Seismic Interaction of Flexural Ductility and Shear ...structurallab.egr.uh.edu/.../files/files/reu-rh-report.pdf · Seismic Interaction of Flexural Ductility and Shear Capacity in

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Seismic Interaction of Flexural Ductility and Shear Capacity in Normal Strength Concrete

Rachel Howser, REU Student A. Laskar, Graduate Mentor

and Y. L. Mo, Faculty Mentor

Final Report Department of Civil and Environmental Engineering

University of Houston Houston, Texas

Sponsored by the National Science Foundation August 2007

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Abstract The seismic performance of reinforced concrete bridge piers is a significant issue because the interaction of flexural ductility and shear capacity of such piers with varied amounts of lateral reinforcement is not well established. Several relationships between flexural ductility and shear capacity have been proposed by various researchers in the past. In this paper a parametric study on reinforced concrete bridge piers is conducted using a nonlinear finite element program, “Simulation of Reinforced Concrete Structures (SRCS),” developed at the University of Houston. SRCS has been previously used to predict the seismic behavior of such piers. The predicted results were compared with the test results obtained from experiments performed at the National Center for Research on Earthquake Engineering, Taiwan. Based on the results of the parametric study performed in this paper a new relationship between flexural ductility and shear capacity of normal strength reinforced concrete structures is proposed. Introduction Shear failure in reinforced concrete bridge columns due to seismic loading has proven to be an issue in both experimental research and post-earthquake investigations. Columns are expected to be the primary elements of energy dissipation in bridges subjected to seismic loads, hence they require high ductility. Failure of bridge columns often results in the collapse of bridge spans, as was the scenario for the majority of the bridge damage during the Chi-Chi Earthquake of Taiwan in 1999. Thus the behavior of columns does play an essential role for earthquake resistant bridge design (Mo 2001). The seismic performance of reinforced concrete bridge columns is of significant importance in places of high seismic activity such as California and Alaska in the U.S., as well as in Chile, Indonesia, Iran, Portugal, New Zealand, Greece, Taiwan, and Japan. Since there is not yet a dependably accurate model that connects flexural ductility and shear strength, reinforced concrete bridge column design is difficult in these regions. A number of models have been proposed to describe the interaction between flexural ductility and shear strength. However, these models have proven to either be gross over- or under-estimates, rendering them unusable in design. When one investigates the relationship between shear strength and flexural ductility, it is obvious that the slope changes over the ductility domain. A major flaw in several of the previous models was assuming that these slope changes occur at a constant ductility; however, this is not the case. Another major error in previous models is the assumption that the shear strength provided by the shear reinforcement increases linearly as a function of the shear reinforcement. This is also untrue. Increasing the stirrup ratio does increase the shear strength of the column; however, this phenomenon does not occur in a linear fashion. This paper presents a number of shear capacity models, including a newly proposed model as part of this work. The newly proposed model is based on the results of an analytical study on reinforced concrete columns performed using a nonlinear finite element program, “Simulation of Reinforced Concrete Structures,” developed at the University of Houston. Previous Shear Strength Models Over the past few decades, many researchers have focused on shear strength models for reinforced concrete. Five existing models were selected for examination in this particular project. They include the ACI 318 model (2005), the California Department of Transportation’s (Caltrans) model (1995), a model proposed by the University of California, San Diego (UCSD) (Priestley et al. 1994), a proposed

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model from University of Southern California (USC) (Xiao et al. 1998), and a model proposed by the University of California, Berkeley (UCB) (Sezen et al. 2004). ACI 318 Model The ACI Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (ACI 318R-02) (2005) provides a conservative model for the nominal shear strength.

scn VVV += (1)

Where: nV = the nominal shear strength

cV = the nominal shear strength provided by the concrete

sV = the nominal shear strength provided by the shear reinforcement For members subjected to shear, flexure, and axial compression, the equation used to find the nominal shear strength provided by the concrete in pounds is:

dbfA

PV wcg

c'

200012 ⎟

⎟⎠

⎞⎜⎜⎝

⎛+= (2)

Where:

cV = nominal shear strength provided by the concrete, lb P = the factored axial load normal to the cross section occurring

simultaneously with the factored shear force at the section, lb gA = the gross area of section, in

'cf = the square root of the specified compressive strength of the concrete, psi

wb = web width, in. d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, in. When shear reinforcement perpendicular to the axis of a member is used, the equation to find the nominal shear strength provided by the shear reinforcement in pounds is:

sdfA

V yvs = (3)

Where:

sV = nominal shear strength provided by the shear reinforcement, lb

vA = the area of shear reinforcement within a the distance of s, in2

yf = the specified yield strength of nonprestressed reinforcement, psi d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, in. s = Spacing of shear reinforcement measured in a direction parallel to

longitudinal reinforcement, in.

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Caltrans Model A similar model was proposed by Caltrans in their 1995 “Memo to Designers Change Letter 02.” The key difference in this model compared to ACI’s model is that Caltrans recognized that ductility plays a major role in the shear capacity of a member. Caltrans’ nominal shear strength model is shown below.

scn VVV += (4)



sV = the nominal shear strength provided by the shear reinforcement

ececc AfAfFFV ''21 4≤= (5)

Where:

cV = nominal shear strength provided by the concrete, lb

1F = Factor 1 2F = Factor 2


eA = 80% of the cross-sectional area, in2

0.367.3150"

3.0 1 ≤−+=≤ μρ yhf

F (6)

Where:

1F = Factor 1 "ρ = the ratio of the transverse reinforcement volume to column core volume yhf = the yield strength of the transverse steel, psi


μ = the displacement ductility

5.110005.00.1 2 ≤+⎟⎟⎠

⎞⎜⎜⎝

⎛=≤

gAPF (7)

Where:

2F = Factor 2 P = the factored axial load normal to the cross section occurring


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shfA

V cyhvs = (8)

Where:

sV = nominal shear strength provided by the shear reinforcement, lb


yhf = the yield strength of the transverse steel, psi

ch = the distance from the centerline of the hoop to the center of the cross section, in

s = Spacing of shear reinforcement measured in a direction parallel to longitudinal reinforcement, in

UCSD Model Priestly et al. proposed a slightly different model. They emphasized the axial component of the shear strength model by defining it as a separate entity than the concrete component. They also factored flexural ductility into the model. Their model is shown below.

spcn VVVV ++= (9)



pV = the nominal shear strength provided by the axial load


ecc AfkV '= (10)

Where: cV = nominal shear strength provided by the concrete, N

k = factor for influence of flexural ductility, μ '

cf = the square root of the specified compressive strength of the concrete, MPa

eA = 80% of the cross-sectional area, mm2

For standard units, the flexural ductility factor can be found as follows:

29.0=k for 0.2<μ (11a) )2(095.029.0 −−= μk for 0.40.2 <≤ μ (11b)

01.0=k for 0.4≥μ (11c)

Where: k = factor for influence of flexural ductility μ = the flexural ductility

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Pa

cDVp 2−

= (12)

Where:

pV = nominal shear strength provided by the axial load, N D = the overall section depth, mm c = the depth of the compression zone, mm a = the ratio of the moment to the shear at the critical section P = the factored axial load normal to the cross section occurring

simultaneously with the factored shear force at the section, N The nominal shear strength provided by the shear reinforcement in a rectangular column can be determined by using the following relationship:

( )°= 30cot'

sDfA

V yhvs (13)

Where:

sV = nominal shear strength provided by the shear reinforcement, N

vA = the area of shear reinforcement within a the distance of s, mm2

yhf = the yield strength of the transverse steel, MPa 'D = the distance between the centers of the peripheral hoop or spiral, mm

s = Spacing of shear reinforcement measured in a direction parallel to longitudinal reinforcement, mm

USC Model The University of Southern California expanded on UCSD’s model to provide a model that would more accurately predict the effects of using high strength concrete. The most significant change was the presentation of a bilinear model for ductilities between one and six. Their model is shown below.

sacn VVVV ++= (14)



aV = the nominal shear strength provided by the axial load


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ecc AfkV '= (15)


k = factor for influence of flexural ductility '


eA = 80% of the cross-sectional area, mm2


29.0=k for 0.2<μ (16a) )2(12.029.0 −−= μk for 0.40.2 <≤ μ (16b) )4(025.005.0 −−= μk for 0.60.4 ≤≤ μ (16c)

0=k for 0.6>μ (16d)


PVDMDcDVa )/(2

−= (17)

Where:

aV = nominal shear strength provided by the axial load, N D = the overall section depth, mm c = the depth of the compression zone, mm VDM / = the shear aspect ratio

P = the factored axial load normal to the cross section occurring simultaneously with the factored shear force at the section, N

The nominal shear strength provided by the shear reinforcement in a rectangular column can be determined by using the following relationship:

( )θcot)(

scdfA

V yhvs

−= (18)

Where:

sV = nominal shear strength provided by the shear reinforcement, N

vA = the area of shear reinforcement within a the distance of s, mm2

yhf = the yield strength of the transverse steel, MPa d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, mm. c = the depth of the compression zone, mm s = Spacing of shear reinforcement measured in a direction parallel to

longitudinal reinforcement, mm θ = Angle of principal shear crack to column axis

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UCB Model UCB proposed yet a different model. This model was similar to UCSD and Caltrans’ models in that flexural ductility was a primary contributor to the model. The model is shown below.

scn VVV += (19)




g

gc

cc A

AfP

daf

kV 8.06

16

'

'

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛+= (20)

Where:

cV = nominal shear strength provided by the concrete,lb k = factor for influence of flexural ductility


a = the ratio of the moment to the shear at the critical section d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, in. P = the factored axial load normal to the cross section occurring



00.1=k for 0.2<μ (21a) )2(075.000.1 −−= μk for 0.40.2 <≤ μ (21b)

70.0=k for 0.4≥μ (21c)


The nominal shear strength provided by the shear reinforcement in a rectangular column can be determined by using the following relationship:

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sdfA

kV yvs = (22)

Where:

sV = nominal shear strength provided by the shear reinforcement, lb k = factor for influence of flexural ductility


yf = the yield strength of the transverse steel, psi d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, in. s = Spacing of shear reinforcement measured in a direction parallel to

longitudinal reinforcement, in “Simulation of Reinforced Concrete Structures” (SRCS) Over the past two decades, the University of Houston (UH) has performed a significant amount of research to predict shear behavior of reinforced concrete structures. This research resulted in a series of constitutive models of which the Cyclic Softened Membrane Model (CSMM) (Mansour and Hsu 2005a, 2005b) is the most recent. This model has been shown to accurately predict the cyclic shear behavior of reinforced membrane elements. CSMM was implemented into OpenSees, a finite element framework developed at University of California Berkeley (Fenves 2005), and a nonlinear finite element computer program, “Simulation for Reinforced Concrete Structures” (SRCS) was developed (Zhong 2005). The program is capable of predicating the nonlinear behavior of reinforced concrete structures subjected to various types of loading. SRCS’s accuracy was confirmed by testing three full-scale reinforced concrete rectangular hollow bridge pears at the National Center for Research on Earthquake Engineering (NCREE) in Taiwan (Mo 2006). The test set up for the piers is shown in Figure 1. The dimensions and properties of each of the piers are shown in Table 1 and Figure 2. SRCS did an excellent job predicting the results of the test. Experimental and predicted load-deformation plots for all three specimens are shown in Figures 3, 4, and 5. Parametric Studies Since SRCS can accurately predict seismic behavior, it can be implemented for the use of amassing a considerable amount of data in order to come up with a more accurate model that predicts the relationship between ductility and shear strength of reinforced concrete structures. Since Specimen PI2 was shear governed, it was chosen as a template for future specimens. Specimen PI2’s physical characteristics were varied for the purposed of effectively building and testing other columns in the developed program. This method is preferred over actually testing columns because of the expense and time involved in testing a real column. The modeled columns provide a quick, free, alternative that produces nearly identical results. In this manner, one can perform many experiments quickly and can analyze many sets of results instead of the few made practical by the high cost of testing actual columns. Initially, two parameters were varied while the rest coincided with Specimen PI2. These two variables were the stirrup ratio and longitudinal steel ratio. Each theoretical specimen’s parameters are shown in Table 2. Each of these specimens was in turn analyzed using SRCS. A curve was developed that enveloped the extreme points of the hystersis loops for each specimen. See Figure 6 for an example of an enveloping curve.

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Determination of Ductility and Shear Capacity After each of these envelope curves was developed, the ductility and shear capacity of the specimens was estimated using a method described by Zahn et al. (1990). The displacement at the first yield point was estimated as the displacement that would be reached at the flexural strength if the stiffness of the member was that of the cracked member in the elastic range. This stiffness is obtained by determining the deflection that corresponds with 75 percent of the flexural strength on the envelope curve. The line passing through this point and the point of zero strength and zero deflection is then obtained and extended to determine the deflection associated with the flexural strength. The displacement corresponding to shear failure, or ultimate displacement, was assumed to occur along the enveloping curve at 80 percent of the flexural strength along the descending part of the curve. The ductility was then taken to be the ratio between the ultimate and first yield point displacements. Figure 7 shows a graphical interpretation of this method. This process was followed for each specimen. The critical points are shown in Table 3. A plot comparing shear strength and ductility is shown in Figure 8. It is evident from Figure 8 that points with a shear strength above or below a certain range do no follow the trends present in the range. Therefore, these points at the beginnings and ends of the series were not considered. Comparison of Results obtained from SRCS with Current Models A comparative study was performed for each of the current models by dividing the ultimate shear strengths of all the specimens, described in Table 2, predicted using SRCS (determined graphically as shown in Figure 7) by the shear strengths predicted using each model. The shear strength ratios were plotted against flexural ductility, stirrup ratio, and longitudinal steel ratio as they were the only varying parameters in this study. Given that USC’s model was intended for high strength concrete, it was not considered. A visual perception of the comparative prediction of shear strengths for each of the models with respect to the results obtained using SRCS can be obtained via Figures 9 through 12. From Figure 9, it can be seen that shear strengths predicted by ACI’s model are close to that predicted from SRCS. However, the variation between the results predicted from the two is quite large. Predictions obtained from Caltrans’ model were always found to lower than the SRCS predictions as shown in Figure 10. The variation between the two predictions is also large. The predictions from the Caltrans model are much lower than the SRCS predictions for low stirrup and longitudinal steel ratios. Predictions from UCSD’s model are higher than that of SRCS. However, the two predictions showed similar trends. UCB’s model also makes higher predictions when compared to SRCS with variations similar to the UCSD predictions. The predictions from both UCSD and UCB’s models were considerably higher than the SRCS predictions as the stirrup ratio increased. Proposed Model Many factors were considered when it came time to propose a model. All previous models agree that the proposed slope should begin at a ductility of two. This seems fairly consistent with the data gathered. However, the current models disagree as to where the slope should stop. The data collected seems to indicate that the stopping point varies and is directly proportional to the stirrup ratio until a limit of approximately one percent. The point remains at a fairly constant ductility at high stirrup ratios. With the data gathered at this time, the relationship can be modeled as follows. However, this relationship is likely also dependant upon the compressive strength of the concrete. This issue will be explored at a later time.

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3.5144 +−= tq ρ for 01.0≤tρ (23a) 85.3=q for 01.0>tρ (23b)

Where:

q = flexural ductility at the point where the slope changes to zero tρ = stirrup ratio, decimal

The next aspect explored was the general shape that the model would follow. It was apparent that the relationships depicted in Figure 8 seem to follow a curve or two lines rather than a single line, especially for low stirrup ratios. The figure is redrawn in Figure 13 with approximated trend lines for small stirrup ratios. It was decided to propose a new bilinear model similar to USC’s model. The ductility at which the slope of the line changes varies rather than being present at a constant ductility; however, the point does seem to coincide with a longitudinal steel ratio of two percent. With the data gathered at this time, the ductility at a longitudinal steel ratio of two percent can be estimated as follows. Once again, this factor is also highly likely to also depend upon the compressive strength of the concrete; therefore, the relationship should only be used for compressive strengths near 32 MPa. 8.224213300 2 ++−= ttr ρρ for 01.0≤tρ (23a) 85.3=r for 01.0>tρ (23b) Where:

r = flexural ductility at a longitudinal steel ratio of 2% tρ = stirrup ratio, decimal

It was discovered the slopes used in USC’s two-part sloped model were very close to the slopes present in the data gathered. USC’s model was therefore modified to fit the collected data.

29.0=k for 0.2<μ (25a) )2(12.029.0 −−= μk for r<≤ μ0.2 (25b)

μ025.0095.053.0 −−= rk for qr ≤≤ μ (25c) qrk 025.0095.053.0 −−= for q>μ (25d)

29.0=k for 0.2<μ (25a) )2(12.029.0 −−= μk for r<≤ μ0.2 (25b) μ025.0095.053.0 −−= rk for qr ≤≤ μ (25c) qrk 025.0095.053.0 −−= for q>μ (25d)

Where: k = factor for influence of flexural ductility μ = the flexural ductility r = Flexural ductility at the point where the slope changes q = Flexural ductility at the point where the slope changes to zero

The proposed model for this work is based off of UCSD’s model. This was chosen because UCSD’s model had a small spread. The model is shown below.

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spcn VVVV ++= (26)



aV = the nominal shear strength provided by the axial load


ecc AfkV '= (27)


k = factor for influence of flexural ductility '



Pa

cDVp 2−

= (28)

Where:

pV = nominal shear strength provided by the axial load, N D = the overall section depth, mm c = the depth of the compression zone, mm a = the ratio of the moment to the shear at the critical section P = the factored axial load normal to the cross section occurring

simultaneously with the factored shear force at the section, N After adding the factor for the influence of flexural ductility, it was discovered that the model was still a significantly high estimate. Either the factor for influence of flexural ductility needed to be further adjusted or a factor needed to be added to lessen the contribution associated with the shear reinforcement. An investigation was carried out and it was revealed that to achieve the desired result, the contribution of the concrete would be negative. Since this was impossible, a factor was needed to reduce the effect of the proposed strength associated with the shear reinforcement. It makes sense that such an issue could come up under earthquake loads since concrete will crack, becoming unattached to the steel. This causes the steel to become ineffective. Therefore, UCSD’s nominal shear strength provided by shear reinforcement model was modified to show this relationship.

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( ) ( )θρρρ cot)(2.11153300 23 cdfbV yhwttts −+−= (39)

Where: sV = nominal shear strength provided by the shear reinforcement, N

tρ = stirrup ratio, decimal

wb = width of the web, mm

yhf = the yield strength of the transverse steel, MPa d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement, mm. c = the depth of the compression zone, mm θ = Angle of principal shear crack to column axis, can be assumed to be 30

degrees Determination of the Accuracy of the Proposed Models As previously, the accuracy of the proposed model can be viewed graphically in Figures 14. As one can see in the figure, the spread of the proposed models is very small and the relationship is near one. To further test the accuracy, the model was analyzed against a series of columns tested by Wight and Sozen (1975). Properties of the tested columns are shown in Table 4 and the accuracy of the model can be viewed in Figures 15. The spread on the actually tested specimens was larger; however, this was to be expected since actual specimens contain impurities. The results do straddle the value of one nicely. This seems to imply that the model works nicely for normal strength rectangular specimens of various sizes, configurations, and steel ratios. Conclusions

1. SRCS is capable of accurately predicting the nonlinear behavior of reinforced concrete structures subjected to various loading situations.

2. There are several models available that predict the relationship between flexural ductility and shear strength; however, the accuracy of these models is doubtful due to either over- or under-estimations. Hence, they cannot be used in practice.

3. The proposed models can accurately predict the relationship between flexural ductility and shear strength when normal strength concrete is used.

Acknowledgements The research study described herein was sponsored by the National Science Foundation under the Award No. EEC-0649163. The opinions expressed in this study are those of the authors and do not necessarily reflect the views of the sponsor.

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Appendix I. References ACI Committee 318. (2005). “Building Code Requirements for Structural Concrete,” Farmington

Hills, Mich., pp. 147-159. Caltrans. (1995) “Memo to Designers Change Letter 02,” California Department of Transportation,

Sacramento, Calif., March. Laskar, A., Hsu, T. T. C., and Mo, Y. L. (2007). “Shear Strengths of Prestressed Concrete Beams,”

ACI Structural Journal (submitted). Mander, J. B., Priestley, M. J. N., and Park, R. (1988). “Theoretical stress-strain model for confined

concrete,” J. Struct. Div., ASCE, 114(8), pp. 1804-1826. Mander, J. B., Priestley, M. J. N., and Park, R. (1988). “Observed stress-strain model for confined

concrete,” J. Struct. Div., ASCE, 114(8), pp. 1827-1849. Mansour, M. and Hsu, T. T. C., (2005). “Behavior of Reinforced Concrete Elements under Cyclic

Shear: Part I—Experiments,” Journal of Structural Engineering, ASCE, Vol. 131, No. 1, January, pp. 44-53.

Mansour, M. and Hsu, T. T. C., (2005). “Behavior of Reinforced Concrete Elements under Cyclic

Shear: Part 2—Theoretical Model,” Journal of Structural Engineering, ASCE, Vol. 131, No. 1, January, pp. 54-65.

Mo, Y.L., C.H. Jeng and S.F. Perng, (2001). “Seismic Shear Behavior of Rectangular Hollow Bridge

Columns,” Structural Engineering and Mechanics, Vol. 12, No.4, pp. 429-448. Mo, Y.L., Y.K. Yeh, J. Zhong and T.T.C. Hsu, (2006). “Seismic Behavior of Shear-Critical Hollow

Bridge Columns,” ASCE Structures Congress, St. Louis, MO, May 18-21. . Priestley, M. J. N., Verma, R., and Xiao, Y., (1994). “Seismic Shear Strength of Reinforced Concrete

Columns,” Journal of Structural Engineering, ASCE, Vol. 120, No. 8, August, pp. 2310-2329. Sezen, H. and Moehle, J. P., (2004). “Shear Strength Model for Lightly Reinforced Concrete

Columns,” Journal of Structural Engineering, ASCE, Vol. 130, No. 11, November, pp. 1692-1703.

Wight, J. K., and Sozen, M. A., (1975). “Strength Decay of RC Columns Under Shear Reversals,” J.

Struct. Div., ASCE, 101(5), pp. 1053-1065. Xiao, Y., and Martirossyan, A. (1998). “Seismic Performance of High-Strength Concrete Columns,”

Journal of Structural Engineering, ASCE, Vol. 124, No. 3, March, pp. 241-251. Zahn, F. A., Park, R., and Priestley, M. J. N. (1990). “Flexural Strength and Ductility of Circular

Hollow Reinforced Concrete Columns without Confinement on Inside Face,” ACI Structural Journal, Vol. 87, No. 2, March-April, pp. 156-166.

Zhong, J. X. (2005), “Model-Based Simulation of Reinforced Concrete Plane Stress Structures,” PhD

dissertation, Dept. of Civil and Environmental Engineering, University of Houston, Houston, TX.

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Appendix II. Notation

eA = 80% of the cross-sectional area

gA = the gross area of section

vA = the area of shear reinforcement within a the distance of s

a = the ratio of the moment to the shear at the critical section wb = web width

c = the depth of the compression zone D = the overall section depth

'D = the distance between the centers of the peripheral hoop or spiral d = the distance from the extreme compression fiber to the centriod of the

longitudinal tension reinforcement 1F = Factor 1, Caltrans Model 2F = Factor 2, Caltrans Model '

cf = the square root of the specified compressive strength of the concrete

yf = the specified yield strength of nonprestressed reinforcement

yhf = the yield strength of the transverse steel

ch = the distance from the centerline of the hoop to the center of the cross section

k = factor for influence of flexural ductility VDM / = the shear aspect ratio

P = the factored axial load normal to the cross section occurring simultaneously with the factored shear force at the section

q = flexural ductility at the point where the slope changes to zero r = flexural ductility at a longitudinal steel ratio of 2% s = spacing of shear reinforcement measured in a direction parallel to

longitudinal reinforcement t = factor to limit the influence of the shear reinforcement aV = the nominal shear strength provided by the axial load

nV = the nominal shear strength


pV = the nominal shear strength provided by the axial load

sV = the nominal shear strength provided by the shear reinforcement μ = the displacement ductility

tρ = the ratio of the transverse reinforcement volume to column core volume

θ = angle of principal shear crack to column axis

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Appendix III. Tables Table 1—Dimensions and Properties of Bridge Piers

Longitudinal Reinforcement

Transverse Reinforcement ID

No. fc’

(MPa) P (kN) P fc’ Ag

L (mm) Dia.

(mm) fy

(MPa) fsu

(MPa) Dia.

(mm) fy

(MPa) Spacing

(mm) PS1 34.0 4000 0.082 6500 22 460.0 647.0 13 343.0 80 PI1 34.0 4000 0.082 4500 22 460.0 647.0 10 510.0 120 PI2 32.0 3600 0.078 3500 22 418.2 626.5 10 420.0 200

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Table 2—Specimen Properties

Specimen Area (mm2)

D (mm)

d (mm)

a (mm)

s (mm)

ρl (%)

ρt (%)

fy (MPa)

fc’ (MPa) P (kN)

PI2 144000 1500 1350 3500 200 1.7 0.24 418.2 32 3600 PI2-32-024-005 144000 1500 1350 3500 200 0.5 0.24 418.2 32 3600 PI2-32-024-010 144000 1500 1350 3500 200 1.0 0.24 418.2 32 3600 PI2-32-024-020 144000 1500 1350 3500 200 2.0 0.24 418.2 32 3600 PI2-32-024-030 144000 1500 1350 3500 200 3.0 0.24 418.2 32 3600 PI2-32-024-040 144000 1500 1350 3500 200 4.0 0.24 418.2 32 3600 PI2-32-024-050 144000 1500 1350 3500 200 5.0 0.24 418.2 32 3600 PI2-32-024-060 144000 1500 1350 3500 200 6.0 0.24 418.2 32 3600 PI2-32-024-070 144000 1500 1350 3500 200 7.0 0.24 418.2 32 3600 PI2-32-024-080 144000 1500 1350 3500 200 8.0 0.24 418.2 32 3600 PI2-32-050-005 144000 1500 1350 3500 200 0.5 0.50 418.2 32 3600 PI2-32-050-010 144000 1500 1350 3500 200 1.0 0.50 418.2 32 3600 PI2-32-050-017 144000 1500 1350 3500 200 1.7 0.50 418.2 32 3600 PI2-32-050-020 144000 1500 1350 3500 200 2.0 0.50 418.2 32 3600 PI2-32-050-030 144000 1500 1350 3500 200 3.0 0.50 418.2 32 3600 PI2-32-050-040 144000 1500 1350 3500 200 4.0 0.50 418.2 32 3600 PI2-32-050-050 144000 1500 1350 3500 200 5.0 0.50 418.2 32 3600 PI2-32-050-060 144000 1500 1350 3500 200 6.0 0.50 418.2 32 3600 PI2-32-050-070 144000 1500 1350 3500 200 7.0 0.50 418.2 32 3600 PI2-32-050-080 144000 1500 1350 3500 200 8.0 0.50 418.2 32 3600 PI2-32-050-090 144000 1500 1350 3500 200 9.0 0.50 418.2 32 3600 PI2-32-075-005 144000 1500 1350 3500 200 0.5 0.75 418.2 32 3600 PI2-32-075-010 144000 1500 1350 3500 200 1.0 0.75 418.2 32 3600 PI2-32-075-017 144000 1500 1350 3500 200 1.7 0.75 418.2 32 3600 PI2-32-075-020 144000 1500 1350 3500 200 2.0 0.75 418.2 32 3600 PI2-32-075-030 144000 1500 1350 3500 200 3.0 0.75 418.2 32 3600 PI2-32-075-040 144000 1500 1350 3500 200 4.0 0.75 418.2 32 3600 PI2-32-075-050 144000 1500 1350 3500 200 5.0 0.75 418.2 32 3600 PI2-32-075-060 144000 1500 1350 3500 200 6.0 0.75 418.2 32 3600 PI2-32-075-070 144000 1500 1350 3500 200 7.0 0.75 418.2 32 3600 PI2-32-075-080 144000 1500 1350 3500 200 8.0 0.75 418.2 32 3600 PI2-32-075-090 144000 1500 1350 3500 200 9.0 0.75 418.2 32 3600 PI2-32-100-005 144000 1500 1350 3500 200 0.5 1.00 418.2 32 3600 PI2-32-100-010 144000 1500 1350 3500 200 1.0 1.00 418.2 32 3600 PI2-32-100-017 144000 1500 1350 3500 200 1.7 1.00 418.2 32 3600 PI2-32-100-020 144000 1500 1350 3500 200 2.0 1.00 418.2 32 3600 PI2-32-100-030 144000 1500 1350 3500 200 3.0 1.00 418.2 32 3600 PI2-32-100-040 144000 1500 1350 3500 200 4.0 1.00 418.2 32 3600 PI2-32-100-050 144000 1500 1350 3500 200 5.0 1.00 418.2 32 3600 PI2-32-100-060 144000 1500 1350 3500 200 6.0 1.00 418.2 32 3600 PI2-32-100-070 144000 1500 1350 3500 200 7.0 1.00 418.2 32 3600 PI2-32-100-080 144000 1500 1350 3500 200 8.0 1.00 418.2 32 3600 PI2-32-100-090 144000 1500 1350 3500 200 9.0 1.00 418.2 32 3600 PI2-32-125-010 144000 1500 1350 3500 200 1.0 1.25 418.2 32 3600 PI2-32-125-017 144000 1500 1350 3500 200 1.7 1.25 418.2 32 3600 PI2-32-125-020 144000 1500 1350 3500 200 2.0 1.25 418.2 32 3600 PI2-32-125-030 144000 1500 1350 3500 200 3.0 1.25 418.2 32 3600

Page 18 of 33

Specimen Area (mm2)

D (mm)

d (mm)

a (mm)

s (mm)

ρl (%)

ρt (%)

fy (MPa)

fc’ (MPa) P (kN)

PI2-32-125-040 144000 1500 1350 3500 200 4.0 1.25 418.2 32 3600 PI2-32-125-050 144000 1500 1350 3500 200 5.0 1.25 418.2 32 3600 PI2-32-125-060 144000 1500 1350 3500 200 6.0 1.25 418.2 32 3600 PI2-32-125-070 144000 1500 1350 3500 200 7.0 1.25 418.2 32 3600 PI2-32-125-080 144000 1500 1350 3500 200 8.0 1.25 418.2 32 3600 PI2-32-125-090 144000 1500 1350 3500 200 9.0 1.25 418.2 32 3600 PI2-32-150-010 144000 1500 1350 3500 200 1.0 1.50 418.2 32 3600 PI2-32-150-017 144000 1500 1350 3500 200 1.7 1.50 418.2 32 3600 PI2-32-150-020 144000 1500 1350 3500 200 2.0 1.50 418.2 32 3600 PI2-32-150-030 144000 1500 1350 3500 200 3.0 1.50 418.2 32 3600 PI2-32-150-040 144000 1500 1350 3500 200 4.0 1.50 418.2 32 3600 PI2-32-150-050 144000 1500 1350 3500 200 5.0 1.50 418.2 32 3600 PI2-32-150-060 144000 1500 1350 3500 200 6.0 1.50 418.2 32 3600 PI2-32-150-070 144000 1500 1350 3500 200 7.0 1.50 418.2 32 3600 PI2-32-150-080 144000 1500 1350 3500 200 8.0 1.50 418.2 32 3600 PI2-32-150-090 144000 1500 1350 3500 200 9.0 1.50 418.2 32 3600 PI2-32-175-010 144000 1500 1350 3500 200 1.0 1.75 418.2 32 3600 PI2-32-175-017 144000 1500 1350 3500 200 1.7 1.75 418.2 32 3600 PI2-32-175-020 144000 1500 1350 3500 200 2.0 1.75 418.2 32 3600 PI2-32-175-030 144000 1500 1350 3500 200 3.0 1.75 418.2 32 3600 PI2-32-175-040 144000 1500 1350 3500 200 4.0 1.75 418.2 32 3600 PI2-32-175-050 144000 1500 1350 3500 200 5.0 1.75 418.2 32 3600 PI2-32-175-060 144000 1500 1350 3500 200 6.0 1.75 418.2 32 3600 PI2-32-175-070 144000 1500 1350 3500 200 7.0 1.75 418.2 32 3600 PI2-32-175-080 144000 1500 1350 3500 200 8.0 1.75 418.2 32 3600 PI2-32-175-090 144000 1500 1350 3500 200 9.0 1.75 418.2 32 3600 PI2-32-175-100 144000 1500 1350 3500 200 10.0 1.75 418.2 32 3600 PI2-32-200-010 144000 1500 1350 3500 200 1.0 2.00 418.2 32 3600 PI2-32-200-017 144000 1500 1350 3500 200 1.7 2.00 418.2 32 3600 PI2-32-200-020 144000 1500 1350 3500 200 2.0 2.00 418.2 32 3600 PI2-32-200-030 144000 1500 1350 3500 200 3.0 2.00 418.2 32 3600 PI2-32-200-040 144000 1500 1350 3500 200 4.0 2.00 418.2 32 3600 PI2-32-200-050 144000 1500 1350 3500 200 5.0 2.00 418.2 32 3600 PI2-32-200-060 144000 1500 1350 3500 200 6.0 2.00 418.2 32 3600 PI2-32-200-070 144000 1500 1350 3500 200 7.0 2.00 418.2 32 3600 PI2-32-200-080 144000 1500 1350 3500 200 8.0 2.00 418.2 32 3600 PI2-32-200-090 144000 1500 1350 3500 200 9.0 2.00 418.2 32 3600 PI2-32-200-100 144000 1500 1350 3500 200 10.0 2.00 418.2 32 3600

Page 19 of 33

Table 3—Critical Points for the Determination of Flexural Ductility and Shear Strength

Specimen Flexural Capacity (FC) (kN)

75% of FC

(kN)

Deflection at 75% of FC (mm)

Deflection at Yield (mm)

80% of FC (kN)

Deflection at Ultimate

(mm) Ductility

PI2 2549 1912 13.3 17.7 2040 62.4 3.52 PI2-32-024-005 2045 1534 14.1 18.8 1636 85.4 4.55 PI2-32-024-010 2274 1705 12.9 17.2 1819 85.0 4.95 PI2-32-024-020 2650 1988 13.4 17.9 2120 59.2 3.31 PI2-32-024-030 2913 2185 13.5 18.0 2330 50.1 2.78 PI2-32-024-040 3116 2337 13.5 18.0 2493 47.0 2.61 PI2-32-024-050 3271 2453 13.4 17.9 2616 44.4 2.49 PI2-32-024-060 3364 2523 13.1 17.5 2691 43.7 2.49 PI2-32-024-070 3429 2572 12.8 17.1 2743 43.6 2.55 PI2-32-024-080 3458 2593 12.4 16.5 2766 43.5 2.64 PI2-32-050-005 2245 1683 20.4 27.2 1796 115.5 4.25 PI2-32-050-010 2467 1850 15.6 20.8 1974 98.4 4.73 PI2-32-050-017 2741 2055 14.1 18.8 2192 73.8 3.94 PI2-32-050-020 2840 2130 14.0 18.7 2273 69.2 3.70 PI2-32-050-030 3205 2404 14.4 19.2 2564 58.1 3.03 PI2-32-050-040 3458 2594 14.4 19.2 2767 52.3 2.73 PI2-32-050-050 3717 2788 14.6 19.5 2974 48.2 2.48 PI2-32-050-060 3928 2946 14.7 19.6 3142 45.0 2.30 PI2-32-050-070 4072 3054 14.6 19.4 3258 42.6 2.19 PI2-32-050-080 4177 3132 14.3 19.1 3341 40.8 2.14 PI2-32-050-090 4249 3187 14.0 18.7 3399 40.5 2.16 PI2-32-075-005 2370 1778 25.3 33.7 1896 119.1 3.53 PI2-32-075-010 2582 1936 18.4 24.6 2066 99.4 4.05 PI2-32-075-017 2849 2137 15.1 20.1 2280 84.5 4.21 PI2-32-075-020 2867 2225 15.0 20.0 2374 77.2 3.86 PI2-32-075-030 3325 2494 14.7 19.7 2660 64.3 3.27 PI2-32-075-040 3641 2731 14.9 19.8 2913 56.0 2.83 PI2-32-075-050 3918 2939 15.1 20.1 3134 52.9 2.64 PI2-32-075-060 4162 3121 15.2 20.3 3330 49.3 2.43 PI2-32-075-070 4372 3279 15.3 20.4 3498 34.6 2.24 PI2-32-075-080 4533 3399 15.2 20.3 3626 42.7 2.10 P12-32-075-090 4649 3487 15.0 20.1 3719 39.1 1.95 PI2-32-100-005 2434 1826 27.7 37.0 1947 126.2 3.42 PI2-32-100-010 2650 1987 20.3 27.1 2120 101.0 3.73 PI2-32-100-017 2925 2194 16.2 21.6 2340 84.6 3.92 PI2-32-100-020 3039 2280 15.5 20.7 2432 80.7 3.91 PI2-32-100-030 3407 2555 15.0 20.0 2726 66.3 3.32 PI2-32-100-040 3744 2808 15.1 20.2 2995 58.2 2.89 PI2-32-100-050 4014 3011 15.2 20.2 3211 52.8 2.66 PI2-32-100-060 4307 3230 15.5 20.7 3446 50.9 2.46 PI2-32-100-070 4546 3410 15.7 20.9 3637 47.3 2.27 PI2-32-100-080 4761 3570 15.8 21.0 3808 42.4 2.02 PI2-32-100-090 4915 3686 15.7 20.9 3932 40.1 1.92 PI2-32-125-010 2675 2006 20.9 27.9 2140 102.5 3.68 PI2-32-125-017 2947 2210 16.4 21.9 2358 85.0 3.87

Page 20 of 33

Specimen Flexural Capacity (FC) (kN)

75% of FC

(kN)

Deflection at 75% of FC (mm)

Deflection at Yield (mm)

80% of FC (kN)

Deflection at Ultimate

(mm) Ductility

PI2-32-125-020 3068 2301 15.7 21.0 2455 79.9 3.81 PI2-32-125-030 3453 2590 15.2 20.2 2762 69.1 3.42 PI2-32-125-040 3811 2858 15.3 20.4 3049 58.8 2.88 PI2-32-125-050 4085 3064 15.2 20.3 3268 53.8 2.65 PI2-32-125-060 4379 3285 15.5 20.7 3504 49.8 2.41 PI2-32-125-070 4640 3480 15.7 21.0 3712 46.6 2.22 PI2-32-125-080 4869 3652 15.9 21.2 3895 43.7 2.06 PI2-32-125-090 5064 3798 15.9 21.2 4051 41.0 1.93 PI2-32-150-010 2694 2021 21.4 28.6 2155 108.9 3.81 PI2-32-150-017 2961 2221 16.6 22.1 2369 86.6 3.92 PI2-32-150-020 3081 2311 15.7 21.0 2465 81.6 3.89 PI2-32-150-030 3466 2599 15.1 20.1 2773 68.8 3.42 PI2-32-150-040 3843 2882 15.3 20.4 3075 60.8 2.98 PI2-32-150-050 4129 3097 15.3 20.4 3303 55.6 2.73 PI2-32-150-060 4431 3323 15.5 20.7 3545 51.7 2.50 PI2-32-150-070 4714 3535 15.8 21.1 3771 48.6 2.31 PI2-32-150-080 4954 3716 15.9 21.3 3963 45.9 2.16 PI2-32-150-090 5162 3871 16.0 21.4 4130 42.3 1.98 PI2-32-175-010 2712 2034 21.9 29.3 2170 108.0 3.69 PI2-32-175-017 2969 2227 16.6 22.1 2375 90.9 4.11 PI2-32-175-020 3088 2316 15.7 21.0 2471 85.2 4.06 PI2-32-175-030 3469 2602 15.0 20.0 2775 69.9 3.50 PI2-32-175-040 3848 2886 15.1 20.2 3078 61.4 3.04 PI2-32-175-050 4171 3128 15.3 20.4 3337 56.1 2.75 PI2-32-175-060 4476 3357 15.6 20.7 3581 52.0 2.51 PI2-32-175-070 4769 3577 15.8 21.1 3815 49.5 2.35 PI2-32-175-080 4999 3749 15.9 21.2 3999 45.8 2.16 PI2-32-175-090 5228 3921 16.1 21.4 4182 43.3 2.02 PI2-32-175-100 5428 4071 16.2 21.5 4342 40.9 1.90 PI2-32-200-010 2728 2046 22.4 29.9 2182 111.6 3.74 PI2-32-200-017 2974 2231 16.6 22.1 2379 91.6 4.14 PI2-32-200-020 3095 2321 15.7 20.9 2476 85.5 4.08 PI2-32-200-030 3476 2607 14.9 19.9 2781 71.2 3.57 PI2-32-200-040 3858 2893 15.1 20.1 3086 62.4 3.10 PI2-32-200-050 4204 3153 15.3 20.5 3363 57.7 2.82 PI2-32-200-060 4512 3384 15.6 20.8 3609 52.8 2.54 PI2-32-200-070 4806 3604 15.8 21.1 2845 49.4 2.34 PI2-32-200-080 5048 3786 15.9 21.2 4038 47.2 2.22 PI2-32-200-090 5281 3961 16.1 21.4 4225 43.9 2.05 PI2-32-200-100 5504 4128 16.2 21.7 4403 41.6 1.92

Page 21 of 33

Table 4—Specimen Properties for Wight and Sozen’s Tests (1975)

Specimen D (mm)

a (mm)

s (mm) ρl (%) ρt (%) fy

(MPa)fc’

(MPa) P (kN) μ V (kN)

40.033 254 876 127 2.4 0.33 344 33.6 178 3.62 97 25.033 254 876 127 2.4 0.33 344 33.6 111 2.65 87 00.033 254 876 127 2.4 0.33 344 32.0 0 3.67 81 40.048 254 876 89 2.4 0.48 344 26.1 178 3.38 95 00.048 254 876 89 2.4 0.48 344 25.9 0 2.45 86

Page 22 of 33

Appendix IV. Figures Figure 1—Test set-up

Actuator

Oil jack

ColumnReaction

Wall

Strong Floor

RC Foundation

Load Cell

Dial gauge

Cross beam

Hinge

Cross beam

Universal joint

Load Cell Universal joint

Page 23 of 33

Figure 2—Dimensions and Reinforcement of the Cross-Section

1500 15

00 900

900

64-#7

#4@80

66 168 132 168 132 168 132 168 132 168 66

300

i j b a

205

115

115

i j

150 a

1385

170

150b 170

1500

1500

900

900

64-#7

#3@120

66 168 132 168 132 168 132 168 132 168 66

300

i j b a

205

115

115

i j

150 a

1385

170

150b 170

1500

1500

900

900

64-#7

#3@200

66 168 132 168 132 168 132 168 132 168 66

300

i j b a

210

125

100

i j

100 a

1410

125

100b 125

unit : mm

Page 24 of 33

Speciment PS1

-3000

-2000

-1000

0

1000

2000

3000

-500 -400 -300 -200 -100 0 100 200 300 400 500

Displacement (mm)

Forc

e (k

N)

ExperimentalPredicted

Figure 3—Predicted versus Experimental Force—Displacement Curves of Specimen PS1

Speciment PI1

-3000

-2000

-1000

0

1000

2000

3000

-500 -400 -300 -200 -100 0 100 200 300 400 500

Displacement (mm)

Forc

e (k

N)


Figure 4—Predicted versus Experimental Force—Displacement Curves of Specimen PI1

Page 25 of 33

Speciment PI2

-3000

-2000

-1000

0

1000

2000

3000

-500 -400 -300 -200 -100 0 100 200 300 400 500

Displacement (mm)

Forc

e (k

N)


Figure 5—Predicted versus Experimental Force—Displacement Curves of Specimen PI2

PI2

-3000

-2000

-1000

0

1000

2000

3000

-1.00E+02 -5.00E+01 0.00E+00 5.00E+01 1.00E+02

Deflection (mm)

Forc

e (k

N)

OriginalExperimental Loops

Figure 6—Hystersis Loops Recorded During Experiment with Enveloping Curve Estimated by SRCS for Specimen PI2

Page 26 of 33

PI2

0

500

1000

1500

2000

2500

3000

0 10 20 30 40 50 60 70 80

Deflection (mm)

Forc

e (k

N)

DataD @ 75%D @ ultimateD @ yield

Figure 7—Determination of the Deflection at First Yield

Ductility Versus Shear Strength

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03

3.00E+03

3.50E+03

4.00E+03

4.50E+03

5.00E+03

0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00

Ductility

Forc

e (k

N)

Stirrup Ratio=0.24%Stirrup Ratio=0.50%Stirrup Ratio=0.75%Stirrup Ratio=1.00%Stirrup Ratio=1.25%Stirrup Ratio=1.50%Stirrup Ratio=1.75%Stirrup Ratio=2.00%

Figure 8—Ductility versus Shear Strength

Page 27 of 33

ACI Accuracy Based on Ductility

0.00

1.00

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ductility

V/V

(pro

pose

d)

ACI Accuracy Based on Stirrup Ratio

0.00

1.00

2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Stirrup Ratio (%)

V/V

(pro

pose

d)

ACI Accuracy Based on Longitudinal Steel Ratio

0.00

1.00

2.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00

Longitudinal Steel Ratio (%)

V/V

(pro

pose

d)

Figure 9—Visual Depiction of the Accuracy of ACI’s Model

Page 28 of 33

CALTRANS Accuracy Based on Ductility

0.00

1.00

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ductility

V/V

(pro

pose

d)

CALTRANS Accuracy Based on Stirrup Ratio

0.00

1.00

2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Stirrup Ratio (%)

V/V

(pro

pose

d)

CALTRANS Accuracy Based on Longitudinal Steel Ratio

0.00

1.00

2.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


V/V

(pro

pose

d)

Figure 10—Visual Depiction of the Accuracy of Caltrans’ Model

Page 29 of 33

UCSD Accuracy Based on Ductility

0.00

1.00

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ductility

V/V

(pro

pose

d)

UCSD Accuracy Based on Stirrup Ratio

0.00

1.00

2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Stirrup Ratio (%)

V/V

(pro

pose

d)

UCSD Accuracy Based on Longitudinal Steel Ratio

0.00

1.00

2.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


V/V

(pro

pose

d)

Figure 11—Visual Depiction of the Accuracy of UCSD’s Model

Page 30 of 33

UCB Accuracy Based on Ductility

0.00

1.00

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ductility

V/V

(pro

pose

d)

UCB Accuracy Based on Stirrup Ratio

0.00

1.00

2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Stirrup Ratio (%)

V/V

(pro

pose

d)

UCB Accuracy Based on Longitudinal Steel Ratio

0.00

1.00

2.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


V/V

(pro

pose

d)

Figure 12—Visual Depiction of the Accuracy of UCB’s Model

Page 31 of 33

Ductility Versus Shear Strength

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

2.50E+03

3.00E+03

3.50E+03

4.00E+03

4.50E+03

0.00E+00 1.00E+00 2.00E+00 3.00E+00 4.00E+00 5.00E+00 6.00E+00 7.00E+00

Ductility

Forc

e (k

N) Stirrup Ratio=0.24%

Stirrup Ratio=0.50%Stirrup Ratio=0.75%Stirrup Ratio=1.00%

Figure 13—Ductility and Shear Relationships with Trend Lines for Small Stirrup Ratios

Page 32 of 33

Proposed Model Accuracy Based on Ductility

0.00

1.00

2.00

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Ductility

V/V

(pro

pose

d)

Proposed Model Accuracy Based on Stirrup Ratio

0.00

1.00

2.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25

Stirrup Ratio (%)

V/V

(pro

pose

d)

Proposed Model Accuracy Based on Longitudinal Steel Ratio

0.00

1.00

2.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


V/V

(pro

pose

d)

Figure 14—Visual Depiction of the Accuracy of the Proposed Model

Page 33 of 33

Proposed Model Accuracy Based on Ductility

0

1

2

0 1 2 3 4 5 6

Ductility

V/V

(pro

pose

d)


0

1

2

0 0.25 0.5 0.75 1 1.25

Stirrup Ratio (%)

V/V

(pro

pose

d)


0

1

2

25 26 27 28 29 30 31 32 33 34 35

fc' (MPa)

V/V

(pro

pose

d)

Figure 15—Visual Depiction of the Accuracy of the Proposed Model using Tested Specimens

Documents

Seismic Interaction of Flexural Ductility and Shear ...structurallab.egr.uh.edu/.../files/files/reu-rh-report.pdf · Seismic Interaction of Flexural Ductility and Shear Capacity in