Behavior Of Two-Span Continous Reinforced Concrete Beams
A thesis presented to
the faculty of the Russ College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Colin Michael McCarty
August 2008 2
This thesis titled
Behavior of Two-Span Continuous Reinforced Concrete Beams
by
COLIN MICHAEL MCCARTY
has been approved for
the Department of Civil Engineering
and the Russ College of Engineering and Technology by
Eric P. Steinberg
Associate Professor of Civil Engineering
Dennis Irwin
Dean, Russ College of Engineering and Technology 3
ABSTRACT
MCCARTY, COLIN MICHAEL, M.S., August 2008, Civil Engineering
Behavior of Two-Span Continuous Reinforced Concrete Beams (127 pp.)
Director of Thesis: Eric P. Steinberg
Nine two-span continuous reinforced concrete beams with point loads 4.5’ on each side
of the center reaction were tested. This tested the effects of both longitudinal and
transverse reinforcement on the shear strength of the beams. Results of testing the beams showed the effects of maximum shear and moment occurring at the center reaction.
Three sets of three beams were constructed to examine the affect varying the amount of longitudinal and transverse reinforcement had on the flexural and shear strength of a beam. Within a set of beams, the longitudinal reinforcement consisted of either No. 3, 8, or 10 reinforcing bars.
Transverse reinforcement was consistent for each set of beams. The first set of beams did not have any shear reinforcement and were used to study the effects of longitudinal reinforcement on the shear strength of a beam. The second set of beams considered the effects maximum transverse reinforcement spacing has on the shear strength of a continuous beam. The third set of beams was used to examine whether further increases in transverse reinforcement were sufficient to preserve the shear strength when the flexural steel yields. 4
Specimens with No. 3 longitudinal bars explored the flexural characteristics of
continuous beams with minimum longitudinal reinforcement ratios. Specimens with
larger longitudinal reinforcement were used to evaluate the effects of more longitudinal reinforcement had on the shear strength and moment distribution. A test was performed to determine deformed welded wire fabrics ability to be used as transverse reinforcement.
Measured values from testing were compared to calculated values from ACI 318-05
Building Code Requirements for Structural Concrete, which provided conservative results in both shear and flexural capacity when shear reinforcement existed.
Approved: ______
Eric P. Steinberg
Associate Professor of Civil Engineering 5
ACKNOWLEDGMENTS
Special thanks to Ohio University faculty & staff: Dr. Steinberg, Dr. Sargand, Dr.
Masada, Dr. Shen, Mike Krumlauf, and Issam Khoury. Further appreciation goes to the
Ohio University graduate students without their help, this project could not have been completed. I would like to extend my gratitude to Dr. Brown and my Parents for there guidance.
6
TABLE OF CONTENTS
Page
Abstract ...... 3
Acknowledgments...... 5
List of Tables ...... 9
List of Figures ...... 10
Chapter 1: Introduction ...... 13
Objectives ...... 13
Outline of Thesis ...... 13
Chapter 2: Literature Review ...... 15
Effects of Continuity on Shear Strength ...... 15
Effects of Transverse Reinforcement and Spacing on Shear Strength ...... 20
Effects of Longitudinal Reinforcement Ratio on Shear Strength ...... 22
Effects of Shear Span to Depth Ratio ...... 23
Effects of Minimum Amounts of Shear Reinforcement ...... 23
Effects of Welded Wire Fabric as Shear Reinforcement ...... 24
Chapter 3: Experimental Program ...... 26
Design of Specimens ...... 26
Description of Set I Beams ...... 28
Description of Set II Beams ...... 29
Description of Set III Beams ...... 31
Instrumentation ...... 33 7
Testing Setup ...... 35
Chapter 4: Beam Test Results Set I, II, & III ...... 43
Test Results I-0-3 ...... 46
Test Results I-0-8 ...... 51
Test Results I-0-10 ...... 56
Test Results II-d/2-3 ...... 59
Test Results II-d/2-8 ...... 65
Test Results II-d/2-10 ...... 73
Test Results III-M-3 ...... 77
Test Results III-F-8 ...... 82
Test Results III-F-10 ...... 84
Summary of Results ...... 87
Chapter 5: Discussions ...... 90
No. 3 Longitudinal Reinforcement Comparison ...... 90
No. 8 Longitudinal Reinforcement Comparison ...... 92
No. 10 Longitudinal Reinforcement Comparison ...... 96
Beams without Shear Reinforcement ...... 97
Beams with Shear Reinforcement ...... 98
Continuity ...... 99
Chapter 6: Summary and Conclusions ...... 103
Conclusions ...... 103
Further Research ...... 104 8
References ...... 106
Appendix: A ...... 109
Appendix: B ...... 110
Appendix: C ...... 111
Appendix: C Continued ...... 112
Appendix: C Continued ...... 113
Appendix: D ...... 114
Appendix: D cont’d ...... 115
Appendix: E ...... 116
Appendix: E cont’d ...... 117
Appendix: F ...... 118
Appendix: F cont’d ...... 119
Appendix: G ...... 120
Appendix: G cont’d ...... 121
Appendix: H ...... 122
Appendix: H cont’d ...... 123
Appendix: I ...... 124
Appendix: I cont’d ...... 125
Appendix: J ...... 126
Appendix: J cont’d ...... 127
9
LIST OF TABLES
Page
Table 1: Beam Properties ...... 43
Table 2: Results of Testing for Beam I-0-3 ...... 47
Table 3: Results of Testing for Beam I-0-8 ...... 51
Table 4: Results of Testing for Beam I-0-10 ...... 57
Table 5: Results of Testing for Beam I-d/2-3 ...... 60
Table 6: Results of Testing for Beam I-d/2-8 ...... 65
Table 7: Results of Testing for Beam I-d/2-10 ...... 75
Table 8: Results of Testing for Beam I-M-3 ...... 78
Table 9: Results of Testing for Beam I-F-8 ...... 82
Table 10: Results of Testing for Beam I-F-10 ...... 86
Table 11: Summary of Results ...... 89
Table 12: Results of Beams without Shear Reinforcement ...... 98
Table 13: Capacity of Beams that Supported Load past Initial Yield ...... 100
Table 14: Capacity of Beams Past Yielding or Hinge Formation ...... 101
10
LIST OF FIGURES
Page
Figure 1: Continuous beam loading pattern ...... 16
Figure 2: Diagram of the elastic shear reaction of the test beam ...... 17
Figure 3: Elastic moment diagram of a test beam ...... 17
Figure 4: Diagram of a simply supported beam ...... 18
Figure 5: Shear diagram for a simply-supported beam ...... 18
Figure 6: Moment diagram for a simply-supported beam ...... 19
Figure 7: Nomenclature for beams ...... 27
Figure 8: Typical beam cross section...... 28
Figure 9: Stirrup bent from No. 3 reinforcing bar ...... 30
Figure 10: Jig for bending deformed Welded Wire Fabric ...... 32
Figure 11: Picture of stirrups made from WWF ...... 33
Figure 12: Test beam dimensions ...... 36
Figure 13: Exploded view of pedestal setup ...... 39
Figure 14: Reaction frame set up ...... 40
Figure 15: Picture of adjustment to test setup ...... 41
Figure 16: Propped cantilever reaction used for calculations ...... 45
Figure 17: Shear crack failure for beam I-0-3 (North Side of Beam) ...... 48
Figure 18: Width of shear crack under load beam I-0-3 (South Side of Beam) ...... 48
Figure 19: Plot of total load versus deflection for I - 0 - 3 ...... 49
Figure 20: Plot of Elastic Theory versus Measured Center Reaction for Beam I-0-3 ....50
11
Figure 21: Shear crack formed at 40 kips beam I-0-8 (South Side of Beam) ...... 52
Figure 22: Spalling over center reaction for beam I-0-8 (North Side of Beam) ...... 53
Figure 23: Failure of beam I-0-8 (South Side of Beam) ...... 53
Figure 24: Plot of total load versus deflection for beam I-0- 8 ...... 54
Figure 25: Plot of elastic theory versus measured center reaction for beam I-0-8 ...... 55
Figure 26: Plot of total load versus deflection for beam I-0-10 ...... 57
Figure 27: Plot of elastic theory versus measured center reaction for beam I-0-10 ...... 58
Figure 28: Shear failure of beam I-0-10 ...... 59
Figure 29: Half inch crack over center support of beam II-d/2-3 ...... 61
Figure 30: Plot of total load versus deflection for beam II-d/2-3 ...... 62
Figure 31: Plot of elastic theory versus measured center reaction for beam II-d/2-3 .....63
Figure 32: Flexural cracks above center support for beam II-d/2-3 ...... 64
Figure 33: Flexural cracks underneath loading for beam II-d/2-3 ...... 64
Figure 34: Spalling on southeast side underneath load for beam II-d/2-8 ...... 66
Figure 35: Plot of total load versus deflection for beam II-d/2-8 ...... 67
Figure 36: Plot of elastic theory versus measured center reaction for beam II-d/2-8 .....68
Figure 37: Shear crack in beam II-d/2-8 ...... 69
Figure 38: Negative moment strain cracking beam II-d/2-8 ...... 72
Figure 39: Close up of negative moment strain cracking of beam II-d/2-8 ...... 73
Figure 40: Spalling near loading for beam II-d/2-10 ...... 74
Figure 41: Plot of total load versus deflection for beam II-d/2-10 ...... 75
Figure 42: Center reaction of beam II-d/2-10 ...... 76
12
Figure 43: Plot of elastic theory versus measured center reaction for beam II-d/2-10 ...77
Figure 44: Failure of beam III-M-3 ...... 79
Figure 45: Ruptured stirrup in beam III-M-3 ...... 79
Figure 46: Plot of total load versus deflection for beam III-M-3 ...... 80
Figure 47: Plot of elastic theory versus measured center reaction for beam III-M-3 .....81
Figure 48: Plot of total load versus deflection for beam III-F-8 ...... 83
Figure 49: Plot of elastic theory versus measured center reaction for beam III-F-8 ...... 84
Figure 50: Cracking pattern after maximum loading for beam III-F-10 ...... 85
Figure 51: Plot of total load versus deflection for beam III-F-10 ...... 86
Figure 52: Plot of elastic theory versus measured center reaction for beam III-F-10 ....87
Figure 53: Negative moment rebar after testing for beam III-M-3 ...... 92
Figure 54: Failure pattern for beam II-d/2-8 ...... 94
Figure 55: Failure pattern for beam III-F-8 ...... 94
Figure 56: Shear crack splitting concrete below the reinforcement for beam III-F-8 ....95
Figure 57: Strain cracking along the longitudinal reinforcement ...... 102
13
CHAPTER 1: INTRODUCTION
In structures the predominant means of testing beams is done using simply supported
beams. The results of which are used to define the codes to which structures are designed
and built. However, looking at most concrete structures one can observe that in one
shape, form, or fashion are not simply supported systems. This research was designed to
test continuous beams in order to observe the results that would more closely represent
structures that have and will be built. Nine two-span continuous beams were constructed
varying both the longitudinal and transverse reinforcement.
Objectives
The project was designed to test a specific set of criteria: concrete contribution to shear
strength on a continuous two-span beam, the effects a minimum amount of shear
reinforcement would have on the shear strength of a continuous beam, analyze if shear
reinforcement at maximum spacing has the ability to control shear forces and prevent a
shear failure, examine the flexural characteristics of a beam with a minimum amount of
longitudinal reinforcement, and if a continuous beam that is heavily reinforced in both
longitudinal and shear displays the characteristics that the Building Code Requirements
for Structural Concrete ACI 318-05 desires.
Outline of Thesis
The report is organized into five chapters. Chapter 2 consists of the literature review
which cites similar reports and past experiments. The third chapter describes the testing
along with the construction and differences between each set of beams. The fourth chapter provides the results of testing. The fifth chapter analyzes the data and tries to use 14 the results to define failure types and present conclusions for each of the above criteria.
The sixth and final chapter summarizes the results and gives direction for further research. 15
CHAPTER 2: LITERATURE REVIEW
Effects of Continuity on Shear Strength
Rodriquez et. al. (1959) acknowledged that structures are built as a continuous system, but the predominant testing method for reinforced concrete is a simply supported system.
The difference between the two is significant because the maximum values for shear and moment occur at the same location for continuous beams. However, Rodriquez et. al.
(1959) found “the manner of diagonal crack formation and modes of failure of two-span continuous beams failing due to shear are similar to those observed in tests of simple beams and of simply supported restrained beams.” The difference was that continuity allowed a redistribution of forces once first yielding occurred and this allowed the beam to handle more load. Failure of the simply supported beams are defined by yielding of the longitudinal reinforcement. The research helped illustrate the differences between simply supported and continuous beams.
Ernst (1957) tested twenty-four nineteen foot two-span continuous beams with eight and a half foot spans. The beams were provided with sufficient transverse reinforcement to allow for a flexural failure. He found that redistribution of forces begins with yielding of longitudinal reinforcement in the first critical section and ended with the yielding of the longitudinal reinforcement in the final critical section regardless of the longitudinal reinforcement ratio. Redistribution at the center support of beams with extended negative longitudinal reinforcement did not exceed 2.44% for beams with rigid supports. 16
However, Ernst (1957) had several series of beams that were initially simply supported with the third support below the bottom surface of the beam. As the beam deflected under load, it came in contact with the third support. The beams with displaced supports far exceeded 2.44% redistribution.
The figures below are given to show the differences in forces for a continuous beam and a simply supported beam. A continuous beam with its shear and moment diagram can be seen in Figures 1-3.
Figure 1. Continuous beam loading pattern.
17
Figure 2. Diagram of the elastic shear reaction of the test beam.
Figure 3. Elastic moment diagram of a test beam.
A simply supported beam with shear and moment diagram can be seen in Figures 4-6. 18
Figure 4. Diagram of a simply supported beam.
Figure 5. Shear diagram for a simply-supported beam.
19
Figure 6. Moment diagram for a simply-supported beam.
As can be seen above, in a continuous beam the maximum values of shear and moment
happen at the same point of the beam, while a simply supported beam has a maximum
moment when the shear force is zero. These figures show the differences of forces in the
two types of beams. As noted above, continuity allows for a redistribution of forces once
first yielding occurs and allows the beam to carry more load. The majority of research
testing has been performed on simply supported beams. Failure occurs when the concrete
crushes after yielding of longitudinal reinforcement in simply supported beams. In
continuous beams as the longitudinal reinforcement yields, a redistribution of forces from
the first critical section to the next occurs. The forces present in the first critical section
could be enough to cause a failure while redistribution takes place. These papers show
that an adequate amount of transverse reinforcement is needed to allow for a
redistribution of forces, otherwise the continuous beam could fail in shear. Shear failure
is a brittle-unpredictable failure which is what ACI 318-05 requirements try to prevent. 20
Effects of Transverse Reinforcement and Spacing on Shear Strength
Mphonde (1989) found that the cracking load that caused an inclined crack to progress to
mid-depth of a beam was the same for beams with or without shear reinforcement. He
claimed that empirically, transverse reinforcement increased shear strength of a beam
enough to warrant their use. Mphonde (1989) believed that a lower bound for shear
strength should be created by testing a series of beams with shear reinforcement. In this
paper, Mphonde (1989) defined an implicit fault in the current ACI 318-05 code which is
summing the concrete and transverse reinforcement shear strength. The two values are
not mutually exclusive and one affects the other.
Russo and Puleri (1997) investigated the problem that Mphonde (1989) claimed the shear
strength of a beam is not the superposition of shear strength from concrete and transverse
reinforcement. Russo and Puleri (1989) found that two different behaviors occur
depending on the shear span-to-depth ratio. If the beam had a shear span-to-depth ratio
of greater than two, then the transverse reinforcement was fully engaged providing the full shear reinforcement contribution of the transverse steel. However, if the shear span- to-depth ratio was less than 2, the transverse reinforcement was not fully engaged therefore not providing its entire contribution to shear strength. Russo and Puleri (1997) developed factors to account for the varied contribution of transverse reinforcement to shear strength and using these factors they were able to better predict the shear strength of a beam.
21
Johnson and Ramirez (1989) studied the effects of a higher strength concrete on the shear
strength of a beam. They found that an increase in concrete strength, above 6000 psi, reduced the relative strength of a beam if the minimum amount of transverse steel was
used. This was due to “the increased shear force at the onset of diagonal tension cracking
and the reduced aggregate interlock contribution, this transfer of forces may cause the first mobilized stirrups to yield and rupture.” A lager load is needed to bring about a diagonal tension crack with higher strength concrete compared to normal strength concrete. However, a greater amount of minimum transverse reinforcement negates this problem.
Typically, transverse reinforcement is made from No. 3 or 4 rebar. To increase the amount of transverse steel in a beam, the spacing between the stirrups (transverse steel) needs to be decreased. Mphonde (1989) found that stirrups increased the shear strength enough to warrant their use, but the “stirrup effectiveness” varied with the shear span-to- depth ratio. Russo and Puleri (1997) were able to define the “stirrup effectiveness” for two different shear span-to-depth ratios which encompassed the variety of beams in use.
The importance of stirrups and their spacing shown by the researchers is to increase the shear strength and allow a flexural failure, which is ductile and more predictable. A flexural failure is the type of failure that ACI 318-05 promotes. A flexural failure is more predictable and allows identification of a problem by the width of flexural cracks, whereas a shear failure is more unpredictable and gives less notice before failure. 22
Effects of Longitudinal Reinforcement Ratio on Shear Strength
Kani (1966) tested 144 simply supported beams without shear reinforcement, varying only the longitudinal reinforcement and the shear span-to-depth ratio. He found that within a shear span-to-depth ratio of two to six, the flexural strength of a beam was lower than what the flexural equations predicted. Since shear failures are similar between simply supported and continuous beams, the failure types apply to continuous beams.
However, the forces that occur in continuous beams are different than in simply supported beams, as can be seen in Figures 1 and 2, which could cause failure at significantly lower strengths because of the concurrent maxima of shear and moment.
Kani (1996) also found that by increasing the amount of longitudinal steel, the relative strength of the beams decreased. This was due to a shear failure that occurred in the beam prior to the longitudinal reinforcement yielding. The flexural strength equations are based on the fact that longitudinal reinforcement yields.
Kani’s (1966) work was important in that it shows that more factors govern a shear failure than just the concrete shear strength and the amount of longitudinal reinforcement.
His work indicated that the most important factor governing the shear strength of a beam is the shear span-to-depth ratio. Kani’s (1966) work also showed that increasing the amount of longitudinal reinforcement actually decreased the relative strength of a beam. 23
Effects of Shear Span to Depth Ratio
Kani (1966) found that for a shear-span to depth ratio of two to six, the relative strength of a beam decreased. Relative strength is defined by the strength of the tested beam
versus that what found using flexural equations. The range given above is a transition
between short-deep beams and flexural ones. Short-deep beams form struts where the
load can be transferred directly to the support. The forces in flexural beams can be
modeled as a truss and the subsequent loads are transferred the supports this way.
Beams that fall into the transition between short-deep beams and flexural ones have a
lower relative strength. Mphonde (1989) and Russo and Puleri (1997) both addressed
this problem which was discussed above. The authors developed methods to deal with
beams that fell within the transition range.
Effects of Minimum Amounts of Shear Reinforcement
Bryant et. al. (1962) tested 21 two-span continuous reinforced concrete beams. The
authors used Equation 1 below to quantify a beam’s shear strength.
Krf y Equation 1 where: = stirrupsupbentforfactorK 1( verticalfor stirrups) A r = v ∗sb
v = stirruponeofareaA = widthbeamb = stirrups spacing
y = yieldf stirrupofstrength
24
The authors claimed that most beams tested had Krfy values greater than 200, while in
practice most beams were under 200. The authors found that “beams with Krfy values of
155 and 200 with 11 loads per span” failed in flexure. Beams with a Krfy value less than
200 failed in shear compression or diagonal tension.
Bryant et. al. (1962) found that a minimum amount of shear reinforcement was found inadequate to control diagonal tension cracking and brittle failures from point loads.
However, as the loading pattern approached uniform loading, 11 point loads, a critical section was difficult to asses. This was due to the fact that a more uniform flexural rigidity existed throughout the beam allowing elastic theory to more closely predict the beam’s strength. This removed the unpredictable nature of the brittle failures and allowed for a lesser amount of shear reinforcement to support larger total loads than what could have been supported if there were fewer loading points.
Effects of Welded Wire Fabric as Shear Reinforcement
Griezic et. al. (1994) constructed two T-beams with deformed welded wire fabric (WWF)
to test WWF’s ability to control cracking and display adequate ductility for a
redistribution of forces to other stirrups. The two beams with WWF used Grade 500
deformed welded wire fabric. The authors also created two T-beams with No. 3 stirrups
to compare against the WWF T-beams. The area of shear reinforcement for the two sets
of beams, with WWF and No. 3 stirrups, were almost identical. Griezic et. al. (1994)
found that WWF stirrups maintained adequate ductility and were able to control cracking
at service loads. 25
Deformed welded wire fabric can control cracking in beams if the area of these stirrups is comparable to the No. 3 stirrups used in test beams. 26
CHAPTER 3: EXPERIMENTAL PROGRAM
Design of Specimens
The beams were designed to study the effect maximum shear and moment forces, occurring at the same location, had on the shear strength for two-span continuous reinforced concrete beams. Three sets of beams were designed to test the effects of varying the longitudinal and shear reinforcement had on the shear strength and moment capacity of reinforced concrete beams. The test setup remained the same for all beams.
Each set of beams contained three longitudinal reinforcement bars that extended the entire length across the top of the beams for negative moment resistance at the center support. Two longitudinal reinforcement bars extended across the bottom of the beam to provide positive moment reinforcement within the spans. The positive moment longitudinal reinforcement had ACI 318-05 (2005) standard hooks that were bent up 90 degrees 2ft. 2 in. beyond the outer edge of the exterior support plate. All longitudinal reinforcing bars were designed such that sufficient development length existed to avoid bond failure.
Each set of beams contained a different amount of shear reinforcement. The first set of three beams contained no shear reinforcement to examine both the contribution of the concrete shear strength and the effects of different amounts of longitudinal reinforcement had on the shear strength of each beam. The second set was designed to examine the effects of maximum shear reinforcement spacing, per ACI 318-05 (2005), along with 27
varying the amount of longitudinal reinforcement. The third and final set of three beams
was designed to fail in flexure by providing a greater amount of shear reinforcement than
what was required at the critical section over the center support.
The concrete was ordered from a local ready mix company with a specified 28-day compressive strength of five ksi. The maximum aggregate size was three eighths of an inch and there were no admixtures. Slump tests and cylinders were made for each set of beams. The results of the cylinder tests can be seen in Table 1.
Figure 7 below indicates the nomenclature used for each beam. The Roman numeral represents which set of beams a beam came from; I, II, or III. The second location dictates the spacing of the shear reinforcement and if the minimum amount of shear
reinforcement allowable was used. The third number states the size of the longitudinal reinforcing bars used.
I – Vs Spacing – No.
Longitudinal Reinforcement No. 3, 8, or 10’s
0 – No Vs d/2- Max Spacing M – Minimum amount of Shear Steel at Max Spacing Set of Beams I, II, III
Figure 7. Nomenclature for beams. 28
The beams for all three sets had the following dimensions: a width of 8 inches, a height
of 16.25 inches, and a length of 32 feet. The cross section of the beams can be seen in
Figure 8 below.
Figure 8. Typical beam cross section.
Description of Set I Beams
The first set of beams consisted of number three’s, eights, or tens for longitudinal reinforcement. Three longitudinal reinforcing bars extended the entire length across the top of the beam for the negative moment reinforcement and two bars extended the entire length on the bottom for positive moment reinforcement. The bottom reinforcing bars 29 were terminated with standard ACI 318-05 (2005) 90 degree hooks at the ends. The hooks, along with the negative moment reinforcement, were terminated outside the bearing plate of the end reactions to ensure proper development length. Set I did not contain any shear reinforcement steel. Construction of this beam began by placing the longitudinal reinforcement inside the form work. Three quarter inch chairs were placed at intervals along the bottom of the formwork to insure no sagging of the rebar and a consistent spacing between the positive moment longitudinal reinforcement. 2X4’s were nailed across the top of the form work to straighten the form walls The negative moment longitudinal reinforcement were hung individually using a thin gage wire to the 2X4’s.
The concrete cover was measured from the top of the rebar to the bottom of the 2X4 that supported it. The concrete cover was an inch and a quarter for the negative moment longitudinal reinforcement. The longitudinal reinforcement had development lengths in accordance with Sections 12.11 and 12.12 of ACI 318-05. This was accomplished by extending both the positive and negative longitudinal reinforcement the entire length of the beam. The positive moment longitudinal reinforcement was designed in accordance with section 12.5 of ACI 318-05 having a standard hook and stirrups spaced at less than
3db and starting within 2db of the outside of the hook. Shear failures for Set I were expected in the area between the point loads and the center reaction.
Description of Set II Beams
The longitudinal reinforcement for the second set of beams was exactly the same as the first set. This set had No.3 reinforcing bar bent to form stirrups, as can be seen in Figure
9 below, that were used for shear reinforcement. 30
Figure 9. Stirrup bent from No. 3 reinforcing bar.
Set II used the maximum spacing, d/2, for shear reinforcement allowed by ACI 318-05
(2005). In all three beams, this spacing was seven inches. The stirrups were cold bent in
accordance to Section 7.2 of ACI 318-05 (2005). The construction of these beams was
similar to Set I. The positive moment longitudinal reinforcement was placed on the
bottom of the beam. The number of stirrups needed was calculated for the spacing and
length of the beam. All three beams in Set II required 49 stirrups for the length of the
beam, and the end development region was designed the same as Set I. The stirrups were
tied to the positive moment longitudinal reinforcement so they would stand vertically. 31
The negative moment longitudinal reinforcement was then run inside the stirrups along
the length of the beam. The negative longitudinal reinforcement was then temporarily
tied in place to supports that stretched across the top of the formwork. The negative
moment longitudinal reinforcement was then tied to the stirrups.
Description of Set III Beams
For beam III-M-3, welded wire fabric was used to form stirrups. First, the WWF was cut
to the length required to make a stirrup. The horizontal elements of the WWF were cut
off using a bolt cutter leaving approximately ¼ inch of the horizontal component of the
WWF on either side of the vertical. A jig was constructed using steel pipe and 2 X 4’s, to
form a bend radius of greater than 4db as required by ACI 318-05 section 7.2.3. Both the
jig and the WWF that was cut to length were placed in a vice so that the WWF could be
bent to the correct shape. A picture of the jig used to create the stirrups can be seen in
Figure 10. An example of WWF stirrups can be seen in Figure 11. Figure 11 shows how
closely the WWF stirrups resemble the stirrups created by the steel fabricator.
32
Figure 10. Jig for bending deformed Welded Wire Fabric.
WWF 2.5 was used because the area it provided was between the minimum required for
ACI 318-05 and AASHTO. Using an effective depth of 14 inches, the area of shear reinforcement required was 0.0495 in2 by ACI 318-05 and 0.0659 in2 for AASHTO at d/2. The deformed WWF provided a cross sectional area of 0.05in2. This beam was designed to see the effects of minimum shear reinforcement and the use of WWF.
However, the yield strength was never determined and was assumed to be 75ksi which was allowed in Section 9.4 of ACI 318-05.
33
Figure 11. Picture of stirrups made from WWF.
Set III beams, containing No.8’s and 10’s were designed with enough shear reinforcement to ensure a flexural failure would occur. Beams III-F-8 and III-F-10 used
the stirrups supplied by the steel fabricator spaced at 4.5 and 3 inches respectively. This
equated to over 120 stirrups for beam III-F-10.
Instrumentation
To measure the reactions, three Honeywell Model 41 load cells were placed on each of
the supports. The load cells at the end supports had 75,000 lb capacities while the load
cell at the center support had a 150,000 lb capacity. The Honeywell Model 41 has a 34
bonded foil strain gage with a 10Vdc excitation voltage. These load cells have a plus minus one percent zero balance tolerance.
A UniMeasure Model PA-15 wire pot was placed under each load point to measure the deflection of the beam. The PA models from UniMeasure are precision potentiometers that measure movement. This particular model had a 15 inch range over which it could measure displacement. Prior to testing the beam, a hook was epoxied to the bottom of the beam. The wire from the wire pot was then connected to the hook and the reading was taken as zero deflection.
Seven type FLA-5-11-3LT-120 3% electronic strain gages were attached to the reinforcing bars at the points of maximum positive and negative moments. The strain gages were applied to the rebar following the Vishay strain gage application guide that accompanied the strain gages. The rebar was prepared using a degreaser to remove any dirt or grease that was present on the rebar. A die grinder was used to remove the deformed ridges and create an area suitably flat for proper installation of the strain gage.
The area was then roughened using 400 grit wet/dry sandpaper. Once the area was suitably clean and smooth, an acid and base conditioner was used to clean the surface of the rebar. After the gage area was clean (no residual dirt visible on the cloth used for cleaning), the strain gage was aligned on the rebar using tape. A zip tie was the placed around the rebar and the strain gage wire to hold everything in place while the cyanoacrylate adhesive was applied. Once the glue was allowed to dry, the tape was 35
removed. An acrylic liquid was then placed on top of the gage to protect it from
moisture. The acrylic was then allowed to dry before the rest of protection was applied to
the gage. The final protection steps included a neoprene pad placed on the acrylic and
secured in place using aluminum foil duct tape.
A MegaDac 5414 AC data collector was used to collect 50 samples per second while the
test was being conducted. The MegaDac 5414 AC collected data from all seven strain
gages, the three load cells, and the two UniMeasure PA-15 wire pots. This sampling rate
was used for the two beams tested, I-0-10 and I-0-8, and was reduced to 25 samples per
second due to the large amount of data that was not necessary. Since the MegaDac 5414
AC data collector was set at such a high rate of data collection, the data collector was
paused while the cracks were marked on the beams.
Testing Setup
A two-span continuous setup was used to examine the effects of indeterminacy on shear
strength. The dimensions of the test beam can be seen in Figure 12. The setup caused maximum shear and moment over the center support. This was critical to examine the progression of flexural to shear cracks.
36
Figure 12. Test beam dimensions.
37
In addition, there was a concern that the maximum shear and moment forces occurring at
the same location would compound each other to cause lower shear strength than determined by using ACI 318-05 (2005). This is due to the width of the flexural cracks reducing the aggregate interlock component discussed in references (ACI-ASCE
Committee 326, 1962, Bresler and Scordelis, 1963, Mphonde, 1989). The two frames
consisted of two W14X30 crosshead beams which were connected to W8X10 columns.
The columns were bolted to the strong floor. Two RRH-6010 Enerpac hydraulic rams
were bolted to the center of each crosshead. The rams had a 10.12 inch stroke with a
12.73 in2 effective cylinder area. The frames were designed to resist 120 kips each, which was the maximum load that could be applied by the hydraulic rams. The setup can be seen in Figure 14.
An Enerpac series 30,000 hydraulic pump was used to power the rams. The Enerpac series 30,000 hydraulic pump had an input and output line. These lines ran into splitter from which lines of equal length ran to the hydraulic rams. Since everything running from the hydraulic pump to the rams was identical, hydraulic pressure difference in the rams was considered minimal. The beams rested upon 20 inch by 20 inch by 14 inch tall concrete pedestals that sat upon a two and a half foot thick concrete strong floor. The concrete supports were plumbed and leveled on the strong floor using Hydrastone.
Hydrastone is a cement and lime mixture that has a very high compressive strength but is
very workable when wet. To make sure that the tops of the supports were at the same
height with respect to one another, a one inch thick plate was placed and leveled on 38
Hydrastone using a laser level. After the hydrastone had set, a final check of the three
supports was made using the laser level which confirmed that three supports were within
a 1/16th inch of each other.
An example of the support setup is shown below in Figure 13. The beam sat upon two 8 by 8 by 1.75 inch thick steel plates. For the end supports, a one inch diameter steel cylinder was placed in between the two plates and acted as a roller. The center support had a two inch diameter rod that was cut in half and welded to the bottom plate. The 8
inch by 8 inch by 1.75 inch thick plate sat on the top side of the load cell. The load cell
sat upon a metal disk that was slightly larger than the diameter of the threaded hole in
center of the load cell. The metal disk was placed on the one inch steel plate that was on
a concrete pedestal. The load cell has an inner and outer ring. The metal disk was used to
separate the rings and allow the load cells to function properly. 39
Figure 13. Exploded view of pedestal setup.
To ensure that the beams were properly supported, the weight of the beam was taken by summing the three load cells readings. The weight was then proportioned according to a two-span continuous beam with a uniform load. The center reaction accounted for 5/8th of the total load while each end reaction supported the remaining 3/16th of the total beam weight. The beams had to be shimmed at the reactions, typically at the ends, in order to create the defined reaction and hence qualify the beam as properly leveled. Rodriguez et. al. (1959) claimed that “a small variation in the center reaction could affect the moment at critical sections appreciably” so care was taken to balance the beam correctly. Once the beam was supported correctly, the load cells were zeroed and the test was ready to 40
begin. To confirm that the reactions were correct while in the elastic range, the values
gathered by the data collector were checked for accuracy during the first loading interval.
The three sets of beams were tested using the setup that can be seen in Figure 14.
Figure 14. Reaction frame set up.
The test setup for beams III-F-8 and 10 was different from the rest of the beams. Beam
II-d/2-10 could not be taken to failure because the center reaction had reached the load
cells capacity prior to the beam showing a failure mode; shear, shear compression, or
flexure, which was seen in the previous five beams. The change was made by removing
the center load cell from the center reaction, and replacing it with a large steel cylindrical
plate that was approximately the same height as the load cell. The reaction frame on the
east side of the beam was altered by raising the crosshead three inches to accommodate 41 for the load cell under the hydraulic ram. As was done in the previous beam setups, the entire weight of the beams was measured by the load cells. The adjustment to the test setup can be seen in Figure 15 below. The white arrows point out the changes to the testing set up in Figure 15.
Figure 15. Picture of adjustment to test setup.
For beams III-F-8 and 10, the two exterior load cells were used to obtain the entire weight. The center support was then introduced and shimmed until the each end reaction totaled 3/16th’s the self weight of the beam. This left 5/8th’s at the center support which meant the beam’s self weight was balanced as a uniform load on the supports. The load 42 cell that had been under the center support was placed under the east loading point. The entire load was considered twice that measured by the load cell. To confirm this assumption, the west reaction was multiplied by 13.5 while in the elastic range. This value was found using elastic theory equations. This was done to verify that the beam was properly supported when compared to twice the load cell. When the west reaction was multiplied by 13.5 under initial loading, it was equal to the load applied to the beam, which was twice the load cell reading. There was a single hydraulic pump that ran to a splitter which then ran to each hydraulic ram. The lengths of the hydraulic hoses were equal and the rams were identical. A hydraulic pressure difference between the two rams was considered minimal. 43
CHAPTER 4: BEAM TEST RESULTS SET I, II, & III
Table 1 shows the particular dimensions and average concrete strengths determined for
each beam. The No. 3 stirrups used in all the beams were identical. The stirrups were
cold bent with an inner diameter of 6db in accordance with ACI 318-05 section 7.2.
Table 1 shows the effective depth of the beams based on the stirrup dimensions and the rebar diameters. This effective depth, shown in Table 1, was used in all strength calculations. The f’c column is the average compressive strength of the cylinders tested the same day of the beam test. To determine f’c for each beam, 8-9 concrete cylinders were made for each beam on the day that concrete was placed. After each beam test, the cylinders were tested per ASTM C39 and the average strength was taken and used for the subsequent calculations for that particular beam.
Table 1
Beam Properties V d f’ A s f V ρ c v spacing yt s Beam ID in. psi in2 ksi kips % in. I - 0 - 3 14.8 5821 N/A N/A 60 0.00 0.28 I - 0 - 8 14.625 6255 N/A N/A 60 0.00 2.06 I - 0 - 10 14.49 5908 N/A N/A 60 0.00 3.34 II - d/2 - 3 14.8 5621 0.22 7 60 27.91 0.28 II - d/2 - 8 14.625 4948 0.22 7 60 27.58 2.06 II - d/2 - 10 14.49 5437 0.22 7 60 27.32 3.34 III - M - 3 14.8 5972 0.05 7 75 7.93 0.28 III - F - 8 14.625 6011 0.22 4.5 60 42.90 2.06 III - F - 10 14.49 6239 0.22 3 60 63.763.34
The Av column of Table 1 is the cross sectional area of two legs of the shear
reinforcement. The Vs spacing column is the spacing of the shear reinforcement. The fyt 44
column is the nominal yield strength of the shear reinforcement. Beam III-M-3 fyt is 75 ksi which is allowed by ACI318-05 Section 11.5.2. As mentioned earlier, the yield strength of the deformed welded wire fabric was not determined. The Vs column represents the shear reinforcement contribution to shear strength. The Vs was determined by Equation 4 discussed below. The final column is the longitudinal reinforcement ratio as shown in Equation 2.
A ρ = s Equation 2 bd
In all the figures that are pictures of individual beam tests, the numbers on the beam
represent the total load, in kips, when loading was halted in order to mark the progression
of cracks.
The test results for each beam can be seen in Tables 2-11. The following is a description
of Tables 2-11. The Vmeasured column in Table 2 represents half the load recorded by the center load cell, which is equivalent to the maximum shear at the center support. The nominal shear strength column is calculated by summing the contribution of the concrete shear strength, using equation 3, and the contribution of the shear reinforcement, using equation 4. The Vc column uses equation 3 along with the average compressive strength
of the cylinders tested the day of the beam test. For beam sets I and II, shear strength was
considered the limiting factor and the moment was calculated based on the anticipated
shear strength of the beams. The nominal shear strength, shown as Eqn. 3 below, (ACI
318-05) was considered the shear reaction at the middle support.
c = '2 wc dbfV Equation 3 45
Equation 4 below, (ACI 318-05), is the Vs column in Tables 2-11.
yv dfA V = Equation 4 s s
Equation 5 is the nominal shear capacity column and was used to calculate the Mv column.
+= VVV scn Equation 5
Figure 16 below shows the equations used to calculate the Mv column and the Mu column of Tables 2-11. The nominal shear capacity, from equation 5, was taken as R2 and the equations in Figure 16 were used to calculate the Mv column.
Figure 16. Propped cantilever reaction used for calculations.
46
The equations for Figure 16 were from the Thirteenth Edition of the Steel Construction
Manual (AISC). The Mn column of Tables 2-11 represents the calculated moment for a
flexural failure using strain compatibility and Whitney stress block. The strain
compatibility calculations can be seen in Appendix C. The Vmeasured result was set equal
to R2 in Figure 16 and the equations were then used to calculate the Mu column. The ∆
columns are the maximum deflection of the beam at the loading locations. The percent
redistribution column is the redistribution of the beam with respect to the center reaction.
The percent redistribution was calculated using the elastic reaction from the applied load
versus the actual recorded value at the center support.
Test Results I-0-3
Beam I-0-3 was very lightly reinforced with a ρ = 0.47% with both top and bottom
longitudinal reinforcement in tension at nominal capacity. The beam showed hairline
flexural cracks at loads less than 10 kips, but there was little displacement until the load
of 36 kips was reached. At this point, a shear crack appeared on east side of the beam
between the middle support and the point load. The crack was not the result of a flexural crack turning into a shear crack. The beam continued to support load, but by 42 kips, five of the six working strain gages in both positive and negative moment regions had surpassed the initial yield of 2000με. There was extreme flexural cracking located above the center reaction and underneath the loads. Flexural cracking extended from beneath the load point to approximately halfway to the outer supports. The beam continued to support additional load, without much deflection, until 44 kips. At this point, a shear crack formed on the west side of the beam between the load and center reaction. The 47 load instantly dropped to 39 kips. The beam was then reloaded to 44 kips where it was found that the beam would not support any additional load. While reloading the beam after the formation of the second shear crack, it was noticed that the strain in the longitudinal rebar in the negative moment region had dropped to 1600-1800με, while the strain in the longitudinal rebar in the positive moment region had increased to 2400με.
A shear crack can be seen in Figures 17 and 18. The results of the testing can be seen in
Table 2.
Table 2
Results of Testing for Beam I-0-3 I-0-3
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu ∆East ∆West % kips kips kips kips k-ft k-ft test k-ft in in Redistribution 17.99 18.06 18.06 0 53.00 24.13 52.8 0.52 0.59 7.16
Table 2 shows that the shear strength, Vn calculated using equation 2, and the measured shear strength, Vmeasured, of the beam were almost identical. This would indicate that the beam did not fail in shear. Table 2 also shows that the moment capacity of the beam was twice that of what was predicted using strain compatibility. The shear crack seen in
Figures 17 and 18 point toward a shear failure, but the beam failed in flexure because the moment capacity was twice that of what was predicted for the beam.
48
Figure 17. Shear crack failure for beam I-0-3 (North Side of Beam).
Figure 18. Width of shear crack under load beam I-0-3 (South Side of Beam). 49
Figure 19 shows a ductile failure where the beam could not support a load more than 44 kips, but as can be seen in Figure 17 there is a definite shear crack which reduced the rigidity of the beam. Figure 19 shows that further attempts to surpass 44 kips only led to further displacement of the beam.
I-0-3
50,000
45,000
40,000
35,000
30,000
25,000 East Deflection West Deflectioin Total Load (lb) Load Total 20,000
15,000
10,000
5,000
0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Deflection (in) Figure 19. Plot of total load versus deflection for I - 0 - 3.
Even though there was a shear crack, there was not a sudden or dramatic drop in load or increase in deflection which indicates a shear failure.
Figure 20 below shows behavior of the beam and how it deviates from elastic behavior.
50
Figure 20 shows that redistribution of forces begins at a third of the ultimate load and continued to increase with larger loads.
I-0-3
20,000
18,000
16,000
14,000
12,000 Elastic vs Measured 10,000 Elastic Theory
8,000
Measured Center Reaction (lbs) Reaction Center Measured 6,000
4,000
2,000
0 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 Theoretical Center Reaction (lbs) Figure 20. Plot of Elastic Theory versus Measured Center Reaction for Beam I-0-3.
To model elastic behavior, the two-span continuous beam was treated as a propped
cantilever beam. The center reaction represented the fixed reaction of a propped
cantilever beam because with loads equidistant from the center reaction, the center was
not allowed to rotate as if it was a pin or roller connection. Figure 20, as well as the rest
of the elastic versus measured plots, use the equations found in Figure 16 to plot the
reaction that should have occurred given the applied load and half the actual reaction at
the center support. 51
Test Results I-0-8
This beam was heavily reinforced in the longitudinal direction, but did not contain any shear reinforcement. The beam showed very little evidence of flexural cracking up until the development of the first shear crack on the west span between the load and the center support which can be seen in Figure 21 below. This shear crack appeared at just over 40
kips. The beam continued to carry load while the width of the shear crack on the west
side grew from a 1/16th inch to 1/8th inch. The beam failed dramatically when the total load reached 64 kips. Significant spalling occurred on the north side of the beam above the center support. This can bee seen in Figure 22. The spalling was not due to rotation,
because several beams surpassed this amount of rotation, or due to total load. The results
for beam I-0-8 can be seen in Table 3.
Table 3
Results of Testing for I-0-8 I-0-8
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu ∆East ∆West % kips kips kips kips k-ft k-ft test k-ft in in Redistribution 25.1 18.5 18.5 0 54.30 153.9 73.7 0.55 0.28 6.62
There was a large amount of redistribution, compared to the other No. 8 specimens, even though the strains in the longitudinal reinforcement did not surpass yield which can be see in Appendix A. The Mv calculated in Table 3 above was based upon equation 4.
52
Figure 21. Shear crack formed at 40 kips beam I-0-8 (South Side of Beam).
After the spalling, seen in Figure 22, the deflection increased 0.2 inches on the east side of the beam, and the load dropped from 60kips to 33kips. The percent redistribution jumped to over 17% based on elastic theory and the measured load for the middle support. This was after spalling and when the load supported by the beam dropped by half.
53
Figure 22. Spalling over center reaction for beam I-0-8 (North Side of Beam).
Figure 23 below shows cracking along the positive and negative moment longitudinal reinforcement.
Figure 23. Failure of beam I-0-8 (South Side of Beam). 54
These are not flexure or shear cracks typically seen in simply supported beams. Figure
23 shows how failure occurred. The inclined cracks along the reinforcement weakened the concrete where spalling eventually occurred.
Figure 24 shows failure once the beam surpassed 62 kips. Figure 24 also shows a definitive change in stiffness of the beam after the formation of the first shear crack at 40 kips. Even though the beam continued to carry a larger load, the deflection increased at a greater rate after the formation of the shear crack.
I - 0 - 8
70,000
60,000
50,000
40,000
30,000 Total Load (lbs) Load Total East Deflection West Deflection 20,000
10,000
0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 Deflection (in)
Figure 24. Plot of total load versus deflection for beam I-0- 8.
The west side showed the first signs of a shear crack seen in Figure 21, but by examination, it is clearly evident that the east side had a greater displacement prior to 40 55 kips and the formations of the shear crack on the west side. By examining Figure 23, failure was caused by the diagonal tension crack extending far enough into the compression zone to cause a dramatic drop in the load the beam was able to support. The load dropped from 58 kips to 34 kips while the deflection on the east side increased from
0.33 inches to 0.55 inches. After this dramatic decrease in load, the amount of redistribution increased from 6.62% to 17.3%. This increase can be seen in Figure 25.
Due to the very little deflection (0.3 inches), along with the crack widths and number of flexural cracks remaining relatively small compared to Beam I-0-3, beam I-0-8 was considered a shear compression failure.
I - 0 - 8
30,000
25,000
20,000
15,000
10,000 Measured Center Reaction (lbs) Elastic vs. Measured Elastic Theory 5,000
0 0 5,000 10,000 15,000 20,000 25,000 30,000 Theoretical Center Reaction (lbs) Figure 25. Plot of elastic theory versus measured center reaction for beam I-0-8. 56
The recorded strains from this beam did not behave like the other beams because the maximum strains recorded during testing did not occur over the center support (see
Appendix A). After the diagonal tension crack, the center strains dropped while the strains under the point loads increased. The maximum strains recorded for the beam occurred under the point loads and they were almost identical. At the maximum load, none of the strain gages were at their maximum strains. The maximum strain for the east side was recorded before the maximum load was reached and actually reduced prior the maximum load. The maximum strains recorded for the west span under the point load were recorded as the beam failed in the east span and then the west span strain gages dropped when the load dropped because of the beam failure.
Test Results I-0-10
This beam was heavily reinforced in the longitudinal direction with no shear reinforcement. The predicted strains for the different diameter longitudinal reinforcement can be seen in Appendix C. As the loading of the beam proceeded, very few flexural cracks were observed. There was very little if any deflection in the beam until 36kips. This can be seen in Figure 26. At 40 kips a shear crack formed on the east side of the beam. By examining Figure 26, the effects of the crack can be seen in the plot. The west side of the beam failed at 42.5kips. A shear crack formed from the center support to the compression zone of the beam. The shear crack then ran along the compression zone up to the loading point. This can be seen in Figure 28. The results and the values calculated for beam I-0-10 can be seen in Table 4. 57
Table 4
Results of Testing for I-0-10 I-0-10
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu ∆East ∆West % kips kips kips kips k-ft k-ft test k-ft in in Redistribution 17.995 17.8 17.8 0 52.24 239 52.8 0.08 0.13 4.75
Table 4 shows that there was very little deflection in this beam even at failure. It also shows that the predicted and measured shear strength were very close.
I - 0 - 10
45,000
40,000
35,000
30,000
East Deflection 25,000 West Deflection
20,000 Total Load (lbs) Load Total
15,000
10,000
5,000
0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Deflection (in) Figure 26. Plot of total load versus deflection for beam I-0-10.
58
Figure 27 shows the comparison of the beam if it behaved as a completely elastic system
versus the measured reaction at the center support. It can be seen that the beam behaves
almost exactly like an elastic beam. However, it can be seen in Table 4 that the percent
redistribution is 4.75%. The reason for this is that beam I-0-10 was reloaded to confirm that it had failed by seeing if it would hold a larger load. The load was increased to 47 kips and the east side failed in a manor almost identical to the west side of the beam.
This information was not included because the data acquisition was turned off in between the east and west failure. It was therefore not included.
I - 0 - 10
20,000
18,000
16,000
14,000
12,000
10,000
8,000 Elastic vs. Measured Elastic Theory
Measured Center Reaction (lb.) Reaction Center Measured 6,000
4,000
2,000
0 0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 Theoretical Center Reaction (lb.) Figure 27. Plot of elastic theory versus measured center reaction for beam I-0-10.
59
Figure 28. Shear failure of beam I-0-10.
Test Results II-d/2-3
This beam was lightly reinforced in the longitudinal direction using No.3 reinforcing bar,
but for this beam, the shear reinforcement was more than twice what was required based
on a load from flexural capacity. This beam was constructed to test the maximum
spacing of shear reinforcement. During testing, the load was steadily increased up to 20
kips, at this point, the load dropped to 18.9 kips and strains in longitudinal steel on the
east side of the beam jumped from 350 and 230με to 850 and 900με, respectively. The strain gages on the longitudinal steel over the center support jumped from around 590 to
850με. A search for the crack that would have caused this could not be found. As loading continued past 38 kips, the length and number of cracks increased dramatically over the center support and under both point loads. The total load that the beam could withstand 60
was slightly more than 56 kips. At this point, the flexural crack over the center support,
which can be seen in Figure 29, reached a ½ inch width. Continued loading only caused
greater deflection without a significant increase in load. This can be seen in Figure 30.
Table 5 shows that the maximum deflection for beam II-d/2-3 was 1.7 inches. Coupled with the number and width of the flexural cracks, this beam was considered to fail in flexure.
Table 5
Results of Testing for II-d/2-3 II-d/2-3
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu ∆East ∆West % kips kips kips kips k-ft k-ft test k-ft in in Redistribution 23.85 45.66 17.75 27.91 134.0 24.13 70.0 1.7 1.37 3.84
61
Figure 29. Half inch crack over center support of beam II-d/2-3.
62
II - d/2 - 3
70,000
60,000
50,000
40,000
East Deflection West Deflection 30,000 Total Load (lbs) Total Load
20,000
10,000
0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 Deflection (in) Figure 30. Plot of total load versus deflection for beam II-d/2-3.
The data collection rate for this beam was 100 samples per second with some of the
loading periods taking as long as 45 seconds. There was an astronomical amount of data
for this beam, most of which occurred in the later part of the test when the deflection increased with very little increase in total load, as can be seen in Figure 30. The data used in Figure 30 was reduced using a program that would erase a chosen amount of rows in a Microsoft Excel spreadsheet, keep a row of data, and then erase the same amount of stated rows again. This is the reason for the irregular appearance of the second half of the
data in Figure 30. As can be seen in Figure 30 above, beam II-d/2-3 could have sustained
a larger load, but with the large amount of deflection and readings from the strain gages
on the longitudinal reinforcement approaching rupture, the test was terminated. 63
II - d/2 - 3
30,000
25,000
20,000
15,000 Elastic vs Measured Elastic Theory
10,000 Measured CenterReaction (lbs)
5,000
0 0 5,000 10,000 15,000 20,000 25,000 30,000 Theorectical Center Reaction (lbs)
Figure 31. Plot of elastic theory versus measured center reaction for beam II-d/2-3.
Figure 31 shows that beam II-d/2-3 had considerably less redistribution, 3.84 versus 7.16, than its counterpart with no shear reinforcement. This seems a little counterintuitive; beam II-d/2-3 deflected an inch more then beam I-0-3 with both beams containing the same longitudinal reinforcement, but beam II-d/2-3 had almost 50% less redistribution than beam I-0-3. Both beams had similar crack widths at failure, the only difference was the type of cracking that occurred. Beam II-d/2-3 had flexural cracks, while beam I-0-3 had shear compression cracks. The flexural cracks allowed for a greater amount of rotation and a greater amount of cracks having a significant width. Figures 32 and 33 below show the negative and positive moment flexural cracks, respectively, that beam II- d/2-3 was able to withstand. 64
Figure 32. Flexural cracks above center support for beam II-d/2-3.
Figure 33. Flexural cracks underneath loading for beam II-d/2-3.
From the data and the figures, rotation only increased strains and deflection but did not cause a significant amount of redistribution as compared to beams without shear 65
reinforcement. The maximum amount strain for each location for all the beams can be
seen in Appendix A.
Test Results II-d/2-8
This test had several problems to start out with, but it was completed successfully. The
first can be seen in Figure 35, where the load data starts at 65kips. The computer that
collected the data malfunctioned causing the initial data to be lost. The loading was then
started from zero again, but since there was already flexural cracking that took place and
the new data would not reflect the original flexural cracking, it was left out. The second problem that occurred was that the demand on the pump caused the electrical breaker to trip resulting in power being lost to the hydraulic rams. The process was then started for a third time. This time the load was increased in 10kip increments and then held constant in order to mark the progression of the cracks. At 160kips, the crack widths were around
1/16th of an inch, but as the load passed 162kips, the widths widened to 1/8th and 1/4th of an inch. This can be seen by the drop in load in Figure 35. At this point, as can be seen in Figure 35, the beam could not support anymore load and it just continued to deflect.
Significant spalling occurred beneath the east point load which can be seen in Figure 34.
Table 6
Results of Testing for II-d/2-8 II-d/2-8
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu test ∆East ∆West % kips kips kips kips k-ft k-ft k-ft in in Redistribution 68.07 44.04 16.46 27.58 129.2 153.9 199.8 0.97 0.38 2.02
66
Figure 34. Spalling on southeast side underneath load for beam II-d/2-8.
Examining Figure 35 below and Appendix A, it can be seen that the longitudinal reinforcement on the east side yielded while the west side did not. Figure 35 below shows a considerable amount of deflection under the east point load due to the yielding of the steel. When the beam failed, the deflection under the east point load increased about a tenth of an inch while the deflection under the west point load actually rebounded. This
increased deflection accounts for the spalling of the concrete that is seen in Figure 34
above. Even though the longitudinal reinforcement yielded on the east side of the beam,
the crack width was significantly less than that of beam II-d/2-3. The load was
considerably larger and the strains were smaller in this beam versus beam II-d/2-3 which
also helps account for the spalling and the crack widths. The lower strains helped restrain
crack propagation but this increased forces in the concrete compression zone. Table 6 67 shows that this beam exceeded the calculated moment and shear strength even though there was very little redistribution.
II - d/2 - 8
180,000
160,000
140,000
120,000
100,000
80,000 Total Load (lbs.) Load Total
60,000
40,000 East Deflection West Deflection
20,000
0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Deflection (in.) Figure 35. Plot of total load versus deflection for beam II-d/2-8.
68
II - d/2 - 8
80,000
70,000
60,000
50,000
40,000 Elastic vs Measured Elastic Theory 30,000 Measured CenterReaction (lbs) 20,000
10,000
0 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 Theoretical Center Reaction (lbs) Figure 36. Plot of elastic theory versus measured center reaction for beam II-d/2-8.
As can be seen in Figure 36 above, very little redistribution took place in beam II-d/2-8.
Beam II-d/2-8, while it deflected almost an inch, had a sudden loss of load carrying capacity. This was due to the width of the shear crack, which can be seen in Figure 37 below, and spalling of concrete. 69
Figure 37. Shear crack in beam II-d/2-8.
This beam surpassed the design strength of 47.2kips at the center support with a shear
reaction of 68kips. At design strength, the percent redistribution was 0.6% and increased
to 2.02% as the load on the beam reached the maximum. The reason for this may be
from the initial balancing of the beam on the load cells. For this beam, which was one of
the first tested, the end reactions were 350lbs more than what elastic reactions predicted.
However, after recording the total load of the beam the load cells were zeroed and the
beam behaved in an elastic manor with the actual reactions deviating from elastic by less
than 1%. This initial variation did not reduce the overall strength of the beam because the beam surpassed the nominal strength by almost 50 k-ft. 70
The predicted strain, seen in Appendix C, was 0.011, is well above the required 0.0075 in
ACI 318-05 Section 8.4.3 that allows moment redistribution. Using the predicted strain would have given a predicted moment redistribution of 11 percent which is far greater than the two percent seen in beam II-d/2-8. This could be because the negative moment strains recorded during testing merely surpassed initial yield. Another indication of redistribution would come from the strains experienced by the rebar. If redistribution were to occur, then the strains recorded for the negative moment longitudinal reinforcement would not change or reduce while the strains under the point loads would increase with an increase in loading. The south strain gage over the center reaction reached a maximum of 1908με and then dropped slowly as the strain recorded on the east side under the point load continued to rise. The strain in the center rebar did not drop, it continued to rise until the maximum load was reached. At 161kips, the maximum strain reading was reached for the center negative moment rebar over the center support. As can be seen in Figure 35, the east side of the beam continued to deflect and the strains under the east point load continued to increase. The strains on the east side of the beam for the positive moment reinforcement increased from 2373 and 2281με to 2376 and
2358με respectively, while the deflection increased from 0.55 to 0.97 inches. After the maximum load had been reached, the strains under the west point load never reached initial yield but came within 100με. At the maximum load, the deflection was 0.38 inches and this was the point of maximum strain for the positive moment reinforcement under the west point load. Also at the maximum load, the end reactions varied by 3.5 percent with the east side supporting more load. The total load surpassed 160kips twice, 71
but in between the two peaks, the west side actually had a period where it supported more
load with the maximum difference of 1.9 percent. The deflection under the east point
load at the maximum load was 0.55 inches. The failure method was shear compression,
but the beam had deflection and crack widths over a quarter of an inch, that would
indicate a flexural failure.
The predicted moment using strain compatibility was 153.9 k-ft while the result from the
test was 202.2 k-ft. The results and the calculations for strain compatibility can be seen in Appendix C. Even though the longitudinal steel did yield, the strains did not surpass
2500με which is less than a quarter of the 11000με that was predicted by strain
compatibility. The concrete along the negative moment longitudinal reinforcement
showed excessive cracking starting at a distance d away from the center reaction and
extending to the point load. This can be seen in Figure 38 below. 72
Figure 38. Negative moment strain cracking beam II-d/2-8.
A close up of this cracking at a higher load is shown in Figure 39 below. Figure 39
shows the distress that the concrete underwent due to the strains in the negative moment
longitudinal reinforcement. In Figure 39 below, the point load was in the top left of the photograph indicating that this is not negative moment flexural cracking because it is past the point of inflection from negative to positive moment. This cracking is most likely due to zone of influence of the larger reinforcing bar in the concrete. It also shows that if there was not ample development length past the point load, and not the point of inflection, a bond splitting failure would have occurred causing a premature failure.
73
Figure 39. Close up of negative moment strain cracking of beam II-d/2-8.
Test Results II-d/2-10
Beam II-d/2-10 behaved very closely to II-d/2-8 but the beam could not reach failure due
to the capacity of the center load cell. This beam showed the same crack pattern as II-
d/2-8, but this beam’s flexural cracks extended further towards the supports and the shear
cracks that formed extended the entire shear span. There was also crushing of concrete
near the loads as can be seen in Figure 40 below. This beam also showed a slightly
greater amount of redistribution than II-d/2-8 which can be seen in Table 8 and in Figure
34.
74
Figure 40. Spalling near loading for beam II-d/2-10.
Even though the beam did not fail, Table 7 shows that the measured shear strength was almost 40% more than what equation 4 above predicted. Again, this beam never showed the typical failure modes that the previous beams had shown so this beam did not fail.
There was no retest because the beam had to be moved out of the lab so that three more beams could be tested using the same testing setup. Moving beam II-d/2-10 caused further, if only minor, damage to it. Upon examining the beam after it was moved, it was noticed that the wires for the strain gage had been cut off at the beam, making it next to impossible to reattach wires to the strain gage in the beam without removing concrete from the beam.
75
Table 7
Results of Testing for II-d/2-10 II-d/2-10
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu test ∆East ∆West % kips kips kips kips k-ft k-ft k-ft in in Redistribution 71.27 44.42 17.1 27.32 130.4 239 209.2 0.52 0.61 2.59
II - d/2 - 10
180,000
160,000
140,000
120,000
100,000
East Deflection 80,000 East Deflection
Total Load (lbs) Load Total West Deflection 60,000 West Deflection
40,000
20,000
0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Deflection (in) Figure 41. Plot of total load versus deflection for beam II-d/2-10.
Figure 41 above, shows that the stiffness of the beam changed around 60kips, which was the start of the first shear crack which can be seen in Figure 42 below on the east (left) side of the picture. Figure 43 shows that there was very little deviation from elastic theory with a maximum of 2.59% redistribution occurring when the test was ended. 76
Figure 42. Center reaction of beam II-d/2-10.
77
II - d/2 - 10
80,000
70,000
60,000
50,000
40,000 Elastic vs Measured Elastic vs Measured 30,000 Elastic Theory Measured Center Reaction (lbs) 20,000
10,000
0 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 Theoretical Center Reaction (lbs) Figure 43. Plot of elastic theory versus measured center reaction for beam II-d/2-10.
Test Results III-M-3
Beam III-M-3 was loaded in five thousand pound increments stopping at each five kip increment to mark the crack progression. After the beam was setup, hairline cracks were noticed around the center reaction and marked after the first loading increment. It was unknown what caused this initial cracking, but it did not seem to affect the overall strength of the beam as can be seen in Table 8. The loads written on the beam correspond to the center load cell versus the total load that was written on the beam for the other beams. This change was due an oversight in the reading data on the computer screen. 78
At ten kips, the beam experienced a number of flexural cracks, however, it was not until
20 kips that the beams experienced major flexural cracking. Table 8 shows the results of
testing for beam III-M-3.
Table 8
Results of Testing for III-M-3 III-M-3
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu ∆East ∆West % kips kips kips kips k-ft k-ft test k-ft in in Redistribution 23.44 26.23 18.3 7.93 72.31 24.13 68.8 2.21 1.43 5.2
The beam continued to support an increasing load with very little deflection until the first
shear crack appeared at 30kips. The deflection of the beam began when the total load
was between 39 and 41kips. However the beam continued to support more load until
horizontal cracking underneath the load at the bottom of the beam began at 43kips. At
this point, the beam was continuously loaded because it deflected considerably while
seeing modest gains in load at the center reaction. This can be seen in plot of load versus deflection in Figure 46 below.
Beam III-M-3 failed in flexure far exceeding the predicted moment capacity. This beam underwent considerable deformation before rupture of both legs of multiple stirrups caused a catastrophic failure. This can be seen in Figure 44 and 45 below.
79
Figure 44. Failure of beam III-M-3.
Figure 45. Ruptured stirrup in beam III-M-3. 80
III-M-3
60,000
50,000
40,000
30,000 Total Load (lbs.) Load Total
20,000
East Deflection West Deflection 10,000
0 0.00 0.50 1.00 1.50 2.00 2.50 Deflection (in.) Figure 46. Plot of total load versus deflection for beam III-M-3.
Figure 46 above shows the total load reached by III-M-3 was very close to that of II-d/2-3
which was provided with three times the amount of shear reinforcement. However, as
noted above, beam II-d/2-3 did not fail in a dramatic fashion which was characteristic of
beam III-M-3.
Figure 47 below shows that redistribution started around 18kips, but only the strain gage
readings on the east side of the beam had surpassed 2000με. As can be seen in figure 47, when the center reaction had reached 19.5kips all the strain gages surpassed 2000με.
After the rupture of the stirrups, the measured center reaction and the center reaction 81
predicted by elastic theory differed by 22.96%. However the total load supported by the
beam dropped from 51kips to 17.5kips after the rupture. Because of the drop in total load, the reading taken right before the collapse was used as the percent redistribution.
III-M-3
25,000
20,000
15,000
10,000 Elastic vs. Measured Elastic Theory Measured Center Reaction (lbs.)
5,000
0 0 5,000 10,000 15,000 20,000 25,000 30,000 Theoretical Center Reaction (lbs.) Figure 47. Plot of elastic theory versus measured center reaction for beam III-M-3.
As can be seen in Figure 46 above, the minimum area for stirrups placed at d/2 allowed
by ACI-318-05 and AASHTO delayed the onset of a brittle failure to allow both sides to
deflect over 1.4 inches. However, the stirrups ruptured while the beam continued to carry
greater amounts of loads but capacity of the beam never reached the nominal shear
strength. 82
Test Results III-F-8
Beam III-F-8 was loaded in 20kip increments. Loading was stopped to mark the crack
propagation. At 20kips, flexural cracking had started on top of the beam over the center
support, but did not continue down the face of the beam. These flexural cracks did not
extend down the face of the beam until the load was between 60 and 80kips. At 60 kips,
a minor shear crack was found. This shear crack extended and widened with the next
increment of loading. Multiple shear cracks were found after the beam was loaded to
100kips. Major flexural cracking occurred around the 140kip interval but the beam
continued to support more load. The maximum load the beam could support was
167kips. The test set-up for the last two beams was changed. The load cell that was
originally at the center reaction was moved under the load point on the east side. The
reading recorded by this load cell was then multiplied by two to obtain the total load that
the beam was able to support. The test was terminated because the beam could not
support more load, as can be seen in Figure 48, and concrete began to spall under the east
load point and at the center reaction. Table 9 shows the results of the test. Figure 49
shows that there was very little redistribution of forces.
Table 9
Results of Testing for III-F-8
III-F-8
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu test ∆East ∆West % kips kips kips kips k-ft k-ft k-ft in in Redistribution 70.5 61 18.1 42.9 179.0 153.9 206.9 1.45 0.65 3.13
83
III - F - 8
180,000
160,000
140,000
120,000
100,000
East Deflection 80,000 West Deflection Total Load (lbs) Load Total
60,000
40,000
20,000
0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 Deflection (in) Figure 48. Plot of total load versus deflection for beam III-F-8.
84
III - F - 8
80,000
70,000
60,000
50,000
40,000 Elastic vs. Measured Elastic Theory 30,000 Measured Center Reaction Center Measured (lbs) 20,000
10,000
0 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 Theoretical Center Reaction (lbs) Figure 49. Plot of elastic theory versus measured center reaction for beam III-F-8.
Test Results III-F-10
Beam III-F-10 was loaded in 20kip increments so that crack propagation could be
marked. The beam showed normal cracking through 80kips with little deflection.
Flexural cracking was noticed closer to the exterior supports than in the other beams. At
100 kips, a shear crack was found. The beam continued to support a greater load until
180kips with less than a half inch of deflection. At 180kips, only one strain gage located above the center support, read above 2000με. The crack pattern can be seen in Figure 50 below.
85
Figure 50. Cracking pattern after maximum loading for beam III-F-10.
Beam III-F-10 and beam II-d/2-10 both showed very little evidence of redistribution as can be seen in Table 11 below. Figure 52 shows that this beam followed the elastic theory until the test was terminated. Testing of Beam III-F-10 was terminated because the limits of the hydraulic system had been reached. The hydraulic lines were rated at
10,000psi and at the max load of 215 kips, the pressure gage read 9800psi. At this point, hydraulic fluid was seen leaking out of a valve used to control the rate of flow before the output connected to the splitter. The valve was not leaking from the threaded areas, but from within the valve itself. Because of this, it was considered unsafe to proceed to a higher load. The results of the test can be seen in Table 10.
86
Table 10
Results of Testing for III-F-10 III-F-10
Vmeasured Vn Vc Vs Mv calc'd Mn calc'd Mu test ∆East ∆West % kips kips kips kips k-ft k-ft k-ft in in Redistribution 90.56 82.24 18.48 63.76 241.4 239 268.8 0.63 0.56 1.94
III - F - 10
250,000
200,000
150,000
East Deflection Total Load (lbs) 100,000 West Deflection
50,000
0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 Deflection (in) Figure 51. Plot of total load versus deflection for beam III-F-10.
87
III - F - 10
100,000
90,000
80,000
70,000
60,000
50,000
40,000 Elastic vs Measured Elastic Theory
Measured Center Reaction (lbs) 30,000
20,000
10,000
0 0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 Theoretical Center Reaction (lbs) Figure 52. Plot of elastic theory versus measured center reaction for beam III-F-10.
Summary of Results
Table 11 lists all the results for ease of comparison. Table 11 contains all the results from
Tables 2-10 which have been discussed earlier. Column Eqn (11-5), equation 6 below,
(ACI 318-05, 2005), is the shear strength of a beam without shear reinforcement.
Equation 6 was utilized to compare the theoretical concrete shear strength to the data gathered from two-span continuous beam. Eqn. 6 is the more complex nominal equation to determine the concrete shear strength per ACI.
⎛ dV ⎞ ⎜ u ⎟ Equation 6 c =⎜ fV c + 2500'9.1 ρ w ⎟ w db ⎝ M u ⎠
88
In this equation Vud/Mu was calculated using the measured reactions over the center
support to determine Vu and Mu. Equation 6 is used to calculate the force needed to cause
diagonal tension cracking in beams without shear reinforcement. The design strength
percent redistribution column was the maximum amount of redistribution when the
design strength of the beam was reached. It was considered the best choice to find what
the percent redistribution was simply because that was what the beam was designed for.
For extreme loads that caused the beam to collapse, the maximum amount of
redistribution, after failure, is included in the final column in Table 11. These were the
maximum redistribution amounts once the beam surpassed the design strength. Beam I-
0-8 and III-M-3 have two values in the second column because the two values represent
before and after collapse of the beam. A dramatic failure was considered for all beams
without shear reinforcement and for the beam with deformed welded wire reinforcement
as stirrups. Beams I-0-3 and I-0-10 while having dramatic failures, did not have a
substantial increase in redistribution as shown in beams I-0-8 and III-M-3. All other
beams with shear reinforcement showed adequate amounts of deflection and crack widths
along with no dramatic increase in deflection with the corresponding drop in load. It should be noted that for the calculated moment, the Mn calc’d column in Table 11, a
value of 5000psi was used for calculations because this was the design strength.
Table 11
Summary of Results
Eqn Mv Mn Vmeasured Vn Vc Vs Mu test ∆East ∆West Design Strength Max % (11-5) calc'd calc'd Beam ID kips kips kips kips k-ft in in % Redistribution Redistribution kips k-ft k-ft
I-0-3 17.99 18.0618.06 0 17.72 53.0024.13 55.1 0.52 0.59 2.72 7.16 I-0-8 25.1 18.5 18.5 0 19.91 54.30 153.9 78 0.55 0.28 N/A 6.62 - 17.3 I-0-10 17.995 17.8 17.8 0 20.81 52.24 239 53.34 0.08 0.13 N/A 4.75 II-d/2-3 23.85 45.6617.75 27.91 17.43 134.0 24.13 71.8 1.7 1.37 5.3 3.14 II-d/2-8 68.07 44.0416.46 27.58 18.07 129.2 153.9 202.2 0.97 0.38 0.6 2.02 II-d/2-10 71.3 44.42 17.1 27.32 20.84 130.4 239 213.5 0.52 0.61 2.26 2.34 III-M-3 23.44 26.2318.3 7.93 17.94 72.31 24.13 71.3 2.21 1.43 2.19 5.2 - 22.96 III-F-8 70.5 61 18.1 42.9 19.63 179.0 153.9 212.4 1.45 0.65 1.96 3.13 III-F-10 90.56 82.24 18.48 63.76 21.27 241.4 239 268.8 0.63 0.56 1.28 1.47
CHAPTER 5: DISCUSSIONS
No. 3 Longitudinal Reinforcement Comparison
The amount of shear steel did not affect the amount of strain recorded at the center
support of the beam. All beams were able to surpass the 3000με limit of the gages.
The concrete contribution to shear strength allowed for the strains in beam I-0-3 to surpass yield but did not have enough reserve shear capacity to allow a redistribution of forces and cause the positive moment longitudinal reinforcement to yield in both spans.
Although observing Appendix A reveals that the strain recorded on the east span surpassed the strain gage limit and the west span fell short of yielding by 30με.
Table 2 shows that the maximum moment supported by the beam II-d/2-3 was only 0.5 k- ft less than III-M-3. The difference in the amount of shear steel was 71.6%, while the difference in nominal shear capacity was 41.1%. This assumed a 75kip tensile strength for the deformed welded wire fabric as allowed by ACI 318-05 Section 11.5.2. The increase in nominal shear strength only equated to a 0.7% increase in flexural strength and a 1.7% increase in nominal shear strength. This indicates that increasing the amount of shear steel while keeping the spacing constant does not appreciably change the capacity of the beam.
A minimum amount of shear steel allowed for complete redistribution of forces causing all strain gages to surpass the limit of 3000με. Minimum shear steel supplied enough reserve shear strength to allow the negative moment longitudinal reinforcement to yield and the allowed an increase in load which caused the positive moment longitudinal 91 reinforcement to surpass 3000με in both spans. Minimum shear steel also supplied enough reserve shear strength to allow 20.5% more total load past the yielding of all the strain gages.
All of the beams with No.3’s longitudinal reinforcement did not meet minimum requirements of equation 7 (Eqn. 10-3, ACI 318-05) if negative moment longitudinal reinforcement is not included.
'3 cf As min = bd Equation 7 f y
The commentary of Section 10.5 ACI 318-05 states “a minimum amount of tensile reinforcement required by 10.5.1 in both positive and negative moment regions.” The design amount of longitudinal reinforcement using a concrete compressive strength of
5000psi is 0.42in2 but if the average concrete compressive strength of the beams is used the minimum amount increases to 0.45 in2. Having both compression and tension steel in this lightly reinforced beam caused the neutral axis to be above the compression steel causing it to be in tension. The strain calculated from strain compatibility, Appendix C, was -970με. If all the longitudinal reinforcement is counted towards minimum longitudinal steel, then the beam meets the requirement of ACI 318-05 Section 10.5.1 with 0.55in2.
This set of beams had a minimum of a half inch deflection and crack widths of at least a half inch before failure. The positive and negative moment reinforcement yielded without rupturing. As can be seen in Figure 53 below, there was not any sign of necking over the center support. 92
Figure 53. Negative moment rebar after testing for beam III-M-3.
The results show that for lightly reinforced beams with compression and tensile
longitudinal reinforcement running the entire length, the beams were able to allow large
cracks to form and undergo significant deflection without sudden failures as indicated
would be the case by ACI 318-05 Section 10.5.1.
No. 8 Longitudinal Reinforcement Comparison
Beam I-0-8 displayed that ACI 318-05 equation 11-5 underestimated the shear strength of the beam at the center support. It also showed that shear steel was needed to keep a sudden failure from happening. 93
Beam II-d/2-8 showed that the maximum spacing allowed by ACI 318-05 Section
11.5.5.1 was sufficient to allow yielding of both positive and negative longitudinal
reinforcement before failure. The beam allowed sufficient shear capacity for a
redistribution of forces while the negative moment longitudinal reinforcement was
yielding to allow the positive moment longitudinal reinforcement to yield.
Comparing beams II-d/2-8 and III-F-8 reveals that a 27.9% increase in the nominal shear
capacity only increased moment carrying capacity by 4.72%. The shear capacity of beam
II-d/2-8 was only 3.4% less than beam III-F-8. Further comparison shows that beam II-
d/2-8 surpassed the nominal shear capacity at the center reaction by 35.3% while beam
III-F-8 only surpassed it by 15.6%. Both beams had shear cracks develop that were not a
result of flexural cracks. Beams II-d/2-8 and III-F-8 both developed hinges on the east
span of the beam. Beam II-d/2-8 at failure showed spalling under the load while beam
III-F-8 showed no signs of spalling while undergoing a larger amount of deformation.
This indicates that a greater amount of shear steel reduces the stress in the compressive
strut of the beam. Both beams had similar crack patterns that can be seen in Figure 54
and 55. Figure 54 is beam II-d/2-8 at a load before failure while Figure 55 is beam III-F-
8 after failure.
94
Figure 54. Failure pattern for beam II-d/2-8.
Figure 55. Failure pattern for beam III-F-8.
95
Figure 55 shows that the shear crack extended to the neutral axis of the beam and caused
a split below the negative moment longitudinal reinforcement which also can be seen in
Figure 56 below.
Figure 56. Shear crack splitting the concrete below the reinforcement for beam III-F-8.
Figure 56 shows that the strains in the negative moment longitudinal reinforcement
caused the 45 degree cracks that cross and extend a couple inches on either side of the
longitudinal reinforcement. These 45 degree cracks were not the result of flexure or
shear in the beam. The 45 degree cracks were able to cause a local bond failure near the
point load increasing the strains in both the positive and negative moment longitudinal 96 reinforcement causing a hinge to form under the point load. This is the manner in which both II-d/2-8 and III-F-8 failed.
No. 10 Longitudinal Reinforcement Comparison
Beam I-0-10 did not meet ACI 318-05 equation 11-5 at the center support. The strains recorded for this beam were much lower than those seen in beam I-0-8 indicating that the cross section of the beam was not sufficient to cause an appreciable strain in the longitudinal reinforcement. The maximum strains of both beams can be seen in
Appendix A.
Beams II-d/2-10 and III-F-10 both surpassed their respective nominal shear capacity at the center reaction.
The set up of the reaction frames for beam II-d/2-10 did not allow the beam to be tested to failure. This led to a change in the reaction frames in order to test beams III-F-8 and
III-F-10 so that these beams could be tested to failure. Under the first setup, beam II-d/2-
10 barely surpassed yield at the center support as can be seen in Appendix J. The only conclusion that can be drawn from beam II-d/2-10 is that it behaved in a similar manner to II-d/2-8. It developed tension strain cracks along the negative moment longitudinal reinforcement as discussed earlier. However, the load was not sufficient to extend a shear crack to the strain tension cracks.
97
Testing beam III-F-10 with the new setup allowed for the redistribution of forces to
begin, but the limit of the hydraulic system was found before the capacity of the beam.
Again, the cracking pattern observed was similar to beams II-d/2-8, III-F-8, and II-d/2-
10.
Beams without Shear Reinforcement
Table 12 shows the results of all the beams without shear reinforcement. To calculate the
flexural strength, equation 2 was used to predict the shear strength of the concrete. This
value was then used as the shear reaction of a propped cantilever beam and the moment
that this reaction created was the Mv calculated (see equations from Figure 16). For the
three beams that were tested without shear reinforcement, this proved to be an accurate way to find the nominal strength of a beam. However, the effects of a/d, shear span to depth ratio, and concurrent maxima of shear and moment when compared with the computed flexural strength of a beam can be seen in Table 12 below. The Vmeasured and
Vn columns represent the measured and predicted shear strength of the beam,
respectively. The Mn column is the moment capacity of the beam calculated using strain equilibrium. The Mu column is moment capacity of the beam calculated from the results
of the test. The Mu/ Mn column is the ratio of tested moment capacity versus the
calculated moment capacity. This column represents how an increase in the amount of
longitudinal reinforcement affects the relative strength of the beam. It is plain to see that
as the amount of longitudinal steel increases, the relative strength of the beam decreases dramatically.
98
Table 12
Results of Beams without Shear Reinforcement
Mn Mu test Beam Vmeasured Vn Mu test ρ calc'd ------ID kips kips k-ft k-ft Mn calc'd %
I-0-3 17.99 18.06 24.13 55.1 2.28 0.28 I-0-8 25.1 18.5 153.9 78 0.507 2.06 I-0-10 17.995 17.8 239 53.34 0.223 3.34
Beams with Shear Reinforcement
All the beams that had shear reinforcement with spacing of d/2 or less had sufficient reserve capacity to allow for a redistribution of forces and fail flexurally. The concurrent maxima of shear and moment did not affect the capacities of any of the beams. The addition of shear reinforcement increased the shear capacity of the beams by an amount greater than what was calculated using equation 7. This can be seen in Table 11 above.
However, comparing beams II-d/2-8 and III-F-8, decreasing the shear reinforcement spacing only increased the shear capacity by 3.4%. This can be explained by the angle of the shear cracks and the number of stirrup legs crossed by the cracks. d/2 spacing was sufficient to control shear cracking and allow for a flexural failure.
The design strength of the beams, determined by strain equilibrium, underestimated the capacity of the beams. Beams II-d/2-10 and III-F-10 did not reach failure, but surpassed design strength. 99
Continuity
For all the beams with shear reinforcement tested, at design strength, five of the six
beams had less than 2.3% redistribution. Beam II-d/2-3 had early redistribution but after
the design strength, the redistribution actually dropped.
Out of the nine beams that were tested, seven surpassed yield of the longitudinal
reinforcement. Five of the seven that yielded, reached a failure mode. The two that did
not reach a failure mode were due to the limits of the testing setup. The maximum load
supported by all of the beams was compared to the load at which the strain in the negative moment longitudinal reinforcement over the center support reached initial yield
to examine a continuous beams ability to support load past yielding of longitudinal
reinforcement. This was considered important since concrete crushing after yielding is
considered failure for simply supported beams and this is a discussion on the effects of
continuity. By definition, yielding is the elongation of the steel without any noticeable
increase in force. Table 13 shows the results of the five beams that surpassed yield and
continued to support load until a failure occurred.
100
Table 13
Capacity of Beams that Supported Load past Initial Yield Load Total at % Capacity Beam ID Load Yield Past Yield kips kips I-0-3 44.1 40.9 7.26 II-d/2-3 57.5 31.9 44.5 II-d/2-8 161.8 136.7 15.5 II-d/2-10* 170.8 170.4 N/A III-M-3 57.1 44.4 22.2 III-F-8 170.0 130.5 23.2 III-F-10* 215.0 188.7 N/A * Did not Fail
As can be seen in Table 13, the smallest percentage was 7.26%, which happened to fail in
shear. The remaining beams in Table 13 failed in a flexural manner with an ability to
support 14.5 to 44.5% more load after yield. From Table 11 earlier, no beam had
redistribution greater than 7.16% before it failed. Again, the 7.16% was beam I-0-3
which failed in shear.
Redistribution of forces occurs internally but the results of which are seen at the supports.
It was further considered necessary to check if the beams were able to support additional load beyond the point at which all strain gage surpassed 2000με, yield of the reinforcement. Three beams had all gages surpass yield while the other two had gages in one span come within 100με of yielding. Those two beams were I-0-3, which failed in shear, and II-d/2-8 which failed by forming a hinge under the point load.
To calculate the capacity of the three beams that had all the gages surpass 2000με, the load which caused yielding was found and that was compared to the load which caused 101 failure to determine the beams surplus capacity. Table 14 below lists the results. The total load column lists the maximum load that the beams were able to support. The load at hinge formation column represents the total load on the beam when a plastic hinge formed on one span. This load was determined when the positive moment longitudinal reinforcement surpassed yield. The load at SG yield column is the load that caused all gages to reach 2000με. Here again, even though these values decreased from Table 13, they are much greater than the percent redistribution seen by the reactions, except for beam I-0-3 which failed in shear.
Table 14
Capacity of Beams Past Yielding or Hinge Formation Load at % Total Load at Beam Hinge Capacity Load SG Yield ID Formation Past kips kips kips Yield I-0-3 44.1 41 N/A 7.03 II-d/2-3 57.5 N/A 43.4 24.52 II-d/2-8 161.8 136.7 N/A 15.51 III-M-3 57.1 N/A 45.4 20.49 III-F-8 170.0 N/A 158.6 6.71
The interesting part of Table 14 is how much load the lightly reinforced beams were able to carry even after all the strain gages have yielded. One explanation is strain hardening, but as seen in Figure 53 above, the longitudinal reinforcement did not neck or look close to rupturing. As the longitudinal reinforcement reached strain hardening, yield propagated along the length of the reinforcement causing the cracking along the beam at the depth of the longitudinal reinforcement as seen in Figure 57 below. 102
Figure 57. Strain cracking along the longitudinal reinforcement.
Table 14 shows that a continuous beam has the capacity to support load well beyond what can be explained by redistribution alone.
103
CHAPTER 6: SUMMARY AND CONCLUSIONS
The research performed was used to examine the characteristic of two-span continuous beams. A series of tests were carried out to examine: 1) the concrete contribution to shear strength, 2) whether a minimum amount of shear reinforcement would adequately supply a nominal shear strength, 3) is the maximum allowable spacing for shear reinforcement, d/2, prevent a shear failure, 4) what are the flexural characteristics with minimum longitudinal reinforcement, 5) do heavily reinforced beams, longitudinally and in shear, behave flexurally?
Conclusions
The following observations were made as a result of the testing.
For lightly reinforced concrete beams, the concrete contribution to shear strength
was sufficient to cause yielding over the center support, but did not ensure a
flexural failure.
For beams without shear reinforcement, as the longitudinal reinforcement ratio
increases, the ratio between tested moment and the calculated moment decreases
significantly.
For beams without shear reinforcement, equation 11-3 of ACI 318-05 if used to
predict the shear strength at the critical section can be used to adequately predict
the moment capacity.
Using deformed welded wire fabric as stirrups provided adequate shear strength
to allow a redistribution of forces sufficient to cause all the strain gages to
surpass 3000με but ultimately four stirrup legs ruptured. 104
Crack widths for deformed welded wire fabric spaced at d/2 where smaller in
width than using No.3’s at d/2.
For all beams stirrup spacing of d/2 was sufficient to allow for negative moment
longitudinal reinforcement to yield.
Minimum longitudinal reinforcement with shear reinforcement was able to
sustain a minimum of 1.37 inches of deflection and quarter in crack width. Beam
II-d/2-3 was able to allow half inch crack widths.
Comparing beams III-M-3 with II-d/2-3 and II-d/2-8 with III-F-8 shows that
increasing the amount of shear reinforcement has a diminishing return for
flexural and shear capacity.
Further Research
As can be seen in Appendices D-J, the design strength computed for the beams
underestimates the tested strength of the beam. For Appendices D-J, the design load was
found using strain compatibility and a concrete compressive strength of 5000psi. As seen
in Table 2, only one beam is below 5000psi and it is the closest to the design concrete
compressive strength. The only time that strain compatibility approached the strength of the beams at yield of the longitudinal reinforcement over the center reaction is for the
No.10’s, and strain compatibility predicted a 7363με at failure of 191.2 kips of total load, prior to phi factors. For all beams with No.3’s as longitudinal reinforcement, strain compatibility greatly underestimated the strength of the beam even when strain compatibility was adjusted for a higher concrete compressive strength. Further research 105 needs to be done to determine if strain compatibility is overly conservative for lightly reinforced beams.
Throughout the literature review, the pattern of cracking observed in the test specimens was not mentioned. It was hypothesized that strain hardening of the longitudinal reinforcement over the center reaction caused yielding of the reinforcement to propagate along the length of the reinforcement until it reached the point load. If this is what happened, was the cracking that occurred a result of yielding?
More samples should be tested to have a statistically significant number to draw accurate conclusions.
106
REFERENCES
1. ACI 318-05, 2005, “Building Code Requirements for Structural Concrete and Commentary,” American Concrete Institute, Farmington Hills, Michigan.
2. ACI-ASCE Committee 326 (now 426), “Shear and Diagonal Tension,” ACI Journal, Proceedings V. 59, No.1, Jan. 1962, pp. 1-30; No.2 , Feb. 1962, pp. 277- 334; and No. 3, Mar. 1962, pp.352-396.
3. ACI-ASCE Committee 426, “Shear Strength of Reinforced Concrete Members (ACI 426R-74) (Reapproved 1980),” Proceedings, ASCE, V.99, No. ST6, June 1973, pp. 1148-1157.
4. ACI-ASCE Committee 445, “Recent Approaches to Shear Design of Structural Concrete,” Journal of Structural Engineering, Proceedings V. 124, No.12, Sept. 1998, pp. 1375-1414.
5. Anderson, N. S., and Ramirez, J. A., “Detailing of Stirrup Reinforcement,” ACI Structural Journal, V. 86, No. 5, Sept.-Oct. 1989, pp. 507-515.
6. Angelakos, D.; Bentz, E. C.; and Collins, M. D., “Effect of Concrete Strength and Minimum Stirrups on Shear Strength of Large Members,” ACI Structural Journal, V. 98, No. 3, May-June 2001, pp. 290-300.
7. Bazant, Z.P., and Kim, J.K., “Size Effect in Shear Failure of Longitudinally Reinforced Beams,” ACI Structural Journal, V. 81, No.5, Sept.-Oct. 1984, pp. 456-468.
8. Bresler, B., Scordelis, A. C., “Shear Strength of Reinforced Concrete Beams,” ACI Structural Journal, Proceedings V. 60, No.1, Jan. 1963, pp.51-74
9. Brown, M.D., Sankovich, C.L., Barak, O., Jirsa, J.O., Breen, J.E., and Wood, S.L., 2006, Design for Shear in Reinforced Concrete Using Strut-and-Tie Models, Doctoral Dissertation, The University of Texas at Austin, 1-310 pp.
10. Bryant, R.H., Bianchini, A.C., Rodriguez, J.J., and Kesler, C.E., 1962, “Shear Strength of Two-Span Continuous Reinforced Concrete Beams with Multiple Point Loading,” ACI Journal, Proceedings V. 59, No.6, September 9, pp. 1143- 1175.
11. Cohn, M.Z., 1959, “Rotational Compatibility in the Limit Design of Reinforced Concrete Continuous Beams,” International Symposium of Flexural Mechanics of 107
Reinforce Concrete. New York, American Society of Civil Engineers, 1965, pp. 359-382.
12. Collins, M.P., “A Rational Approach to Shear-Design-The 1984 Canadian Code Provisions,” ACI Journal, American Concrete Institute, V. 83, No. 80, Nov.-Dec 1985, pp. 925-933.
13. Ernst, G.C., 1957, “Moment and Shear Redistribution in Two-Span Continuous Reinforced Concrete Beams,” ACI Journal, Proceedings V. 55, No.5, October 16, pp. 573-588.
14. Griezic, A., Cook, W. D., and Mitchell, D., “Tests to Determine Performance of Deformed Welded-Wire Fabric Stirrups,” ACI Structural Journal, V. 91, No. 2, Mar.-Apr. 1994, pp. 211-220.
15. Johnson, M.K., and Ramirez, J.A., “Minimum Amount of Shear Reinforcement in High Strength Concrete Members,” ACI Structural Journal, V. 86, No. 4, July- Aug. 1989, pp. 376-382.
16. Kani, G. N. J., “Basic Facts Concerning Shear Failure,” ACI JOURNAL, Proceedings V. 63, No. 6, June 1966, pp. 675-692.
17. Kani, G. N. J., “How Safe Are Our Large Reinforced Concrete Beams,” ACI JOURNAL, Proceedings V. 64, No. 3, Mar. 1967, pp. 128-141.
18. Kotsovos, M.D., “Behavior of Beams with Shear Span-to-Depth Ratios Greater Than 2.5,” ACI Journal, American Concrete Institute, V. 83, No. 93, Sep.-Oct 1987, pp. 1026-1034.
19. Marti, P., “Basic Tools of Reinforced Concrete Beam Design,” ACI JOURNAL, Proceedings V. 82, No. 1, Jan.-Feb. 1985, pp. 46-56.
20. Mattock, A. H.; Johal, L.; and Chow, H. C., “Shear Transfer in Reinforced Concrete with Moment or Tension Acting Across the Shear Plane,” Journal of the Prestressed Concrete Institute, V. 20, No. 4, July-Aug. 1975, pp. 76-93.
21. Mphonde, A.G., “Use of Stirrup Effectiveness in Shear Design of Concrete Beams,” ACI Structural Journal, V. 86, No. 5, Sep.-Aug. 1989, pp.541-545.
22. Rodriguez, J.J., Bianchini, A.C., Viest, I.M., and Kesler, C.E., 1959, “Shear Strength of Two-Span Continuous Reinforced Concrete Beams,” ACI Journal, Proceedings V. 55, No.10, February 26, pp. 1089-1130.
108
23. Roller, J. J., and Russell, H. G., “Shear Strength of High-Strength Concrete Beams with Web Reinforcement,” ACI Structural Journal, V. 87, No. 2, Mar.- Apr. 1990, pp. 191-198.
24. Russo, G., and Puleri, G., “Stirrup Effectiveness in Reinforced Concrete Beams under Flexure and Shear,” ACI Journal, Proceedings V. 94, No.3, May-June 1997, pp. 227-238.
25. Schlaich, J.; Schafer, K.; and Jennewein, M., “Toward a Consistent Design of Structural Concrete,” Journal of the Prestressed Concrete Institute, V. 32, No. 3, May-June 1987, pp. 74-150.
26 Steel Construction Manual. American Institute of Steel Construction Inc., Thirteenth Ed. 2005
27. Victor, D.J., and Ferguson, P.M., 1968, “Beams Under Distributed Load Creating Moment, Shear, and Torsion,” ACI Journal, Proceedings V. 65, No.23, April, pp. 295-308.
109
APPENDIX: A
East Side Center of Beam West Side Stain Stain Stain Stain Stain Stain Stain Beam Gage Gage Gage Gage SE Gage NE Gage SW Gage SW South Center North I-0-3 3000 3000 2144 3000 3000 N/A 1970 I-0-8 1031 911 418 668 N/A 1010 1050 I-0-10 448 388 461 261 499 437 394 II-d/2-3 3000 3000 3000 3000 3000 3000 3000 II-d/2-8 2376 2358 1908 2314 N/A 1976 1907 II-d/2-10 1605 1437 545** 1953 1944 1516 1365 III-M-3 3000 3000 3000 3000 3000 3000 3000 III-F-8 3000 3000 3000 3000 3000 2384 2406 III-F-10 1602 1776 1896 2690 2261 1887 1629
110
APPENDIX: B
Initial Yield - 2000με East West Total Center Rxn Deflection Deflection Load (lbs) (lbs) (lbs) (lbs) I-0-3 40,900 16,800 0.26 0.29 I-0-8 N/A N/A N/A N/A I-0-10 N/A N/A N/A N/A II-d/2-3 31,900 13,200 0.13 0.15 II-d/2-8 136,700 57,750 0.24 0.24 II-d/2-10 170,350 71,000 0.49 0.54 III-M-3 44,400 18,550 0.41 0.29 III-F-8 130,500 55,500 0.4 0.36 III-F-10 188,700 80,350 0.44 0.46
111
APPENDIX: C
Predicted Strains for No. 3’s – Flexural Failure:
d’ εcs= -0.00097 d' = 1.31 in dt = 14.69 in
dt
εs = 0.0416
β1= 0.8 εs1= -0.0009737 =ε's
f'c= 5 ksi εs2= 0.04156 =εs c= 0.989 in
dt= 14.69 in fs= 60.00 ksi if εs<εy then fs = Es*εs
εcu= 0.003 if εs>εy then fs = fy b= 8 in
a= 0.7912 in f's = ε'(Es) - β1(f'c)
Cc= 26.9008 kips f's = -32.2376 ksi if ε's<εy then fs' = Es*ε's
d'= 1.31 in if ε's>εy then fs' = fy
As'= 0.22 in2 Fs' = -7.09227 kips F's = f's*A's As= 0.33 in2 Fs = 19.8 kips Fs = fs*As Es= 29,000 ksi Fy= 60 ksi Mn= 289.58 k-in 24.13 k-ft Cc+Cs-T= 0.01
112
APPENDIX: C CONTINUED
Predicted Strains for No. 8’s – Flexural Failure:
d’ εcs= 0.00142 d' = 1.625 in dt = 14.375 in
dt
εs = 0.0110
β1= 0.8 εs1= 0.001417 =ε's
f'c= 5 ksi εs2= 0.011002 =εs c= 3.08 in
dt= 14.375 in
εcu= 0.003 fs= 60.00 ksi if εs<εy then fs = Es*εs
b= 8 in if εs>εy then fs = fy a= 2.464 in
Cc= 83.776 kips f's = ε'(Es) - β1(f'c)
d'= 1.625 in fs' = 37.10 ksi if ε's<εy then fs' = Es*ε's
if ε's>εy then fs' = fy As'= 1.58 in2 As= 2.37 in2 Fs' = 58.62 kips F's = f's*A's Es= 29,000 ksi Fs = 142.2 kips Fs = fs*As Fy= 60 ksi Mn= 1847.22 k-in Cc+Cs-T= 0.19 153.94 k-ft
113
APPENDIX: C CONTINUED
Predicted Strains for No. 10’s – Flexural Failure:
d’ εcs= 0.00172 d' = 1.76 in dt = 14.25 in
dt
εs = 0.00736
β1= 0.8 εs1= 0.00172 =ε's f'c= 5 ksi εs2= 0.00736 =εs c= 4.125 in
dt= 14.25 in
εcu= 0.003 fs= 60.00 ksi if εs<εy then fs = Es*εs
b= 8 in if εs>εy then fs = fy a= 3.3 in
Cc= 112.2 kips f's = ε'(Es) - β1(f'c)
d'= 1.76 in f's = 45.88 ksi if ε's<εy then fs' = Es*ε's
if ε's>εy then fs' = fy As'= 2.54 in2 As= 3.81 in2 F's = 116.53 kip F's = f's*A's Es= 29,000 ksi Fs = 228.6 kip Fs = fs*As Fy= 60 ksi Mn=2868.416 k-in Cc+F's-Fs= 0.14 239.03 k-ft
114
APPENDIX: D
0.70 0.60 East Deflection East West Deflectioin 0.50 0.40 Deflection (in) Deflection I-0-3 0.30 0.20 Design Strength = 17.38kips Strength Design 0.10 Total Load = 40.9kips Yield @ Initial 0.00 0
5,000
50,000 45,000 40,000 35,000 30,000 25,000 20,000 15,000 10,000 Total Load (lb) Load Total
115
APPENDIX: D CONT’D Elastic vs Measured Elastic Theory Design StrengthDesign = 7.4kips I-0-3 Theoretical Center Reaction (lbs) Reaction Center Theoretical Center Reaction @ Initial Yield = 16.8kips
0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 16,000 18,000 20,000 0
8,000 6,000 4,000 2,000
20,000 18,000 16,000 14,000 12,000 10,000 Measured Center Reaction (lbs) Reaction Center Measured 116
APPENDIX: E East DeflectionEast West Deflection Deflection (in) II - d/2 - 3 TotalLoad @ InitialYield =31.9kips
Design Strength = 17.38kips 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 0
70,000 60,000 50,000 40,000 30,000 20,000 10,000 Total Load (lbs) Load Total 117
APPENDIX: E CONT’D 30,000 25,000 Elastic vs Measured Theory Elastic 20,000 15,000 II - d/2 - 3 Design Strength = 7.4kips Theorectical Center Reaction (lbs) 10,000 5,000
Center Reaction @ Initial Yield = 13.2kips 0 0
5,000
30,000 25,000 20,000 15,000 10,000 Measured Center Reaction (lbs) Reaction Center Measured 118
APPENDIX: F East Deflection West Deflection Deflection (in.) Deflection II - d/2 8 II Total @ Load Initial Yield = 136.7kips Design Design Strength = 110.8kips
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0
80,000 60,000 40,000 20,000
180,000 160,000 140,000 120,000 100,000 Total Load (lbs.) Load Total 119
APPENDIX: F CONT’D Elastic vsMeasured Elastic Theory II - d/2 - 8 Theoretical Center Reaction(lbs) Design StrengthDesign = 47.2kips Center Reaction @ Initial Yield = 57.8kips
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 0
80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 Measured Center Reaction (lbs) Reaction Center Measured 120
APPENDIX: G 0.70 0.60 East Deflection East Deflection East West Deflection West Deflection 0.50 0.40 Deflection (in) Deflection II - d/2 10 II 0.30 Design Strength = 172kips Strength Design Total @ Load Initial Yield = 170.4kips 0.20
0.10 0.00 0
80,000 60,000 40,000 20,000
180,000 160,000 140,000 120,000 100,000 Total Load (lbs) Load Total 121
APPENDIX: G CONT’D Elastic vs Measured Elastic vs Measured Elastic Theory Design Strength = 73.26kips Strength Design = 71kips Yield Initial @ Reaction Center II - d/2 - 10 d/2 - II Theoretical Center Reaction (lbs)
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 0
80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 Measured Center Reaction (lbs) Reaction Center Measured 122
APPENDIX: H East Deflection West Deflection III-M-3 Deflection (in.) Deflection Total @ Load Initial Yield = 44.4kips
Design StrengthDesign = 17.38kips 0.00 0.50 1.00 1.50 2.00 2.50 0
60,000 50,000 40,000 30,000 20,000 10,000 Total Load (lbs.) Load Total 123
APPENDIX: H CONT’D Elastic vs. Measured Elastic Elastic Theory III-M-3 DesignStrength = 7.4kips Theoretical CenterReaction (lbs.)
CenterReaction @ Initial Yield= 18.55kips
0 5,000 10,000 15,000 20,000 25,000 30,000 0
5,000
25,000 20,000 15,000 10,000 Measured Center Reaction (lbs.) Reaction Center Measured 124
APPENDIX: I 1.60 1.40 1.20 East Deflection West Deflection 1.00 0.80 Deflection (in)Deflection III - F - 8 0.60 Total @ Load Initial Yield = 130.5kips Design Design Strength = 110.8kips 0.40
0.20 0.00 0
80,000 60,000 40,000 20,000
180,000 160,000 140,000 120,000 100,000 Total Load (lbs) Load Total 125
APPENDIX: I CONT’D Elastic vs. Measured Elastic Theory III -III F - 8 Theoretical Center Reaction (lbs) Center Reaction @ Initial Yield = 55.5kips Design Strength = 47.2kips Strength Design
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 0
80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 Measured Center Reaction (lbs) Reaction Center Measured 126
APPENDIX: J East Deflection Deflection West Deflection (in)Deflection III - F - 10 Design StrengthDesign = 172kips Total @ Load Initial Yield = 188.7kips
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0
50,000
250,000 200,000 150,000 100,000 Total Load (lbs) Load Total 127
APPENDIX: J CONT’D Elastic vsElastic Theoretical Elastic Theory III -III F - 10 Design Strength = 73.26kips Center Reaction @Initial Yield =80.35kips Theoretical CenterRxn (lbs)
0 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 0
90,000 80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000
100,000 Measured Center Rxn (lbs) Rxn Center Measured