-sf =efEftsc
Side CFRP Tensile Force
Tsf=E^^Eft5(h-c)
Bottom CFRP Tensile Force
The sum of these forces on the section must be zero
zL F = Cc + Csf - Tsf - Tf = 0
(31)
(32)
(33)
(34)
(35)

Ill
Substituting the forces from Equations (31) - (34) into Equation (35) yields a quadratic equation
in c, which when solved yields
c = -
2hejEfts + ejEftb^
+ 1
f 4he|,Efts +2ejEftb^
2
A
-2ecEfh(tsh + tb)
l fcb J
_ 2 y
l f> J
1
«T*
o*
i
(36)
where the positive root is the only one physically possible.
Knowing the value of the neutral axis location c allows calculation of the bending
moment M2. Referring to Figure 56(b), and summing moments about the neutral axis gives
M2 =
3 3
The load P2 is easily determined from
f Cbc2 +2ecEftsc2 + 2ecEfts(h-c)3 + ecEftb(h-c)2
3c
(37)
p _4M2
r2 ~ T
(38)
In Region 2, the bending stiffness (EI)2 is determined again from the parallel axis theorem, with
the tensile concrete area removed so that
(El) = E
v 'i c
2E.
*♦*{2
12 V2.
+ 2E.
^ + ct(‘
12 s\2
ts(h-c)3 ..
h -
2'
bt3 , , ,
— —+ (h-c)t
12 s
2
+
— +bt(h-c)2
12 ’
(39)
and the slope of the load - deflection curve is determined from the relationship
!), =?(E,)>
and the incremental increase in displacement from Point 1 to Point 2 is determined by
P2-P.
(40)
A2 - â– 
(I).
(41)
so the total displacement at Point 2 becomes

112
52=5,+A2 (42)
Region 3 - Cracked Tension Concrete. Inelastic Compression Concrete. All Other Elastic
In Region 3, in order to determine a neutral axis position, the concrete compressive
stress distribution is based on an equivalent rectangular distribution as described by Winter and
Nilson [24]. The equivalent stress distribution shown in Figure 57(b) uses the empirical
Figure 57. Stress, Strain, and Load Distribution at Concrete Failure
parameter P to describe the equivalent concrete compressive force Cc and its position relative to
the top of the beam. This empirical parameter is a function of the unconfmed compressive
strength f c [24], For an fc = 5500 psi (37.93 MPa) its value is
P = 0.388
(43)

113
The position of the concrete compressive force is shown also in Figure 57(b), and its magnitude
is given by
Cc =0.85f'c2(3cb (44)
In addition to the equivalent concrete compressive stress distribution, it is also assumed
that the concrete strain at the compression face of the beam will have reached some ultimate
strain su of approximately 0.004, which is calculated from the stress-strain curve given in Figure
8. This assumption establishes a known parameter needed to determine the strains in the side
and bottom CFRP. Using the experimentally observed linear strain distribution [21], even at
peak beam response, and referring to Figure 57(a), the strains are determined as
sc Sf = eu = 0.004
t,sf ~ ef “ Eu
(45)
Referring again to Figure 57(b), the forces acting on the section may be written as
Compressive Concrete Force
Cc = 0.85fc2(3bc
Compressive Side CFRP Force
Csf— suEfCts
Tensile Side CFRP Force
Trf=eu(^)Ef(h-c)ts
Tensile Bottom CFRP Force
(46)
(47)
(48)
Tf =Mh-c)Eftb (49)
Again assuming flexural response, the sum of the normal forces acting on the cross - section is
zero.

114
XF = Cc+Csf-Tsp-Tf=0 (50)
Substitution of these forces, represented by Equations (46) - (49) into Equation (50) again yields
a quadratic equation for the location c of the neutral axis. Utilizing the quadratic formula yields
the solution for c which is given by
-ec,uEf(2tsh + tb) | l
0.85^ 4pb “ 2
In Region 3, concrete crushing may occur prior to failure of the CFRP, and the CFRP
may also debond due to inadequate peel strength between the CFRP and adhesive or low shear
strength at the concrete/adhesive interface. Calculation of the ultimate bending moments, loads,
displacement of concrete reinforced only with steel is in itself difficult, and neither
straightforward nor highly accurate. Replacing the steel with external CFRP simply compounds
the assumptions and problems, and increases the difficulty of the section analysis.
Acknowledging these shortcomings of the analysis, the assumption is made that the neutral axis
is based on Equation (51) above, and that failure of the beam is attained when the strain in the
concrete reaches an ultimate strain of 0.004.
The moment M3 is based on the assumed concrete compressive force Cc shown in Figure
57(b), along with the other forces in the section, and is given by
M3=^ec,uEftsc2+0.85f;2pbc2(l-P) + ^EcuEfts^¿ + ecuEftbfc^- (52)
3 3 c c
The bending/flexural stiffness for Region 3 is of the same form as that for Region 2; all regions
of the section are active except the tension concrete area. The only difference then between the
flexural stiffness in Region 2 versus Region 3 is the difference in the value of c, or the position
of the neutral axis. The value of c from Equation (51) is then substituted into Equation (39)
sc,uEf(2tsh + tb)
0.85fc2pb
-4
-£c,uEfh(tsh + tb)
0.8 5 fé 2pb
(51)

115
which yields the value for (EI)3. Calculation of the moment M3 from Equation (52) allows
determination of the load from the load - moment relationship
P3=-
4M,
(53)
and the load - displacement slope is given by
(54)
Utilizing the load - displacement slope from Equation (54) and the load from Equation (53)
allows calculation of the increase in displacement, given by
A3 =
(55)
and the displacement at Point 3 is finally determined as
^3 — ^2 + ^3
(56)
The results of the analyses yield three sets of load - displacement points, plus the origin,
which when plotted define a multilinear load - displacement curve for each of the six beam
types. The equations were then programmed using the MATHCAD software. A complete
listing of the computer program developed for the analytical model is presented as Appendix E.
The section analysis portion of the analytical model is embedded within the overall analytical
model, as several of the calculated parameters from the section analysis are needed prior to
execution of the dynamic portion of the analytical model.
Comparison to MTS® Test Data
Analytical load - displacement curves were calculated for all 16 beams tested statically
in the MTS load frame. Figure 58(a) shows a comparison of the section analysis results with a
plain concrete beam, and Figure 58(b) shows a comparison with a beam reinforced with one ply

116
LOAD vs DISPLACEMENT
BEAM LW5-38, PLAIN CONCRETE
(a) Plain Concrete
LOAD vs DISPLACEMENT
BEAM LW2-16,1 PLY CFRP BOTTOM ONLY
(b) One Ply CFRP Bottom
Figure 58. Comparison of Calculated and Experimental Load - Deflection Curves

117
CFRP. If the modulus of rupture is determined correctly, the section analysis does very well
predicting Point 1, the termination of elastic behavior, as indicated in Figure 58(a). Both the
load and displacement at failure are within 10 percent of the experimental values. For the one
ply beam, the analysis again correctly predicts Point 1, the end of elastic behavior, but the
section analysis overpredicts the stiffness in Region 2, as well as overpredicting the load and
displacement at failure. Ross et al. [22], using similar assumptions, showed excellent agreement
between the section analyses and experimental results. However, the beams analyzed by Ross et
al. [22] were much larger (8.0 x 8.0 x 108.0 in/0.2 x 0.2 x 2.7 m), had internal tension steel
reinforcement as well as three ply bottom CFRP, and were loaded in third point bending, which
produces a region of constant moment between the load points, rather than the three or center
point bending as the beams in this study were loaded. It is unclear at this point why the section
analysis overpredicts load and displacement in Regions 2 and 3 for the one ply beams, and as
will be shown later, for the other four beam types as well. It is quite possible that shear forces
may also need to be included in the cracked section.
Figure 59 compares the two and three ply CFRP beam section analyses with their
respective experimental results. Figure 59(a) shows the results for one of the two ply CFRP
beams. Similar to the plain concrete and one ply CFRP beams, the elastic behavior is fairly well
predicted by the analysis, but the stiffnesses in Regions 2 and 3 are overpredicted. The
displacement at failure, however, is within a few percent of the experimental result. Figure 59(b)
shows the results for the beam reinforced with three ply CFRP. The results are remarkably
similar to the two ply beam, with the elastic region correctly predicted, and the stiffness in
Regions 2 and 3 again too high. The displacement at failure is not quite as accurately predicted
as in the two ply CFRP beam. Qualitatively, it appears that as the beam stiffness increases, the
difference between the predicted and actual stiffnesses in Regions 2 and 3 decreases.

118
LOAD vs DISPLACEMENT
BEAM LW4-30,2 PLY CFRP BOTTOM ONLY
(a) Two ply CFRP Bottom
LOAD vs DISPLACEMENT
BEAM LW9-67, 3 PLY CFRP BOTTOM ONLY
(b) Three Ply CFRP Bottom
Figure 59. Comparison of Calculated and Experimental Load - Deflection Curves

119
Figure 60(a) shows the results for one of the fibrous concrete beams reinforced with
three ply CFRP. Unfortunately, there is no provision in the section analysis to account for the
fibers in the concrete, and there was no stress - strain curve available for this type of concrete.
So the section analysis was run using only the unconfined compressive strength, which in turn
determined the elastic modulus and the modulus of rupture. The assumptions for the strains at f c
and 0.85f c were the same as for the plain lightweight concrete beams. The results are consistent
with the plain lightweight concrete reinforced with three ply CFRP; the elastic behavior is well -
reproduced, and the stiffnesses in Regions 2 and 3 are again too high. The displacement at
failure is overpredicted by about 10 percent.
Figure 60(b) shows the analytical and experimental results for a beam reinforced on all
three sides with three ply CFRP. These beams had the highest loads and displacements to failure
of all beams tests, and therefore, the highest fracture energies. Neville [23] shows that for
concrete under a general biaxial state of stress, the compressive strength of the concrete may be
as much as 25 percent higher than when loaded solely in uniaxial compression. Biaxial tensile
strength, however, is no different than uniaxial tensile strength.
When beams reinforced on all three sides with CFRP are three point loaded, the side
CFRP, in effect, places the concrete into a state of biaxial compressive stress in the region above
the neutral axis of the beam. In an attempt to take this biaxial stress state into account in the
section analysis and to effectively bound the problem, a parameter entitled confinement factor
was introduced into the section analysis to adjust the unconfined compressive strength. Figure
60(b) shows two curves plotted for the section analysis; one curve uses a confinement factor of
1.0, and the other curve uses a confinement factor of 1.25. When the confinement factor is set at
1.25, this merely adjusts the unconfined compressive strength upward by 25 percent, which has
the effect of increasing both the load and displacement for Points 1, 2, and 3. In either case,

120
LOAD vs DISPLACEMENT
BEAM LWF11-85, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY
(a) Fibrous Concrete - Three ply CFRP Bottom
LOAD vs DISPLACEMENT
BEAM LW9-70, 3x3 PLY CFRP
(b) Three Ply CFRP Bottom and Sides
Figure 60. Comparison of Calculated and Experimental Load - Deflection Curves

121
the section analysis again overpredicts the stiffness of the beam. However, the total
displacement predicted by the analysis is within 10 percent of the experimental result, which
implies that the predicted loads are too high. As with the other beam types, the transition point
from elastic to inelastic behavior, Point 1, compares very well to the experimental transition
point.
Determination of Beam Equivalent Mass
The calculation of the beam’s equivalent mass is essential not only for determination of
the experimentally derived inertial loads, but also for modeling the beam as an equivalent single
degree of freedom system, subjected to an impulsive loading condition. The starting point for
the derivation of the equivalent mass presented here utilizes the concept of a traveling plastic
hinge, and is an extension of work conducted by Johnson [44], The assumption is that when a
simply supported beam is dynamically loaded at midspan by a rigid striker or tup, the deflections
induced are everywhere assumed small, and that from the moment of impact, a plastic hinge
forms under the tup and two plastic hinges travel outward from it, with the portions of the beam
in between the hinges remaining rigid, as shown in Figure 61(a). The outward moving hinges
move to a position approximately one quarter span from beam midspan where they become
stationary, as shown in Figure 61(b). At this point, the “bent” portion of the halfspan OAB
merely straightens, with section OA and AB rotating as rigid bodies, shown in Figure 61(c).
Eventually in time, the beam develops into a characteristic plastic state shown in Figure 61(d).
Interestingly enough, these four stages of motion were subsequently reproduced by finite
element method (FEM) simulations. Plots of the beam deformed shape every 100 psec were
produced and then compared to position and time information from the traveling hinge theory of
Johnson [44], The FEM simulations conducted as part of this investigation will be discussed in
further detail later in Chapter 4.

122
Figure 61. Impulsive Load Applied at the Center of a Simply Supported Beam
Assuming that section AB remains nearly straight and undeflected during the early
stages of deformation, Johnson [44] shows that the equation for the velocity of the outward
traveling plastic hinge may be written as
(57)

123
where
x
M
m
Velocity of the Hinge
Position of the Hinge
Mass of the Tup
Mass per Unit Length of the Beam
Plastic Moment
v0 Tup Impact Velocity
Integrating Equation (57) for time as an explicit function of hinge position x yields
Mmv M2 + (M + mx)2
t(x) = '
12Mp m2(M + mx)
If x is allowed to vary over the halfspan, say from 0 to L/2, Equation (58) then allows
determination of hinge position, and the time at which the hinge is at that location.
Biggs [45] shows that the equivalent mass for a single degree of freedom system may be
determined from
(58)
me = J^V(x)dx
(59)
where
m Mass of Beam
L Length of Beam
and cp(x) is the assumed shape function on which the equivalent system is based. It can be shown
that for a concentrated load at midspan and for the fully developed plastic state shown in Figure
61(d), the shape function is of simple linear form and is given by
, . 2x L
A
where the magnitude of cp(x) is set by the requirement that

124
(61)
In the plastic regime, (p(x) is not the actual shape, but the shape of the incremental deflection
after the plastic moment has been attained. Substitution of the shape function represented by
Equation (60) into Equation (59) and integrating over the length of the beam yields an expression
for the equivalent mass in terms of the original beam mass
me=y (62)
where
me Equivalent Mass
However, this equivalent mass is only valid for the linear displacement field represented by
Equation (60), and as depicted in Figure 61(d). Since it takes a finite amount of time for this
state to develop in the beam, it is inappropriate to use Equation (62) for the equivalent mass
directly, without further consideration of the temporal nature of the beam’s displacement
behavior.
Johnson [44] further shows that the traveling plastic hinge becomes stationary at a point
approximately 0.59 - 0.67L/2 from beam midspan, at which point section OAB “straightens”,
and the beam assumes a fully developed plastic state. Knowing the velocity at which the hinge
travels and the point at which the hinge becomes stationary then allows determination of the
displacement field along the length of the beam as a function of time, and therefore, using
Equation (62), the beam’s equivalent mass as a function of time.
Integration of the calculated equivalent mass versus time curve over the length of the
load pulse, and subsequent division by the length of the load pulse then allows determination of
the average equivalent beam mass during the time the loading was being applied. The average
equivalent beam mass, individually calculated for each beam tested, is a function of beam mass,

125
plastic moment, tup mass, drop height, and the length of the load pulse. Considering the beam as
a single degree of freedom system, and calculation of the average equivalent mass using the
above procedure allowed determination of the beam’s resistance to motion, otherwise known as
inertial force, presented in Chapter 2. If the load pulse was short, the plastic hinge did not have
enough time to travel outward and become stationary; that portion of the beam involved in the
localized bending was small in comparison to the original span length, hence the equivalent mass
was small. Conversely, longer load pulses allowed the plastic hinge to travel farther, possibly
becoming stationary, allowing the beam enough time to assume the fully developed plastic state,
at which point the equivalent mass becomes one third of the total beam mass.
The equations necessary to calculate the equivalent mass were programmed following
the above described methodology using the MATHCAD software, and are presented in
Appendix E. Since the equivalent mass calculations require certain parameters such as the
plastic moment and tup impact velocity, it necessarily follows the section analyses, where these
parameters have already been calculated.
Single Degree of Freedom Representation of a Beam Subject to a Half - Sine Pulse
Impulsive Load
Rigorous or exact analysis of structures and structural elements subjected to dynamic
loading conditions is only possible for relatively simple structures, and only when the loading
function is a convenient mathematical function. Therefore, in order to develop a simple
analytical model which would predict the displacement-time behavior of beams reinforced with
CFRP when subjected to an intense, impulsive load, it was necessary to idealize the beams and
the loading functions to some degree. With this motivation, the beams were idealized as single
degree of freedom (SDOF) systems, subject to a half - sine pulse impulsive load. Fortunately,

126
the loads measured by the tup are easily idealized as half - sine pulses. It was also demonstrated
that no damping forces are present, which also helps to simplify the analyses as well.
On the other hand, the beam’s effective mass, me is quite difficult to calculate, since it
changes during the time the loading is being applied. Furthermore, the beams do not remain
elastic until failure; in fact, their load-displacement behavior is nonlinear. Therefore, their
equivalent stiffness, ke is nonlinear. To complicate matters even further, there is no clear
methodology to account for failure in the beam SDOF model in order to conveniently stop the
analysis at some physically meaningful termination point.
Figure 62(a) shows the idealized SDOF system having the parameters Iq,, me, and P(t).
Here, P(t) is the idealized half - sine pulse impulsive load, defined as
m,
'e
x
777777
(a) Idealized SDOF System
(b) Free Body Diagram
Figure 62. Equivalent Single Degree of Freedom System

127
P(t) = P0 sin to t
where
P(t) Applied (Forcing) Load, lbs
P0 Amplitude of Half-Sine Pulse, lbs
to Driving or Forcing Frequency, rad/sec
t Time, sec
From the free body diagram of Figure 62(b), the equation of motion is quite simply
mex + kex = P(t) = P0 sintnt
or dividing by the equivalent mass
k P
x h—— x — —— sin TiJ t
me me
It is convenient to define the “equivalent” natural circular frequency co as
which allows Equation (65) to be rewritten as
2 P
x + co x = —— sin rnt
me
The complementary solution of the homogeneous portion of Equation (67) is
xc(t) = A coscot + B sin cot
and the particular solution is assumed to be of the form
xp(t) = C sin in t
which upon substitution into Equation (64) yields the particular solution
P
x (t) = ° , sin tut
P ke(l-p2)
(63)
(64)
(65)
(66)
(67)
(68)
(69)
(70)
Where P is defined as the ratio of the driving frequency to the natural circular frequency. When

128
added to the complementary solution, the general solution then becomes
x(t)=Acosco t + Bsincot +
-sinrat
(71)
k.O-PO
The initial or “at rest” conditions allow determination of the constants A and B. Assuming no
initial deflection nor velocity, i.e.
x(0) = 0
x(0) = 0
(72)
the general solution becomes
P
x(t) = 2—— [sin rrrt - Bsincot] (73)
ke(l~P2)
The Phase 1 response, shown in Figure 63, is the forced response of the SDOF system during the
time the impulsive load is being applied, while the Phase 2 response is the unforced or free
Figure 63. Phase 1 and Phase 2 Response

129
SDOF system response which depends on the initial conditions x(t)) and x(t,). Resonance,
defined as the condition when P = 1, i.e. the driving frequency is the same as the SDOF natural
circular frequency, makes the solution indeterminate, and may be resolved using L’HopitaPs
rule. Fortunately, this does not occur for the idealized beam SDOF system.
Since there is no applied load in Phase 2, the equation of motion for Phase 2 is simply
the homogeneous ordinary differential equation
mex+ ke x = 0
or (74)
x + co2 x = 0
which was taken to have the solution
x(t) = A cos co t + B sin cot (75)
Using the initial conditions from Equation (72) allows us to rewrite the constants A and B in
Equation (75) simply as
.. ... x(0) .
x(t) = x(0) cos cot + sin co t
to
(76)
In Phase 2, it is convenient to define the relation
t = t-t,
(77)
which when substituted into Equation (76) yields the general form of the solution for Phase 2.
(78)
x(tj)
x(t) = x(t |) cos cot H — sin cot
co
x(t,)
The initial conditions for Phase 2, x(fi) and —, are obtained from the solution for Phase 1,
co
given by Equation (73), at t = tj, the end of the load pulse. It can be shown that the complete
solution for the Phase 2 response is given by
x(t) =
-PoP
. TC 71 . —
sin —cosco t + (1 + cos—)sinco t
Mi-POL P P
(79)

130
Equations (73) and (79) completely define the displacement-time response of the SDOF system
in Phases 1 and 2, given that the amplitude and duration of the load pulse, as well as the beam
equivalent mass and stiffness are all known. These equations, along with other required
parameters, were programmed as the third and final portion of the analytical model using the
MATHCAD software, and are presented as Appendix E.
Comparison to Drop Weight Impact Test Data
Figure 64 shows two typical analytical derived displacement-time curves compared to
the corresponding experimental data. Figure 64(a) shows the results for a plain concrete beam
with a 2.0 in (5.08 cm) drop height. The analytical model generally overpredicts the
displacement; however, up until the onset of beam failure occurs at approximately 660 psec, the
model gives reasonable results. For this beam, the load pulse was 1034 psec long and had an
amplitude of 2530 lbs (11.25 kN). Figure 64(b) shows results for a beam with three ply CFRP
with a 18.0 in (45.72 cm) drop height. The load duration was 822 psec, and the load amplitude
was 7970 lbs ( 35.45 kN). This beam began to fail at about 230 psec. Again, the model does
fairly well predicting the early time displacement behavior, and then similar to the plain concrete
beam, overpredicts at later times.
There are several reasons why the model may be overpredicting the displacements. The
data from the dropweight tests showed that the beam’s peak bending loads are much higher
under dynamic loading conditions. Conversely, the displacements at failure were much less.
This implies that the beams are much stiffer dynamically than they are statically, due to strain
rate effects in the concrete. Hence, a stiffer beam does not displace as much. Beam stiffnesses
are calculated in the model using section analyses, which are quasistatic values. The change in
stiffness from Region 1 to Region 2 and then to Region 3 is also calculated in the model using
the quasistatic displacements 81 and 82 as “triggers” to indicate a change in slope has occurred.

131
BEAM LW6-43 DISPLACEMENT vs TIME
x LW6-43 DATA
to
BEAM LW9-66 DISPLACEMENT vs TIME
x LW9-66 DATA
(b)
Figure 64. Analytical Displacement - Time Curves versus Experiment

132
These changes in stiffness are responsible for the slight discontinuities in the displacement-time
curves shown in Figures 64(a) and (b). In addition, since the dynamic failure displacements are
typically 1.5 - 10 times smaller than the static failure displacements, this implies the triggers to
change the stiffness in the model are probably occurring too late temporally. Currently, there is
no analytical method available to calculate a “dynamic” beam stiffness.
As discussed previously, the beam’s equivalent mass is also changing as a function of
time, due to the fact that only a localized portion of the beam under the tup is initially displacing.
These so - called plastic hinges move outward and become stationary, at which point the beam
assumes the characteristic plastic state. The equivalent mass, of course, is based upon this final
displaced state. The analytical model has assumed a constant equivalent mass, which has been
adjusted to account for the position of the plastic hinge during the time of the loading.
Obviously, both of these shortcomings in calculating the stiffness and mass affect other
parameters which are derived from them such as the natural circular frequency co and the
frequency ratio (3. A smaller equivalent mass would have the same effect as a higher stiffness; it
would reduce the displacement.
Finally, there is no provision in the analytical model to account for failure. It was
observed during the experimental portion of this research that in nearly all of the beams tested,
the onset of failure began about halfway through the duration of the load pulse, or about halfway
through Phase 1. Notwithstanding the definitional problems associated with what failure means,
there is no present method available in the model to account for flexural cracking in the beam,
which would tend to reduce its overall stiffness. However, a reduction in stiffness implies an
increase in displacement, which is counter to the required trend in analytical model results to
better match the data.
In summary, the analytical model does a reasonable job predicting the early
displacement-time behavior of the beams. However, it generally overpredicts the displacements

133
after about halfway through the duration of the applied load. The overprediction in displacement
is probably due to the fact that the beams are much stiffer dynamically than they are statically,
due to strain rate effects in the concrete. The model calculates static stiffness values from
section analysis, since no method is available to predict dynamic stiffnesses.

CHAPTER 4
FINITE ELEMENT METHOD CALCULATIONS
In order to study the dynamic behavior of the beams in a more fundamental and rigorous
computational method unavailable in an analytical model, a series of finite element method
(FEM) calculations using the Automatic, Dynamic, Incremental, Nonlinear Analysis (ADINA)
[46] code were conducted. The ADINA code has been used extensively in nonlinear concrete
analyses, and is well-suited to handle concrete beam bending problems of the type being studied
in this investigation.
Description of the ADINA FEM Computer Code
The ADINA FEM code, developed in 1975 at the Massachusetts Institute of Technology
by Bathe [47], is an evolutionary extension of the original Structural Analysis Program (SAP)
series developed in the late 1960s at the University of California, Berkeley. With the wide use of
plain, reinforced, and prestressed concrete as a structural material and because of the
development of powerful FEM procedures, interest in nonlinear analysis of concrete structures
has increased. However, a number of factors have prevented wider acceptability of nonlinear
FEM procedures in analysis of concrete structures.
Concrete exhibits a complex structural response with several important nonlinearities;
nonlinear stress-strain behavior, tensile cracking and compression crushing material failures, and
temperature dependent creep strains [48], All of these nonlinearities depend on the triaxial state
of stress in the concrete. As a consequence, the constitutive properties of concrete have not been
completely identified, and there is no generally accepted material law available to model the
concrete behavior. Additional nonlinearities may also be introduced into the behavior when
134

135
reinforcing materials such as steel and CFRP are used in the structural element. Steel is typically
modeled as an elastic-perfectly plastic or elastic-work hardening material, while CFRP, at least
in this study, is modeled as a purely elastic material.
Additionally, nonlinear FEM analysis can be very costly, largely due to the difficulties
encountered in the stability and accuracy of the solutions. These difficulties are a direct result of
the specific numerical implementation of the concrete nonlinearities.
ADINA, as with other FEM codes, assumes that the structure may be idealized as an
assemblage of finite elements. The stress analysis process can generally be understood to consist
of essentially three phases:
1) Calculation of the structure matrices K, the stiffness matrix; M, the mass matrix;
C, the damping matrix; and R, the external load vector.
2) Solution of the equilibrium equations.
3) Evaluation of the element stresses.
When step two involves the inclusion of acceleration and velocity dependent forces, i. e. inertial
and damping forces, the equation of motion (or equilibrium) is given by
MÜ + CÜ + KU = R (80)
where
U Displacement Vector
Ü Velocity Vector
Ü Acceleration Vector
Mathematically, Equation (80) represents a system of linear differential equations of second
order, and in principle, the solution of the equations can be obtained by standard procedures for
the solution of differential equations with constant coefficients. However, these solution
procedures can become cost prohibitive if the matrices M, C, K are large. Therefore, in the

136
present study, a direct numerical integration was utilized. Rather than solving Equation (80) for
any time t, a direct numerical integration scheme only attempts to satisfy (80) at discrete time
intervals At apart. This means that equilibrium, to include inertial and damping forces (if
applicable), is only sought at discrete points in time within the solution interval. Therefore, all
solution techniques used in static analysis can also be used in direct numerical integration.
Careful selection of the time step for numerical integration in a wave propagation
problem is crucial to the stability and accuracy of the solution. In the present study, the
Newmark Method of implicit time integration with a consistent mass formulation was employed.
Since the Newmark Method is implicit, hence unconditionally stable, selection of the time step
may be based solely on accuracy considerations. In a wave propagation problem, the maximum
time step is related to wave speed in the material, and to the element size. The maximum time
step is selected so that the stress wave propagates the distance between element integration
points within that time increment. The maximum time step is defined by
(81)
where Lc is the length of an element in the direction of the wave propagation and c is the velocity
of propagation of a longitudinal wave in a bounded media (sometimes referred to as the bar
velocity) which is given by
(82)
in which E is the elastic modulus and p is the mass density of the material. Tedesco et al. [49]
has determined that a time step of
At <-At
3
max
(83)

137
yields accurate results. In the present study, a time step of 1 psec was used in the FEM dynamic
analyses. The solution of the equilibrium equations represented by Equation (80) yields all
element nodal point displacements U. Once the nodal point displacements are obtained, the
element strains are obtained from
e = BU (84)
where B is the strain - displacement transformation matrix. Finally, the stresses in the elements
are obtained from the strains using the appropriate material laws.
The Concrete Material Model
The concrete material model used in ADINA is a hypoelastic model based upon a
uniaxial stress strain relation, shown in Figure 65, which has been generalized to account for
Figure 65. Uniaxial Stress - Strain Relation Used in Concrete Model

138
biaxial and triaxial states of stress. The model employs three basic features to describe the
concrete’s behavior: (1) a nonlinear stress-strain relation including strain softening to allow for
weakening of the material in compression, (2) failure envelopes that define cracking in tension
and crushing in compression, (3) a three part strategy to model the post-cracking and crushing
behavior of the material.
To establish the uniaxial stress - strain law of Figure 65 which accounts for multiaxial
stress conditions, and to identify whether tensile fracture or compression crushing has occurred,
failure envelopes are employed. Since the fracture stress of concrete is roughly 10 percent of the
compressive crushing stress, failure of concrete is tension dominated. The tensile failure
envelope used in the ADINA concrete model is shown in Figure 66. To identify whether the
material has failed, the principal stresses are used to locate the current stress state in the failure
envelope. The tensile strength of the material in a principal direction does not change with the
introduction of tensile stresses in the other principal stress directions, but the compressive
stresses in the other directions change this tensile strength.
Tensile failure occurs when the tensile stress in a principal stress direction exceeds the
fracture stress. The code assumes a plane of failure develops perpendicular to the principal
stress direction. The effect of this material failure is that the normal and shear stiffnesses across
the plane of failure and the corresponding normal and shear stresses are reduced.
The compressive triaxial failure envelope employed in ADINA is quite complex and
requires considerably more data input that the tensile failure model. The envelope may be used
to represent a large number of different biaxial and triaxial envelopes, since it is implemented in
ADINA as 24 discrete stress values. The shape of the default compressive failure surface in the
code is based on experimental evidence [48], and was subsequently used in this investigation.

139
or’ «UNIAXIAL CUT-OFF TENSILE STRESS UNDER
MULTIAXIAL CONDITIONS
?t = UNIAXIAL CUT-OFF TENSILE STRESS
SV «UNIAXIAL COMPRESSIVE FAILURE STRESS UNDER
MULTI AX IAL CONDITIONS
t»Di't°-B2 .t pl pZ p3 AND 3 AT TIME T
Figure 66. 3-D Tensile Failure Envelope Used in Concrete Model [46]
The post failure material behaviors considered in ADINA include: (1) post tensile cracking, (2)
post compression crushing, and (3) strain softening behavior.
Once a tensile plane of failure has formed, it is checked in each subsequent solution step
to see whether the failure is still “active”. The failure is considered to be “inactive” provided the

140
normal strain across the failure plane becomes negative and less than the strain at which the last
failure occurred. Otherwise, the tensile failure is still considered to be active.
If a material has crushed in compression, it is assumed that the material strain softens in
all directions. Consider the uniaxial stress conditions shown in Figure 65. For a strain larger
than s'c, the material has crushed, and softens with increasing compressive strain, i.e. the
modulus becomes negative. Under multiaxial conditions, the compression crushing is identified
using the multiaxial failure envelope. Once the material has crushed, isotropic conditions are
assumed using the uniaxial stress-strain law, with the constants f'c and e'c corresponding to the
multiaxial conditions at crushing.
The pertinent material parameters for the tensile and compressive failure envelopes and
the uniaxial stress-strain relation are summarized in Table 17.
Table 17. Concrete Material Model Parameters
PARAMETER
VALUE
E„
Initial Tangent Modulus
3.47 x 106 psi
u
Poisson’s Ratio
0.2 (unitless)
f;
Uniaxial Cut - Off Tensile Strength
630 psi
í
Uniaxial Maximum Compressive Strength
6729 psi
<*u
Uniaxial Ultimate Compressive Strength
5720 psi
ec
Uniaxial Compressive Strain at £
0.003 in/in
Su
Uniaxial Compressive Strain at au
0.004 in/in
P
Mass Density
1.77 x 10'4 lbs - sec2/in4

141
The CFRP Material Model
The CFRP is assumed to behave elastically under both static and dynamic loading
conditions until fiber tensile strength is achieved, at which point the carbon fibers break and
failure abruptly occurs. In fact, the CFRP 0° tensile strength is so high (320 ksi/2.2 GPa), fiber
breakage rarely occurred during dynamic testing. Since the fibers in the CFRP were all oriented
and subsequently loaded in the same direction, a linear elastic material model was employed,
whereby the total stress is uniquely determined by the total strain. Table 18 lists the parameters
used to model the CFRP.
Table 18. CFRP Material Model Parameters
PARAMETER
VALUE
E Elastic Modulus, Axial (0°) Direction
20 x 106 psi
p Mass Density
1.475 x 10"4 lbs sec2/in4
Concrete Beam with CFRP - Finite Element Model
Two nonlinear dynamic analyses were conducted; one analysis simulated a plain
concrete beam tested at a drop height of 2.0 in (5.08 cm), the other analysis simulated a concrete
beam reinforced with three ply CFRP on the bottom only, tested at a drop height of 18.0 in
(45.72 cm). Figure 67 shows the finite element mesh employed in the analyses. The concrete
was modeled as 30 eight node, two dimensional, plane stress elements. The elements were
isoparametric displacement-based finite elements, with eight node elements being chosen for
their efficacy in modeling thick, rectangular structures such as a prismatic beam. The individual
element sizes were 1.5 in (3.81 cm) in the Y direction, and 1.0 in (2.54 cm) in the Z direction.
To model the simple support, node six is constrained in the Z direction, and since there is a plane

142
Figure 67. Finite Element Mesh
of symmetry at beam midpoint, nodes one through four are constrained in the Y direction. All
other nodes are considered free.
For the beam with three ply CFRP on the bottom, 10 three node truss elements were used
to model the CFRP, since the three node truss element is compatible with the two - dimensional
eight node plane stress element. The area of the truss element was taken as the width of the
CFRP panel (3.0 in/7.62 cm) times the ply thickness. The CFRP was assumed to be perfectly
attached to the concrete, since the epoxy bond line was not modeled. This is a limitation in the
FEM model employed for this analysis, and is an area of potential future research.
Dynamic Loading Calculations
Nonlinear dynamic analyses were conducted for two beam types; one plain concrete
beam and one beam reinforced with three ply CFRP. Figure 68(a) shows the load pulse

143
O
•J
0*
P
H
(a)
0.0s 1.0ms 2.0«s 3.0ms
TIME
(b)
Figure 68. Tup Load - Time Curves for (a) Beam LW6-43, and (b) Beam LW9-66
Used in ADINA Simulations

144
recorded from plain concrete beam LW6-43, which was tested at a 2.0 in (5.08 cm) drop height.
The load amplitude was 2530 lbs (11.25 kN), with a duration of 1034 psec. The load pulse was
discretized in 50 psec increments in order to build a load-time function for use in ADINA. The
code assumes a linear variation in load between two sets of load-time data. Figure 68(b) shows
the experimental load pulse recorded for the three ply reinforced beam LW9-66, which was
tested at 18 in (45.72 cm). The load amplitude was 7970 lbs (35.45 kN) with a temporal duration
of 822 psec. This load pulse was also discretized at 50 psec time increments in order to build a
load - time function for use in the code. The concrete material properties used for the
calculations are given in Table 17, and the CFRP properties are given in Table 18. There was
some variation in the unconfmed compressive strength between the two beams tested which
resulted in slight variations in the initial tangent modulus and the uniaxial ultimate compressive
strength. Otherwise, the material properties were the same in both calculations. Appendix F
lists the input and plot files for plain concrete beam LW6-43, and Appendix G contains the input
and plot files for the three ply CFRP reinforced beam LW9-66.
The calculations were conducted on a two processor Cray Y-MP located at Eglin Air
Force Base, Florida. The calculations were carried out to 1.5 msec, which was experimentally
determined to be sufficient time for the beams to fail in flexure and/or shear. The time step was
previously determined to be 1 psec; therefore, there were 1500 solution steps calculated. Both
calculations required approximately 1050 sec of CPU time, using about 25 percent of both
processor’s capacity.
Results and Comparison to Test Data
The ADINA calculational results for the three ply beam, LW9-66, are presented in
Figures 69-71. Figure 69 shows the temporal variation in the displacement field at 100 psec
increments. The original finite element mesh is shown as the dashed lines, while the deformed

145
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN CONCRETE BEAM w/3 PLY CFRP
ADINA ORIGINAL DEFORMED XVMIN -0.1183
LOAD STEP l__j i i XVMAX 15.14
TIME_0.0001000 0.9168 0.0007158 YVMIN -0.9168
YVMAX 3.016
ADINA ORIGINAL DEFORMED XVMIN -0.1183
LOAD STEP l__j , i XVMAX 15.14
TIME 0.0002000 0.9168 0.0007158 YVMIN -0.9168
YVMAX 3.016
LOAD_STEP l _ _ j i i XVMAX 15.14
TIME 0.0003000 0.9216 0.01018 YVMIN -0.9216
YVMAX 3.032
ADINA ORIGINAL DEFORMED XVMIN -0.1678
LOAD_STEP L__j i i XVMAX 15.14
TIME 0.0004000 0.9216 0.01018 YVMIN -0.9216
YVMAX 3.032 •
Figure 69. ADINA Displacement Fields at 100 psec Intervals for Beam LW9-66

146
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN CONCRETE BEAM w/3 PLY CFRP
ADINA ORIGINAL DEFORMED XVMIN -0.1815
LOAD STEP l__j i > XVMAX 15.08
TIME~0.0005000 0.9303 0.03187 YVMIN -0.9303
YVMAX 3.060
TIME_0.0006000 0.9303 0.03187 YVMIN -0.9303
YVMAX 3.060
ADINA ORIGINAL DEFORMED XVMIN -0.1997
LOAD_STEP L__J , I XVMAX 15.04
TIME 0.0007000 0.9429 0.05854 YVMIN -0.9429
YVMAX 3.102
TIME 0.0008000 0.9429 0.05854 YVMIN -0.9429
YVMAX 3.102 .
Figure 69-continued

147
mesh is shown as solid lines. The beam’s centerline is located on the right hand edge of the
plots. Note that the scale for the deformed shape varies by almost two orders of magnitude
between the initial plot at 100 psec and the final plot at 800 psec. The FEM calculation shows
that only a small portion of the beam is initially involved in bending, indicating that the
assumptions made in the analytical model to calculate the beam’s effective mass are quite
reasonable. In fact, looking at the displacement fields at 300 and 400 psec, the traveling plastic
hinge appears to have become stationary at about 8.25 in (20.96 cm) from beam midpoint, which
is 0.61 L/2. This is consistent with the range in which Johnson [44] predicts the traveling plastic
hinge to become stationary (0.59 - 0.67 L/2 ) as well as the time, 350 psec. The stationary hinge
is even more visible in the plots at 500 - 600 psec, where it appears as a discontinuity in the
upper and lower surfaces of the beam. By 700 psec, the outer portion of the beam has rotated
around the support, and the beam now assumes a characteristic (linear) plastic shape.
Figure 70 shows the crack patterns which develop in the beam during the loading and
subsequent failure process. Cracks are predicted to occur when the maximum principal stress
exceeds the pre - determined tensile or fracture stress in the concrete, which for this calculation
was 570 psi ( 3.93 MPa). ADINA predicts that flexural cracking (vertical cracks appearing at the
bottom center of the beam) begins between 200 and 300 psec, and by 400 psec, the crack(s) have
run to the upper surface of the beam. Data from strain gage number three, located on the bottom
center of the beam, indicated an abrupt change of slope in the strain versus time curve at about
230 psec, indicating crack initiation has occurred. Strain gage number two, located on the
original neutral axis of the beam 1.5 in (3.81 cm) from the bottom, indicated failure also at about
230 psec. Strain gage number one, located 2.75 in (7.0 cm) from the bottom of the beam,
showed a change of slope in strain versus time at about 355 psec, indicating the crack(s) had

148
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN CONCRETE BEAM w/3 PLY CFRP
ADINA ORIGINAL XVMIN 0.000
, , XVMAX 15.00
1.544YVMIN 0.000
YVMAX 3.000
100 usee
ADINA ORIGINAL XVMIN 0.000
i , XVMAX 15.00
1.544YVMIN 0.000
YVMAX 3.000
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, i XVMAX 15.00
1.544 YVMIN 0.000
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1.544YVMIN 0.000
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500 usee
ADINA ORIGINAL XVMIN 0.000
i i XVMAX 15.00
1.544 YVMIN 0.000
YVMAX 3.000
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Figure 70. ADINA Crack Patterns versus Time for Beam LW9-66

149
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN CONCRETE BEAM w/3 PLY CFRP
ADINA ORIGINAL XVMIN 0.000
« , XVMAX 15.00
1.544YVMIN 0.000
YVMAX 3.000
H H l-i|[ 8
m
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ADINA ORIGINAL XVMIN 0.000
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1.544YVMIN 0.000
YVMAX 3.000
HI
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, i XVMAX 15.00
1.544YVMIN 0.000
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7-^
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Figure 70—continued
reached the upper surface of the beam. A post-test damage assessment indicated a single
flexural crack had formed at the beam’s midpoint
The calculation shows that shear crack(s) in the concrete appear at a later time, starting
at about 400 psec. By 800 psec, the crack(s) are located about 7.5 - 9.0 in (19.1 - 22.9 cm), as
measured outward from the bottom from the beam’s centerline. A post-test damage assessment
conducted on beam LW9-66 indicated two major shear cracks had formed in the beam, one
located at 8.0 in (20.32 cm), and the other at 3.0 in (7.62 cm), as measured from bottom center of
the beam. Both of these cracks ran up to the upward-shifted neutral axis at 45 degrees, where

150
they subsequently joined and ran to beam midpoint. Without the CFRP, the beam merely would
have failed by a single flexural crack.
Figure 71 shows the predicted midspan displacement versus time compared to data from
the standoff displacement gage. Even though the calculation is unable to predict additional
Figure 71. ADINA Midpoint Displacement versus Time for Beam LW9-66
complexities such as CFRP debonding, the code clearly shows that the displacements, hence the
velocities and accelerations are correctly computed. The excellent agreement exceeded all
expectations, considering the complexity of the problem and the assumptions made in the
material models. Experimental data indicated failure occurred at about 730 psec, at a
corresponding displacement of 0.064 in (1.63 mm), indicating that the code also correctly
reproduces the post - failure displacement - time behavior. Since the FEM code is able to adjust
the mass and stiffness matrices at each time step, it is able to account for reductions in stiffness
due to cracking, as well as computing the changing inertial forces and natural circular
frequencies, all of which the simple analytical model was unable to accomplish.

151
Figures 72 - 74 present the corresponding FEM results for the plain concrete beam,
LW6-43. Similar to the three ply CFRP reinforced beam, the displacement is initially localized
under the tup, which is coincident with beam centerline. Again, the centerline is shown as the far
right hand portion of the beam, since there is a plane of symmetry through the beam’s center.
Figure 72 shows the temporal variation in displacement up to 800 psec, in 100 psec
increments. As with the previous displacement field plots, there are two orders of magnitude
difference in scale between the first and last plots. Again, the original undeformed beam is
shown in dashed lines. AD1NA predicts that it takes about 600 psec for the entire beam to
become involved in the motion.
Figure 73 shows the crack pattern in the beam, which starts at about 700 psec. Since
there is no reinforcing steel nor CFRP on this beam, the flexural crack simply initiates at the
bottom of the beam once the fracture stress has been exceeded, and quickly runs to the top of the
beam. The code predicts that by about 900 psec, the beam is broken into two pieces.
Experimental evidence shows that strain gage number three, on the bottom of the beam,
suffers an abrupt change of slope at about 660 psec, indicating a crack has initiated. Strain gage
number two, located 1.5 in (3.81 cm) from the bottom of the beam, changes slope abruptly at 680
psec, indicating a crack is beginning to traverse across the active face of the gage. Strain gage
number one, located 2.75 in (7.0 cm) from beam bottom, recorded a change in slope at 855 psec,
indicating the crack had almost reached the upper surface of the beam, for a total transit time of
about 195 psec. The FEM code again does well, not only predicting the time at which the crack
initiates, but the propagation velocity as well. The predicted crack velocity was 1250 ft/sec
(381.1 m/sec), and the experimentally derived crack velocity was 1175 ft/sec (358.3 m/sec).
Figure 74 shows the ADINA displacement-time curve compared to data recorded by the
standoff displacement gage. The FEM code did not predict the displacement-time behavior as

152
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN PLAIN CONCRETE BEAM
ADINA ORIGINAL DEFORMED XVMIN -0.1490
LOAD_STEP L__J . , XVMAX 15.14
TIME 0.0001000 0.9177 2.112E-05 YVMIN -0.9177
YVMAX 3.019
ADINA ORIGINAL DEFORMED XVMIN -0.1490
LOAD_STEP l _ — j i i XVMAX 15.14
TIME 0.0002000 0.9177 2.112E-05 YVMIN -0.9177
YVMAX 3.019
ADINA ORIGINAL DEFORMED XVMIN -0.1554
LOAD STEP t__j i i XVMAX 15.15
TIME 0.0003000 0.9205 0.0006649 YVMIN -0.9205
YVMAX 3.028
ADINA ORIGINAL DEFORMED XVMIN -0.1554
LOAD_STEP i__j i i XVMAX 15.15
TIME 0.0004000 0.9205 0.0006649 YVMIN -0.9205
YVMAX 3.02a
Figure 72. ADINA Displacement Fields at 100 psec Intervals for Beam LW6-43

153
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 199b
DYNAMIC ANALYSIS OF A 3x3x30 IN PLAIN CONCRETE BEAM
ADINA ORIGINAL DEFORMED XVMIN -0.1335
LOAD STEP l__j i i XVMAX 15.13
TIME_0.0005000 0.9261 0.003457 YVMIN -0.9261
YVMAX 3.047
ADINA ORIGINAL DEFORMED XVMIN -0.1335
LOAD STEP l__j i i XVMAX 15.13
TIME_0.0006000 0.9261 0.003457 YVMIN -0.9261
YVMAX 3.047
TIMELO.0007000 0.9440 0.009350 YVMIN -0.9440
YVMAX 3.106
TIME_0.0008000 0.9440 0.009350 YVMIN -0.9440
YVMAX 3.106
Figure 72~continued

154
ADINA-PLOT VERSION 4.0.3, 10 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN PLAIN CONCRETE BEAM
ADINA ORIGINAL XVMIN 0.000
. . XVMAX 15.00
1.544 YVMIN 0.000
YVMAX 3.000
700 usec
#
•
•1 1
* ) 1
1 t »
*
1
ADINA ORIGINAL XVMIN 0.000
. , XVMAX 15.00
1.544 YVMIN 0.000
YVMAX 3.000
. . .
- - -
800 ysec
i ' t
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, i XVMAX 15.00
1.544 YVMIN 0.000
YVMAX 3.000
900 Msec
. , ¿
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1 t 1
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1 1 »
♦ 1 I
1 * *
« « ♦
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♦ i t
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Figure 73. ADINA Crack Patterns versus Time for Beam LW6-43
ADINA-PLOT VERSION 4.0.3, 18 DECEMBER 1995
DYNAMIC ANALYSIS OF A 3x3x30 IN PLAIN CONCRETE BEAM
2
Figure 74. ADINA Midpoint Displacement versus Time for Beam LW6-43

155
well as it did previously with the three ply CFRP reinforced beam. Here, it slightly
overpredicted the displacement, beginning at about 600 psec. The beam failed experimentally
(bending load went to zero) at about 825 psec, which is consistent with the strain gage data.
This indicates that even prior to failure, the FEM code is predicting a “less stiff’ beam, allowing
it to displace more than the experimental evidence indicates it should.
In summary, the ADINA code does an excellent job at predicting the displacement-time
behavior of the beams reinforced with three ply CFRP, which was significantly stiffer and loaded
at a higher load rate than the plain concrete beam. The bottom strain gage measured a peak
tensile strain rate of 17.5/sec at 345 psec. The FEM code predicted two distinct types of cracks;
flexural and shear. The predicted time of incipient flexural cracking compared well to the
experimental failure times of the strain gages. The code predicted that the flexural cracks appear
several 100 psec prior to the shear cracks. The predicted location of the shear cracking was in
agreement with the experimental evidence. The FEM code also correctly predicted the
displacements for an additional 750 psec after failure occurred, which was when the calculation
was terminated.
The ADINA code did a reasonably good job predicting the displacement - time behavior
of the less stiff plain concrete beam which was loaded at a slightly lower load rate; the bottom
strain gage indicated a peak tensile strain rate of 16.3/sec at 725 psec. The code however, did an
excellent job predicting incipient flexural crack initiation at the bottom of the beam, as well as
predicting the correct transit time through the beam. Since there was no internal nor external
reinforcement, the beam failed by a single flexural crack at beam midpoint.
The FEM calculations also gave insight into the traveling plastic hinge phenomena. The
calculation of the beam’s equivalent mass in this study would have otherwise been quite

156
problematic, since data analysis and SDOF modeling relied upon reasonable estimation of this
parameter.

CHAPTER 5
SUMMARY AND DISCUSSION
The objective of this research was to gain a fundamental understanding of the dynamic
behavior of laboratory scale plain and fibrous concrete beams which had been externally
reinforced with various thicknesses of high performance CFRP panels. An innovative vacuum
fixturing technique was developed to apply the thin CFRP panels to the beams, and a patent is
pending. The technique has been shown to work for any size beam.
In order to characterize the concrete used in the beam test specimens, a series of static
and dynamic compression and splitting tension tests were conducted. The static tests were
conducted on a load frame, and the dynamic tests were conducted on a 2.0 in (5.08 cm) diameter
Split Hopkinson Pressure Bar (SHPB). Test results for the lightweight concrete used in the beam
samples are compared to normal weight concrete of varying unconfined compressive strength.
The lightweight concrete was shown to behave quite differently in tension dynamically than its
normal weight counterpart.
A total of seventy-two beams in six different configurations were tested, both statically
in a load frame, and dynamically, in a drop weight impact machine. All testing was conducted in
three (center) point bending mode, and the beams were simply supported. The dynamic tests
were conducted at various drop heights to vary the loading rate, hence the strain rate in the test
samples. Comparison of measured load duration to calculated beam period showed the loading
to be purely impulsive. Loads, displacements, impact velocities, accelerations, strains and high
speed film data were all recorded, and detailed analyses were subsequently conducted. Various
157

158
measures of merit were used to compare and contrast the six different beam types; peak tup,
inertial, and bending loads, peak displacement, and fracture energies were calculated for all
beams. Strains and strain rates were also determined at up to three locations in the midspan
cross section of each beam.
An analytical model was developed, whereby the beams were idealized as a single
degree of freedom (SDOF) system subjected to a half sine pulse impulsive load. It was shown
that damping forces may be assumed negligible in the SDOF analyses, since the loading was
impulsive. Results from the analytical SDOF model are compared to experimental data from
two different beam types.
Finally, a series of finite element method (FEM) analyses were conducted using the
ADINA code. The code has been used extensively in both static and dynamic nonlinear
structural analyses, and is well-suited to handle concrete beam bending problems of the type
studied in this investigation. A description of the finite element mesh, material models, failure
envelopes, and assumptions made in the calculations are presented. Results from two
calculations are compared to the experimental evidence.
Characterization of the Concrete
The results from all of the quasistatic and SHPB direct compression tests indicate that in
compression, the lightweight concrete behaves identically, at least within the scatter of the data,
to its normal weight counterpart, when loaded either quasistatically or dynamically. In
compression, lightweight concrete shows moderate strain rate sensitivity, similar to its normal
weight counterpart, with increases in strength up to 2.3 times the static value at strain rates from
100 to 300/sec.
In tension, the lightweight concrete does not appear to fall within the scatter of the data
presented for five different strength normal weight concrete mixes. In fact, the data show that

159
the lightweight concrete is less strain rate sensitive than its normal weight counterpart, being
shifted about a half decade in strain rate for a constant dynamic increase factor. In other words,
lightweight concrete must be loaded about three times faster in order to achieve the same
dynamic tensile strength as normal weight concrete. In either case, the normal and lightweight
concrete have a higher strain rate sensitivity in tension than in compression. It is believed that
this data represents the only information reported to date on strain rate sensitivity of lightweight
concrete.
Static Beam Bending Experiments
Results from the static three (center) point bending experiments show that beams with
three ply CFRP on the sides as well as the bottom are clearly able to take the most load prior to
failure, and have the largest energy absorption or fracture energy capacity of all six different
beam types tested.
Beams with two ply CFRP on the bottom only have the next highest energy absorption
or fracture energy capacity, due to their high ductility and relatively high load carrying
capability. Conversely, the experimental evidence indicates little to no benefit is realized using
nylon fibers in the concrete mix as a potential technique to increase load, displacement, or
fracture energy capacity, when compared to similarly reinforced beams without the nylon fibers
added to the mix. It was thought that the nylon fibers should provide some additional energy
absorption capacity by providing “bridges” across cracks thereby attenuating cracking.
There are still tremendous gains over the plain concrete beams in terms of load and
displacement capacity, even with the bottom only one, two, and three ply CFRP reinforced
beams. Increases of two to four in load and 11 to 17 in displacement are quite easily achieved
with the addition of the CFRP. Consequently, increases in energy absorption or fracture energy
from 30 to 80 are also achieved, when compared to the baseline plain concrete beams.

160
Dynamic Beam Bending Experiments
Results from drop weight impact testing show that the measured peak amplitude of the
tup load increases with an increase in drop height, along with corresponding increases in the
calculated peak inertial load and peak bending load. Most of the load recorded by the tup is
inertial in nature, therefore it is not surprising to see the tup load increase with drop height. The
increase in peak bending load with increasing drop height implies an increase in beam impact
strength with an increase in the load or stress rate, hence strain rate. Data from strain gages
located on the bottom or tension side of the beams indicated an increase in strain rate with drop
height. The peak bending load increase was highest for the “less reinforced” or less stiff beams,
gradually declining in value as beam stiffness increased.
For the plain concrete beams, the maximum displacement (at failure) decreases with an
increase in drop height. However, for those beams reinforced with CFRP, the displacement at
failure increases with an increase in drop height.
Fracture energy consistently increases with an increase in drop height, which is in
consonance with nonlinear fracture mechanics theory. This theory purports that immediately
ahead of a moving crack is a zone of microcracking called the process zone. Since the size of the
process zone is dependent upon crack velocity, a faster crack has a larger process zone. Higher
stress rates propagate cracks more quickly, thereby creating a larger process zone. This zone of
increased microcracking may explain the increasing fracture energies recorded at increasing drop
heights.
The tup load pulse also foreshortens with increasing drop height. This would seem to
indicate that the failure is occurring more quickly as the drop height is increased. Since the tup
measures the total resistance the beam offers to include both bending and inertial forces, a

161
foreshortening of the bending load versus time curve would indicate a foreshortening in the time
to failure.
Static versus Dynamic Beam Bending Experiments
The average static peak bending load was always less than the dynamic peak bending
load, even at the lowest drop heights. The increase in peak bending load with drop height is
attributed to strain rate effects in the concrete.
The dynamic fracture energy is larger than the static fracture energy for the plain
concrete beams. However, for the remainder of the beams all reinforced with external CFRP, the
dynamic fracture energies were all consistently less than the static fracture energies. The
addition of external CFRP to the beam significantly stiffens the beam, thereby enhancing the
beam’s brittle behavior when loaded dynamically. This implies that a high strength plain
concrete beam is “dynamically equivalent” or has the same energy absorption capacity, as a
normal strength concrete beam which has been externally reinforced with CFRP.
The peak displacements were always less under dynamic loading when compared to the
quasistatic loading case. Except for the plain concrete beams, the dynamic fracture energies
were always less than the static values, even though the peak dynamic bending loads were
typically two to three times higher than the peak static bending loads. This would seem to
indicate that displacement is the limiting parameter in determining dynamic fracture energy.
This implies that for a given drop height, i.e. strain rate, a beam has a fixed capacity to absorb
energy, dictated by the concrete’s impact strength and limited by displacement, emphasizing the
brittle nature of concrete.
The mechanism by which the beams failed dynamically did not change appreciably from
the static loading case; failure of all beams with only tensile CFRP reinforcement was one of
shear failure in the concrete at approximately one quarter span from beam midpoint, followed by

162
delamination and peeling of the CFRP. Failure of beams with CFRP on the sides as well as the
bottom was concrete shear failure followed by CFRP side panel splitting and buckling along the
major shear crack in the concrete. Failure of the plain concrete beams was one of flexure, with a
single flexural crack at beam midpoint merely breaking the beam into two pieces, creating two
new fracture surfaces.
Analytical Model
Results from the section analysis indicate that if the concrete modulus of rupture is
accurately determined, the section analysis does very well predicting Point 1, the termination of
elastic behavior, for all six beam types. For the plain concrete beam, the predicted load and
displacement was within 10 percent of the experimental value. However, for all of the beams
reinforced with CFRP, the section analysis generally overpredicted the stiffnesses in Regions 2
and 3. The displacements at the predicted failure point, Point 3, were generally within 10
percent or less of the experimentally determined failure displacements, which implies that the
predicted loads are too high. These calculated stiffnesses were subsequently used in the SDOF
model.
Results from the SDOF idealization of the beams subjected to a half sine pulse impulsive
load show that the analytical model generally overpredicts the displacement - time behavior of
the beams. However, up until beam failure occurs, the model gives reasonable results. There are
several reasons why the model may be overpredicting the displacements; data from the
dropweight tests show that the beams are much stiffer dynamically than they are statically, due
to strain rate effects in the concrete. Beam stiffnesses in the model are calculated from section
analyses, which uses quasistatic material property values. There is currently no methodology
available to calculate a dynamic stiffness. Secondly, the beam’s effective mass is changing as a
function of time, since only a localized portion of the beam under the applied load is initially

163
displacing. This is due to the impulsive nature of the loading. Eventually, as the plastic hinges
form, travel outward and become stationary, the beam assumes a final displaced state and a
known shape function upon which the equivalent mass in the SDOF model is based. The SDOF
idealization assumes a constant equivalent mass, which has been adjusted to account for the
changing displacement field during the time of loading. Lastly, the analytical model contains no
provisions to account for failure. It was observed in the experimental portion of this research
that in nearly all of the beams tested, the onset of failure began about halfway through the
duration of the load pulse, or the Phase 1 response. Failed or cracked sections in the beam
reduce the beam’s overall flexural stiffness.
Finite Element Method (FEM) Calculations
The FEM code ADINA generally did very well predicting the displacement-time
behavior of the two beam types studied in detail; a plain concrete beam and a beam reinforced
with three ply CFRP on the bottom. The applied loads used in the calculations were discretized
from actual experimental tup loads. Key concrete material properties were also matched to
experimental data; the CFRP material properties came from the supplier.
The ADINA code predicted two types of cracking in the beam reinforced with three ply
CFRP on the bottom or tension side of the beam; flexural and shear. For the plain concrete
beam, the code predicted only flexural cracking. In both cases, the code not only correctly
predicted the type and location of the cracks, but the times at which they initiated. In addition,
the code also correctly predicted the transit time of the crack(s) through the beam to within 10
percent of the experimentally determined times from the strain gages.
The FEM calculations also gave insight into the localized displacement behavior of the
beams when subjected to an intense, impulsive load, as well as the traveling plastic hinge
phenomena. The determination of the beam’s equivalent mass, used extensively throughout this

164
investigation, would have otherwise been quite problematic, since data analyses and analytical
SDOF modeling relied upon reasonable estimation of this quantity.
Future Research
As with most research efforts, additional opportunities for further study manifested
themselves during the course of this work. These ideas are presented below, in no particular
order.
In the section analysis portion of the analytical model, no shear forces were included in
the force balance. Clearly, if the shear and moment diagrams are drawn for a simply supported
prismatic beam loaded at its midpoint, both shear and bending stresses are present in the section.
Further work should attempt to include the effects of shear stress in the section analysis. This
may address why the predicted stiffnesses are too high in Regions 2 and 3 in the current section
analysis.
Most concrete structural elements contain internal reinforcing steel in the form of round
bars or wires, in order to provide tensile strength and a certain amount of ductility to an
otherwise brittle structural material. The concrete beams fabricated for this study did not have
internal reinforcing steel due to the small cross-sectional area of the beams, since beam size was
dictated by the support platforms of the drop weight impact machine and the load frame.
Inclusion of tensile reinforcing steel would mandate a larger beam size, which in turn, would
dictate substantial modifications to the test equipment. However, realistic concrete structures
contain internal reinforcement. The effect of this internal steel, along with the external CFRP on
both the static and dynamic behavior of structural elements should be studied and understood, as
its effect will be pervasive.
Since it is unclear whether the results obtained in this study are directly scaleable to
large beams, it is recommended that dynamic testing be conducted on larger scale beams to

165
investigate potential scaling effects. Duplication of controlled, large scale center point loading
may however, prove to be problematic. Distributed loadings such as those generated by
explosives may offer a potential solution to this problem.
Realistic concrete structures also contain slab elements such as floors and walls. When
loaded elastically, such slabs bend into a dished surface, meaning that at any point, the slab is
curved in both principal directions. Since bending moments are proportional to curvatures,
moments must also exist in both directions; to resist these moments, the slab must be reinforced
in both directions. Externally applied composite materials offer unique possibilities to solve the
problem, since the laminae properties and orientation may be tailored or matched to the expected
structural loads
The effects of external reinforcement on mitigating the effects of blast on a structure is
not only an interesting problem, but a more complicated one, since typical external walls on a
structure are of such proportions and are supported in such a way that two-way action results.
Under blast loading, concrete slabs of moderate thickness and conventionally reinforced with
steel usually deform by yield line motion at approximately 45° with respect to the slab edges.
Symmetric angle-ply laminates, where some of the plies are oriented normal to the yield lines,
may provide the best ply arrangement. As with the initial studies on beams, it is recommended
that small scale slabs be studied initially, in order to gain a fundamental understanding of their
behavior. Large scale dynamic studies using distributed loadings, such as those generated by
explosives, should also be investigated.
Those beams reinforced on the sides as well as the bottom with CFRP proved to have the
highest energy absorption capacity of all beam types tested. In the present study, three ply
unidirectional CFRP was used on the sides of the beams. Since the failure of this beam type was
one of shear followed by side CFRP buckling and splitting, a [± 45°]s lamina stacking sequence

166
on the side CFRP should be studied, as it may provide even higher fracture toughness values.
Further investigation into these preferred fiber orientations is certainly warranted.
This investigation focused solely on one external reinforcing material; pre - preg CFRP.
Recently, the Japanese have also investigated the use of CFRP to retrofit several full scale
concrete structures in Japan. The Japanese work is different from the present work in that it
makes use of dry fiber type CFRP rather that the pre-preg CFRP plates or strips used in this
study. Two petrochemical industries, the Tonen Corporation and Mitsubishi Chemical, have
aggressively pursued this technology, and have developed proprietary processes to impregnate
the dry CFRP sheets with various epoxy resins directly onto the concrete, thus eliminating the
requirement for mechanical or vacuum fixturing when applying the CFRP to the structure. This
new material and application process, along with other more traditional materials such as
Kevlar , S-glass, and E-glass also warrant investigation, as they may provide a more cost
effective solution to the external post - strengthening problem.

CHAPTER 6
CONCLUSIONS
Static and dynamic material property testing indicated that in compression, lightweight
concrete behaves identically to its normal weight counterpart. It shows moderate strain rate
sensitivity, with increases in strength up to 2.3 times the static value at strain rates from 100 to
300/sec.
In tension, however, lightweight concrete does not behave the same as normal weight
concrete. It is less strain rate sensitive, and must be loaded about three times faster in order to
achieve the same dynamic strength as normal weight concrete. In either case, normal and
lightweight concrete have a higher strain rate sensitivity in tension than in compression.
Static three (center) point bending experiments showed that beams with three ply CFRP
on the sides as well as the bottom are clearly able to take the most load prior to failure, and have
the largest fracture energy capacity of all six different beam types tested. Conversely, little to no
benefit is realized introducing nylon fibers into the concrete mix, when compared to similarly
reinforced beams without the nylon fibers added to the mix.
Drop weight impact testing showed that the peak amplitude of the tup load increases
with an increase in drop height, along with corresponding increases in the peak inertial and peak
bending loads. Most of the load recorded by the tup was inertial in nature.
For the plain concrete beams, the maximum displacement (at failure) decreased with an
increase in drop height. However, for those beams reinforced with CFRP, the displacement at
failure increased with an increase in drop height.
167

168
The average static peak bending load was always less than the dynamic peak bending
load, even at the lowest drop heights. The increase in peak bending load with drop height is
attributed to strain rate effects in the concrete.
The peak displacements were always less under dynamic loading when compared to the
quasistatic loading case. Except for the plain concrete beams, the dynamic fracture energies
were always less than the static values, even though the peak dynamic bending loads were
typically two to three times higher than the peak static bending loads. This implies that for a
given drop height, i.e. strain rate, a beam has a fixed capacity to absorb energy, dictated by the
concrete’s peak bending load and limited by displacement, emphasizing the brittle nature of
concrete.
The mechanism by which the beams failed dynamically did not change appreciably from
the static loading case; failure of all beams with only tensile CFRP reinforcement was one of
shear failure in the concrete at approximately one quarter span from beam midpoint, followed by
delamination and peeling of the CFRP. Failure of beams with CFRP on the sides as well as the
bottom was concrete shear failure followed by CFRP side panel splitting and buckling along the
major shear crack in the concrete. Failure of the plain concrete beams was one of flexure, with a
single flexural crack at beam midpoint merely breaking the beam into two pieces, creating two
new fracture surfaces.
Section analysis did very well predicting Point 1, the termination of elastic behavior, for
all six beam types. However, for all beams reinforced with CFRP, the section analysis generally
overpredicted the stiffnesses in Regions 2 and 3. The displacements at the predicted failure
point, Point 3, were within 10 percent or less of the experimental failure displacements, which
implies that the predicted loads are too high.

169
Results from the SDOF idealization of the beams subjected to a half sine pulse impulsive
load showed that the analytical model generally overpredicted the displacement-time behavior of
the beams. However, up until beam failure occurred, the model gave reasonable results.
The FEM code ADINA did very well predicting the displacement - time behavior of two
beam types studied in detail; a plain concrete beam and a beam reinforced with three ply CFRP
on the bottom. The ADINA code also predicted two types of cracking in the beam reinforced
with three ply CFRP; flexural and shear. For the plain concrete beam, the code predicted only
flexural cracking. In both cases, the code not only correctly predicted the type and location of
the crack(s), but the times at which they initiated and their transit time through the beams.
The FEM calculations gave insight into the localized displacement behavior of the
beams when subjected to an intense impulsive load, as well as the traveling plastic hinge
phenomena, which helped to determine the beam’s equivalent mass. Data analyses and SDOF
modeling relied upon reasonable estimation of this quantity.

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APPENDIX A
SUMMARY OF CONCRETE UNCONFINED COMPRESSIVE STRENGTHS
(Courtesy of D. Wahlstrom)
Date
Date
Age in
Batch
Load
fc
Poured
Tested
Days
Number
lbs
psi
16-NOV-94
21-Jun-95
217
LW1-1
71560
5695
16-Nov-94
21-Jun-95
217
LW1-2
76700
6104
16-Nov-94
21-Jun-95
217
LW1 Average
74130
5899
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fC
psi
29-Nov-94
10-Jul-95
223
LW2-1
71020
5652
29-Nov-94
10-Jul-95
223
LW2-2
72720
5787
29-Nov-94
10-Jul-95
223
LW2-3
74020
5890
29-Nov-94
10-Jul-95
223
LW2-4
73220
5827
29-Nov-94
10-Jul-95
223
LW2 Average
72745
5789
Date
Date
Age in
Batch
Load
fc
Poured
Tested
Days
Number
lbs
psi
29-Nov-94
10-Jul-95
223
LW2A-1
77240
6147
29-Nov-94
10-Jul-95
223
LW2A-2
76920
6121
29-Nov-94
10-Jul-95
223
LW2A Average
77080
6134
NOTE: The LW2A Cylinders Were From the SHPB Sample Batch.
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
6-Dec-94
10-Jul-95
216
LW3-1
88500
7043
6-Dec-94
10-Jul-95
216
LW3-2
87720
6981
6-Dec-94
10-Jul-95
216
LW3-3
85640
6815
6-Dec-94
10-Jul-95
216
LW3-4
88460
7039
6-Dec-94
10-Jul-95
216
LW3 Average
87580
6969
174

175
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
14-Dec-94
10-Jul-95
208
LW4-1
82720
6583
14-Dec-94
10-Jul-95
208
LW4-2
84220
6702
14-Dec-94
10-Jul-95
208
LW4-3
76280
6070
14-Dec-94
10-Jul-95
208
LW4-4
77280
6150
14-Dec-94
10-Jul-95
208
LW4 Average
80125
6376
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
17-Jan-95
10-Jul-95
174
LW5-1
79560
6331
17-Jan-95
10-Jul-95
174
LW5-2
75060
5973
17-Jan-95
10-Jul-95
174
LW5-3
79940
6361
17-Jan-95
10-Jul-95
174
LW5-4
74700
5944
17-Jan-95
10-Jul-95
174
LW5 Average
77315
6153
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
19-Jan-95
6-Jun-95
138
LW6-1
88980
7081
19-Jan-95
6-Jun-95
138
LW6-2
84440
6720
19-Jan-95
6-Jun-95
138
LW6-3
81360
6474
19-Jan-95
6-Jun-95
138
LW6-4
83460
6642
19-Jan-95
6-Jun-95
138
LW6 Average
84560
6729
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
23-Feb-95
6-Jun-95
103
LW7-1
72380
5760
23-Feb-95
6-Jun-95
103
LW7-2
73240
5828
23-Feb-95
6-Jun-95
103
LW7-3
69320
5516
23-Feb-95
6-Jun-95
103
LW7-4
72340
5757
23-Feb-95
6-Jun-95
103
LW7 Average
71820
5715

176
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
28-Feb-95
7-Jun-95
99
LW8-1
71420
5683
28-Feb-95
7-Jun-95
99
LW8-2
68060
5416
28-Feb-95
7-Jun-95
99
LW8-3
75820
6034
28-Feb-95
7-Jun-95
99
LW8-4
74960
5965
28-Feb-95
7-Jun-95
99
LW8 Average
72565
5775
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
2-Mar-95
7-Jun-95
97
LW9-1
69680
5545
2-Mar-95
7-Jun-95
97
LW9-2
72940
5804
2-Mar-95
7-Jun-95
97
LW9-3
72560
5774
2-Mar-95
7-Jun-95
97
LW9-4
73440
5844
2-Mar-95
7-Jun-95
97
LW9 Average
72155
5742
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
7-Mar-95
10-Jul-95
125
LWF10-1*
59160
4708
7-Mar-95
10-Jul-95
125
LWF10-2*
60200
4791
7-Mar-95
10-Jul-95
125
LWF10-3*
63180
5028
7-Mar-95
10-Jul-95
125
LWF10-4*
63160
5026
7-Mar-95
10-Jul-95
125
LWF10 Average
61425
4888
Date
Poured
Date
Tested
Age in
Days
Batch
Number
Load
lbs
fc
psi
9-Mar-95
10-Jul-95
123
LWF11-1*
71720
5707
9-Mar-95
10-Jul-95
123
LWF11-2*
74580
5935
9-Mar-95
10-Jul-95
123
LWF11-3*
70660
5623
9-Mar-95
10-Jul-95
123
LWF11-4*
74760
5949
9-Mar-95
10-Jul-95
123
L WF 11 Average
72930
5804
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard

APPENDIX B
MTS® LOAD - DISPLACEMENT CURVES
(Tests Conducted July 11- 12, 1995)

LOAD vs DISPLACEMENT
BEAM LW5-33, PLAIN CONCRETE
Thousandths
DISPLACEMENT, INCHES
12 July 1995

LOAD vs DISPLACEMENT
BEAM LW5-36, PLAIN CONCRETE
Thousandths
DISPLACEMENT, INCHES
12 July 1995

LOAD vs DISPLACEMENT
0
12 July 1995
0.005
DISPLACEMENT, INCHES
0.01
0.015

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW2-16, 1 PLY CFRP BOTTOM ONLY
12 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW2-9, 1 PLY CFRP BOTTOM ONLY
12 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW3-19, 1 PLY CFRP BOTTOM ONLY
12 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW4-30, 2 PLY CFRP BOTTOM ONLY
11 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW4-27, 2 PLY CFRP BOTTOM ONLY

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW3-23, 2 PLY CFRP BOTTOM ONLY
11 July 1995
0.05
0.1 0.15 0.2
DISPLACEMENT, INCHES
0.25
0.

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW8-62, 3 PLY CFRP BOTTOM ONLY
11 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LW9-67, 3 PLY CFRP BOTTOM ONLY
11 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LWF11-85, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY
11 July 1995

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LWF10-80, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY

LOAD, POUNDS
LOAD vs DISPLACEMENT
BEAM LWF10-76, FIBROUS CONCRETE, 3 PLY CFRP BOTTOM ONLY
11 July 1995

LOAD vs DISPLACEMENT
BEAM LW9-70, 3x3 PLY CFRP
11 July 1995

LOAD vs DISPLACEMENT
BEAM LW7-53, 3x3 PLY CFRP
11 July 1995

APPENDIX C
SUMMARY OF STATIC THREE POINT BENDING EXPERIMENTS

Beam Designator
No. of
Plies CFRP
Unconfined
Compressive
Strength, fc
psi (MPa)
Peak Load
lbs (N)
Max
Displacement
in (mm)
Fracture
Energy
ft lbs (Nun)
Load
Increase
Ratio
Displacement
Increase
Ratio
Fracture
Energy
Increase
Ratio
LW5-33
”0
6153
(42.43)
648
(2810)
0.007
(0.178)
0300
(0.407)
LW5-36
—0-
6153
(42.43)
621
(2769)
(TÜÜ6
(0.152)
0*240
(0.325)
LW5-38
—0“
6153
(42.43)
643
(2867)
0^014
(0.356)
0.392
(0.531)
LW2-16
~r~
5789
(39.92)
1226
(5467)
0.189
(4.801)
147809
(20.077)
im—
030
37778
LW2-9
5789
(39.92)
1100
(4905)
0*139
(3.531)
9.740
(13.205)
1773
933
24.85—
LW3-19
~r~
6969
(48.06)
1121
(4999)
0*153
(3.886)
10*692
(14.495)
—076—
10.93
—2738—
LW4-30
~T~
6376
(43.97)
2538
(11,317)
0*257
(6.528)
35*489
(48.113)
—JM—
036
90.53
LW4-27
~T~
6376
(43.97)
2170
(9676)
0.227
(5.766)
26*842
(36.390)
3.40
16.21
—6837—
LW3-23
~T~
6969
(48 06)
2187
(9752)
0.240
(6.096)
29417
(39.475)
3.43
17.14
—7338—
LW9-67
5742
(39.60)
2642
(11,781)
0.206
(5.232)
28485
(38.211)
4.15
14.71
—7090—
LW8-62
5775
(39.83)
2254
(10051)
0.154
(3.912)
18*809
(25.500)
3.54
11.00
”"47.98—
LWF11-85
3*
5804
(40-03)
2529
(11 277)
020l
(5.105)
28*115
(38.116)
TW1—
1436
—7T772—
LWF10-80
3*
4888
(33.71)
2362
(10 532)
0.173
(4.394)
22*801
(30.912)
3.71
12.36
—58T7—
LWF10-76
3*
4888
(33.71)
2157
(9618)
0452
(3.861)
17238
(24.048)
3.38
1036
45.25—
LWF9-70
3 - Bottom
3 - Sides
5742
(39.60)
4060
(18,104)
0.205
(5.207)
40347
(54.971)
6.37
14.64
103.44
LWF7-53
3 - Bottom
3 - Sides
5715
(39.41)
5297
(23,620)
0284
(7.214)
73492
(100.042)
—831—
2039
188.24
* Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard

APPENDIX D
DYNAMIC BEAM BENDING EXPERIMENTAL DATA - TABULAR SUMMARY

Beam
Designator
Compressive
Strength, f c
psi (MPa)
No. of
Plies
CFRP
Drop
Height
in (cm)
Peak Tup
Load
lbs (kN)
Tup Load
Pulse Length
jxsec
Peak Inertial
Load
lbs (kN)
Peak
Bending Load
lbs (kN)
Peak
Displacement
in (mm)
Fracture
Energy
fit-lbs (kN-m)
LW6-43*
6729
(46.41)
0
2
(5.08)
2530
(11.25)
1034
1600
(7.12)
1600
(7.12)
0.009
(0.229)
1.5
(2.03)
LW5-35
6153
(42.43)
0
2
(5.08)
1940
(8.63)
987
1400
(6.23)
1600
(7.12)
0.01
(0.254)
0.9
(1.15)
LW5-34*
6153
(42.43)
0
4
(10.16)
4050
(18.02)
730
3500
(15.57)
1900
(8.45)
0.009
(0.229)
2
(2.71)
LW5-37
6153
(42.43)
0
4
(10.16)
3580
(15.93)
852
3600
(16.01)
2300
(10.23)
0.007
(0.178)
1.2
(1.63)
LW6-45
6729
(46.41)
0
4
(10.16)
3470
(15.44)
1006
2000
(8.90)
2500
(11.12)
0.0065
(0.165)
1.2
(1.63)
LW5-39*
6153
(42.43)
0
8
(20.32)
3975
(17.68)
840
3000
(13.35)
4000
(17.79)
0.0063
(0.160)
3.2
(4.34)
LW6-41
6729
(46.41)
0
8
(20.32)
4450
(19.80)
957
4500
(20.02)
2200
(9.79)
0.0067
(0.170)
2.6
(3.53)
LW5-40
6153
(42.43)
0
8
(20.32)
4300
(19.13)
1015
4500
(20.02)
2500
(11.12)
0.006
(0.152)
1.8
(2.44)
LW2-15*
5789
(39.92)
1
6
(15.24)
4370
(19.43)
926
4000
(17.79)
2800
(12.46)
0.0105
(0.267)
1.4
(1.90)
LW2-10
5789
(39.92)
1
6
(15.24)
4900
(21.80)
920
3700
(16.46)
2900
(12.90)
0.026
(0.660)
2.8
(3.80)

Beam
Designator
Compressive
Strength, f c
psi (MPa)
No. of
Plies
CFRP
Drop
Height
in (cm)
Peak Tup
Load
lbs (kN)
Tup Load
Pulse Length
jisec
Peak Inertial
Load
lbs (kN)
Peak
Bending Load
lbs (kN)
Peak
Displacement
in (mm)
Fracture
Energy
fit-lbs (kN-m)
LW3-18
6969
(48.06)
1
6
(15.24)
4930
(21.93)
965
4900
(21.80)
3350
(14.90)
0.007
(0.178)
1.2
(1.63)
LW2-13*
5789
(39.92)
1
8
(20.32)
4740
(21.09)
983
2650
(11.79)
2250
(10.00)
0.028
(0.711)
2.5
(3.39)
LW2-12
5789
(39.92)
1
8
(20.32)
6380
(28.38)
679
5700
(25.36)
3850
(17.13)
0.022
(0.559)
6.5
(8.82)
LW3-20
6969
(48.06)
1
8
(20.32)
6160
(27.40)
744
4300
(19.13)
2600
(11.57)
0.0222
(0.564)
3.6
(4.88)
LW3-17*
6969
(48.06)
1
12
(30.48)
5000
(22.24)
970
4750
(21.13)
3700
(16.46)
0.0115
(0.292)
2.4
(3.25)
LW2-11
5789
(39.92)
1
12
(30.48)
6220
(27.67)
987
5200
(23.13)
3000
(13.35)
0.03
(0.762)
3.8
(5.15)
LW2-14
5789
(39.92)
1
12
(30.48)
6930
(30.83)
678
6450
(28.69)
4500
(20.02)
0.0265
(0.673)
9.1
(12.34)
LW4-32*
6376
(43.97)
2
7
(17.78)
5480
(24.38)
875
4300
(19.13)
2900
(12.90)
0.012
(0.305)
3.6
(4.88)
LW4-31
6376
(43.97)
2
7
(17.78)
4340
(19.31)
975
4300
(19.13)
3150
(14.01)
0.0092
(0.234)
1.9
(2.58)
LW3-22*
6969
(48.06)
2
8
(20.32)
7570
(33.67)
590
6950
(30.92)
5300
(23.58)
0.023
(0.584)
7.3
(9.90)
LW3-21
6969
(48.06)
2
8
(20.32)
6140
(27.31)
837
4500
(20.02)
2350
(10.45)
0.028
(0.711)
(6.0)
(8.14)
LW4-29
6376
(43.97)
2
8
(20.32)
6130
(27.27)
730
5150
(22.91)
4700
(20.91)
0.024
(0.610)
6.1
(8.27)
LW4-28*
6376
(43.97)
2
12
(30.48)
8500
(37.81)
586
5400
(24.02)
5550
(24.69)
0.028
(0.711)
11.3
(15.33)

Beam
Designator
Compressive
Strength, f c
psi (MPa)
No. of
Plies
CFRP
Drop
Height
in (cm)
Peak Tup
Load
lbs (kN)
Tup Load
Pulse Length
psec
Peak Inertial
Load
lbs (kN)
Peak
Bending Load
lbs (kN)
Peak
Displacement
in (mm)
Fracture
Energy
ft-lbs (kN-m)
LW4-25
6376
(43.97)
2
12
(30.48)
5830
(25.93)
1015
5050
(22.46)
3000
(13.35)
0.049
(1.245)
8.1
(10.99)
LW3-24
6969
(48.06)
2
12
(30.48)
8610
(38.3)
615
6800
(30.25)
5000
(22.24)
0.036
(0.914)
9.8
(13.29)
LW9-72*
5742
(39.60)
3
8
(20.32)
5390
(23.98)
961
4650
(20.69)
3700
(16.46)
0.045
(1.143)
7.6
(10.31)
LW9-71
5742
(39.60)
3
8
(20.32)
4460
(19.84)
982
4450
(19.8)
2950
(13.12)
0.034
(0.864)
5.4
(7.32)
LW8-63
5775
(39.83)
3
8
(20.32)
6000
(26.69)
905
3850
(17.13)
2650
(11.79)
0.044
(1.118)
7.9
(10.71)
LW9-68*
5742
(39.60)
3
12
(30.48)
7160
(31.85)
900
6750
(30.03)
4100
(18.24)
0.033
(0.838)
8.6
(11.66)
LW8-60
5775
(39.83)
3
12
(30.48)
5330
(23.71)
1008
4500
(20.02)
3500
(15.57)
0.049
(1.245)
8.7
(11.80)
LW8-61
5775
(39.83)
3
12
(30.48)
6500
(28.91)
927
4550
(20.24)
2300
(10.23)
0.047
(1.194)
8.2
(11.12)
LW9-66*
5742
(39.60)
3
18
(45.72)
7970
(35.45)
822
5450
(24.24)
4800
(21.35)
0.064
(1.626)
17.2
(23.33)
LW9-65
5742
(39.60)
3
18
(45.72)
8600
(38.26)
800
5850
(26.02)
5600
(24.91)
0.07
(1.778)
18.6
(25.23)
LW9-69
5742
(39.60)
3
18
(45.72)
8730
(38.83)
893
6200
(27.58)
5150
(22.91)
0.076
(1.93)
19.3
(26.18)
LWF10-73*
4888
(33.71)
3**
8
(20.32)
5690
(25.31)
983
4550
(20.24)
2700
(12.01)
0.027
(0.686)
6.1
(8.27)
LWF11-86
5804
(40.03)
3**
8
(20.32)
4550
(20.24)
900
3200
(14.23)
2350
(10.45)
0.046
(1.168)
6.3
(8.54)

Beam
Designator
Compressive
Strength, f c
psi (MPa)
No. of
Plies
CFRP
Drop
Height
in (cm)
Peak Tup
Load
lbs (kN)
Tup Load
Pulse Length
psec
Peak Inertial
Load
lbs (kN)
Peak
Bending Load
lbs (kN)
Peak
Displacement
in (mm)
Fracture
Energy
fit-lbs (kN-m)
LWF10-78
4888
(33.71)
3**
8
(20.32)
6080
(27.05)
902
4150
(18.46)
2600
(11.57)
0.032
(0.813)
6.9
(9.36)
LWF10-75*
4888
(33.71)
3**
12
(30.48)
7090
(31.54)
908
5200
(23.13)
3450
(15.35)
0.053
(1.346)
9.8
(13.29)
LWF 11-87
5804
(40.03)
3**
12
(30.48)
7940
(35.32)
632
5500
(24.47)
6150
(27.36)
0.051
(1.295)
10.3
(13.97)
LWF 10-77
4888
(33.71)
3**
12
(30.48)
6350
(28.25)
1055
4600
(20.46)
2500
(11.12)
0.052
(1.321)
9.6
(13.02)
LWF 10-74*
4888
(33.71)
3**
18
(45.72)
9140
(40.66)
657
6450
(28.69)
6750
(30.03)
0.0605
(1.537)
17.3
(23.46)
LWF 10-79
4888
(33.71)
3**
18
(45.72)
8030
(35.72)
758
5600
(24.91)
4350
(19.35)
(0.070)
(1.778)
15.3
(20.75)
LWF11-88
5804
(40.03)
3**
18
(45.72)
9600
(42.70)
596
6800
(30.25)
6300
(28.02)
(0.070)
(1.778)
16.9
(22.92)
LW8-58*
5775
(39.83)
3 - Bottom
3 - Sides
16
(40.64)
9000
(40.04)
720
6900
(30.69)
6750
(30.03)
0.055
(1.397)
17.1
(23.19)
LW6-47
6729
(46.41)
3 - Bottom
3 - Sides
16
(40.64)
8550
(38.03)
801
6700
(29.80)
6500
(28.91)
0.061
(1.549)
15.7
(21.29)
LW7-55*
5715
(39.41)
3 - Bottom
3 - Sides
18
(45.72)
9330
(41.5)
742
6300
(28.02)
6650
(29.58)
0.083
(2.108)
22.3
(30.24)
LW7-50
5715
(39.41)
3 - Bottom
3 - Sides
18
(45.72)
9220
(41.01)
793
5550
(24.69)
5900
(26.25)
0.075
(1.905)
18.8
(25.50)
LW7-52
5715
(39.41)
3 - Bottom
3 - Sides
18
(45.72)
8410
(37.41)
1040
5500
(24.47)
3800
(16.90)
0.069
(1.753)
20.0
(27.12)
LW7-49*
5715
(39.41)
3 - Bottom
3 - Sides
24
(60.96)
10200
(45.37)
825
6750
(30.03)
6600
(29.36)
0.0725
(1.842)
23.5
(31.87)
200

Beam
Designator
Compressive
Strength, f c
psi (MPa)
No. of
Plies
CFRP
Drop
Height
in (cm)
Peak Tup
Load
lbs (kN)
Tup Load
Pulse Length
p.sec
Peak Inertial
Load
lbs (kN)
Peak
Bending Load
lbs (kN)
Peak
Displacement
in (mm)
Fracture
Energy
fit-lbs (kN-m)
LW6-48
6729
(46.41)
3 - Bottom
3 - Sides
24
(60.96)
10470
(46.57)
854
6600
(29.36)
6350
(28.25)
0.085
(2.159)
27.3
37.02
LW6-46
6729
(46.41)
3 - Bottom
3 - Sides
24
(60.96)
9530
(42.39)
867
5250
(23.35)
4950
(22.02)
0.074
(1.880)
24.9
33.77
* Highly Instrumented Beams (3 Strain Gages/Beam and High Speed Framing Camera in Addition to Usual Instrumentation)
** Fibrous Concrete Beams with 3 lbs (1.36 kg) Nylon Fibers/Cubic Yard

APPENDIX E
ANALYTICAL MODEL COMPUTER PROGRAM

RIGOROUS ANALYSIS OF SINGLE DEGREE OF FREEDOM REPRESENTATIONS OF 1, 2,
AND 3 PLY CFRP REINFORCED CONCRETE BEAMS SUBJECTED TO A HALF - SINE WAVE
CENTER POINT IMPULSIVE LOAD, WITH VARIABLE STIFFNESS, AND NO DAMPING
INPUT VALUES FOR LW9-66
Po := 7970 Amplitude of the Load Pulse, lbs (From Test Data)
tl := 0.000822 Duration of the Load Pulse (Also End of Phase 1), sec (From Test Data)
tf = 0.000894 Time of Fracture / End of Bending Load (From Test Data), sec
te := 0.0015 Desired Ending Time of the Analysis (Also End of Phase 2), sec
xtpl := 13.5 Hinge Location, As Measured From Beam Center, at End of Load Pulse tl
(Determined Graphically from Hinge Location vs Time Graph), in
tsl := .000331 Time of Change in Stiffness Between Region 1 and Region 2
(Determined Graphically from Displacement vs Time Graph), sec
ts2 := 0.000894 Time of Change in Stiflhess Between Region 2 and Region 3
(Determined Graphically from Displacement vs Time Graph), sec
dh := 18 Drop Height of Tup, in
t:= 0.0195 CFRP Thickness, in. (Note: 1 Ply = 0.0085, 2 Ply = 0.014, 3 Ply = 0.0195)
ts := 0.0 Side CFRP Thickness, in. (Note: No Side CFRP = 0.0, 3 Ply = 0.0195)
con = 1.0 Confinement Factor (1.25 for Beams with Side CFRP, 1.0 Otherwise), unitless
fc := con-5742 Concrete Compressive Strength (From Test Data), psi fc = 5.742* 103
ft = 7.5- Jfc ACI Equation for Splitting Tensile Strength of Concrete, psi ft = 568.319892
1^0^ ft Modulus of Rupture for Lightweight Concrete Using
0.57 the ACI Equation for Splitting Tensile Strength, psi
MOR =997.052443
ec := 0.003 Concrete Compressive Strain at fc, in/in (From WL/FIVCO Stress-Strain Curve)
eu := 0.004 Concrete Ultimate Compressive Strain at y f in/in
a := 0.66 Empirical Parameters Used to Define the Equivalent
0 -0 3875 Concrete Compressive Force at y fc, unitless
203

204
OTHER NECESSARY PARAMETERS
a Empirical Parameter Used to Define the Concrete
^ 2 /3 Compressive Force at eu, unitless
g - 386.4 Acceleration of Gravity, in / sec2
L := 27 Supported Span Length of the Beam, in
p := 118 Specific Weight of the Beam, lbs / ft3
p ,, i.5 LT Secant Modulus of Elasticity of Concrete
^ ” (Based Upon ACI Equation), psi
Secant Modulus of Elasticity of Concrete at fc
(Based Upon ACI Equation), psi
7=0.851613
Ec
Ec2 := 0
2
Ec =3.205299-10°
Ec2:=Ec
Ec2= 3.205299-106
Ef ;= 20.0-10 Modulus of Elasticity of CFRP (From WL/ML Data), psi
A := 9 Cross Sectional Area of the Beam, in2
b = 3 Width of the Beam, in
h .= 3 Depth of the Beam, in
„ 48 Constant Obtained in the Derivation of Mid-Point
Displacement for 3 - Point Bending, 1 / in3
K =0.002439 L3 = 1.9683-104
W := 95.2 Weight of Tup, lbs
W
M := — Mass of the Tup, lbs - sec2 / in M = 0.246377
g
w := ——A L Weight of the Beam Between the Supports, lbs
1728
w = 16.59375
w
ml Mass Per Unit Length of the Beam, lbs-sec2 / in2 ml=0.001591
27-g
vo ;= ^2-0.9-g-dh Impact Velocity of the Tup, in / sec vo = 111.889946
dh = 18
tobar := — The Driving or Forcing Frequency (tl From Test Data), rads/sec cobar =3.821889-103
tl

205
SECTION ANALYSIS
REGION 1
cl :=
b h2 +■ 2 —h (ts h + b t)
Ec
2 b h + 2 ^ (2 ts h + b t)
Ec
Location of the Neutral Axis (As Measured From
the Top Surface of the Beam) for Region 1, in
cl =1.558466
kl
h
kl =0.519489
bcl3
/cl\2
ts-cl3 / cl\2
Ell 1 := Ec-
+ bcl-
+ 2Ef
+ cl'ts-
+ Ec-
12
U /
12 \21
b (h-Cl)3+b.(h-cl) /h-Cl'2
12
EI12 := 2 Ef-
ts (h - cl)
12
+ ts (h - cl)-
+ Ef
bt3 . /
t\2l
+ b t- h- cl +-
\ 2 j
12 \
2/
Ell = Ell 1 +EI12 Flexural Stiffness for Region l,lbs-in2 Ell =2.41987-10?
II
Ell
Ec-
bh
â– Ef
bh + 2tsh + tb/ \b h + 2 ts h + t b
2 ts h + t b
Planar Moment of Inertia for the T,
II =7.302491
Composite Section in Region 1, in'* â– 
— =0.213416
II
PI := 4 I1MOR Load for Point 1, lbs PI =748.273222
L(h-cl)
PI . ...
61:= Displacement for Point 1, in 61=0.01268
K-EI1
K1 : = — Slope or Stiffness for Region 1, lbs / in K1 = 5.901222• 104
61
PI L T
Mpl := Plastic Moment for Region 1, in - lbs Mpl = 5.050844-10

206
REGION 2
c21
-(4 h ec Efts + 2 ec Eft b)
2fcb
c22
^4 h ec Ef ts + 2-ccEftb^2
-2-ec-Ef h (ts h + t b)
l fob /
fc b
. .. Location of the Neutral Axis (As Measured From .
c2:=c21+c22 v „ . c2 =0.920556
the Top Surface of the Beam) for Region 2, in
k2:=— k2 =0.306852
h
v „ fc b c22 2ecEftsc22 2 ec Efts (h - c2)3 ec-Eft b (h - c2)2 plastic Moment for
Mp2 := + _ + — + Region 2, in - lbs
3c2
c2
Mp2 =2.135328-10
4Mp2 3
P2 : = Load for Point 2, lbs P2 = 3.16345 • 10
EI21 = Ec2
bc23
/c2\2]
+ 2 Ef-
ts c23
f-)1
+ bc2
—
+ c2ts-
12
2
12
\ 2 / .
EI22 := 2 Ef-
ts (h - c2)
12
+• ts (h - c2)-
M2'
ibt3 /
t\2l
+ Ef
+ b1- h - c2 + —
\ 2 )
12 \
2/
EI2 - EI21 + EI22 Flexural Stiffness for Region 2, lbs - in2 EI2 = 7.607227-10°
K2 := K EI2 Slope or Stiffness for Region 2, lbs / in K2 = 1 855139-10
A2 :=
P2 - PI Incremental Increase in Displacement in
K2 Region 2 From Point 1 to Point 2, in
A2 =0.130188
62 := 61 + A2 Total Displacement at Point 2, in 62 = 0.142868

207
REGION 3
c31
-eu-Ef (2-tS'h + tb)
Y'fc-4/Sb
c32 := -
2
eu-Ef-(2-ts-h + tb)
y-fe-2-0 b
- 4-
- eu-Ef-h-(ts-h + tb)
y-fc-2/3-b
c3 := c31 + c32
Location of the Neutral Axis (As Measured From
the Top Surface of the Beam) for Region 3, in
2
w . ,2 ,, 2-eu-Efts-c3
Mp31 = y fc-2-j8-b-c3 -(1 — j3) +-
c3 =0.924349
k3= — k3 =0.308116
h
.. 2 eu-Ef-ts-(h - c3)3 eu-Ef t-b-(h - c3)2
Mp32 := +
3-c3
c3
,, . ,, ,, ,, Plastic Moment for ... . ,.4
Mp3 := Mp31 + Mp32 Region 3 ln ibs Mp3 =2.776302-10
p3 4vMp3 Loadforpoint3)lbs P3 =4.11304* 103
EI31 := 2-Ef-
ts-c33
/c3\2
+ c3-ts-
12
W /
EI32 := 2-Ef-
ts-(h - c3)
12
ts-(h - c3)-
+ Ef-
ib-t3 /
t\21
+ b-1- h - c3 +-
\ 2 )
12 \
2/
EI3 := EI31 + EI32 Flexural Stiffness for Region 3, lbs - in2 EI3 = 5.088248-10
K3 := K-EI3 Slope or Stiffness for Region 3, lbs / in K3 = 1.240847* 10
P3 - P2 Incremental Increase in Displacement in
K3 Region 3 From Point 2 to Point 3, in
A3 =0.076528
63 := 62 + A3 Total Displacement at Point 3, in
63 =0.219396

208
GRAPH OF LOAD vs DISPLACEMENT AS PREDICTED FROM SECTION ANALYSIS
i = 0,1.. 3 Increment 1 Set of 4 Data Points for Load vs Displacement Plot
po
:= 0.0
P1
:= PI
P2
:=P2
P3
:= P3
50
:= 0.0
«1
:= 61
«2
:= 62
«3
= 63
Data Set Generated From Section Analysis
CALCULATION OF STATIC FRACTURE ENERGY
Sfract = 0.5 P1Ó1 + 0.5 ((P2 - PI) (62 - 61)) + Pl(62 - 61) + 0.5((P3 - P2) (63 - 62)) + P2 (63 - 62)
sfract fc = 5.742* 103 con = 1 PI =748.273222 P2 =3.16345-103 P3 =4.11304-103
—— =44.816724 Static Fracture Energy, ft - lbs61 =0.01268 62 =0.142868 63 =0.219396
K1 =5.901222-104 K2 = 1.855139-104 K3 = 1.240847-104 £C
eu =0.004
LOAD vs DISPLACEMENT, LW9-66, CON=1.0

209
CALCULATION OF THE EFFECTIVE MASS BASED UPON THE JOHNSON EQUATION
AND CALCULATED PARAMETERS FROM THE SECTION ANALYSIS
x := 0.01,0.02..-
2
Incrementing Over the Supported Half-Length of the Beam, in
xdotl(x) :=
12-Mpl (M + mix)2
M-ml-vo x-(2-M + ml-x)
The Johnson Equation for Velocity of the
Traveling Plastic Hinge in Region 1, in / sec
tjl(x) :=
2 2
M ml vo M + (M + mlx)
12'Mp! (M + ml-x)-ml2
. . Integrated Equation for Time as an Explicit Function of x,
U. 14UyUo/oJ TT. . , X-- - i
Using the Plastic Moment From Region 1, sec
i := 1.2..6
:= 0.000100
xPl
:= 4.5
:= 0.000300
xp3 := 9
t5 = 0.000500
xPj := 12
= 0.000200
xp2
:= 6.5
l4
= 0.000400
xp4 := 10.5
t6 = 0.000600
xp6 := 13.5
5*

PERCENT EQUIVALENT MASS FACTOR, UNITLESS HINGE LOCATION FROM BEAM CENTER, INCHES
210
HINGE LOCATION vs TIME
0 1 1 1 1 1 I I 1 I
6.63622e-012 TIME, SECONDS 0.000492228

211
CALCULATING THE AVERAGE EQUIVALENT MASS DURING THE LOADING PULSE
•xtpl
Area := xtpl tl -
0.01
2 2
M ml vo M + (M +â–  mi x)
12-Mpl (M + mlx)ml2
xtpl = 13.5
0.140906763 dx
Area Under the Hinge Location vs
Load Pulse Curve, in-sec
, Area
avgxtpl â–  =
tl
Area = 0.008836
avgxtpl = 10.749514 Average Hinge Location at End of Load Pulse tl, in
avgmf := av8xtPQ 3333 avgmf = 0.39611
0.67-
2
i | | If the Average Hinge Location During the Load Pulse tl is Greater
avgmf := if avgmf <-, avgmf,- Than 0.67 L/2, the Average Equivalent Mass Factor is Set at 0.3333
3 3
avgmf = 0.333333 Average Equivalent Mass Factor During the Load Pulse, unitless
SINGLE DEGREE OF FREEDOM ANALYSIS, HALF SINE WAVE IMPULSIVE LOAD,
NO DAMPING, AND VARIABLE STIFFNESS
ADDITIONAL NECESSARY PARAMETERS
me := avgmf w The SDOF Equivalent Weight, lbs me = 5.53125 w = 16.59375
0)1 :=
KL386.4 ,
The Natural Circular Frequency Using Stiffness Kl, rads/sec 0)1 =2.030383-10
me
K2-386.4
0)2:= The Natural Circular Frequency Using Stiffness K2, rads/sec 0)2 = 1.138401 - 10J
me
„ K3-386.4 _ , .
0)3:= The Natural Circular Frequency Using Stiffness K3, rads/sec 0)3 =931.034992

212
cobar
0)1
The Ratio of the Driving Frequency to the Natural Circular Frequency
Calculated Using Stiffness Kl, unitless
(31 =1.882349
cobar
co2
The Ratio of the Driving Frequency to the Natural Circular Frequency
Calculated Using Stiffness K2, unitless
(32 =3.357244
cobar
0)3
The Ratio of the Driving Frequency to the Natural Circular Frequency
Calculated Using Stiffness K3, unitless
j33 = 4.10499
td = 0.0,0.000001.. tl Incrementing the Time in Phase 1
P(td) := Po-sin (cobar-td) The Forcing or Driving Function, lbs
THE GENERAL SOLUTION FOR PHASE 1
ts 1 := 0.000331 ts2 : = 0.000894 tl =8.22-10 4
tpxl = 0.0,0.000001.. tsl - 0.000001 Incrementing the Time in Phase 1 From 0 to tsl
tsx2 := if(ts2 tpx2 := tsl,tsl + 0.000001.. tsx2 - 0.000001 Incrementing the Time in Phase 1 From tsl to ts2
tpx3 := tsx2,tsx2 + 0.000001.. tl Incrementing the Time in Phase 1 From ts2 to tl
xl(tpxl) :=
x2(tpx2) :=
x3(tpx3) :=
Po
Kl-
(.-
/312)
Po
K2-
(1-
022)
Po
K3-
(.-
/S32)
•( sin (cobar-tpxl) - /3Tsin(o)Ttpxl))
â– (sin(cobartpx2) - /32-sin(co2-tpx2))
(sin(cobar-tpx3) - /83-sin(co3-tpx3))
Displacement as a Function of Time
for Phase 1 Using Stiffness K1
xl(tsl) =0.011594
61 =0.01268
Displacement as a Function of Time
for Phase 1 Using Stiffness K2
x2(ts2) =0.130857 ts2 =8.94-10~4
62=0.142868 , „ „„ -4
tl =8.22-10
Displacement as a Function of Time
for Phase 1 Using Stiffness K3
x3(tpx3) =0.115234

213
DISPLACEMENT vs TIME IN PHASE 1
REGION 1, STIFFNESS K1
— REGION 2, STIFFNESS K2
REGION 3, STIFFNESS K3
CALCULATION OF ACCELERATION IN PHASE 1
Acceleration as a Function of Time for Phase 1 Using Stiffness K1
axl(tpxl) :=
Po
Mi-/?!2)]
(-sin(wbartpxl) wbar2 + 01 sin(tol tpxl)-a)l2J
Acceleration as a Function of Time for Phase 1 Using Stiffness K2
ax2(tpx2) :=
Po
•(-sin (tobar-tpx2)-tobar2 + 02-sin(to2tpx2)to22)
[K2-(l -022)]
Acceleration as a Function of Time for Phase 1 Using Stiffness K3
Po / 2 2
ax3(tpx3) := - —• sin(tobar^px3)-tobar + 03 sin(to3 tpx3) to3
K3
â– (l - 032) ]

214
THE GENERAL SOLUTION FOR PHASE 2 tl =8.22-10 4 tsl =3.3 MO 4 ts2 =8.94-10 4
tpy2 := tl ,tl + 0.000001.. ts2 - 0.000001
tpy3 := tl ,tl + 0.000001.. ten
tpy3 := ts2,ts2 + 0.000001.. te
Incrementing the Time in Phase 2 From tl, the End of Phase 1
to ts2, Where the Stiflhess Changes to K3
Incrementing the Time in Phase 2 From tl, the End of Phase 1
to te, the End of the Analysis (Note: Only Valid in Region 3)
Incrementing the Time in Phase 2 From ts2, to te,
yi(tpyi) :=â– 
the End of the Analysis
Displacement as a Function of Time for Phase 2 Using Stiflhess K1
-Po-jSl
ts2 =8.94-10
te =0.0015
y2(tpy2) :=
sini—]-cos(col tpyl - wl tl) + (1 + cosf — 1 ]• sin(col tpyl - col tl)
Kl\l-0l*)l l *1/ l W
Displacement as a Function of Time for Phase 2 Using Stiffness K2
^ ^ sin|— jcos(o>2-tpy2 - co2-tl) + |l + cos|—j |sin(co2tpy2 - co2-tl)
x2(ts2) =0.130857
y2(ts2) =0.130988
62=0.142868
y3(tpy3)
K2-(l - 022)
Displacement as a Function of Time for Phase 2 Using Stiffness K3
-Po/33
K3-U - 0T
sinj — j cos((i)3 tpy3 - co3-tl) + íl + cosí—])■ sin(ca3 tpy3 - co3-tl)
CALCULATION OF ACCELERATION IN PHASE 2
Acceleration as a Function of Time for Phase 2 Using Stiffness K1
(81 I . / IT \ , 2
-sin — cos(-col tpyl -i- col tl)-col
[ki (i - /312)] V
ayll(tpyl) := -Po
ayl2(tpyl) := - Po
ay 1 (tpyl) := ayll(tpyl) + ayl2(tpyl)
Acceleration as a Function of Time for Phase 2 Using Stiffness K2
01
/ In \\ 2]
1 + cos — smí-cül tpyl + col tl) col
[kl(i-012)1
[\ \l81 II
Po-
02
cos(-w2 tpy2 + co2-tl)-co22 |
K2-(1 - 022)]
\/82
/
Po-
02
1 + cos
— 1) - sin(-co2-tpy2 + (c2-tl)-w22
K2-(l - 022)\
\
1(32//
ay2(tpy2) := ay21(tpy2) + ay22(tpy2)

215
Acceleration as a Function of Time for Phase 2 Using Stiffness K3
ay31(tpy3) = -Po
ay32(tpy3) := -Po
03
[k3-(i - 032)] \ l'33
03
-sin — cos(-w3tpy3 +0)3tl)w3'
[k3-(i - 032)]
ay3(tpy3) := ay31(tpy3) + ay32(tpy3)
1 + cos(— 1 )sin(-w3tpy3 + io3-tl)w3
DISPLACEMENT vs TIME DATA FOR BEAM LW9-66
j = 0,1.. 15
l0 0
do
:= 0
l5
-.= 0.000500
d5
:= 0.0329527
l10
:= 0.00100
dio
:= 0.0956403
tj := 0.000100
dl
:= 0.0005939
:= 0.000600
d6
:= 0.0480166
lll
:= 0.00110
dll
:= 0.104388
^ := 0.000200
d2
:= 0.00370417
b
:= 0.000700
d7
:= 0.0615329
l12
:= 0.00120
d12
:= 0.112518
:= 0.000300
d3
:= 0.0100704
b
:= 0.000800
d8
:= 0.074592
l13
:= 0.00130
d13
:= 0.120663
t. = 0.000400
4
d4
:= 0.0197027
*9
:= 0.000900
d9
:= 0.0857216
l14
:= 0.00140
d14
= 0.129486
tjj := 0.00150 d15 := 0.139495
BEAM LW9-66 DISPLACEMENT vs TIME
— PHASE 1, STIFFNESS K2
“ PHASE 1, STIFFNESS K3
PHASE 2, STIFFNESS K2
— PHASE 2, STIFFNESS K3
D LW9-66 DATA

216
DRIVING AND INERTIAL LOADS vs TIME
INERTIAL LOAD, PHASE 1, STIFFNESS K2
-- INERTIAL LOAD, PHASE 1, STIFFNESS K3
— INERTIAL LOAD, PHASE 2, STIFFNESS K2
— INERTIAL LOAD, PHASE 2, STIFFNESS K3
— DRIVING LOAD
CALCULATION OF THE BENDING LOAD
Pl(tpxl) = Po-sin(u)bartpxl)
P2(tpx2) = Po-sin (cobar tpx2)
P3(tpx3) := Po-sin(cobartpx3)
P4(tpy2) := 0 The Forcing
tpy3 = ts2,ts2 + 0.000001.. tf
P5(tpy3) := 0 The Forcing
The Forcing or Driving Function in Phase 1 From 0 to tsl, lbs
The Forcing or Driving Function in Phase 1 From tsl to ts2, lbs
The Forcing or Driving Function in Phase 1 From ts2 to tl, lbs
or Driving Function in Phase 2 From tl to ts2, lbs
Incrementing the Time in Phase 2 From ts2, to tf, ts2 = 8.94* 10 4
the Time of Fracture tf = g 94.1()-4
or Driving Function in Phase 2 From ts2 to tf, lbs

217
BLI(tpxl) :=Pl(tpxl) - axl(,Pxl) me
386.4
The Bending Load in Phase 1 From 0 to tsl, lbs
___ _ _ ax2(tpx2) me
BL2(tpx2) := P2(tpx2)
386.4
The Bending Load in Phase 1 From tsl to ts2, lbs
(If ts2 < tl)
BL3(tpx3) , P3(,px3) - ""
386.4
The Bending Load in Phase 1 From ts2 to tl, lbs
(Ifts2 < tl)
BL4(.py2) := P4(tpy2) - *y2 386.4
The Bending Load in Phase 2 From tl to ts2, lbs
(If ts2 > tl)
BL5(lpy3)P4(,py3) - ay3(*P33) me
386.4
The Bending Load in Phase 2 From ts2 to tf, lbs
(Ifts2>tl)
BENDING LOAD vs TIME
— PHASE 1, STIFFNESS K1
PHASE 1, STIFFNESS K2
" PHASE 1, STIFFNESS K3
— PHASE 2, STIFFNESS K2
PHASE 2, STIFFNESS K3

218
BENDING LOAD vs DISPLACEMENT
PHASE 1, STIFFNESS K2
- - PHASE 1, STIFFNESS K3
— PHASE 2, STIFFNESS K2
PHASE 2, STIFFNESS K3
CALCULATION OF DYNAMIC FRACTURE ENERGY
Dfractl := 0.5 (BLl(tsl - 0.000001)-xl(tsl - 0.000001)) Dfractl =3.899743
Dfract21 = 0.5 ((BL2(tl - 0.000001) - BL2(tsl))(x2(tl - 0.000001) - x2(tsl)))
Dfract22 := BL2(tsl)(x2(tl - 0.000001) - x2(tsl))
Dfract2 := Dfract21 + Dfract22 Dfract2 = 116.725981

219
Dfract31 := 0.5 ((BL3(tl) - BL3(ts2))(x3(tl) - x3(ts2)))oDfract31 := 0.0
Dfract32 := BL3(ts2) (x3(tl) - x3(ts2))n Dfract32 = 0.0
Dfract3 := Dfract31 + D£ract32 Dfract3 =0
Dfract41 := 0.5 ((BL4(ts2 - 0.000001) - BL4(tl)) (y2(ts2 - 0.000001) - y2(tl)))
Dfract42 := BL4(tl) (y2(ts2 - 0.000001) - y2(tl))
Dfract4 := Dfract41 + Dfract42 Dfract4 =40.013505
Dfract51 := 0.5 ((BL5(tf) - BL5(ts2))(y3(tf) - y3(ts2)))
Dfract52 := BL5(ts2)(y3(tf) - y3(ts2))
Dfract5 := Dfract51 + Dfract52 Dfract5 =0
Dfract â– = Dfiactl + Dfract2 + Dfract3 + Dfract4 + Dfract5
5—ÜÍ = 13.386602 Dynamic Fracture Energy, ft - lbs
Sfract
=3.347879 Ratio of Static to Dynamic Fracture Energy
Dfract
CHECK STIFFNESSES OF DYNAMIC BENDING LOAD vs DISPLACEMENT GRAPH
BLl(tsl - 0.000001)
xl(tsl - 0.000001)
KD1 =5.901222' 104 lbs/in ComparetoKl: K1 = 5.901222» 104 lbs/in
BL2(tl - 0.000001) - BL2(tsl)
x2(tl - 0.000001) - x2(tsl)
KD2 = 1.855139-104 lbs/in ComparetoK2: K2 = 1.855139-104 lbs/in
m3 . BL3(tsl)-BL3(ts2)
x3(tsl) - x3(ts2)
KD3 =1.240847-104 lbs/in ComparetoK3: K3 = 1.240847-104 lbs/in

220
KD4
BL4(ts2) - BL4(tl)
y2(ts2) - y2(tl)
KD4 =1.855139-104 lbs/in ComparetoK2: K2 = 1.855139-104 lbs/in
KD5 :=
BL5(tf) - BL5(ts2)
y3(tf) - y3(ts2)
KD5 =0
lbs/in Compare to K3: K3 =1.240847-10 lbs/in
CALCULATION OF ELASTIC WAVE SPEEDS IN CONCRETE
P =
Density of Concrete, lbs-sec2 / in4 p = 1.767263 • 10
1728-g
v = 0.2 Poisson's Ratio for Concrete, unitless
Ec
P
X :=
2(1 + V)
Ecu
(1 + v)-(l - 2 v)
Lame's Elastic Constant (Shear Modulus), psi p = 1.335541 • 10
Lame's Elastic Constant, psi X = 8.903608-10
X + 2p . j
CL := Longitudinal Wavespeed (For an Unbounded, Isotropic Material), in / sec CL = 1.419588-10
CT := — Transverse Wavespeed (For an Unbounded, Isotropic Material), in / sec CT =8.693168-104
^JP
1 10
CL
= 7.044295 Longitudinal Wavespeed, psec / in
1 10°
CL
•3 =21.132886 Time for P-Wave toTraverse 3 inch Deep Beam, psec
M0U
CT
= 11.503286 Transverse Wavespeed, psec / in
1 10
CT
•13.5 = 155.294363 Time for S-Wave toTraverse 13.5 inches to Simple Support, psec

221
CALCULATION OF DAMPING COEFFICIENTS
ij := 0.05 Assumed Damping Factor for the Concrete Beams with CFRP, unitless
Critical Damping Factor for Region 1 with Stiffness Kl, lbs - sec / in
Critical Damping Factor for Region 2 with Stiffness K2, lbs - sec / in
Critical Damping Factor for Region 3 with Stiffness K3, lbs - sec / in
Cl := £-Ccritl Damping Coefficient for Region 1 with Stiffness Kl, lbs - sec / in Cl
C2 := £ Ccrit2 Damping Coefficient for Region 1 with Stiffness K2, lbs - sec / in C2
C3 := £ Ccrit3 Damping Coefficient for Region 1 with Stiffness K3, lbs - sec / in C3
Ccritl =
Ccrit2 :=
Ccrit3 :=
2-me-ci)l
386.4
2-me- 0)2
386.4
2-me- 0)3
386.4
2.906458
1.629601
1.332761

APPENDIX F
ADINA INPUT AND PLOT FILES FOR BEAM LW6-43
* A D IN A - IN 3.0 INPUT FILE
*
* DYNAMIC ANALYSIS OF A PLAIN CONCRETE BEAM, LW6-43
*
FILEUNITS LIST=8 LOG=7 ECHO=7
FCONTROL HEADING=UPPER ORIGIN=UPPERLEFT
CONTROL PLOTUNIT=PERCENT HEIGHT=1.25
*
DATABASE CREATE
*
HEADING 'DYNAMIC ANALYSIS OF A 3x3x30 IN PLAIN CONCRETE BEAM'
*
MASTER IDOF=100111 REACTIONS=YES NSTEP=1500 DT=l.E-06 IRINT=1
ANALYSIS TYPE=DYN MASS=CONSISTENT METHOD=NEWMARK
ITERATION METHOD=FULL-NEWTON LINE-SEARCH=YES
AUTOMATIC-ATS NUMBER=10
TOLERANCES TYPE=EF RNORM=2.530 ETOL=1.0E-06 ITEMAX=500 PRINT=2
PRINTOUT VOLUME=MAXIMUM IPRIC=0 IPRIT=0 IPDATA=3 CARDIMAGE=NO
PORTHOLE FORMATTED=YES FILE=60
PRINTSTEPS 1 1 1 10 1500 10
*
* USING LOAD vs. TIME DATA FROM BEAM LW6-43
TIMEFUNCTION 1
0.E-06
0.000
50.E-06
0.024
100.E-06
0.067
150.E-06
0.189
200.E-06
0.397
250.E-06
0.645
300.E-06
0.903
350.E-06
1.202
400.E-06
1.568
450.E-06
1.925
500.E-06
2.173
550.E-06
2.324
600.E-06
2.457
650.E-06
2.530
700.E-06
2.385
222

223
750.E-06
1.192
800.E-06
1.537
850.E-06
1.189
900.E-06
0.893
950.E-06
0.540
1000.E-06
0.199
1050.E-06
0.000
1600.E-06
0.000
COORDINATES
ENTRIES NODE Y Z
1 15.0 3.0
2 0.0 3.0
3 0.0 0.0
4 15.0 0.0
5 1.5 3.0
6 1.5 0.0
*
LINE STRAIGHT 1 4 EL=3 MID=1
LINE STRAIGHT 1 5 EL=9 MID=1 RATIO=1.0
LINE STRAIGHT 5 2 EL=1 MID=1 RATIO=1.0
LINE STRAIGHT 4 6 EL=9 MID=1 RATIO=1.0
LINE STRAIGHT 6 3 EL=1 MID=1 RATIO=1.0
*
FIXB 2 TYPE=LINES
1 4
FIXB 3 TYPE=NODES
6
*
EGROUP 1 TWODSOLID STRESS2 MATERIALS INT=3
GSURFACE 1 5 6 4 EL 1=9 EL2=3 NODES=8
GSURFACE 5 2 3 6 EL1=1 EL2=3 NODES=8
EDATA
ENTRIES EL PRINT SAVE THICK
1 YES YES 3.0
STEP 1 TO
27 YES YES 3.0
28 NO YES 3.0
STEP 1 TO
30 NO YES 3.0
MATERIAL 1 CONCRETE E0=.3468573E4 NU=.2 SIGMAT=.630 SIGMAC=-6.729 ,
EPSC=-.3E-2 SIGMAU=-5.720 EPSU=-.4E-2 BETA=.75 C1=0.0,
C2=0.0 XSI=0.0 STIFAC=0.0 DENSITY=. 176726E-6 OPTION=INPUT,
SP11=0.0 SP12=.25 SP13=.5 SP14=.75 SP15=1.00 SP16=1.2,
SP311=1.0 SP321=1.35 SP331=1.75 SP341=2.15 SP351=2.5 SP361=2.8,
SP312=1.25 SP322=1.7 SP332=2.1 SP342=2.55 SP352=2.95 SP362=3.3,
SP313=1.2 SP323=1.6 SP333=2.0 SP343=2.4 SP353=2.8 SP363=3.1
LIST COORDINATES 1350

224
*
LOADS CONCENTRATED
1 3 -.5 10 0
*
WORKSTATION SYSTEM=12 COLORS=RGB BACKGROUND=WHITE
FRAME
MESH NODES=l 1 SUBFRAME=2211
PLOTAREA 1 5 95 5 60
MESH NODES=10 ELEMENT=2 BCODE=ALL PLOTAREA=l SUBFRAME=2111
EVECTOR VAR=PFORCE TIME=l.E-06 TEXT=NO OUTPUT=ALL
*
ADINA
*
END

225
* ADINA-PLOT 4.0 INPUT FILE
*
* DYNAMIC ANALYSIS OF A PLAIN CONCRETE BEAM, LW6-43
*
FILEUNITS LIST=8 LOG=7 ECHO=7
FCONTROL HEADING=UPPER ORIGIN=UPPERLEFT
CONTROL PLOTUNIT=PERCENT HEIGHT=1.25
*
* DATABASE COMMANDS TO LOAD OR OPEN THE ADINA-PLOT DATABASE
*
DATABASE CREATE FORMATTED=YES
♦DATABASE OPEN
*
* ORIGINAL MESH
*
WORKSTATION SYSTEM=12 COLORS=RGB BACKGROUND=WHITE
*
FRAME
MESH OR=l DE=0 NODES=l 1 SUBFRAME=2211
PLOTAREA 1 5 95 5 60
LIST PLOTAREA
MESH OR=l DE=0 NODES=10 ELEMENT=2 BCODE=ALL PLOTAREA=l,
SUBFRAME=2111
EVECTOR VAR=PFORCE OUTPUT=ALL
*
* DEFORMED PLOTS AND EVECTOR PLOTS, SHOWING CRACKS
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0001 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0002 DM=-1.0 SUBFRAME=2111,
♦ WINDOW=-l PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0003 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0004 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0005 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=1 TIME=.0006 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—I
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0007 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0008 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0009 DM=5.0 SUBFRAME=2211

226
*MESH 0RIGINAL=2 DEF0RMED=1 TIME=.0010 DM=-1.0 SUBFRAME=2111,
* WINDOW— 1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0011 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0012 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0001 STATE=YES OUTPUT=ALL,
* LENGTH=1
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0002 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=1 DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0003 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦FRAME
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARI ABLE=CRACK_STRES S TIME=.0004 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0005 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STR£SS TIME=.0006 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦FRAME
♦MESH ORIGINAL=1 DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0007 STATE=YES OUTPUT=ALL,
* LENGTH=1
♦MESH ORIGINALA DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARI ABLE=CRACK_STRES S TIME=.0008 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0009 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦FRAME
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0010 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0011 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRES S TIME=.0012 STATE=YES OUTPUT=ALL,

227
LENGTH—1.0
* FORCE VS DISPLACEMENT GRAPH, TIME HISTORY GRAPHS
*
* DATA FROM LW6-43
*
USERDATA JEROME TIME DISPLACEMENT
0.000
0.000
0.0001
0.0000866
0.0002
-0.00018504
0.0003
-0.00013902
0.0004
0.00026929
0.0005
0.00112005
0.0006
0.00291368
0.0007
0.00546444
0.0008
0.00884522
0.0009
0.0133247
0.0010
0.0190277
0.0011
0.0242555
0.0012
0.0301099
0.0013
0.0350674
0.0014
0.0393628
0.0015
0.0424749
NPOINT MIDSPAN NODE=4
LIST NPOINT
*EPOINT CFRP MID SPAN GROUP=l ELEMENT=9 POINT=l
*LIST EPOINT
*
RESULTANT MIDSPANDISPLACEMENT '-'
RESULTANT MID_SPAN_VELOCITY '-'
RESULTANT MIDSPANACCELERATION '-'
♦RESULTANT CFRP_MID_SPAN_STRESS '-'
LIST RESULTANT
*
ALIAS MSD MID SPAN DISPLACEMENT
ALIAS MSV MID SPAN VELOCITY
ALIAS MSA MID SPAN ACCELERATION
♦ALIAS MSS CFRP MID SPAN STRESS
LIST ALIAS
*
PMAX MID SPAN VAR=MSD MSV MSA
*
AXIS 1 0.0 0.0 0.0 0.0 0.0015 'TIME (SEC)'
AXIS 2 0.0 0.0 0.0 0.0 0.06 'BEAM MIDSPAN DISPL (IN)'
LIST AXIS
*
FRAME
GRAPH TIME NULL MSD MID_SPAN XAXIS=1 YAXIS=2 OUTPUT=ALL,

228
SYMBOL=0 SSKIP=0 SUBF=2211
UGRAPH JEROME XAXIS=-1 YAXIS=-2 OUTPUT=ALL SYMBOL=-2
TEXT XP=25 YP=75 COLOR=GREEN STRING='<0> ADINA'
TEXT XP=25 YP=70 COLOR=GR£EN STRING='<2> LW6-43 DATA'
GRAPH TIME NULL MSA MID_SPAN OUTPUT=ALL SYMBOL=0 SSKIP=0,
SUBF=2111
*
FRAME
GRAPH TIME NULL MSV MID_SPAN OUTPUT=ALL SYMBOL=0 SSKIP=0,
SUBF=2211
*
* LIST CRACKS
*
EGZONE CONCRETE
1
LIST ZZONE CONCRETE
ZLIST CONCRETE TSTART=.0016 VARIABLE=CRACK_FLAG
*
* CHECK LISTING
*
CONTROL EJECT=NO LINPAG= 10000
FILEUNITS LIST=9
PLIST MID SPAN VAR=MSD MSV MSA
*PLIST VAR=MSD MSV MSA
END

APPENDIX G
ADINA INPUT AND PLOT FILES FOR BEAM LW9-66
* A D IN A - IN 3.0 INPUT FILE
*
* DYNAMIC ANALYSIS OF A CONCRETE BEAM REINFORCED WITH 3 PLY CFRP,
LW9-66
*
FILEUNITS LIST=8 LOG=7 ECHO=7
FCONTROL HEADING=UPPER ORIGIN=UPPERLEFT
CONTROL PLOTUNIT=PERCENT HEIGHT=1.25
*
DATABASE CREATE
*
FIEADING 'DYNAMIC ANALYSIS OF A 3x3x30 IN CONCRETE BEAM w/3 PLY CFRP'
*
MASTER IDOF=100111 REACTIONS=YES NSTEP=1500 DT=l.E-06 IRINT=1
ANALYSIS TYPE=DYN MASS=CONSISTENT METHOD=NEWMARK
ITERATION METHOD=FULL-NEWTON LINE-SEARCH=YES
AUTOMATIC-ATS NUMBER=10
TOLERANCES TYPE=EF RNORM=7.970 ETOL=1.0E-06 ITEMAX=500 PRINT=2
PRINTOUT VOLUME=MAXIMUM IPRIC=0 IPRIT=0 IPDATA=3 CARDIMAGE=NO
PORTHOLE FORMATTED=YES FILE=60
PRINTSTEPS 1 1 1 10 1500 10
*
* USING LOAD vs. TIME DATA FROM BEAM LW9-66
*
TIMEFUNCTION 1
0.E-06
0.000
50.E-06
0.669
100.E-06
2.178
150.E-06
4.168
200.E-06
5.912
250.E-06
6.950
300.E-06
7.430
350.E-06
7.763
400.E-06
7.970
450.E-06
7.545
500.E-06
6.090
550.E-06
3.955
600.E-06
2.027
650.E-06
0.892
229

230
700.E-06
0.423
750.E-06
0.205
800.E-06
0.062
850.E-06
0.000
1600.E-06
0.000
COORDINATES
ENTRIES NODE Y Z
1
15.0
3.0
2
0.0
3.0
3
0.0
0.0
4
15.0
0.0
5
1.5
3.0
6
1.5
0.0
*
LINE STRAIGHT 1 4 EL=3 MID=1
LINE STRAIGHT 1 5 EL=9 MID=1 RATIO=1.0
LINE STRAIGHT 5 2 EL=1 MID=1 RATIO=1.0
LINE STRAIGHT 4 6 EL=9 MID=1 RATIO=1.0
LINE STRAIGHT 6 3 EL=1 MID=1 RATIO=1.0
*
FIXB 2 TYPE=LINES
1 4
FIXB 3 TYPE=NODES
6
*
EGROUP 1 TRUSS MATERIALS
GLINE 6 4 EL=9 NODES=3
GLINE 3 6 EL=1 NODES=3
EDATA
ENTRIES EL AREA SAVE
1 .0585 YES
STEP 1 TO
10 .0585 YES
MATERIAL 1 ELASTIC E=2.0E4 DENSITY=1.475E-7
*
EGROUP 2 TWODSOLID STRESS2 MATERIAL=2 INT=3
GSURFACE 1 5 6 4 EL 1=9 EL2=3 NODES=8
GSURFACE 5 2 3 6 EL1=1 EL2=3 NODES=8
EDATA
ENTRIES EL PRINT SAVE THICK
1 YES YES 3.0
STEP 1 TO
27 YES YES 3.0
28 NO YES 3.0
STEP 1 TO
30 NO YES 3.0
MATERIAL 2 CONCRETE E0=.3203495E4 NU=.2 SIGMAT=.630 SIGMAC=-5.742 ,
EPSC=-.3E-2 SIGMAU=-4.880 EPSU=-.4E-2 BETA=.75 Cl =0.0,

231
C2=0.0 XSI=0.0 STIFAC=0.0 DENSITY=.176726E-6 OPTION=INPUT,
SP11=0.0 SP12=.25 SP13=5 SP14=75 SP15=1.00 SP16=1.2,
SP311=1.0 SP321=1.35 SP331=1.75 SP341=2.15 SP351=2.5 SP361=2.8,
SP312=1.25 SP322=1.7 SP332=2.1 SP342=2.55 SP352=2.95 SP362=3.3,
SP313=1.2 SP323=1.6 SP333=2.0 SP343=2.4 SP353=2.8 SP363=3.1
*
LIST COORDINATES 1 350
*
LOADS CONCENTRATED
1 3 -.5 10 0
*
WORKSTATION SYSTEM=12 COLORS=RGB BACKGROUND=WHITE
FRAME
MESH NODES=l 1 SUBFRAME=2211
PLOTAREA 1 5 95 5 60
MESH NODES=10 ELEMENT=2 BCODE=ALL PLOTAREA=l SUBFRAME=2111
EVECTOR VAR=PFORCE TIME=l.E-06 TEXT=NO OUTPUT=ALL
*
ADINA
*
END

232
* ADINA-PLOT 4.0 INPUT FILE
*
* DYNAMIC ANALYSIS OF A CFRP REINFORCED CONCRETE BEAM, LW9-66
*
FILEUNITS LIST=8 LOG=7 ECHO=7
FCONTROL HEADING=UPPER ORIGIN=UPPERLEFT
CONTROL PLOTUNIT=PERCENT HEIGHT=1.25
*
* DATABASE COMMANDS TO LOAD OR OPEN THE ADINA-PLOT DATABASE
*
DATABASE CREATE FORMATTED=YES
♦DATABASE OPEN
*
* ORIGINAL MESH
*
WORKSTATION SYSTEM=12 COLORS=RGB BACKGROUND=WHITE
*
FRAME
MESH OR=l DE=0 NODES=l 1 SUBFRAME=2211
PLOTAREA 1 5 95 5 60
LIST PLOTAREA
MESH OR=l DE=0 NODES=10 ELEMENT=2 BCODE=ALL PLOTAREA=l,
SUBFRAME=2111
EVECTOR VAR=PFORCE OUTPUT=ALL
*
* DEFORMED PLOTS AND EVECTOR PLOTS, SHOWING CRACKS
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0001 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0002 DM=-1.0 SUBFRAME=2111,
* WINDOW=-1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0003 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0004 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0005 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0006 DM=-1.0 SUBFRAME=2111,
* WINDOW— 1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0007 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0008 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0009 DM=5.0 SUBFRAME=2211

233
*MESH 0RIGINAL=2 DEF0RMED=1 TIME=.0010 DM=-1.0 SUBFRAME=2111,
* WINDOW—1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0011 DM=5.0 SUBFRAME=2211
♦MESH ORIGINAL=2 DEFORMED=l TIME=.0012 DM=-1.0 SUBFRAME=2111,
♦ WINDOW— 1 PLOTAREA—1
*
♦FRAME
♦MESH ORIGINAL=1 DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0001 STATE=YES OUTPUT=ALL,
* LENGTH=1
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0002 STATE=YES OUTPUT=ALL,
* LENGTH— 1.0
♦MESH ORIGINAL=1 DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0003 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦FRAME
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0004 STATE=YES OUTPUT=ALL,
* LENGTH— 1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0005 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0006 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦FRAME
♦MESH ORIGINAL=1 DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0007 STATE=YES OUTPUT=ALL,
* LENGTH=1
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0008 STATE=YES OUTPUT=ALL,
* LENGTH— 1.0
♦MESH ORlGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0009 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
*
♦FRAME
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3311 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STR£SS TIME=.0010 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3211 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS TIME=.0011 STATE=YES OUTPUT=ALL,
* LENGTH—1.0
♦MESH ORIGINAL=l DEFORMED=0 SUBFRAME=3111 MARGIN=YES
♦EVECTOR VARIABLE=CRACK_STRESS T1ME=.0012 STATE=YES OUTPUT=ALL,

234
LENGTH—1.0
* FORCE VS DISPLACEMENT GRAPH, TIME HISTORY GRAPHS
*
* DATA FROM LW9-66
*
USERDATA JEROME TIME DISPLACEMENT
*
0.0000
0.00000
0.0001
0.0005939
0.0002
0.00370417
0.0003
0.0100704
0.0004
0.0197027
0.0005
0.0329527
0.0006
0.0480166
0.0007
0.0615329
0.0008
0.074592
0.0009
0.0857216
0.0010
0.0956403
0.0011
0.104388
0.0012
0.112518
0.0013
0.120663
0.0014
0.129486
0.0015
0.139495
NPOINT MIDSPAN NODE=4
LISTNPOINT
*EPOINT CFRP MID SPAN GROUP=l ELEMENT=9 POINT=l
*LIST EPOINT
*
RESULTANT MIDSPANDISPLACEMENT ’-'
RESULTANT MID_SPAN_VELOCITY '-'
RESULTANT MIDSPANACCELERATION '-’
* RESULTANT CFRP_MID_SPAN_STRESS '-'
LIST RESULTANT
*
ALIAS MSD MID SPAN DISPLACEMENT
ALIAS MSV MID SPAN VELOCITY
ALIAS MSA MID SPAN ACCELERATION
* ALIAS MSS CFRPMIDSPANSTRESS
LIST ALIAS
*
PMAX MID SPAN VAR=MSD MSV MSA
*
AXIS 1 0.0 0.0 0.0 0.0 0.0015 TIME (SEC)'
AXIS 2 0.0 0.0 0.0 0.0 0.15 'BEAM MIDSPAN DISPL (IN)'
LIST AXIS
FRAME

235
GRAPH TIME NULL MSD MID_SPAN XAXIS=1 YAXIS=2 OUTPUT=ALL,
SYMBOL=0 SSKJP=0 SUBF=2211
UGRAPH JEROME XAXIS=-1 YAXIS=-2 OUTPUT=ALL SYMBOL=-2
TEXT XP=25 YP=75 COLOR=GREEN STRING='<0> ADINA’
TEXT XP=25 YP=70 COLOR=GREEN STRING='<2> LW9-66 DATA'
GRAPH TIME NULL MSA MID_SPAN OUTPUT=ALL SYMBOL=0 SSKIP=0,
SUBF=2111
*
FRAME
GRAPH TIME NULL MSV MID_SPAN OUTPUT=ALL SYMBOL=0 SSKIP=0,
SUBF=2211
*
* LIST CRACKS
*
EGZONE CONCRETE
2
LIST ZZONE CONCRETE
ZLIST CONCRETE TSTART=.0016 VARIABLE=CRACK_FLAG
*
* CHECK LISTING
*
CONTROL EJECT=NO LINPAG= 10000
FILEUNITS LIST=9
PLIST MID SPAN VAR=MSD MSV MSA
END

BIOGRAPHICAL SKETCH
David Mark Jerome was bom in November 1952 in Ashtabula, Ohio, and attended
primary and secondary school in Olmsted Falls, Ohio. He entered The Ohio State University in
September 1971 and graduated with a Bachelor of Science in Agriculture in December 1975.
After a brief tour with the Navy, he re - entered The Ohio State University and graduated with a
Bachelor of Science in Mechanical Engineering in August 1981. Subsequently, he was
employed as a research engineer by the Air Force at Eglin Air Force Base, Florida, in September
1981.
After obtaining a fellowship from the Air Force to continue his education, he moved to
Gainesville, Florida, and entered the graduate school at the University of Florida in August 1986.
He graduated with a Master of Science in engineering mechanics in August 1987.
David obtained another Air Force fellowship to continue his education at the University
of Florida’s Graduate Engineering and Research Center at Eglin Air Force Base, Florida, in
August 1993. While there, he completed the requirements for the Doctor of Philosophy degree.
David is married to the former Elisabetta Lidia Landi. They have twin sons, Matthew
Allen and Nathan Kelley.
236

I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate
for the degree of Doctor of Philosophy.
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
, in scope and quality, as a dissertation
Oís
' A I Inn D rtfin 1-» r* a *•
C. Allen Ross, Chair
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy. ¿y
James E. Milton, Cochair
Engineer of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
Edward K. Walsh
Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate^in-scope and quality, as a dissertation
for the degree of Doctor of Philosophy. . ) si, A , _., ,
(_A/.L W
DavyM/Belkf-
Assistant Professor of Aerospace
Engineering, Mechanics, and
Engineering Science
I certify that I have read this study and that in my opinion it conforms to acceptable
standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation
for the degree of Doctor of Philosophy.
Christopner S. Anderson
Assistant Professor of Electrical
and Computer Engineering

This dissertation was submitted to the Graduate Faculty of the College of Engineering
and to the Graduate School and was accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
May 1996
Winfred M. Phillips
Dean, College of Engineering
Karen A. Holbrook
Dean, Graduate School

LD
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UNIVERSITY OF FLORIDA
3 1262 08554 4426



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