Simplified Shear Design of Structural Concrete Members (2005)

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5CHAPTER 1 INTRODUCTION AND RESEARCH APPROACH The goal of this project was to develop proposed simpli- fied shear design provisions for the AASHTO LRFD Bridge Design Specifications that would overcome perceived diffi- culties with using the current shear design provisions, which are the provisions of the Sectional Design Model (A5.8.3). This Sectional Design Model constitutes the general shear design requirements in the first three editions of the AASHTO LRFD Bridge Design Specifications (1, 7, and 17). Section 1.1 describes the problem that led to this project and begins with a summary of the LRFD Sectional Design Model (A5.8.3), followed by a brief description of the basis of this model, and a discussion of the differences between the AASHTO LRFD and Standard Specifications (AASHTO, 2002) shear design provisions. Section 1.2 summarizes the information that was available to develop the proposed simplified provisions. This information con- sists of an overview of what is known about the mecha- nisms of shear resistance, a summary of code provisions, and descriptions of available experimental test data and analysis methods for shear. Section 1.3 defines project objectives, the approach used for meeting these objectives, and project tasks. 1.1 THE AASHTO LRFD SHEAR DESIGN SPECIFICATIONS 1.1.1 Summary of the LRFD Sectional Design Model (S5.8.3) The AASHTO LRFD Section Design Model for Shear (A5.8.3) is a hand-based shear design procedure derived from the Modified Compression Field Theory (MCFT). Prior approaches focused on expressions for shear strength that were then modified for the effect of other forces. This is a comprehensive design approach for structural concrete members in which the combined actions of axial load, flexure, and prestressing are taken into account when com- pleting the shear design of any section of any member. In this approach, the nominal shear capacity is taken as a sum of a concrete component, a shear reinforcement com- ponent, and the vertical (or transverse) component of the prestressing: (Eq. 1)V V V Vn c s p= + + The concrete contribution is controlled by the value of the coefficient β as follows:. where f ′c is in ksi units (Eq. 2) The coefficient of 0.0316 is and is used to con- vert the relationship for Vc from psi to ksi units. A variable angle truss model is used to calculate the con- tribution of the shear reinforcement. See Equation 3 where the angle of the field of diagonal compression, θ, is used in calculating how many stirrups, [dvcot(θ)/s], are included in the transverse tie of the idealized truss. (Eq. 3) where dv ≥ 0.9d or 0.72h, whichever is greater. (Eq. 4) The values for β and θ are obtained from Table 1 for mem- bers that contain at least the minimum required amount of shear reinforcement (See Equation 5) and from Table 2 for members that contain less than that amount. where f ′c and fy are in ksi units (Eq. 5) To obtain values for β and θ from Table 1 (Av < Av,min), the designer selects the row in which to enter the table from the shear design stress ratio (v/f ′c ) and the column by the longi- tudinal strain
x at mid-depth, which may be taken as one-half of the strain in the longitudinal tension reinforcement,
t. This strain is equal to the force in the longitudinal tension reinforcement divided by the axial stiffness of the tension reinforcement. As shown in Equation 6 and illustrated in Figure 1, the effects of all demands on the longitudinal rein- forcement are taken into account: (Eq. 6) Equation 6 assumes that the member is cracked and, there- fore, only the axial stiffness of the reinforcement need be

x t u v u u p ps poM d N V V A f = = + + − − 2 0 5 0 5 2 / . . ( )cot( )θ ( )E A E As s p ps+ A f b sfv c v y ,min .≥ ′0 0316 V A f d s s v y v = cot( )θ 1 1000/ V f b dc c v v= ′0.0316β

considered when evaluating
t and
x. If
x is negative, then the member is uncracked and the axial stiffness of the uncracked concrete needs to be considered per Equation 7. (Eq. 7) where Act is the area of the concrete beneath mid-depth. Alternatively, the designer can conservatively take
x = 0 if Equation 6 yields a negative value. Table 1 shows that as the longitudinal strain becomes larger, the values for β decrease and the values for θ increase. This means that as the moment and longitudinal strain increase, both the magnitude of the concrete and shear rein- forcement contributions to shear resistance decrease. To obtain values for β and θ when Av < Av,min, Table 2 is used. As for members containing at least minimum shear reinforcement, the column by which the designer enters Table 2 is based on the value of the longitudinal strain at mid- depth,
x. To determine the row, the spacing of the layers of crack control reinforcement is used, sxe (see Equation 8 and Figure 2). (Eq. 8)s s a xe x g = + 1 38 0 63 . .

x t u v u u p ps poM d N V V A f = = + + − − 2 0 5 0 5 2 / . . ( )cot( )θ ( )E A E A A Es s p ps ct c+ + 6 where ag is the maximum aggregate size in inches and taken equal to 0 when f ′c ≥ 10 ksi. Table 2 shows that as sxe and
x increase, the value of β decreases and θ increases. The result is that, as the member becomes deeper and the value of the moment increases, the contributions of the concrete and shear reinforcement decrease. The LRFD Sectional Design Model introduced a new requirement into shear design provisions—the direct consid- eration of shear in determining the required capacity of the longitudinal reinforcement at any point along the length of the member (see Equation 9). (Eq. 9) In the end regions of prestressed concrete members, the development length of the strands at the location of the first diagonal crack must be taken into consideration when satis- fying the requirements of Equation 9. In the design of a member by the LRFD Sectional Design Model, the member can be considered to be divided into design spans of length dvcot(θ) as shown in Figure 3. Each design span can be designed for the shear force midway along the length of the span. If the load is applied to the top of the member, then a staggered shear design concept may be used in which each design span is designed for the lowest value of shear occurring within the design span. T N V M d A fu u u v ps psmin . . cot≥ + + −0 5 0 5 θ TABLE 1 Values of
and  for members with at least minimum shear reinforcement

The Sectional Design Model was developed for regions in which engineering beam theory applies and there is a uni- form flow of the diagonal compressive stresses. However, the LRFD specifications also permit the end region of mem- bers (the distance between the support and dvcot(θ)/2 from the support) that are subject to a complex state of stress to be 7 designed by the Sectional Design Model for the shear force at dvcot(θ)/2 from the support. Figure 4 is a flowchart of the entire procedure for use of the LRFD Sectional Design Model. To further illustrate this procedure, a brief example is given for the design of a sec- tion of the 72-inch-deep bulb-tee girder in Figure 5. (This TABLE 2 Values of
and  for members with less than minimum shear reinforcement Figure 1. Effects of axial load, moment, shear, and prestressing on longitudinal strain in non-prestressed member.

example was extracted from a design of a 120-foot single- span AASHTO-PCI bulb-tee beam bridge with no skew. The example briefly illustrates the shear design procedure in LRFD specifications. The critical section is taken at 0.06L from centerline of a support.) 1. Compute shear stress ratio v fu c′ = 0.115 v V V b du u p v v 0.7473 ksi= − = φ φ v fu c′ b dv v= = =6 in, 73.14 in, 0.9φ 28500 ksEp = A A f fs ps po c= = = ′ =0, 5.508 in 189.0 ksi, 6.52 , ksi, 23.4 kipsVp = V M Nu u u= = =316.2 kips, 2134.0 ft-kips, 0 kips, 8 2. Assume
x as , then obtain θ = 22.8° and β = 2.94 from Table 1 (S5.8.3.4-1). 3. Compute
x Given that
x is negative, recalculate if
x satisfies the assumed range, then θ = 22.8° and β = 2.94 are O.K. = − × −0.080 10 3
x u v u u p ps po c c M d N V V A f E A E = + + − − + 0 5 0 5 2 . . ( )cot ( θ s s p psA E A+ ) = − × ≤−1.091 10 0.0023
x u v u u p ps po s s M d N V V A f E A E = + + − − + 0 5 0 5 2 . . ( )cot ( θ p psA ) − × ≤ ≤ − ×− −0.10 10 0.05 103 3
x Flexural tension side εx h/2 h/2 bw Act As sz 0.003b szv Area ≥ sz = dv Figure 2. Evaluation of crack spacing parameter Sx. Figure 3. Design regions and shear demand using the sectional design model. dvcotθ 0.5dvcotθ0.5dvcotθ 0.5dvcotθ2 0.5dvcotθ2 dvcotθ2 Sh e a r Location Design Span Design Span Vu Vr Design Section Design Section dvcotθ dvcotθ2 Sh e a r Location Design Span Design Span Vu Vr Design Section Design Section

9From , and set fy = 60.0 ksi, then Use #4 bar double legs @12 in., This provides 5. Compute maximum limit check: , O.K. 6. Compute longitudinal reinforcement check at the end of beam , O.K.A f A fs y ps ps+ = ≥460.1 kips 456.4 kips M d N V V Vu v u u s pφ φ φ θ+ + − −     =0 5 0 5. . cot 456.4 kips A f A f M d N V V Vs y ps ps u v u u s p+ ≥ + + − −φ φ φ θ0 5 0 5. ( . )cot V V f b dc s c v v+ = ≤ ′ =448.5 kips 713.1 kips0 25. V V f b dc s c v v+ ≤ ′0 25. V A f d s s v y v = + = (cot cot )sinθ α α 344.6 kips >/in 0.021 in /in2 A s v = 0.033 in /i2 A s V f d v s y v = = cot θ 0.021 in /in2 V A f d s s v y v = +(cot cot )sinθ α α Figure 4. Flowchart for LRFD design procedure. Figure 5. Design example implementing the LRFD sectional design model. Start Determine b and d Eq. 5.8.2.9 v v Calculate Vp Calculate shear stress ratio v/fc′, Eq. 5.8.2.9-1 If the section is within the transfer length of any strands, calculate the average effective value of fpo If the section is within the development length of any reinforcing bars, calculate the effective value of As Assume value of εx and take θ and β from corresponding cell of Table 1. Calculate εx Eq. 5.8.3.4.2-1 Is calculated εx less than assumed value? Is assumed εx too conservative? ( too high?) Can longitudinal reinforcement resist required tension? Eq.5.8.3.5 Can you use excess shear capacity to reduce the longitudinal steel requirements in Eq.5.8.3.5-1? Choose values of θ and β corresponding to large εx , Table 1 Provide additional longitudinal reinforcement Determine transverse reinforcement to ensure V < φVn Eq. 5.8.3.3 Yes No Yes End No No No Yes Yes 4. Determine shear reinforcement , V V V Vu c ps 224.0 kips= − − =φ V f b dc c v v= ′ =0.0316 103.9 kipsβ

1.1.2 Basis of the LRFD Sectional Design Model The LRFD Sectional Design Model is derived from the MCFT, a behavioral model that can be used to predict the shear-stress versus shear-strain response of an element sub- jected to in-plane shear and membrane forces. The theory consists of constitutive, compatibility, and equilibrium rela- tionships that enable determination of the state of stress (fx, fy, vxy) in structural concrete corresponding to a specific state of strain (
x,
y,
xy) as shown in Figure 6. The full implementation of the MCFT is possible in a two-dimensional continuum analysis tool, such as that done in program VecTor2 (18). The MCFT is also implemented in Response 2000, a multilayer sectional analysis tool that can predict the response of a section to the simultaneously occurring actions of axial load, prestressing, moment, and shear. In Response 2000, the plane section assumption is used which constrains the distribution of shear stress over the depth of the section. For each layer, an equivalent dual 10 section analysis is performed that uses the MCFT to solve for the angle of diagonal compression, longitudinal stress, and shear stress in each layer (19). In a typical analysis, the cross section will be divided into more than 100 layers. The LRFD Section Design Model is also derived from the MCFT, but developing this hand-based general shear design method (20) required several additional simplifications and assumptions to be made. The most significant of these was that the distribution of shear stress over the depth of the sec- tion was taken as the value at mid-depth as calculated by the MCFT using the designer-calculated longitudinal strain,
x, at mid-depth. Additional assumptions that were made in the development of the LRFD Sectional Design Model were that the shape of the compressive stress-strain response of the concrete was parabolic with a strain at peak stress of −0.002, and, for mem- bers with Av ≥ Av,min, that the spacing of the cracks was 12 inches and the size of the maximum aggregate was 0.75 inches. Figure 6. MCFT for predicting shear response of an element.

Although the LRFD specifications were derived from the MCFT, because of the significant simplification and assump- tions used in developing this method, the shear capacity determined using the LRFD Sectional Design Model should not be considered equivalent to the shear capacity calculated by the MCFT. 1.1.3 Comparison of AASHTO LRFD and AASHTO Standard Specifications The LRFD Sectional Design Model provides a complete shear design approach for structural concrete in which the actions of axial loading, moment, and prestressing are con- sidered explicitly. This approach is a significant departure from the shear design procedures of the AASHTO Standard Specifications and ACI318-02. The key differences between the AASHTO LRFD and Standard Specifications are as follows: • LRFD Eliminates Approach of Evaluating Vc Based on the Diagonal Cracking Load In the AASHTO Standard Specifications, the concrete contribution to shear resistance, Vc, is taken as the load at which diagonal cracking is expected to occur. In this approach, Vc is taken as the lesser of the force required to cause web-shear cracking, Vcw, or flexure-shear cracking, Vci. In the LRFD approach, Vc is taken as a measure of the con- crete contribution at ultimate. A significant effect of this dif- ference is that with LRFD the state of shear cracking in a member cannot be used to estimate the force that it has sup- ported nor can the designer evaluate whether or not the mem- ber is likely to be cracked in shear under service loads. • LRFD Introduces Use of a Variable Angle Truss Model In the Standard Specifications, the contribution of the shear reinforcement to capacity is determined using a 45-degree parallel chord truss model. In this way, the number of stirrups considered to lift the diagonal compres- sion across a single shear crack is taken as d/s where d is the depth of the member and s is the spacing of the shear reinforcement. In the LRFD Sectional Design Model, the angle of diagonal compression can be taken as ranging from 18.1 to 43.9 degrees and where the number of stirrups considered to lift the diago- nal compression force is taken as dvcotθ/s. Because cot(18.1 degrees) is 3.06, a given number of stirrups can be calculated by the LRFD Specifications to provide about three times as much shear capacity as would be calculated by the Standard Specifications. • Evaluation of Shear Depth In the LRFD Specifications, the shear depth is taken as dv, rather than d, to overcome a previous simplification in the Standard Specifications. In accordance with the parallel chord truss model, the shear depth is equal to the distance from the centroid of the longitudinal tension reinforcement 11 to the centroid of the compression block (i.e., the flexural level arm). In developing the Standard Specifications, d was used rather than the flexural lever arm for the sake of sim- plicity and also because the provisions still proved to be conservative with the use of d. In the LRFD specifications, dv is used as the flexural lever arm and is typically taken as 0.9d. • LRFD Raises Minimum Shear Reinforcement Requirement The LRFD shear design provisions require a substantially larger amount of minimum shear reinforcement (typically 50 percent more), than do the Standard Specifications, as shown in Figure 7. This difference is particularly important for prestressed concrete members for it is common that large portions of the length of prestressed concrete members require minimum shear reinforcement only. • LRFD Introduces Longitudinal Reinforcement Require- ment Check In the Standard Specifications, anchorage rules for longi- tudinal reinforcement have been used to account for the demands that shear imposes on the longitudinal reinforcement requirements. In the LRFD Specifications, the demand that shear imposes on longitudinal reinforcement requirements is taken into account directly. The difference between these approaches is particularly significant at the ends of simply supported prestressed members where the horizontal compo- nent of the diagonal compression force can be large and yet, by the LRFD Specifications, only the developed portion of the strands may be considered to provide the required resis- tance (see Figure 8). • LRFD Enables Design for Much Higher Shears One of the greatest differences with the LRFD Specifications is that it enables members to be designed for shear stresses that can exceed 2.5 times those permitted by the Standard Specifi- cations. In the Standard Specifications, the contribution of the shear reinforcement is limited to so as to guard against′f b dc w8 Figure 7. Minimum required amount of shear reinforcement.

the member being overly reinforced in shear and failing by diagonal crushing of the concrete or another means before yielding of the shear reinforcement. According to the MCFT, and based on the results of shear tests on elements (21, 22), such failure mechanisms do not occur until design shear stresses are in excess of 0.25 f ′c. The difference between these limits is shown in Figure 9. • LRFD Requires an Iterative Shear Design Procedure The LRFD shear design procedure requires the evaluation of the longitudinal strain at mid-depth,
x, in order to obtain values for β and θ from Table 1 and Table 2. Because
x is a function of θ (see Equations 1-6 and 1-7), the design proce- dure is iterative. The angle θ is first assumed and then
x is evaluated for the given value of θ. The value of θ is obtained from Table 1 or Table 2, and then
x is checked to confirm that is not significantly changed by using the new value of θ. If it is, then it may be necessary for a different column to be used for obtaining β and θ. 12 • Further Iteration Required for Capacity Evaluation In the LRFD Sectional Design Model,
x and thus β and θ are functions of Vu. Thus, the shear design force must be known in order to evaluate Vc, Vs, and the nominal shear strength. As a result, the procedure for evaluating capacity is iterative and requires the engineer to guess the capacity, evaluate model parameters and Vn, and then check that the calculated capacity is close to the factored load. • Empirical versus Model-Based Justification The Standard Specifications justify the relationship for Vc by experimental test data (23) which indicates that the mea- sured shear capacity of prestressed and non-prestressed test beams is conservatively predicted by the sum of Vc (lesser of Vci and Vcw) and the contribution of the shear reinforcement, Vs, as calculated using a 45-degree parallel chord truss model. The LRFD Sectional Design Model shear provisions are derived from a comprehensive behavioral model (the MCFT); therefore, the basis of this model is the MCFT. The calculated capacities by the LRFD Sectional Design Model were illus- trated by experimental test data (24) to provide conservative estimates of shear capacity. • Difference in Shear Reinforcement Requirements and Capacity Ratings The LRFD shear design requirements different consider- ably from those of the Standard Specifications. This leads to significant differences in required amounts of shear rein- forcement and rated capacities of existing structures. Because the structure of the design provisions is so different, it cannot be readily said when one set of provisions will be more con- servative than the other. Further, with use of the Standard Specifications it is easy to perform independent checking of designs. However, the opposite is true with use of the LRFD Specifications. 1.2 INTRODUCTION TO SHEAR BEHAVIOR AND DESIGN PRACTICES This section summarizes the resources considered and used to develop the proposed simplified provisions. This subsection presents the development of U.S. code provisions and compression field approaches for shear design and dis- cusses the factors that influence the primary mechanisms of shear resistance; lists other code provisions warranting consideration; and presents an overview of available experi- mental test data, analysis tools, and design data. 1.2.1 Development of Traditional U.S. Code Provisions for Shear The basic model for how shear is carried in structural concrete is the parallel chord truss model that was first proposed by Ritter in 1899 (25). In this model, the load is carried in reinforced con- Figure 8. Shear demands on longitudinal reinforcement at end of prestressed girder. Figure 9. Maximum allowable design shear stress.

crete in the same manner as load flows in a truss with the load zigzagging its way to the support. The load flows down the concrete diagonal struts and then is lifted to the compression chord by transverse tension ties on its way to the support. Equilibrating the flow of forces puts tension in the bottom chord and compression in the top chord of the truss. Although the model is traditionally shown as one truss with stirrups at a longi- tudinal spacing of “d,” such as given in Figure 10a, it was cor- rectly understood by Ritter that there was a continuous band of diagonal compression carried up and over cracks by a band of stirrups, Figure 10b. For a 45-degree truss, the capacity provided by the shear reinforcement is equal to the capacity of an individ- ual stirrup multiplied by the number of stirrups over the length, “d” which is approximately equal to “d/s.” See Equation 10. (Eq. 10)V A f d s s v y = 13 When the 45-degree parallel chord truss model was intro- duced in the United States in the early 1900s, researchers at the University of Illinois (26) and the University of Wisconsin (27, 28) observed through experimental research that the shear capacity of beams was greater than that predicted by this truss model by nearly a constant amount (see Figure 11). Thus, the idea of a concrete contribution to shear resistance was intro- duced. This contribution was originally taken as equal to a shear stress of between 2 and 3 percent of f ′c multiplied by the shear area (b × d). However, over time that contribution became linked to the diagonal cracking strength because this provided a better fit with test data. The most commonly used relationship in U.S. design practice for the diagonal cracking load, and thus the concrete contribution to shear resistance in reinforced concrete members, is given by Equation 11: where f ′c is in psi units (Eq. 11)V f b dc c v= ′2 Figure 10. Parallel chord truss model. Figure 11. Shear strength of RC beams with shear reinforcement.

There is no mechanical reason to suggest that the concrete contribution to shear resistance at ultimate is equal to the diagonal cracking load, but experimental test data supported the argument that the sum of the diagonal cracking strength plus a shear reinforcement contribution calculated using a 45-degree truss provided a reasonably conservative estimate of shear capacity. Over time, additional expressions for the diagonal cracking strength were developed to account for the influence of prestressing, flexure, and other axial loads. However, as noted in University of Illinois Bulletin No. 493 (29), where the data that forms the basis for the prestressed concrete shear design concepts of the Standard Specifications and ACI 318-05 (30) are reported, the equat- ing of the concrete contribution at ultimate to the shear at inclined cracking is a convenience justified by the simplicity of the result and not by a rational theoretical model. 1.2.2 Compression Field Approaches for Modeling Shear Behavior When the parallel chord truss model was developed, Mörsch (31, 32) argued in 1920 and 1922 that it was not possible to calculate the angle of diagonal compression for there were four unknowns and only three equations (see Figure 12). This dilemma was overcome by Mitchell and Collins in the Compression Field Theory (33) through the introduction of a compatibility relationship made possible by the assumption that the direction of principal compres- sive stress was equal to the direction of principal compres- sive strain. In addition, within the compression field theory, the concept of compression softening was introduced. The principal tensile strain,
1, is considered to decrease the stiffness and strength of concrete in compression. In the MCFT, the average tensile stress in the concrete after 14 cracking was considered. The MCFT can predict the com- plete response of an element subjected to shear and mem- ber forces as described in Figure 6 and more fully explained in Appendix A (Appendix A is available on line as part of NCHRP Web-Only Document 78). Since the development of the MCFT, three other com- pression field behavioral models developed worth noting have been developed: the variable-angle softened truss model introduced by Belarbi and Hsu (34–37), the fixed- angle softened truss model by Pang and Hsu (38), and the disturbed stress field model by Vecchio (39). 1.2.3 Other Approaches and Design Provisions The MCFT provides a clear model for the flow of forces in both prestressed and non-prestressed (reinforced) concrete members and for calculating the angle of diagonal compres- sion and the concrete contribution based on the average ten- sile stress in the concrete. However other ways of looking at shear resistance remain. Another approach for evaluating the angle of diagonal compression is based on plasticity theory and an assumption that the diagonal compressive stress is limited to a fraction of the uniaxial compressive strength; 0.6f ′c is common. This model is used in some European design approaches. Methods for calculating the concrete contribution to shear resistance are far more varied because the concrete contribu- tion at ultimate is really the sum of several mechanisms of resistance as described in Figure 13. These mechanisms are shear in the uncracked compression zone, aggregate inter- lock or interface shear transfer across cracks, dowel action, and residual tensile stresses normal to cracks. In prestressed concrete members, such as bulb-tee girders, the bottom bulb Figure 12. Free body diagrams for development of shear relationship. jd jd . cos θ s . sin θ M M = 0 0.5Nv 0.5Nv s Av Av fv f2 f2 V

may also provide significant shear capacity. Additional com- ponents are the vertical component of the force of draped prestressing strands and the shear transmitted directly to the support by arch action. The relative magnitude of each of these components to the total resistance depends on many factors but it is generally agreed that the dominant concrete components to shear resistance in beams with transverse reinforcement are shear in uncracked compression zones and interface shear transfer. Although researchers agree on the foregoing mechanisms of shear resistance, the structure of code provisions and the amount of shear reinforcement required by different codes for the same design situation vary because of the complexity of shear resistance mechanisms, the factors that influence these mechanisms, and the different methods used to evalu- ate the contributions of the shear reinforcement. The discussion presents some of the complexities of devel- oping a model for shear resistance and to show how different codes have chosen dramatically different approaches. Those approaches have then lead to the development of different infrastructures for design equations and different ways of thinking about shear. For this development of proposed AASHTO simplified shear design provisions, primary resources were underlying models for shear resistance and behavior, shear design equations in current national codes of practices, and expressions for calculating shear capacity that are promoted by individual researchers. 1.2.4 Factors Influencing Shear Resistance Different factors can have surprising effects on shear resis- tance. Shear is complex, there are potential safety concerns 15 with traditional approaches, and developing simplified pro- visions may require making conservative assumptions. Influence of Depth A core assumption in the ACI 318 and AASHTO Stan- dard Specifications is that the shear capacity is proportional to the depth of the member. This assumption was investi- gated in a landmark study conducted by Shioya et al. (40) in which they tested reinforced concrete members that ranged in depth from 4 to 118 inches. All members were simply supported, did not contain shear reinforcement, were lightly reinforced in flexure (0.4%), and subjected to a uniformly distributed load. In Figure 14, the normalized shear stress at failure is plotted versus the depth of the member. The horizontal line corresponds to the shear strength calculated using the traditional shear design expression of the ACI and AASHTO Standard Specifica- tions. The results show that the shear stress at failure decreases as the depth of the member increases. Of partic- ular concern is that members greater than 36 inches deep failed under stresses approximately one-half of the strength calculated by these codes of practice. However, although this depth effect is marked for beams without transverse reinforcement, available test data show little if any depth effect for beams with transverse reinforcement (41). Influence of Concrete Strength In traditional U.S. design practice, and in the LRFD Sectional Design Model, the contribution of the concrete to Figure 13. Mechanism of shear resistance.

shear resistance is taken as proportional to the square root of the cylinder compressive strength f ′c. Figure 15 presents some of the test data by Moody et al. in 1954 (42) from which the permissible design stress limit of was devel- oped. The test beams were typically around 14 inches deep, overly reinforced in flexure, and contained large aggregates. Also shown in this plot are the results from a series of tests by Angelakos in 2001 (43) conducted at the University of Toronto on larger and more lightly reinforced members cast using smaller size aggregates. As the results in Figures 14 and 15 show, the apparent safety of the traditional equation for as used in U.S. practice for beams without shear reinforcement is also dependent on the parameters of beam depth, concrete strength and maximum aggregate size, not considered in that expression. Influence of Axial Loads The influence of axial compression and tension on shear capacity is examined in Figures 16 (44) and 17 (45). As shown, traditional U.S. design practice expressions can be both conservative and unconservative. Part of the explana- tion for these shortcomings is the assumption that the angle 2 ′fc 2 ′fc 16 of diagonal compression is at 45 degrees whereas, as these figures illustrate, axial compression increases the number of stirrups that carry the shear across diagonal cracks while axial tension decreases the number of stirrups that are avail- able to carry the shear across cracks. 1.2.5 Experimental Test Data The previous examples illustrate the importance of evalu- ating and calibrating any potential simplified provisions with extensive experimental data. Professors Reineck and Kuchma (46), and their research assistants have assembled what is probably the largest available database of results from shear tests on structural concrete members. The data- base contains more than 2000 test results. This database can be mined to assess the accuracy and limitation of all prospec- tive code approaches. 1.2.6 Analysis Tools In addition to experimental test data, analytical tools can be used to predict the capacity of prestressed and non- prestressed concrete members. These tools are particularly Figure 14. Influence of depth on shear capacity.

useful for predicting the capacity for the types of members for which no experimental test data is available. Before the use of any analytical tool, the accuracy and reliability of the tool must first be assessed by making comparisons with existing experimental test data. A further consideration is the 17 effort required to use these tools to obtain an evaluation of the shear capacity. Some of the most promising available tools are Response 2000 (15), ABAQUS (47), VecTor2, DIANA (48), and ATENA (49). 1.2.7 Design Cases A further way to evaluate design methods is to compare the required strengths of shear reinforcement (pvfy ≡ Avfy /bvs) by the different design methods for a large database of design cases. Ideally, these cases would represent the range and fre- quency of members built using the given design provisions. Comparing the required amount of shear reinforcement by dif- ferent design approaches for each design case can reveal where prospective provisions may be unconservative or overly con- servative. It is also useful to compare these required strengths of shear reinforcement (pv fy) with the strength determined using analysis tools such as Response 2000. Figure 15. Influence of concrete strength on shear capacity. Figure 16. Influence of axial compression on shear capacity. Sh ea r S tr ee s, V /(b d ) ( MP a) v v 5 10 15 20 25 0.5 1 1.5 2 Axial Tension Stress, N/(b d ) (MPa)v v Experimental CSA 1994 0 0 ACI code As b dw = 1.95% M/V = 0.635 m Figure 17. Influence of axial tension on shear capacity.

1.3 PROJECT OBJECTIVES AND TASKS 1.3.1 Project Motivation and Objectives The LRFD shear design provisions provide some defi- nite advantages over the methods in the AASHTO Standard Specifications. The Sectional Design Model provides a comprehensive approach for shear design of sections subjected to the actions of axial load, prestressing, and moment while the strut-and-tie method provides a completely general design method for regions in which the flow of forces is more complex, such as near geometric discontinuities or near concentrated forces and reactions. However, the Sectional Design Model requires an iterative design procedure that involves selecting β and θ values from tables. Some designers consider this procedure complex to use and difficult to understand, with the effect that some design engineers lose a feel for what they are evaluating. With the strut-and-tie method, concerns have been expressed that solutions require an iterative approach and are non-unique. The overall objective of this research was to provide sim- plified procedures for the shear design of the most common concrete structures, including reinforced concrete T-beams; prestressed concrete I girders continuous for live load; pre- stressed concrete box beams; and cast-in-place post-tensioned box girders, hammerhead piers, and concrete bents. These simplified provisions were expected to be in a form similar to the standard specifications and to be applicable for both pre- stressed and precast members up to concrete strengths of 18 ksi and cast-in-place concrete strengths up to 10 ksi. Although there are recognized challenges to the application of the strut-and-tie method, there was no project objective to refine the strut-and-tie design provisions. 1.3.2 Research Approach and Project Tasks There are many approaches for shear design, underlying theories for explaining how shear is carried in structural concrete, and the primary factors that influence the mecha- nisms of resistance. The approach on this project was to investigate and then select the most suitable simplified shear design provisions based on a detailed review of existing shear design approaches and an evaluation of these approaches by comparison with both experimental test data and with the predictions from numerical methods. The members of the research team were selected so that differ- ent points of view and experiences were represented. Sev- eral of the important, if not essential, attributes of the research team were as follows: • Leadership experience in the developing code provi- sions for shear, including the AASHTO LRFD specifi- cations, the AASHTO Standard Specifications and the ACI 318-05 provisions; 18 • Detailed knowledge of a broad base of mechanistic models for shear including U.S., Canadian, and Euro- pean approaches; • Detailed understanding of the Modified Compression Field Theory and how the LRFD provisions were derived from this theoretical model for behavior; • Custodians of the largest and more detailed shear test database yet assembled; • Not committed to a single line of thinking on the final structure of the simplified provisions or on the mecha- nistic model (or models) on which these provisions should be based; and • Familiarity with the use of non-linear numerical tools for predicting the capacity of members in a design testbed. There were 8 tasks listed in the request for proposals that were required for meeting project objectives. The researcher’s approach on each of these tasks follows the description of each of these individual tasks. The researchers conducted a survey of practicing engi- neers concerning their experiences in the use of AASHTO Standard and LRFD specifications, collected codes of prac- tice, and then conducted a preliminary review and assess- ment of different shear design approaches using an extensive experimental database of shear test results. A refined work plan was established for developing and refining the selected proposed simplified provisions. This plan included the use of a design database to assess the effect of different potential approaches on the efficiency and con- servatism of codes. The researchers produced a tentative list of design examples for consideration by the project review panel from which the final design examples were selected. An Interim Report was submitted and then, following a request by the Project Panel, a more comprehensive interim report was submitted containing an initial proposal by the contractor for the simplified provisions. These proposed sim- plified provisions were essentially the same as those devel- oped by Michael Collins, a developer of the MCFT, for the 2004 Canadian Standards Association “Design of Concrete Structures.” These simplified provisions consisted of equa- tions for β and θ and an elimination of the dependency of
x on θ, thereby eliminating the iterative nature of the LRFD design procedure. The researchers conducted the plan approved by the proj- ect panel. This plan consisted of: • Reviewing shear design provisions additional to those examined in the Interim Report; • Developing a refined experimental database of shear test results of large members with shear reinforcement; • Developing an expanded member design database; • Developing alternative provisions to the CSA method proposed in the Interim Report; and • Developing detailed criteria for selection and verifica- tion of the simplified specifications.

Based on the results of their analytical and design investi- gations, the researchers (1) developed a new simplified shear design procedure for members with minimum shear rein- forcement, (2) verified the need for the existing limit on the required minimum amount of shear reinforcement, (3) veri- fied the need for a new lower limit on the maximum shear stress that can be used in design if members are not supported over their full depth at the ends, and (4) developed modifi- cations to simplify the existing General Procedure for sectional shear design of Article 5.8.3.4.2 of the LRFD Specifications. Based on the final form of the proposed simplified spec- ifications, the goal of the regression testing was the setting 19 of only a few parameters and limits. The tuning of these parameters was performed by considering the fit of the pro- posed simplified provisions with the test results in the refined experimental database and by comparing the required amounts of shear reinforcement for members in the design database with the requirements by other design methods, including the current LRFD Sectional Design Model, the AASHTO Standard Specifications, and Response 2000. The research team prepared eight design examples that covered both prestressed and non-prestressed members, simple span and continuous members, different types of structural components and both stocky and slender members.