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Post-cracking shear modulus of reinforced concrete membrane elements
Engineering Structures 32 (2010) 218–225
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Post-cracking shear modulus of reinforced concrete membrane elements Khaldoun N. Rahal ∗ Civil Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
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Article history: Received 16 October 2008 Received in revised form 29 August 2009 Accepted 3 September 2009 Available online 18 September 2009 Keywords: Deformation Membrane Mode of failure Modulus of rigidity Reinforced concrete Shear Strength
abstract The shear modulus of reinforced concrete membrane elements subjected to monotonic in-plane shearing stresses is typically calculated using the elastic theory. Upon cracking, however, the elements suffer a significant loss in rigidity and hence the shear deformations at service load levels cannot be accurately calculated using the pre-cracking modulus. This paper presents a simple empirical equation for the calculation of the post-cracking shear modulus, given in the form of the tangent slope of the shear stress–strain curve. The proposed empirical equation is based on the experimental results from forty membrane specimens, and relates the cracked modulus to a ‘‘combined’’ index of the reinforcement in the orthogonal directions. In spite of its empirical nature and simplicity, the equation is capable of capturing the effect of different amounts of reinforcement and concrete strength. Hence it applies to both underreinforced and over-reinforced membranes, and to normal and high strength concrete. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The design of many reinforced concrete (RC) structures requires satisfying a serviceability limit state where the calculated deflections are checked against acceptable limits. In RC beams subjected to transverse loading, for example, vertical deflections are calculated based on the flexural moments and stiffness, while shear deflections are neglected. The significant reduction in the flexural stiffness after cracking in beams has long been recognized, and many building design codes provide provisions to calculate a reduced cracked flexural stiffness. For example, the ACI code [1] includes an equation to calculate an ‘‘effective’’ cracked moment of inertia and in some cases it allows the use of a value equal to a specific percentage of the gross moment of inertia. Since shear deformations are typically neglected, design codes do not recognize the considerable reduction in shear modulus after diagonal cracking. However, many studies have shown the importance of accounting for the shear deformations, especially in the post-cracking range of the behavior of shear-critical elements. Based on an experimental and theoretical investigation, Vecchio and Emara [2] concluded that transverse shear strains reduced the lateral stiffness of RC frames and increased the (P–∆) effects. Debernardi and Taliano [3] tested thin-webbed I-beams and observed that the shear deformations contributed about 25% of
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the total vertical deflections. A detailed nonlinear analysis of the I-beams led to the same conclusion. Hence, an accurate calculation of the deformation of some shear-critical RC structures can only be achieved by adequate consideration of the post-cracking behavior of these members. There are numerous behavioral models that are capable of calculating the full response of shear-critical reinforced concrete elements [4–10], including the post-cracking behavior. However, these advanced models require the use of computers to solve for the solution using an iterative procedure. The development of simpler procedures has also attracted considerable attention. Zhu et al. [11] listed thirteen nonlinear constitutive relationships for cracked concrete in shear which can be used to calculate the post-cracking modulus of membranes. The level of the complexity of these relationships varies, but they all require the use of an iterative procedure to calculate the post-cracking modulus. This paper presents a simple non-iterative empirical equation for the post-cracking modulus of membrane elements subjected to pure shear. This equation is based on the experimental results from forty elements subjected to pure shear, and provides a step towards a simple modeling of the full shear stress–strain relationship of membrane and beam elements. 2. Response of membrane elements subjected to pure shear Fig. 1 shows an orthogonally reinforced membrane element and the experimentally observed relationship between the shearing stress (v ) and the shearing strain (γ ) for the membrane specimen A3 tested by Pang and Hsu [12]. These results can be used to identify the key stages of the behavior of such an element. Before cracking, the response was mainly linear, and the slope of the curve in
K.N. Rahal / Engineering Structures 32 (2010) 218–225
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Nomenclature fc0 fy−x fy−y Gcr Gcr–calc Gcr–exp
vn κ ρx ρy ωx ωy v
compressive strength of concrete yield stress of x-direction reinforcement yield stress of y-direction reinforcement post-cracking shear modulus calculated post-cracking shear modulus experimentally observed post-cracking shear modulus ultimate shearing strength calculated using SMCS model ‘‘balanced’’ reinforcing index and concrete crushing factor reinforcement ratio in x direction reinforcement ratio in y direction reinforcement index in x direction reinforcement index in y direction applied shear stress
this region is referred to as the shear modulus. Typically, the theory of elasticity is considered adequate to calculate the shear modulus as a function of the modulus of elasticity of concrete tested in uniaxial compression. Upon cracking at (vcr ), the mechanism of resistance changed significantly leading to a transition zone between the pre- and the post-cracking parts of the behavior. The shape of the transition zone depends mainly on whether the loading machine is operated using force or displacement control. In the postcracking zone, there was a significant decrease in slope (tangent modulus), which remained nearly constant until the reinforcement yielded at v = vy . A sudden reduction in the modulus was observed after the yielding of the steel. The post-yielding strength was not significant in this specimen because the reinforcement in the two orthogonal directions yielded. In this paper, the slope of the precracking part of the response is referred to as the uncracked modulus, while the post-cracking and before steel yielding/concrete crushing is referred to as the post-cracking modulus (Gcr ). The latter is the subject of this study. Fig. 1 also shows the nominal shear strength calculated using the ACI code [1], and the shear stress at service conditions (estimated at about 75% of the nominal calculated capacity). The service load typically fits in the post-cracking region of the response, between the cracking and the yielding shear stresses. The use of the pre-cracking modulus under-estimates the shear strains at service load considerably. The post-cracking modulus is required to accurately calculate the strain at service load and to model an important part of the response that cannot be accurately obtained using elastic theory. A recently developed model named the simplified model for combined stress resultants (SMCS) relates the ultimate shear capacity and mode of failure of the membranes to the concrete compressive strength and the amount and strength of the orthogonal reinforcement [13]. SMCS elegantly combines these factors in the orthogonal reinforcement indexes. This model was shown to be able to accurately calculate the strength and the mode of failure of membranes made of normal and high strength concrete. Full details of SMCS can be found elsewhere [13]. It is proposed that the post-cracking modulus Gcr depends on the same main factors that influence the strength of the membrane elements. The SMCS model is briefly described in the following section to introduce the reinforcement indexes and the balanced reinforcement limits, and then the equation for Gcr is related to the same factors.
Fig. 1. Experimentally observed shear stress–strain response of a reinforced concrete membrane.
3. SMCS model The nominal shear strength vn is calculated as follows:
vn fc0
=
√ ωx ωy ≤ κ
(1)
where the reinforcement indexes ωx and ωy in the x- and ydirections respectively are:
ωx = ωy =
ρx fy−x fc0
ρy fy−y fc0
≤κ
(2a)
≤ κ.
(2b)
The upper limit (κ ) set on ωx and ωy is an over-reinforcement limit, and leads to a crushing limit on vn /fc0 (as shown in Eq. (1)). This limit depends on the concrete compressive strength fc0 and is given by the following equation:
κ = 1/3 − fc0 /900 0
(3)
where fc is in MPa. Eq. (3) is based on a review of the test results from 50 specimens subjected to in-plane shearing and tensile normal stresses [13]. The reduction of the normalized shear stress at higher concrete strengths is well recognized and has been quantified in numerous studies such as the theory of plasticity [14, 15], the modified compression field theory (MCFT) [16] and the softened truss model (STM) [17]. In addition, it is implied in the ACI code’s upper limit on the design shear force inpwhich the maximum permissible shear design stress is related to fc0 . If the mechanical reinforcing index calculated using Eq. (2) is limited by the over-reinforcement limit κ , this steel does not yield when the ultimate shear strength is reached. This leads to
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three possible modes of failure at ultimate conditions: (1) fully under-reinforced membrane where both x- and y-reinforcement yields; (2) partially under-reinforced membrane where only the x- or the y-reinforcement yields; and (3) fully over-reinforced membrane where crushing takes place before yielding in any of the reinforcement. Experimental evidence showed that increasing the amount of reinforcement increases the shear strength. However, this increase becomes less significant when the reinforcement is increased above a ‘‘balanced’’ reinforcement level [13]. Similar to the flexure theory, the ‘‘balanced’’ level refers to the amount of reinforcement required to ensure that first yielding of the steel occurs simultaneously with concrete crushing in compression, leading to ultimate conditions. The limited increase in strength by the reinforcement in excess of the balanced reinforcement is neglected to maintain the simplicity of the model. This is achieved by imposing the limit κ on the ‘‘usable’’ reinforcement indexes in Eq. (2). This same limit is translated into the upper limit on shear strength when concrete crushes prior to yielding in fully overreinforced membranes. See Eq. (1). The calculations of the model were checked against the observed ultimate shear strengths from eighty four test specimens subjected to in-plane shearing and normal stresses [13]. The average of the observed to the calculated strength was 1.03 and the coefficient of variation was 11.3%. The experimental mode of failure was available in forty nine membranes, and the SMCS model was capable of calculating the correct mode of failure in forty five of them.
a
b
4. Post-cracking modulus The post-cracking tangent modulus Gcr can be obtained from the v –γ response observed in pure shear tests. Results from forty tests [4,5,12,17–19] on membrane elements were analyzed, and Gcr was calculated as the slope of the best straight line fit of the post-cracking, pre-yielding behavior. A consistent approach was followed in selecting the data measurements which are considered to be within the postcracking part of the response of a specimen. The experimental readings in the pre-cracking (v ≤ vcr ) and the post-yielding (v ≥ vy ) parts of the response were not considered. To exclude the transition zone between the pre-cracking and the post-cracking response, the experimental measurements between vcr and 1.2vcr were not considered. This margin was found to be suitable for this purpose in all the specimens. A regression analysis of the data points which fit within the post-cracking part of the response was used to find Gcr . The results of this exercise for the forty test specimens are listed in Table 1. The SMCS showed that the main properties affecting the shear behavior of the membrane can be elegantly combined in a single √ factor ωx ωy , where each of the reinforcement indexes is limited to the balanced reinforcement ratio given as shown in Eq. (2). The relationship between Gcr and this factor is investigated. Of the 40 tests available, 21 specimens were tested at the University of Toronto (UT) [4,18,19] and 19 at the University of Houston (UH) [5,12,17]. The available data is split into two groups; one is used to develop the equation for the modulus and the other to provide an independent evaluation. The development set is taken as the 21 UT tests. This provides a good challenge to the adequacy of the equation since the two sets of data differ significantly. For instance, the concrete strength fc0 ranged from 11.6 to 66.1 MPa in the UT tests and from 41.3 to 103.1 MPa in the UH tests. The average fc0 in the two sets of tests was 30 MPa and 69 MPa respectively. The size of the specimens also changed significantly. Twenty of the UT specimens were 890 × 890 × 70 mm and one was 1626 × 1626 × 287 while the 19 UH specimens were
Fig. 2. (a) Proposed relation based on the UT tests and (b) corroboration between observed and calculated modulus for all tests.
1400 × 1400 × 178 mm. In addition, the UT specimens were tested under stress control loading conditions while the UH specimens were tested under deformation control conditions. √ Fig. 2(a) shows a plot of the factor Gcr /fc0 and the factor ωx ωy for the 21 UT test results. An obvious trend is observed. The best √ linear fit is Gcr /fc0 = 153.4 ωx ωy − 3.9, which gives a ratio of observed to calculated modulus of 1.00 and a coefficient of variation of 17.8%. The best-fit line which passes through the origin √ of the axes is Gcr /fc0 = 135.4 ωx ωy with an average of observed to calculated modulus of 0.97 and a coefficient of variation of 18.0%. The latter equation provides a simpler relationship without a significant loss of accuracy. Hence, it is proposed that the postcracking modulus is calculated using the following equation:
√
Gcr /fc0 = 135 ωx ωy
(4)
where Gcr carries the same units as the concrete compressive strength fc0 . Table 1 compares the calculations of Eq. (4) against the experimental results obtained from the (UH) evaluation data set. The average of the experimental to the calculated values of the modulus is 1.03 and the coefficient of variation is 12.5%. The coefficient of variation in the evaluation set of data was slightly better than in the development set, which enhances the confidence in the performance of the equation. Fig. 2(b) shows a plot of the ratio Gcr–exp /Gcr–calc versus the concrete strength for the 40 tests. Eq. (4) performed adequately in specimens made of higher strength concrete though it was developed based on data from specimens made of normal strength concrete.
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Table 1 Details of specimens and experimental verification of shear modulus. ρx fy−x
ρy fy−y
fc0
fc0
0.250 0.120 0.096 0.163 0.159 0.700 0.340 0.269 0.523 0.430 0.419 0.419 0.417 0.382 0.385
27.0 49.9 43.0 66.1 58.4 55.9
Size (mm)
Specimen
fc0 (MPa)
[4]
890 × 890 × 70
PV1 PV3 PV4 PV5 PV6 PV9 PV10 PV11 PV12 PV19 PV20 PV21 PV22 PV26 PV27
34.5 26.6 26.6 28.3 29.8 11.6 14.5 15.6 16.0 19.0 19.6 19.5 19.6 21.3 20.5
[18]
1626 × 1626 × 287
PP1
890 × 890 × 70
PA1 PA2 PHS2 PHS3 PHS8
Source
[19]
21 University of Toronto tests
Gcr–exp (GPa)
Gcr–calc (GPa)
Gcr– exp Gcr–calc
0.235 0.120 0.096 0.163 0.159 0.700 0.190 0.197 0.076 0.112 0.134 0.201 0.327 0.220 0.385
1.10 0.29 0.42 0.44 0.71 0.54 0.41 0.70 0.38 0.42 0.52 0.64 0.90 0.69 0.88
1.13 0.43 0.35 0.62 0.64 0.50 0.48 0.49 0.33 0.48 0.54 0.66 0.82 0.75 0.86
0.98 0.67 1.20 0.71 1.10 1.08 0.84 1.43 1.14 0.87 0.97 0.98 1.09 0.92 1.02
0.345
0.116
0.66
0.68
0.97
0.173 0.202 0.296 0.335 0.350
0.086 0.100 0.032 0.073 0.116
0.69 0.75 0.59 1.08 1.27
0.82 0.82 0.82 1.10 1.34
0.85 0.91 0.72 0.98 0.95
Average Coefficient of variation (%)
0.97 18.0
[12]
1400 × 1400 × 178
A2 A3 A4 B1 B2 B3 B4 B5 B6
[5]
1400 × 1400 × 178
HB3 HB4
66.8 62.9
0.120 0.223
0.040 0.042
0.72 0.86
0.62 0.83
1.15 1.04
1400 × 1400 × 178
VA1 VA2 VA3 VA4 VB1 VB2 VB3 VB4
95.1 98.2 94.6 103.1 98.2 97.6 102.3 96.9
0.056 0.100 0.173 0.239 0.100 0.167 0.266 0.085
0.056 0.100 0.173 0.239 0.054 0.055 0.052 0.028
0.69 1.37 2.14 2.90 1.06 1.27 1.74 0.53
0.72 1.32 2.21 3.05 0.98 1.26 1.48 0.63
0.96 1.04 0.97 0.95 1.08 1.01 1.17 0.84
[17]
41.3 41.7 42.5 45.3 44.1 44.9 44.8 42.9 43.0
0.134 0.192 0.330 0.123 0.181 0.178 0.313 0.326 0.326
0.134 0.192 0.330 0.059 0.126 0.059 0.059 0.130 0.186
0.89 1.25 1.65 0.46 0.95 0.58 0.59 1.34 1.59
0.75 1.08 1.64 0.52 0.90 0.62 0.78 1.11 1.34
1.19 1.16 1.01 0.90 1.05 0.92 0.75 1.21 1.19
19 University of Houston tests
Average Coefficient of variation (%)
1.03 12.5
All 40 tests
Average Coefficient of variation (%) Root-mean square error
1.00 15.6 0.117
Table 1 shows that for the 40 specimens, the average of observed-to-calculated ratios is 1.00 and the coefficient of variation is 15.6%. The root-mean square error is 0.117. Comparison with Eq. (1) shows that Eq. (4) is numerically equivalent to: Gcr = 135vn .
(5)
This equation can be used to estimate the post-cracking modulus before the design of the reinforcement. 5. Evaluation of the proposed equation This section presents an evaluation of the ability of the proposed equation to capture the effects of the main variables that influence the post-cracking modulus of membrane elements. These are namely the amount of reinforcement and the concrete strength. The results of the proposed model are also checked against the experimental results from a zone of RC beam subjected to shear and a relatively negligible bending moment.
5.1. Membrane elements The ability of the model to capture the influence of different levels of concrete strength on Gcr of under-reinforced and overreinforced membranes is investigated and the results are shown in Fig. 3. The modulus from tests of three under-reinforced specimens PV20 [4], B3 [12] and VB4 [17] is shown. These specimens contained similar levels of reinforcement (ρx fy−x = 8.1 MPa, ρy fy−y = 2.7 MPa) and variable concrete strength. The test results showed that the x- and y-reinforcement in B3 and VB4 and the y-reinforcement in PV20 yielded at ultimate conditions. The observed modulus is shown to be relatively independent of the concrete strength. This is adequately reflected in Eq. (4) which, for fully under-reinforced membranes, can be reduced to: Gcr = 135
q
ρx fy−x
ρy fy−y .
(6)
Fig. 3 shows that Eq. (4) or (6) slightly over-estimates the modulus of this series of tests but the results reflected the fact that
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K.N. Rahal / Engineering Structures 32 (2010) 218–225
Fig. 3. Effect of concrete strength and mode of failure on post-cracking modulus.
varying the concrete strength has a limited effect on Gcr of underreinforced membranes. Specimens PV9, PV27 and PV1 [4] were reinforced with nearly similar orthogonal reinforcement (ρx fy−x = ρy fy−y = 8.1 MPa) and failed in a fully-over-reinforced mode. The experimental results in Fig. 3 show that the shear modulus is proportional to the concrete strength. This is also reflected in Eq. (4), which, for fully overreinforced membranes, can be rearranged to give: Gcr = 135 1/3 − fc0 /900 fc0 .
(7)
In general, the proposed equation captures the trend observed in the tests, but slightly over-estimates the effect of higher concrete strength. The ability of the model to capture the effects of different amounts of reinforcement on Gcr of normal and high strength concrete membranes is also investigated. The modulus obtained from two series of tests on normal strength concrete membranes is shown in Fig. 4. In the first series, specimens PV12, PV19 to PV22, PV26 and PV27 [4] had an average concrete compressive strength of about 19.4 MPa and an average ρ fy /fc0 of x-direction reinforcement of about 0.43. The y-reinforcement was variable and covered partially under-reinforced levels in five specimens PV12, PV19 to PV21, PV26, and fully over-reinforced levels in two specimens (PV22 and PV27). The figure shows that Gcr increased with the increase in ρy fy−y /fc0 in under-reinforced elements only. Hence, neglecting the contribution of reinforcement in excess of the balanced amounts is justified in this series of tests. The results of Eq. (4) showed a good agreement with the observed modulus. In the second series, specimens B4, B5, B6 and A4 [12] had a nearly equal concrete compressive strength averaging 43.3 MPa and a ρx fy−x /fc0 value of about 0.32. The y-reinforcement was variable and yielded in all four specimens before ultimate conditions. The x-reinforcement yielded in specimen A4 only. Fig. 4(a) shows that Eq. (4) adequately models the reduction in the rate of increase of modulus at higher reinforcement levels, and the overall behavior of normal strength concrete specimens. Fig. 4(b) shows the modulus values obtained from two series of tests on high strength concrete specimens [17]. The average concrete strength and ρy fy−y /fc0 values of specimens VB1, VB2 and VB3 were 99 MPa and 0.053 respectively while the x-reinforcement was variable. VB1 and VB2 failed in a fully under-reinforced mode while only the y-steel yielded in VB3. The post-cracking modulus showed a significant dependence on the amount of y reinforcement
Fig. 4. Effect of reinforcement indexes on post-cracking modulus of (a) Normal strength concrete and (b) High strength concrete.
and the results of Eq. (4) were accurate in general. The reduction in the rate of increase in modulus for the partially overreinforced membrane VB3 was not observed, possibly because the reinforcement in this specimen was only slightly larger than the calculated balanced amount of reinforcement. Specimens VA1 to VA4 [17] contained equal amounts of orthogonal reinforcement (ρx fy−x /fc0 = ρy fy−y /fc0 ) and the average concrete strength was about 98 MPa. Specimens VA1 to VA3 failed in a fully under-reinforced mode while VA4 failed in a fully over-reinforced mode. A slight reduction in modulus is observed as the amount of reinforcement exceeded the ‘‘balanced’’ level. Series VA specimens contained significantly larger amounts of reinforcement than the VB specimens, which reflected on the shear modulus. In both series, the proposed equation captured the observed trends and accurately calculated the required modulus. It is to be noted that all specimens used in Fig. 4 were part of the validation set of data, and were not used in the development of Eq. (4). Fig. 5 shows the calculated post-cracking modulus in comparison with the observed shear stress–strain response of six specimens which represent the various modes of failure, and normal as well as high strength concrete. The line representing the calculated modulus is started from the point corresponding to the observed
K.N. Rahal / Engineering Structures 32 (2010) 218–225
223
Fig. 5. Observed response and calculated post-cracking shear modulus in six membrane elements.
cracking, and is continued to the value of the ultimate strength as calculated using the SMCS model (Eqs. (1)–(3)). A good agreement is observed in the modulus and in strains at the different levels of load, including the estimated service load levels. 5.2. Reinforced concrete beams In its present form, Eq. (4) is not applicable to beams because it does not take into consideration the effects of bending moments and the non-uniform distribution of the longitudinal
reinforcement. However, it can be applied in zones of negligible bending moments of beams with symmetrical longitudinal steel. Rahal and Collins [20] tested a series of RC beams subjected to combined shear and torsion. The specimen subjected to shear without torsion (RC2-2) is shown in Fig. 6. The loading was arranged in such a way as to create a zone of negligible bending moment at the center of the 1.66 m long test region. The beam was loaded via transverse ‘‘wing’’ beams to reduce the disturbance which is typically caused by the application of the load on the top side of the beam. The central part of the test region can hence be considered a zone of ‘‘pure shear’’.
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K.N. Rahal / Engineering Structures 32 (2010) 218–225
loading (force versus displacement control), modulus of elasticity of concrete and size and spacing of the reinforcement. However, it is suggested that these factors are not significant in comparison with those taken into account in Eq. (4), and will not significantly influence the overall accuracy of the model. On the other hand, the yield strength in the reinforcement is not significant because the post-cracking modulus in question does not extend to the yielding point and hence is obtained for the part of the response where the steel does not ‘‘know’’ how far away it is from yielding. Hence the yield point stress need not be included in the equation for Gcr . The yield strength can be removed from Eq. (4), and the constant (135) will be divided by an average yield strength (say 420 if MPa units are used). Similar changes affect the limit κ . However, the yield stress does not vary considerably in practical cases and its omission or inclusion in the equation does not affect the accuracy of the model. To maintain the simplicity of the model, allow easier implementation of Eq. (4) in the SMCS model and to enable users to take advantage of Eq. (5), the yield strength is maintained in the equation of the shear modulus.
a
7. Future work
b
245 70
640
545
340
Fig. 6. Observed response and calculated post-cracking shear modulus in ‘‘pure shear’’ zone of RC beam.
The test region was reinforced with No. 10 double legged stirrups (fy−y = 466 MPa, bar area = 100 mm2 ) spaced at 125 mm. Near the center, the developed longitudinal bars were ten No. 25 bars (fy−x = 480 MPa, bar area = 500 mm2 ). Full details are found elsewhere [20]. Due to the symmetric longitudinal reinforcement and the absence of a significant flexural moment, the central part of the test region can be considered a ‘‘pure shear’’ region and the proposed equation for Gcr can be applied without modification. The shear strains were measured on the two vertical faces of the test region using concrete surface targets. The measured shear strains over the central 600 mm width by 600 mm height on both sides are averaged and the results shown in Fig. 6. The cracked shear modulus was calculated to be 664 MPa which compares well with the observed value of 712 MPa. The ratio of the experimental to the calculated modulus is 1.07. The post-cracking behavior is constructed in a manner similar to that in Fig. 5, and is compared to the experimental results in Fig. 6. A good agreement is observed, in particular at estimated service load level. 6. Other factors affecting Gcr It is to be noted that factors other than those accounted for in Eq. (4) influence the post-cracking behavior of membranes. Such factors include the size of the specimens, rate of loading, type of
An accurate calculation of the strains at service load requires an accurate modeling of the pre-cracking response. The cracking stress depends on many factors such as fc0 , the age of the specimen on the day it was tested and on the curing conditions at early ages [21]. In its present form, the proposed equation is applicable to RC membrane elements. The applicability of the SMCS in its original graphical format [22] was extended to symmetrically and unsymmetrically reinforced beams subjected to shear, bending and axial load [23] using the concept of superposition or reinforcement. Hence, the proposed method has the potential of being extended to RC beams. Work towards incorporating the pre-cracking behavior and extending the applicability of the method to beams is in progress. A reduced cracked shear modulus can be used in the structural analysis to eliminate most or all iterations required to calculate the shear deformations in structural systems such as those reported in Refs. [2,3]. 8. Conclusions Based on this study, the following conclusions can be presented: 1. A simple equation is proposed to calculate the post-cracking modulus of reinforced concrete membranes subjected to monotonic shear loading. It relates the modulus to a ‘‘combined’’ reinforcement index of the orthogonal reinforcement. 2. In spite of its simplicity and empirical nature, the proposed equation is capable of capturing the trends observed in the tests such as the influence of variable concrete strength (normal and high strength concrete) and variable amounts of reinforcement. This is due to the ability of the combined reinforcement index √ ωx ωy and the over-reinforcement limit κ to model the key aspects of the shear behavior. Further work is required to integrate the model with a precracking response equation to allow a simple modeling of the full response of membranes and to generalize it to cover the case of RC beams subjected to flexure and shear. References [1] ACI-318. Building code requirements for reinforced concrete and commentary ACI 318M-08, American Concrete Institute, Committee 318, 2008. [2] Vecchio FJ, Emara MB. Shear deformation in reinforced concrete frames. ACI Struct J 1992;89:46–56.
K.N. Rahal / Engineering Structures 32 (2010) 218–225 [3] Debernardi PG, Taliano M. Shear deformation in reinforced concrete beams with thin webs. Mag Concrete Res 2006;58:157–71. [4] Vecchio FJ, Collins MP. Modified compression field theory for reinforced concrete elements subjected to shear. ACI J 1986;83:219–31. [5] Hsu TTC, Zhang L. Nonlinear analysis of membrane elements by fixed angle softened-truss model. ACI Struct J 1997;94:483–92. [6] Soltani M, An X, Maekawa K. Computational model for post cracking analysis of RC membrane elements based on local stress–strain characteristics. Eng Struct 2003;25:993–1003. [7] Kwak HG, Kim DY. Material nonlinear analysis of RC shear walls subject to monotonic loadings. Eng Struct 2004;26:1517–33. [8] Navarro Gregori J, Miguel Soza P, Fernández Prada MA, Filippou FC. A 3D numerical model for reinforced and prestressed concrete elements subjected to combined axial, bending, shear and torsion loading. Eng Struct 2007;29: 3404–19. [9] Mostafaei H, Vecchio FJ, Kabeyasawa T. Nonlinear displacement-based response prediction of reinforced concrete columns. Eng Struct 2008;30: 2436–47. [10] Oliver J, Linero DL, Huespe AE, Manzoli OL. Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach. Comput Methods Appl Mech Eng 2007;197:332–48. [11] Zhu RRH, Hsu TTC, Lee JY. Rational shear modulus for smeared-crack analysis of reinforced concrete. ACI Struct J 2001;98:443–50. [12] Pang X, Hsu TTC. Behavior of reinforced concrete membranes in shear. ACI Struct J 1995;92:665–79.
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[13] Rahal KN. Simplified design and capacity calculations of shear strength in reinforced concrete membrane elements. Eng Struct 2008;30:2782–91. [14] Bræstrup MW. Plastic analysis of shear in reinforced concrete. Mag Concrete Res 1974;26:221–8. [15] Nielsen MP. Limit analysis and concrete plasticity, 2nd ed. Englewood Cliffs (NJ): Prentice Hall; 1999. [16] Porasz A. An investigation of the stress–strain characteristics of high strength in shear. MASc thesis. Dept. of Civil Engineering, University of Toronto; 1989. 173p. [17] Hsu TTC, Zhang L. Behavior and analysis of 100 MPa concrete membrane elements. J Struct Division 1998;124:24–34. [18] Marti P, Meyboom J. Response of prestressed concrete elements to in-plane shear forces. ACI Struct J 1992;89:503–13. [19] Vecchio FJ, Collins MP, Aspiotis J. High-Strength concrete elements subjected to shear. ACI Struct J 1994;91:423–33. [20] Rahal KR, Collins MP. Experimental evaluation of ACI and AASHTO-LRFD design provisions for combined shear and torsion. ACI Struct J 2003;100: 277–82. [21] Bentz EC. Sectional analysis of reinforced concrete members. Ph.D. thesis. Dept. of Civil Engineering, University of Toronto; 2000. 310p. [22] Rahal KN. Shear strength of reinforced concrete, part I: Membrane elements subjected to pure shear. ACI Struct J 2000;97:86–93. [23] Rahal KN. Shear strength of reinforced concrete, part II: Beams subjected to shear, bending moment and axial loads. ACI Struct J 2000;97:219–24.
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Post-cracking shear modulus of reinforced concrete membrane elements Khaldoun N. Rahal ∗ Civil Engineering Department, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
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Article history: Received 16 October 2008 Received in revised form 29 August 2009 Accepted 3 September 2009 Available online 18 September 2009 Keywords: Deformation Membrane Mode of failure Modulus of rigidity Reinforced concrete Shear Strength
abstract The shear modulus of reinforced concrete membrane elements subjected to monotonic in-plane shearing stresses is typically calculated using the elastic theory. Upon cracking, however, the elements suffer a significant loss in rigidity and hence the shear deformations at service load levels cannot be accurately calculated using the pre-cracking modulus. This paper presents a simple empirical equation for the calculation of the post-cracking shear modulus, given in the form of the tangent slope of the shear stress–strain curve. The proposed empirical equation is based on the experimental results from forty membrane specimens, and relates the cracked modulus to a ‘‘combined’’ index of the reinforcement in the orthogonal directions. In spite of its empirical nature and simplicity, the equation is capable of capturing the effect of different amounts of reinforcement and concrete strength. Hence it applies to both underreinforced and over-reinforced membranes, and to normal and high strength concrete. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction The design of many reinforced concrete (RC) structures requires satisfying a serviceability limit state where the calculated deflections are checked against acceptable limits. In RC beams subjected to transverse loading, for example, vertical deflections are calculated based on the flexural moments and stiffness, while shear deflections are neglected. The significant reduction in the flexural stiffness after cracking in beams has long been recognized, and many building design codes provide provisions to calculate a reduced cracked flexural stiffness. For example, the ACI code [1] includes an equation to calculate an ‘‘effective’’ cracked moment of inertia and in some cases it allows the use of a value equal to a specific percentage of the gross moment of inertia. Since shear deformations are typically neglected, design codes do not recognize the considerable reduction in shear modulus after diagonal cracking. However, many studies have shown the importance of accounting for the shear deformations, especially in the post-cracking range of the behavior of shear-critical elements. Based on an experimental and theoretical investigation, Vecchio and Emara [2] concluded that transverse shear strains reduced the lateral stiffness of RC frames and increased the (P–∆) effects. Debernardi and Taliano [3] tested thin-webbed I-beams and observed that the shear deformations contributed about 25% of
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the total vertical deflections. A detailed nonlinear analysis of the I-beams led to the same conclusion. Hence, an accurate calculation of the deformation of some shear-critical RC structures can only be achieved by adequate consideration of the post-cracking behavior of these members. There are numerous behavioral models that are capable of calculating the full response of shear-critical reinforced concrete elements [4–10], including the post-cracking behavior. However, these advanced models require the use of computers to solve for the solution using an iterative procedure. The development of simpler procedures has also attracted considerable attention. Zhu et al. [11] listed thirteen nonlinear constitutive relationships for cracked concrete in shear which can be used to calculate the post-cracking modulus of membranes. The level of the complexity of these relationships varies, but they all require the use of an iterative procedure to calculate the post-cracking modulus. This paper presents a simple non-iterative empirical equation for the post-cracking modulus of membrane elements subjected to pure shear. This equation is based on the experimental results from forty elements subjected to pure shear, and provides a step towards a simple modeling of the full shear stress–strain relationship of membrane and beam elements. 2. Response of membrane elements subjected to pure shear Fig. 1 shows an orthogonally reinforced membrane element and the experimentally observed relationship between the shearing stress (v ) and the shearing strain (γ ) for the membrane specimen A3 tested by Pang and Hsu [12]. These results can be used to identify the key stages of the behavior of such an element. Before cracking, the response was mainly linear, and the slope of the curve in
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Nomenclature fc0 fy−x fy−y Gcr Gcr–calc Gcr–exp
vn κ ρx ρy ωx ωy v
compressive strength of concrete yield stress of x-direction reinforcement yield stress of y-direction reinforcement post-cracking shear modulus calculated post-cracking shear modulus experimentally observed post-cracking shear modulus ultimate shearing strength calculated using SMCS model ‘‘balanced’’ reinforcing index and concrete crushing factor reinforcement ratio in x direction reinforcement ratio in y direction reinforcement index in x direction reinforcement index in y direction applied shear stress
this region is referred to as the shear modulus. Typically, the theory of elasticity is considered adequate to calculate the shear modulus as a function of the modulus of elasticity of concrete tested in uniaxial compression. Upon cracking at (vcr ), the mechanism of resistance changed significantly leading to a transition zone between the pre- and the post-cracking parts of the behavior. The shape of the transition zone depends mainly on whether the loading machine is operated using force or displacement control. In the postcracking zone, there was a significant decrease in slope (tangent modulus), which remained nearly constant until the reinforcement yielded at v = vy . A sudden reduction in the modulus was observed after the yielding of the steel. The post-yielding strength was not significant in this specimen because the reinforcement in the two orthogonal directions yielded. In this paper, the slope of the precracking part of the response is referred to as the uncracked modulus, while the post-cracking and before steel yielding/concrete crushing is referred to as the post-cracking modulus (Gcr ). The latter is the subject of this study. Fig. 1 also shows the nominal shear strength calculated using the ACI code [1], and the shear stress at service conditions (estimated at about 75% of the nominal calculated capacity). The service load typically fits in the post-cracking region of the response, between the cracking and the yielding shear stresses. The use of the pre-cracking modulus under-estimates the shear strains at service load considerably. The post-cracking modulus is required to accurately calculate the strain at service load and to model an important part of the response that cannot be accurately obtained using elastic theory. A recently developed model named the simplified model for combined stress resultants (SMCS) relates the ultimate shear capacity and mode of failure of the membranes to the concrete compressive strength and the amount and strength of the orthogonal reinforcement [13]. SMCS elegantly combines these factors in the orthogonal reinforcement indexes. This model was shown to be able to accurately calculate the strength and the mode of failure of membranes made of normal and high strength concrete. Full details of SMCS can be found elsewhere [13]. It is proposed that the post-cracking modulus Gcr depends on the same main factors that influence the strength of the membrane elements. The SMCS model is briefly described in the following section to introduce the reinforcement indexes and the balanced reinforcement limits, and then the equation for Gcr is related to the same factors.
Fig. 1. Experimentally observed shear stress–strain response of a reinforced concrete membrane.
3. SMCS model The nominal shear strength vn is calculated as follows:
vn fc0
=
√ ωx ωy ≤ κ
(1)
where the reinforcement indexes ωx and ωy in the x- and ydirections respectively are:
ωx = ωy =
ρx fy−x fc0
ρy fy−y fc0
≤κ
(2a)
≤ κ.
(2b)
The upper limit (κ ) set on ωx and ωy is an over-reinforcement limit, and leads to a crushing limit on vn /fc0 (as shown in Eq. (1)). This limit depends on the concrete compressive strength fc0 and is given by the following equation:
κ = 1/3 − fc0 /900 0
(3)
where fc is in MPa. Eq. (3) is based on a review of the test results from 50 specimens subjected to in-plane shearing and tensile normal stresses [13]. The reduction of the normalized shear stress at higher concrete strengths is well recognized and has been quantified in numerous studies such as the theory of plasticity [14, 15], the modified compression field theory (MCFT) [16] and the softened truss model (STM) [17]. In addition, it is implied in the ACI code’s upper limit on the design shear force inpwhich the maximum permissible shear design stress is related to fc0 . If the mechanical reinforcing index calculated using Eq. (2) is limited by the over-reinforcement limit κ , this steel does not yield when the ultimate shear strength is reached. This leads to
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three possible modes of failure at ultimate conditions: (1) fully under-reinforced membrane where both x- and y-reinforcement yields; (2) partially under-reinforced membrane where only the x- or the y-reinforcement yields; and (3) fully over-reinforced membrane where crushing takes place before yielding in any of the reinforcement. Experimental evidence showed that increasing the amount of reinforcement increases the shear strength. However, this increase becomes less significant when the reinforcement is increased above a ‘‘balanced’’ reinforcement level [13]. Similar to the flexure theory, the ‘‘balanced’’ level refers to the amount of reinforcement required to ensure that first yielding of the steel occurs simultaneously with concrete crushing in compression, leading to ultimate conditions. The limited increase in strength by the reinforcement in excess of the balanced reinforcement is neglected to maintain the simplicity of the model. This is achieved by imposing the limit κ on the ‘‘usable’’ reinforcement indexes in Eq. (2). This same limit is translated into the upper limit on shear strength when concrete crushes prior to yielding in fully overreinforced membranes. See Eq. (1). The calculations of the model were checked against the observed ultimate shear strengths from eighty four test specimens subjected to in-plane shearing and normal stresses [13]. The average of the observed to the calculated strength was 1.03 and the coefficient of variation was 11.3%. The experimental mode of failure was available in forty nine membranes, and the SMCS model was capable of calculating the correct mode of failure in forty five of them.
a
b
4. Post-cracking modulus The post-cracking tangent modulus Gcr can be obtained from the v –γ response observed in pure shear tests. Results from forty tests [4,5,12,17–19] on membrane elements were analyzed, and Gcr was calculated as the slope of the best straight line fit of the post-cracking, pre-yielding behavior. A consistent approach was followed in selecting the data measurements which are considered to be within the postcracking part of the response of a specimen. The experimental readings in the pre-cracking (v ≤ vcr ) and the post-yielding (v ≥ vy ) parts of the response were not considered. To exclude the transition zone between the pre-cracking and the post-cracking response, the experimental measurements between vcr and 1.2vcr were not considered. This margin was found to be suitable for this purpose in all the specimens. A regression analysis of the data points which fit within the post-cracking part of the response was used to find Gcr . The results of this exercise for the forty test specimens are listed in Table 1. The SMCS showed that the main properties affecting the shear behavior of the membrane can be elegantly combined in a single √ factor ωx ωy , where each of the reinforcement indexes is limited to the balanced reinforcement ratio given as shown in Eq. (2). The relationship between Gcr and this factor is investigated. Of the 40 tests available, 21 specimens were tested at the University of Toronto (UT) [4,18,19] and 19 at the University of Houston (UH) [5,12,17]. The available data is split into two groups; one is used to develop the equation for the modulus and the other to provide an independent evaluation. The development set is taken as the 21 UT tests. This provides a good challenge to the adequacy of the equation since the two sets of data differ significantly. For instance, the concrete strength fc0 ranged from 11.6 to 66.1 MPa in the UT tests and from 41.3 to 103.1 MPa in the UH tests. The average fc0 in the two sets of tests was 30 MPa and 69 MPa respectively. The size of the specimens also changed significantly. Twenty of the UT specimens were 890 × 890 × 70 mm and one was 1626 × 1626 × 287 while the 19 UH specimens were
Fig. 2. (a) Proposed relation based on the UT tests and (b) corroboration between observed and calculated modulus for all tests.
1400 × 1400 × 178 mm. In addition, the UT specimens were tested under stress control loading conditions while the UH specimens were tested under deformation control conditions. √ Fig. 2(a) shows a plot of the factor Gcr /fc0 and the factor ωx ωy for the 21 UT test results. An obvious trend is observed. The best √ linear fit is Gcr /fc0 = 153.4 ωx ωy − 3.9, which gives a ratio of observed to calculated modulus of 1.00 and a coefficient of variation of 17.8%. The best-fit line which passes through the origin √ of the axes is Gcr /fc0 = 135.4 ωx ωy with an average of observed to calculated modulus of 0.97 and a coefficient of variation of 18.0%. The latter equation provides a simpler relationship without a significant loss of accuracy. Hence, it is proposed that the postcracking modulus is calculated using the following equation:
√
Gcr /fc0 = 135 ωx ωy
(4)
where Gcr carries the same units as the concrete compressive strength fc0 . Table 1 compares the calculations of Eq. (4) against the experimental results obtained from the (UH) evaluation data set. The average of the experimental to the calculated values of the modulus is 1.03 and the coefficient of variation is 12.5%. The coefficient of variation in the evaluation set of data was slightly better than in the development set, which enhances the confidence in the performance of the equation. Fig. 2(b) shows a plot of the ratio Gcr–exp /Gcr–calc versus the concrete strength for the 40 tests. Eq. (4) performed adequately in specimens made of higher strength concrete though it was developed based on data from specimens made of normal strength concrete.
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Table 1 Details of specimens and experimental verification of shear modulus. ρx fy−x
ρy fy−y
fc0
fc0
0.250 0.120 0.096 0.163 0.159 0.700 0.340 0.269 0.523 0.430 0.419 0.419 0.417 0.382 0.385
27.0 49.9 43.0 66.1 58.4 55.9
Size (mm)
Specimen
fc0 (MPa)
[4]
890 × 890 × 70
PV1 PV3 PV4 PV5 PV6 PV9 PV10 PV11 PV12 PV19 PV20 PV21 PV22 PV26 PV27
34.5 26.6 26.6 28.3 29.8 11.6 14.5 15.6 16.0 19.0 19.6 19.5 19.6 21.3 20.5
[18]
1626 × 1626 × 287
PP1
890 × 890 × 70
PA1 PA2 PHS2 PHS3 PHS8
Source
[19]
21 University of Toronto tests
Gcr–exp (GPa)
Gcr–calc (GPa)
Gcr– exp Gcr–calc
0.235 0.120 0.096 0.163 0.159 0.700 0.190 0.197 0.076 0.112 0.134 0.201 0.327 0.220 0.385
1.10 0.29 0.42 0.44 0.71 0.54 0.41 0.70 0.38 0.42 0.52 0.64 0.90 0.69 0.88
1.13 0.43 0.35 0.62 0.64 0.50 0.48 0.49 0.33 0.48 0.54 0.66 0.82 0.75 0.86
0.98 0.67 1.20 0.71 1.10 1.08 0.84 1.43 1.14 0.87 0.97 0.98 1.09 0.92 1.02
0.345
0.116
0.66
0.68
0.97
0.173 0.202 0.296 0.335 0.350
0.086 0.100 0.032 0.073 0.116
0.69 0.75 0.59 1.08 1.27
0.82 0.82 0.82 1.10 1.34
0.85 0.91 0.72 0.98 0.95
Average Coefficient of variation (%)
0.97 18.0
[12]
1400 × 1400 × 178
A2 A3 A4 B1 B2 B3 B4 B5 B6
[5]
1400 × 1400 × 178
HB3 HB4
66.8 62.9
0.120 0.223
0.040 0.042
0.72 0.86
0.62 0.83
1.15 1.04
1400 × 1400 × 178
VA1 VA2 VA3 VA4 VB1 VB2 VB3 VB4
95.1 98.2 94.6 103.1 98.2 97.6 102.3 96.9
0.056 0.100 0.173 0.239 0.100 0.167 0.266 0.085
0.056 0.100 0.173 0.239 0.054 0.055 0.052 0.028
0.69 1.37 2.14 2.90 1.06 1.27 1.74 0.53
0.72 1.32 2.21 3.05 0.98 1.26 1.48 0.63
0.96 1.04 0.97 0.95 1.08 1.01 1.17 0.84
[17]
41.3 41.7 42.5 45.3 44.1 44.9 44.8 42.9 43.0
0.134 0.192 0.330 0.123 0.181 0.178 0.313 0.326 0.326
0.134 0.192 0.330 0.059 0.126 0.059 0.059 0.130 0.186
0.89 1.25 1.65 0.46 0.95 0.58 0.59 1.34 1.59
0.75 1.08 1.64 0.52 0.90 0.62 0.78 1.11 1.34
1.19 1.16 1.01 0.90 1.05 0.92 0.75 1.21 1.19
19 University of Houston tests
Average Coefficient of variation (%)
1.03 12.5
All 40 tests
Average Coefficient of variation (%) Root-mean square error
1.00 15.6 0.117
Table 1 shows that for the 40 specimens, the average of observed-to-calculated ratios is 1.00 and the coefficient of variation is 15.6%. The root-mean square error is 0.117. Comparison with Eq. (1) shows that Eq. (4) is numerically equivalent to: Gcr = 135vn .
(5)
This equation can be used to estimate the post-cracking modulus before the design of the reinforcement. 5. Evaluation of the proposed equation This section presents an evaluation of the ability of the proposed equation to capture the effects of the main variables that influence the post-cracking modulus of membrane elements. These are namely the amount of reinforcement and the concrete strength. The results of the proposed model are also checked against the experimental results from a zone of RC beam subjected to shear and a relatively negligible bending moment.
5.1. Membrane elements The ability of the model to capture the influence of different levels of concrete strength on Gcr of under-reinforced and overreinforced membranes is investigated and the results are shown in Fig. 3. The modulus from tests of three under-reinforced specimens PV20 [4], B3 [12] and VB4 [17] is shown. These specimens contained similar levels of reinforcement (ρx fy−x = 8.1 MPa, ρy fy−y = 2.7 MPa) and variable concrete strength. The test results showed that the x- and y-reinforcement in B3 and VB4 and the y-reinforcement in PV20 yielded at ultimate conditions. The observed modulus is shown to be relatively independent of the concrete strength. This is adequately reflected in Eq. (4) which, for fully under-reinforced membranes, can be reduced to: Gcr = 135
q
ρx fy−x
ρy fy−y .
(6)
Fig. 3 shows that Eq. (4) or (6) slightly over-estimates the modulus of this series of tests but the results reflected the fact that
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Fig. 3. Effect of concrete strength and mode of failure on post-cracking modulus.
varying the concrete strength has a limited effect on Gcr of underreinforced membranes. Specimens PV9, PV27 and PV1 [4] were reinforced with nearly similar orthogonal reinforcement (ρx fy−x = ρy fy−y = 8.1 MPa) and failed in a fully-over-reinforced mode. The experimental results in Fig. 3 show that the shear modulus is proportional to the concrete strength. This is also reflected in Eq. (4), which, for fully overreinforced membranes, can be rearranged to give: Gcr = 135 1/3 − fc0 /900 fc0 .
(7)
In general, the proposed equation captures the trend observed in the tests, but slightly over-estimates the effect of higher concrete strength. The ability of the model to capture the effects of different amounts of reinforcement on Gcr of normal and high strength concrete membranes is also investigated. The modulus obtained from two series of tests on normal strength concrete membranes is shown in Fig. 4. In the first series, specimens PV12, PV19 to PV22, PV26 and PV27 [4] had an average concrete compressive strength of about 19.4 MPa and an average ρ fy /fc0 of x-direction reinforcement of about 0.43. The y-reinforcement was variable and covered partially under-reinforced levels in five specimens PV12, PV19 to PV21, PV26, and fully over-reinforced levels in two specimens (PV22 and PV27). The figure shows that Gcr increased with the increase in ρy fy−y /fc0 in under-reinforced elements only. Hence, neglecting the contribution of reinforcement in excess of the balanced amounts is justified in this series of tests. The results of Eq. (4) showed a good agreement with the observed modulus. In the second series, specimens B4, B5, B6 and A4 [12] had a nearly equal concrete compressive strength averaging 43.3 MPa and a ρx fy−x /fc0 value of about 0.32. The y-reinforcement was variable and yielded in all four specimens before ultimate conditions. The x-reinforcement yielded in specimen A4 only. Fig. 4(a) shows that Eq. (4) adequately models the reduction in the rate of increase of modulus at higher reinforcement levels, and the overall behavior of normal strength concrete specimens. Fig. 4(b) shows the modulus values obtained from two series of tests on high strength concrete specimens [17]. The average concrete strength and ρy fy−y /fc0 values of specimens VB1, VB2 and VB3 were 99 MPa and 0.053 respectively while the x-reinforcement was variable. VB1 and VB2 failed in a fully under-reinforced mode while only the y-steel yielded in VB3. The post-cracking modulus showed a significant dependence on the amount of y reinforcement
Fig. 4. Effect of reinforcement indexes on post-cracking modulus of (a) Normal strength concrete and (b) High strength concrete.
and the results of Eq. (4) were accurate in general. The reduction in the rate of increase in modulus for the partially overreinforced membrane VB3 was not observed, possibly because the reinforcement in this specimen was only slightly larger than the calculated balanced amount of reinforcement. Specimens VA1 to VA4 [17] contained equal amounts of orthogonal reinforcement (ρx fy−x /fc0 = ρy fy−y /fc0 ) and the average concrete strength was about 98 MPa. Specimens VA1 to VA3 failed in a fully under-reinforced mode while VA4 failed in a fully over-reinforced mode. A slight reduction in modulus is observed as the amount of reinforcement exceeded the ‘‘balanced’’ level. Series VA specimens contained significantly larger amounts of reinforcement than the VB specimens, which reflected on the shear modulus. In both series, the proposed equation captured the observed trends and accurately calculated the required modulus. It is to be noted that all specimens used in Fig. 4 were part of the validation set of data, and were not used in the development of Eq. (4). Fig. 5 shows the calculated post-cracking modulus in comparison with the observed shear stress–strain response of six specimens which represent the various modes of failure, and normal as well as high strength concrete. The line representing the calculated modulus is started from the point corresponding to the observed
K.N. Rahal / Engineering Structures 32 (2010) 218–225
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Fig. 5. Observed response and calculated post-cracking shear modulus in six membrane elements.
cracking, and is continued to the value of the ultimate strength as calculated using the SMCS model (Eqs. (1)–(3)). A good agreement is observed in the modulus and in strains at the different levels of load, including the estimated service load levels. 5.2. Reinforced concrete beams In its present form, Eq. (4) is not applicable to beams because it does not take into consideration the effects of bending moments and the non-uniform distribution of the longitudinal
reinforcement. However, it can be applied in zones of negligible bending moments of beams with symmetrical longitudinal steel. Rahal and Collins [20] tested a series of RC beams subjected to combined shear and torsion. The specimen subjected to shear without torsion (RC2-2) is shown in Fig. 6. The loading was arranged in such a way as to create a zone of negligible bending moment at the center of the 1.66 m long test region. The beam was loaded via transverse ‘‘wing’’ beams to reduce the disturbance which is typically caused by the application of the load on the top side of the beam. The central part of the test region can hence be considered a zone of ‘‘pure shear’’.
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loading (force versus displacement control), modulus of elasticity of concrete and size and spacing of the reinforcement. However, it is suggested that these factors are not significant in comparison with those taken into account in Eq. (4), and will not significantly influence the overall accuracy of the model. On the other hand, the yield strength in the reinforcement is not significant because the post-cracking modulus in question does not extend to the yielding point and hence is obtained for the part of the response where the steel does not ‘‘know’’ how far away it is from yielding. Hence the yield point stress need not be included in the equation for Gcr . The yield strength can be removed from Eq. (4), and the constant (135) will be divided by an average yield strength (say 420 if MPa units are used). Similar changes affect the limit κ . However, the yield stress does not vary considerably in practical cases and its omission or inclusion in the equation does not affect the accuracy of the model. To maintain the simplicity of the model, allow easier implementation of Eq. (4) in the SMCS model and to enable users to take advantage of Eq. (5), the yield strength is maintained in the equation of the shear modulus.
a
7. Future work
b
245 70
640
545
340
Fig. 6. Observed response and calculated post-cracking shear modulus in ‘‘pure shear’’ zone of RC beam.
The test region was reinforced with No. 10 double legged stirrups (fy−y = 466 MPa, bar area = 100 mm2 ) spaced at 125 mm. Near the center, the developed longitudinal bars were ten No. 25 bars (fy−x = 480 MPa, bar area = 500 mm2 ). Full details are found elsewhere [20]. Due to the symmetric longitudinal reinforcement and the absence of a significant flexural moment, the central part of the test region can be considered a ‘‘pure shear’’ region and the proposed equation for Gcr can be applied without modification. The shear strains were measured on the two vertical faces of the test region using concrete surface targets. The measured shear strains over the central 600 mm width by 600 mm height on both sides are averaged and the results shown in Fig. 6. The cracked shear modulus was calculated to be 664 MPa which compares well with the observed value of 712 MPa. The ratio of the experimental to the calculated modulus is 1.07. The post-cracking behavior is constructed in a manner similar to that in Fig. 5, and is compared to the experimental results in Fig. 6. A good agreement is observed, in particular at estimated service load level. 6. Other factors affecting Gcr It is to be noted that factors other than those accounted for in Eq. (4) influence the post-cracking behavior of membranes. Such factors include the size of the specimens, rate of loading, type of
An accurate calculation of the strains at service load requires an accurate modeling of the pre-cracking response. The cracking stress depends on many factors such as fc0 , the age of the specimen on the day it was tested and on the curing conditions at early ages [21]. In its present form, the proposed equation is applicable to RC membrane elements. The applicability of the SMCS in its original graphical format [22] was extended to symmetrically and unsymmetrically reinforced beams subjected to shear, bending and axial load [23] using the concept of superposition or reinforcement. Hence, the proposed method has the potential of being extended to RC beams. Work towards incorporating the pre-cracking behavior and extending the applicability of the method to beams is in progress. A reduced cracked shear modulus can be used in the structural analysis to eliminate most or all iterations required to calculate the shear deformations in structural systems such as those reported in Refs. [2,3]. 8. Conclusions Based on this study, the following conclusions can be presented: 1. A simple equation is proposed to calculate the post-cracking modulus of reinforced concrete membranes subjected to monotonic shear loading. It relates the modulus to a ‘‘combined’’ reinforcement index of the orthogonal reinforcement. 2. In spite of its simplicity and empirical nature, the proposed equation is capable of capturing the trends observed in the tests such as the influence of variable concrete strength (normal and high strength concrete) and variable amounts of reinforcement. This is due to the ability of the combined reinforcement index √ ωx ωy and the over-reinforcement limit κ to model the key aspects of the shear behavior. Further work is required to integrate the model with a precracking response equation to allow a simple modeling of the full response of membranes and to generalize it to cover the case of RC beams subjected to flexure and shear. References [1] ACI-318. Building code requirements for reinforced concrete and commentary ACI 318M-08, American Concrete Institute, Committee 318, 2008. [2] Vecchio FJ, Emara MB. Shear deformation in reinforced concrete frames. ACI Struct J 1992;89:46–56.
K.N. Rahal / Engineering Structures 32 (2010) 218–225 [3] Debernardi PG, Taliano M. Shear deformation in reinforced concrete beams with thin webs. Mag Concrete Res 2006;58:157–71. [4] Vecchio FJ, Collins MP. Modified compression field theory for reinforced concrete elements subjected to shear. ACI J 1986;83:219–31. [5] Hsu TTC, Zhang L. Nonlinear analysis of membrane elements by fixed angle softened-truss model. ACI Struct J 1997;94:483–92. [6] Soltani M, An X, Maekawa K. Computational model for post cracking analysis of RC membrane elements based on local stress–strain characteristics. Eng Struct 2003;25:993–1003. [7] Kwak HG, Kim DY. Material nonlinear analysis of RC shear walls subject to monotonic loadings. Eng Struct 2004;26:1517–33. [8] Navarro Gregori J, Miguel Soza P, Fernández Prada MA, Filippou FC. A 3D numerical model for reinforced and prestressed concrete elements subjected to combined axial, bending, shear and torsion loading. Eng Struct 2007;29: 3404–19. [9] Mostafaei H, Vecchio FJ, Kabeyasawa T. Nonlinear displacement-based response prediction of reinforced concrete columns. Eng Struct 2008;30: 2436–47. [10] Oliver J, Linero DL, Huespe AE, Manzoli OL. Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach. Comput Methods Appl Mech Eng 2007;197:332–48. [11] Zhu RRH, Hsu TTC, Lee JY. Rational shear modulus for smeared-crack analysis of reinforced concrete. ACI Struct J 2001;98:443–50. [12] Pang X, Hsu TTC. Behavior of reinforced concrete membranes in shear. ACI Struct J 1995;92:665–79.
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[13] Rahal KN. Simplified design and capacity calculations of shear strength in reinforced concrete membrane elements. Eng Struct 2008;30:2782–91. [14] Bræstrup MW. Plastic analysis of shear in reinforced concrete. Mag Concrete Res 1974;26:221–8. [15] Nielsen MP. Limit analysis and concrete plasticity, 2nd ed. Englewood Cliffs (NJ): Prentice Hall; 1999. [16] Porasz A. An investigation of the stress–strain characteristics of high strength in shear. MASc thesis. Dept. of Civil Engineering, University of Toronto; 1989. 173p. [17] Hsu TTC, Zhang L. Behavior and analysis of 100 MPa concrete membrane elements. J Struct Division 1998;124:24–34. [18] Marti P, Meyboom J. Response of prestressed concrete elements to in-plane shear forces. ACI Struct J 1992;89:503–13. [19] Vecchio FJ, Collins MP, Aspiotis J. High-Strength concrete elements subjected to shear. ACI Struct J 1994;91:423–33. [20] Rahal KR, Collins MP. Experimental evaluation of ACI and AASHTO-LRFD design provisions for combined shear and torsion. ACI Struct J 2003;100: 277–82. [21] Bentz EC. Sectional analysis of reinforced concrete members. Ph.D. thesis. Dept. of Civil Engineering, University of Toronto; 2000. 310p. [22] Rahal KN. Shear strength of reinforced concrete, part I: Membrane elements subjected to pure shear. ACI Struct J 2000;97:86–93. [23] Rahal KN. Shear strength of reinforced concrete, part II: Beams subjected to shear, bending moment and axial loads. ACI Struct J 2000;97:219–24.