Simplified Shear Design of Structural Concrete Members: Appendixes (2005)

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A-1 Appendix A: Models for Shear Behavior A.1 Introduction A.1.1 The Problem of Shear Transfer A flexural member supports loads by internal moments and shear forces. Classical beam theory, in which plane sections are assumed to remain plane, provides an accurate, simple, and effective model for designing a member to resist bending in combination with axial forces. The simplicity and rationality of beam theory can be kept even after cracking for several reasons. The first reason is that flexural cracks form perpendicular to the axis of bending so that the traditional “plane sections remain plane” assumption is valid. The second reason is the weakness of concrete in tension, so that tensile stresses can be effectively neglected at a crack. The third reason is that flexural failure occurs at the maximum moment location so that consideration of conditions at the maximum moment section is sufficient for flexural design. Shear failure is initiated by inclined cracks caused not only by shear force but also by shear force in combination with moments and axial loads. The shear failure load depends on numerous factors such as the dimensions, geometry, loading and structural properties of members. Because shear cracks are inclined and the shear failure load depends on a large number of factors, shear design— unlike flexural design—must consider the response of a finite length of the member, (B-region), rather than the response of a single section. Due to the complications of shear behavior and the difficulties of shear design, the shear behavior and shear strength of members have been major areas of research in reinforced and prestressed concrete structures for decades. This chapter provides information on mechanisms of shear transfer, influencing parameters, shear failure modes, and various key approaches to analyzing shear behavior. A.1.2 Shear Transfer Actions and Mechanisms Shear transfer actions and mechanisms in concrete beams are complex and difficult to clearly identify. Complex stress redistributions occur after cracking, and those redistributions have been shown to be influenced by many factors. Different researchers impose different levels of relative importance to the basic mechanisms of shear transfer. Fig. A-1 shows the basic mechanisms of shear transfer that are now generally accepted in the research community. In 1973 the ASCE-ACI Committee 426 and, again in 1998, its current counterpart the ASCE-ACI Committee 445, reported five important shear transfer actions for beams with shear reinforcement: shear in the uncracked compression zone of the beam; interface shear transfer due to aggregate interlock or surface roughness along inclined cracks; dowel action of the longitudinal reinforcement; residual tensile stresses across inclined cracks; and shear transfer of the shear reinforcement. Each of these actions is depicted in Fig. A-2 and is more fully described below. Shear in the Uncracked Concrete Zone: The uncracked compression zone contributes to shear resistance in a cracked concrete member (i.e. a beam or a slab). The magnitude of that shear resistance is limited by the depth of the compression zone. Consequently, in a relatively slender beam without axial compression, the shear contribution by the uncracked compression zone becomes relatively small due to the small depth of the compression zone.

A-2 Interface shear transfer: Local roughness in the crack plane provides resistance against slip and thus there is shear transfer across shear cracks. The contribution of interface shear transfer to shear strength is a function of the crack width and aggregate size. Thus, the magnitude decreases as the crack width increases and as the aggregate size decreases. Consequently, this component is also called “aggregate interlock”. However, it is now considered more appropriate to use the terminology “interface shear transfer” or "friction" since this action still exists even if crack propagation occurs through the aggregate as it does in high strength concrete where the matrix is of a similar strength to the aggregates. The relatively smooth crack plane in high strength concrete can reduce interface shear transfer compared to the rough crack plane of normal strength concrete. Dowel Action: When a crack forms across longitudinal bars, the dowelling action of the longitudinal bars provides a resisting shear force. The contribution of dowel action to shear resistance is a function of the amount of concrete cover beneath the longitudinal bars and the degree to which vertical displacements of those bars at the inclined crack are restrained by transverse reinforcement. Typically, little dowel action can be provided by reinforcement that is near the tension face of a member without transverse reinforcement because that action is then limited by the tensile strength of the concrete. Residual Tensile Stresses: In concrete tensile stresses can be transmitted directly across cracks because small pieces of concrete bridge the crack. As shown in Fig. A-3, even when concrete is cracked and loaded in uniaxial tension, it can transmit tensile stresses until crack widths reach 0.06 mm to 0.16 mm. Due to the presence of these tensile stresses the cracked concrete, in the vicinity of the tips of inclined and flexural cracks, can also carry shear stresses that add to the shear capacity of the concrete. When the crack opening is small, the resistance provided by residual tensile stresses is significant. However, in a large member, the contribution of crack tip tensile stresses to shear resistance is less significant due to the large crack widths that occur before failure in such members. Shear Reinforcement: In members with shear reinforcement, a large portion of the shear is carried by the shear reinforcement after diagonal cracking occurs. The contribution of shear reinforcement to shear resistance is typically modeled either with a 45 degree truss plus a concrete term, or a variable angle truss without a concrete term. Shear reinforcement also provides a certain level of restraint against the growth of inclined cracks and thus helps to ensure a more ductile behavior. Finally, shear reinforcement provides dowelling resistance to shear displacements along the inclined crack. For these reasons, the presence of shear reinforcement changes the relative contributions of the different shear resisting mechanisms. The minimum amount of shear reinforcement required to affect such changes becomes important and that minimum amount is taken as a function of the concrete strength in most major design codes. Such is the case in both the AASHTO Standard and LRFD specifications. Longitudinal reinforcement, bent up at 30 degrees or more to the longitudinal axis of the beam, extended across the web and anchored on the compression side, was used effectively for many years as shear reinforcement in reinforced concrete beams. Therefore, in the early years of prestressed concrete construction, it was believed that draped prestressing tendons would also be effective in all situations as shear reinforcement. However, University of Illinois tests (Bulletin493) demonstrated that such reinforcement was effective only in delaying shear cracks formed in the web due to

A-3 principal tensile stresses. Draping the prestressing tendons did not delay the formation of inclined cracks that developed out of flexural cracks. Thus, in the ACI 318 Code, and in the AASHTO Standard Specifications, the term for the vertical component of the prestressing force appears only in expressions for the shear strength for web-shear cracking and does not appear in either the expressions for the shear strength for flexure-shear cracking or the shear strength contributed by shear reinforcement. In addition to the shear transfer actions mentioned above, arch action is a dominant shear transfer mechanism in deep members. This is because as members become deeper, a larger portion of the shear is transmitted directly to the support by an inclined strut. A.1.3 Significant Parameters for Members without Transverse Reinforcement Several parameters have been identified as significantly influencing the relative contributions of the different shear resistance mechanisms and thus the ultimate shear capacity. The influence of the most dominant mechanisms is described in the following in accordance with the findings of the state- of-the-art reports by ASCE-ACI Committee 426 (1973) and ASCE-ACI Committee 445 (1998). Concrete Strength: As concrete strength increases, the shear strength also increases. Some researchers believe that concrete compressive strength has a large influence on the shear resistance while others believe that concrete tensile strength has a greater influence than compressive strength on shear strength. The concrete contribution to shear, in ACI 318-02, for example, is regarded as being that due to diagonal cracking shear, and therefore dependent on the tensile strength of the concrete. The concrete compressive strength, f’c, is generally used to estimate the tensile strength because direct tension tests are difficult to conduct, require interpretation of the results, and usually show more scatter than compression test results. In most major design codes, the shear strength of a member is taken as directly proportional to 25.0' )( cf or 33.0' )( cf or 5.0' )( cf . Those power values indicate that the concrete tensile strength is being used as the governing parameter. Recent test results have illustrated that the presumed effect of concrete tensile strength on shear capacity is strongly influenced by the characteristics of the tests conducted to examine the effect. Fig. A-4 presents results from several beam test series that show very different trends. The ACI 318-02 shear design approach in which the shear strength is taken as proportional to the square root of 'cf is also shown in the same figure. The shear failure stresses of the beams tested by Moody et al.(1954) increase as the concrete compressive strength increases. ACI 318-02 is shown to provide a reasonable estimate of the influence of 'cf for these beams which were small, heavily reinforced, and cast with low-to-medium-strength concrete. Similarly, the ACI provision is only slightly unconservative for the moderately reinforced and mid-sized members that were tested by Yoon and Cook (1996). However, Collins, Angelakos, Kuchma et al.(1999, 2001) did not find a similar increase in shear strength with concrete strength for their tests of larger, more lightly reinforced beams, and cast with high strength concretes with a small maximum aggregate size. The explanation offered by some researchers for why the shear stress at failure does not increase as greatly, or not at all, with increasing concrete compressive strength is that the smoother shear cracks in high-strength concrete members reduce the effectiveness of the interface shear transfer mechanism.

A-4 Size Effect: The shear strength of reinforced and prestressed beams without shear reinforcement decreases as the member depth increases; this is called the “size effect” in shear. Both the tests by Kani on size effect in 1967, and the tests by Shioya et al. (1989), effectively demonstrated this effect. Shioya et al. tested beams with depths ranging from 4 to 120 inches (102 mm to 3.0 m), Fig. A-5. The shear stress at failure of the largest beam was only about one-third of that of the smallest beam, and the ultimate shear stress of the largest beam was less than one half of the value calculated using ACI 318-02. In 1956, several US Air Force warehouse beams collapsed under a shear force less than one half of the ACI design value, as indicated in Fig. A-5. The depth of these beams was 36 inches (914 mm). Investigators examining these failures conducted experiments with one-third scale models of the warehouse beams at the Portland Cement Association (PCA) and the failure strengths for those model beams are also shown in Fig. A-5. Due to the much higher failure strength of the PCA test beams than that of the warehouse beams, the investigators concluded that axial tensile stresses due to shrinkage restraints by columns were the primary cause for the low failure strengths. However, it seems more reasonable to explain the results in terms of the size effect in shear. In models used to account for the size effect in shear, some researchers (Bazant and Kim, 1984) explain the size effect by fracture mechanics and suggest that the large amount of energy that is released in the cracking of large members leads to the faster propagation of inclined cracks and lower shear failure stresses. Other researchers, like Collins and Kuchma (1998) and Reineck (1990, 1991), explain the size effect as due to a reduction in interface shear transfer due to the larger crack widths that occur in larger members. Shear Span to Depth Ratio: The shear span is the distance, a, between a support and a point of concentrated load. As the shear span to depth ratio (a/d) decreases, the shear strength increases. Many empirical formulas for calculating shear strength include an a/d ratio to account for the influence of this parameter. The increase in strength is significant in members with a/d ratios less than about 2.5 to 3.0, because a significant portion of the shear may be transmitted directly to the support by an inclined strut. This mechanism is frequently referred to as arch action and the magnitude of the direct load transfer increases with decreasing a/d-ratios. For deep members and the ends of beams, it is therefore more appropriate to use strut-and-tie models than sectional design approaches. The key characteristic of the a/d-ratio is obvious for simple beams subject to point loads. The term relates the maximum moment and the maximum shear force, since Mmax = Vmax×a and thus the moment to shear force ratio is Mmax /Vmax d = a/d. For distributed loading this term is also significant, as has already been pointed out by Kani (1964, 1967), and it gives Mmax /Vmax d = l / 4d, which means that “a” is the distance to the resultant of the loads in one half of the span. Therefore, the a/d-ratio characterizes the slenderness of a simple beam and the value influences the relationships between the different shear transfer actions. The value, a, also relates the flexural and shear capacities, i.e. the shear force at flexural failure can be calculated by dividing by “a” or the moment at mid-span corresponding to shear failure can be calculated by multiplying by “a”. Longitudinal Reinforcement Ratio: For the same magnitude of loading, as the longitudinal reinforcement ratio decreases, flexural stresses and strains increase. Thus, crack widths increase and the shear strength is lowered. Further, as the longitudinal reinforcement ratio decreases, dowel action decreases. It has also been reported that for members having longitudinal bars distributed over their

A-5 height crack spacings are smaller and that improves shear strength significantly (Collins and Kuchma, 1999). Axial Force: When members are subjected to axial tension, the shear strengths of such members decrease. Since axial tension makes the crack angle steeper over almost the full depth of the member, longitudinal reinforcement needs to be provided in both the top and bottom of the member. Once appropriate amounts of longitudinal reinforcement are provided, the failure of such members may occur in a relatively ductile manner. By contrast, axial compression increases the depth of the uncracked compression zone, decreases the width of the shear cracks, and thus the interface shear transfer is increased. All of these factors lead to an increase in shear capacity with increase in axial compression. However, for members subjected to significant axial compression, brittle failures are common. Other Significant Parameters: In a simply supported member, high shear and high moment do not coexist. Thus, in high shear regions of such members, the effect of moment is relatively small. However, in a continuous beam, the negative moment region is subject to both high shear and high moment and, thus, the effect of moment can be significant. Further, in a negative moment region, the compression and tension sides are reversed over those in a positive moment region. Consequently, the size of the uncracked compressive zone in the negative moment regions of a T-beam differs from that for the positive moment region because there is no longer a wide slab in compression. When a member is supported on the bottom and loads are applied on the top, applied forces can be transmitted directly to the supports through inclined struts. However, when the member is supported on the bottom and loaded on the bottom, or supported from the top and loaded from the top, those struts cannot form. The member may have a lower shear capacity than that of a beam supported on the bottom and loaded on the top. A.1.4 Shear Failures of Members without Transverse Reinforcement Shear failures of members without transverse reinforcement depend on several factors as discussed in this section. Shear failures are initiated by inclined cracks, and these cracks are typically divided into two types, i.e., web-shear cracks and flexure-shear cracks as shown in Fig. A-6. More detailed explanations of these two types of inclined cracks are provided in Appendix B. Kani (1964) conducted a very large experimental study on shear and reported relationships between the beam capacity and the a/d ratio. “Kani’s Valley of Shear Failures” is presented in Fig. A- 7 (McGregor, 1988). Kani tested a large number of rectangular beams without shear reinforcement and having various a/d ratios, while the rest of the beam details remained the same, as shown in Fig. A-7(a). The moment and shear at inclined cracking and failure were observed to be as shown in Fig. A-7(b). The flexural capacity, nM , is the horizontal line while the shaded area represents the reduction in flexural strength due to shear. From this figure, beams can be classified into four types depending on their a/d ratio: very short, short, slender, and very slender beams. Fig. A-7(c) can be obtained by dividing the moment in Fig. A-7(b) by the shear span, “a”, since the moment is aVM ×= for beams with two point loads. Kani also tested beams subjected to uniformly distributed load and used the a/d ratio as a quarter of the span length, i.e., L/4.

A-6 The shear failure modes of beams without shear reinforcement were also discussed by ASCE- ACI Committee 426 (1973). They also classified beam types by their a/d-ratios. The failure modes of simply supported rectangular beams without shear reinforcement were described as follows; a) In very slender beams (a/d>6), the members will likely fail in flexure even before the formation of inclined cracks. b) In slender beams (2.5<a/d<6), some of the flexural cracks grow and may become flexure- shear cracks. The diagonal cracks may continue to propagate towards the top and bottom of the beam and cause yield of the tension steel (Fig. A-8). The beam may split into two pieces at failure. This is called as diagonal tension failure. c) In short beams (1<a/d<2.5), a diagonal crack may propagate along the tension steel causing splitting between the concrete and the longitudinal bars (Fig.A-9(a)). This is called a shear- tension failure. The diagonal crack may propagate toward the top of the beam resulting in crushing of the compression zone. This is called a shear-compression failure (Fig.A-9(b)). d) In very short beams (a/d<1), inclined cracks occur along the line between load and reaction. Thus, most of the shear force is transferred by arch action with a structural system as shown in Fig. A-10. Anchorage failure of the tension steel may occur at the end of a tension tie. Bearing failure may occur by the crushing of concrete above a support. Flexural failure is also possible due to the yielding of the tension steel or crushing of the compression zone. Tension failure of the “arch-rib” near the top of an edge may occur due to the eccentricity of the thrust of the compressive stresses in the inclined strut. Compression strut failure is also possible by crushing of the web along the line of the crack. The failures of I-shaped beams are somewhat different from those of rectangular beams because the shear stresses in the webs are much higher than in rectangular beams. Web-crushing failures are the most common failure mode for I-beams, as shown in Fig. A-11, although all the failure modes described previously for rectangular beams are also possible for I-beams. A.1.5 Shear Failures of Members with Transverse Reinforcement and Significant Parameters If a member contains transverse reinforcement (or shear reinforcement) then, after the appearance of the first inclined crack, the behavior changes considerably from that for a beam without transverse reinforcement. Depending on the amount of the transverse reinforcement more inclined cracks may develop until the stirrups yield. After yielding of the stirrups the load may also increased due to flatter inclined struts crossing the cracks. That action is possible due to interface shear or friction along the crack faces. While failure is then initiated by the break-down of the interface shear, the primary cause is still the yielding of the stirrups so that the final failure is relatively ductile. For large amounts of transverse reinforcement the concrete in the inclined struts may fail, the so- called “web-crushing” failure. This failure mode is very likely in thin webbed members such as the I- beams shown in Fig. A-11. The main parameter affecting the behavior and failure mode of webs is the amount of the transverse reinforcement. For that mode, apart from interface shear and some shear transfer in the

A-7 compression zone, the shear transfer actions discussed previously for beams without shear reinforcement do not play a dominant role. For these reasons, the terms for the shear strength provided by the concrete are differentiated as follows: - Vct for members without transverse reinforcement, in order to indicate that the failure is governed by the concrete tensile strength, and - Vc for members with transverse reinforcement. Axial compression influences the shear capacity of members with transverse reinforcement because the depth of the uncracked compression zone increases. However, axial compression also influences the behaviour in the web: shear crack widths decrease, and crack angles decrease, so that the angle of diagonal compression is flattened and this change increases the effectiveness of the shear reinforcement. Concrete strength influences the transfer of forces across cracks, because for high strength concrete, the inclined cracks may run through the aggregates and thus interface shear transfer is reduced. The size effect plays only a minor role for members with transverse reinforcement because the crack widths are mainly controlled by the transverse reinforcement. A.2 Shear Strength Models A.2.1 Overview The truss model was widely used to understand the shear behavior of reinforced concrete beams with transverse reinforcement in the early 1900’s. Ritter (1899) developed a 45o truss model for the analysis of the post-cracking behavior of a reinforced concrete beam and Mörsch (1920, 1922) refined that 45o truss model. Mörsch also used the flexure-shear crack to derive the shear stress distribution for a reinforced concrete beam, i.e., jdb V w =τ . This stress has for many years been called the “nominal” shear stress. In 1907, Withey introduced the 45o truss model into American literature. Withey reported that the 45o truss model gave a conservative result when compared to experimental test results, a statement that was confirmed by Talbot (1909). In the USA in the late 1950’s the 45o truss model was also adopted for prestressed concrete based on the work reported in University of Illinois Engineering Experiment Station Bulletin 493. That bulletin reported that while the 45o model provided conservative results compared to experimental results, its use was recommended because “the design procedure should be no more complicated than would be justified by the certainty of the theory and the economy of the end result”. In 1950s and 1960s, a large amount of experimental research was conducted to study the contribution of aggregate interlock and dowel action on shear resistance. Zwoyer and Siess (1954), Bresler and Pister (1958), Guralnick (1959), and Walther (1962) studied the stress conditions in the concrete above flexural cracks of beams without transverse reinforcement, assuming that all shear would be carried in the flexural compression zone. In 1964, Kani introduced the “comb” model in which the concrete between the flexural cracks is considered as the teeth of the comb and uncracked concrete above the flexural cracks as the backbone of the comb. After studying the large amount of

A-8 available experimental results, the ACI-ASCE shear committee (1962) recommended the use of an empirical expression for the shear stress at inclined cracking as the shear failure load. That expression first appeared in the 1963 ACI 318 Code and is still present in ACI 318-02 as Equation 11-5. That expression is: '' 5.325009.1 cwccr fM Vdfv ≤+= ρ (psi) In the 1950s and 1960s extensive work at the University of Illinois (Bulletin 493) and at the PCA resulted in the development of the shear strength provisions for prestressed concrete beams that continue to be incorporated into ACI 318-02 and the AASHTO Standard Specification. While the model used for the shear strength provided by the transverse reinforcement was the same as that for reinforced concrete beams, the models used for the shear at inclined cracking differed considerably from those for reinforced concrete beams Fenwick and Paulay (1968) suggested that shear resistance carried by the compression zone is only about 25% of the total shear and that “aggregate interlock” and dowel forces carried the remainder of the shear In 1964, Kupfer took a step forward for members with transverse reinforcement by predicting the strut angle using minimum energy principles, and Baumann (1972) further pursued this concept, presenting dimensioning diagrams for plate elements with reinforcement in two and three directions. Inspired by Baumann’s work, and also by Wagner’s tension field theory (1929), Collins and Mitchell (1981) developed the compression field theory (CFT) where equilibrium conditions, compatibility conditions, and constitutive relationships are considered for shear stress conditions The CFT assumes that the direction of the inclined compression field, (i.e. the strut angle and the crack angle), and the principal compressive stress coincide, similar to the assumptions of Kupfer and Baumann. The model is a pure truss and the cracked concrete carries no shear. Thürlimann et al. (1983) and Nielsen (1984) introduced plasticity methods for predicting shear strength. In 1986, Vecchio and Collins proposed the modified compression field theory (MCFT). That theory modifies the CFT by considering the contribution of the concrete in tension. The MCFT provides a behavioral model for predicting the complete load-deformation response in shear. From 1992 through 1995, Hsu et al. developed the rotating-angle softened-truss model (RA- STM) and the fixed-angle softened truss model (FA-STM). While the RA-STM assumes that the direction of both principal stress and strain coincide, the FA-STM considers that after cracking, the direction of the principal stress in the concrete struts does not coincide with the direction of the crack. Unlike the MCFT where the stress conditions at a crack are checked, both the RA-STM and FA-STM reduce the average strength of the reinforcement to account for the local stress effects at a crack. Both the MCFT and FA-STM predict similar strengths in most cases. Descriptions of several of the more important models for determining the behavior in shear of beams with shear reinforcement are presented in the next five sections.

A-9 A.2.2 45o Truss Model Truss models were widely used to understand the shear behavior of reinforced concrete beams in the early 1900’s. Ritter (1899) was the first to use a 45o truss model for the analysis of the post- cracking behavior of a reinforced concrete beam. In his model, diagonal concrete struts were considered to be the diagonal members of the truss, the stirrups were the vertical members of the truss, the longitudinal reinforcement served as the bottom chord of the truss, and the flexural compression zone served as the top chord of the truss. In 1902 Mörsch improved this model by assuming that the diagonal struts extended across more than one stirrup. The tensile stresses in cracked concrete were neglected in this model and diagonal compression stresses were assumed to remain at 45o after the concrete cracked. Equilibrium equations for this model, assuming an angle of diagonal compression of °= 45θ , are shown in Fig. A-12. For a uniform distribution of shear stresses in the effective web area, jdbw , the vertical component of the diagonal compressive force must be balanced by the applied shear: jdb Vf w 2 2 = (A-1) The horizontal component of the diagonal compressive force also must be balanced by the tension in longitudinal reinforcement: VNv = (A-2) From Fig. A-12, the vertical component of the diagonal compression force must be balanced by the tension in the stirrups over the length jd⋅cot45° = jd: jd V s fA vv = (A-3) Eq. (A-3) is used to design the required amount of stirrups. Eq. (A-1) is used to check the compressive stresses in the concrete and this determines the upper limit of the shear force or capacity. In the middle of the 1960´s the 45o truss model was re-examined because it gives overly conservative results for predictions of the shear strength of members with shear reinforcement. The model lowers the effectiveness of the stirrups and, consequently, efforts were directed towards predicting the actual strut angle, which may be flatter than the angle of the inclined cracks. A.2.3 Variable-Angle Truss Model The variable-angle truss model is a version of the 45o truss model modified by assuming flatter strut angles, θ ≤ 45°. In this model, the three equilibrium equations can be derived in the same manner as for the 45o truss model. The equilibrium conditions for this model are shown in Fig. A-13 and Eqs. A-4(a), (b), and (c):

A-10 )cot(tan cossin 1 2 θθθθ +== jdb V jdb Vf ww (A-4a) θcotVNv = (A-4b) θtan jd V s fA vv = (A-4c) However, these three equilibrium equations are not sufficient to solve member forces, because there are four unknowns; the principal compressive stress, 2f ; the tension in the longitudinal direction, vN ; the stresses in the shear reinforcement, vf ; and the strut angle or inclination of the principal compressive stresses, θ . There have been different approaches to solve for the strut angle. In 1964, Kupfer used minimum energy principles to determine the crack angle θ while assuming linear elastic behavior of both reinforcement and concrete. Baumann (1972) continued this work, which was later taken up by Collins and Vecchio in their compression field theory. The traditional truss model assumes that the stirrups yield (i.e., yv ff = ) and θ =45 o, and uses Eq. (A-4c). Plasticity methods assume yielding of the stirrups (i.e., yvx fff == ) and that the maximum compressive stress, 2f , is attained. Such methods result in lower limits for the angle θ . A.2.4 Compression Field Theory The compression field theory uses the same approach for equilibrium conditions as described in the variable-angle truss model. Eqs. A-4 (a), (b), and (c) can be expressed respectively in terms of the stresses as shown below. These equilibrium equations can also be derived from Fig. A-14(a and b): )cot(tan2 θθ += vf (A-5a) θρ cotvfsxx = (A-5b) θρ tanvfsyv = (A-5c) For determining the crack angle, θ , in the variable-angle truss model, Wagner (1929) contributed an important fundamental concept in his “tension field theory”. In his shear design of thin “stressed-skin” aircraft, he assumed shear was carried by a diagonal tension field after buckling of the thin metal web. Then, he considered the deformations of the system by assuming that the angles of inclination of the diagonal tensile stresses would coincide with the angles of the inclination of the principal tensile strains. Similar to Wagner’s tension field theory model, the compression field theory model developed by Collins and Mitchell utilizes the deformations for reinforced concrete by assuming that a diagonal compression field carries shear after cracking. The compatibility conditions used in the compression field theory can be derived from Mohr’s circle for strains as shown in Fig. A-14(c and d).

A-11 From Fig. A-14(c and d), the relation between θ and the strains can be expressed as follows: xy xy εε γθ −=2tan (A-6) Since )tan1/(tan22tan 2 θθθ −= , Eq. (A-6) can be expressed as: ( )22)(1tan xyxyyx xy γεεεεγθ +−±−= (A-7) From Fig. A-14(c and d), the relation between θ and the strains can also be expressed as: R xy xyxy xy 2)( 2sin 22 γ γεε γθ =+−= (A-8) where 22 εεε −+= yxR . Therefore, the relationship between the strains is: 2 22 2)( εεεγεε −+=+− yxxyxy (A-9) From Eqs. A-7 and A-9, the relation between θ and the strains can be simplified as: ( )22tan εεγθ −= xxy (A-10) Thus, ( ) θεεγ cot2 2−= xxy (A-11) From Eq. (A-9), the following expression can be formed: ))(( 2 22 2 εεεεγ −−=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ yx xy (A-12) From Eqs. A-10, A-11, and A-12, the following expression can be derived: 2 22tan εε εεθ − −= y x (A-13) Eq. (A-13) is Wagner’s and Baumann’s compatibility equation, which can be applied in cracked concrete using average strains. From Eq. (A-13), the influence of θ on strains is readily apparent. For steep crack angles, the longitudinal strain is high and for flat crack angles, the transverse strain is high.

A-12 As shown in Fig. A-14(e), the stress-strain relationships for both longitudinal and transverse reinforcement were assumed as bilinear for the CFT. Stress-strain relationships for cracked concrete in compression were proposed by Collins (1978), based on experimental test results, as follows: ' 21 ' max22' ' 2 /)(21 6.3 c c c c ffff εεεεε ++=≤⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= (A-14) where 'cε =strain corresponding to 'cf in a cylinder test. In this equation, the softening of the compressive strength in cracked concrete is expressed in terms of the principal tensile strain, 1ε . The principal tensile strain, 1ε , can be derived from Fig. A- 14(d) and Eq. (A-14) as: θεεεεεεε 2221 cot)( −+=−+= xxyx (A-15) From Eq. (A-15), the longitudinal strain, xε , can be expressed as: )tan1/()tan( 22 2 1 θεθεε ++=x (A-16) A similar expression for yε can be derived from Eqs. A-15 and A-16 as: )tan1/()tan( 2221 θθεεε ++=y (A-17) Once the compression field theory was developed, the governing stress-strain relationships were studied by researchers in a large series of experiments. The result from a typical test (Vecchio and Collins, 1982) is shown in Fig. A-15. Based on the experimental findings, Eq. (A-14) was refined by Vecchio and Collins in 1986 to be as follows: ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= 2 ' 2 ' 2 max22 2 cc ff ε ε ε ε (A-18) where 0.1 1708.0 1 1 ' max2 ≤+= εcf f . This relationship is shown in Fig. A-14(f) and can also be visualized in terms of both 1ε and 2ε as shown in Fig. A-16. Because the compression theory provides the equilibrium conditions, compatibility conditions, and constitutive relationships for both reinforcement and cracked concrete, it can predict shear behavior for any load level as well as the shear strength of members. However, since the compression field theory neglects the tensile stresses in cracked concrete, it gives conservative results for the shear behavior of members, meaning that it underestimates both the shear stiffness and the shear strength.

A-13 A.2.5 Modified Compression Field Theory The tensile stresses in cracked concrete provide significant shear resistance. The modified compression field theory (MCFT) accounts for the influence of tensile stresses on the post-cracking shear behavior. The equilibrium equations for the MCFT can be derived in a similar manner to those for CFT with a concrete tensile stress term added. For the conditions shown in Fig. A-17 (a and b), equilibrium equations are: 1cot fvfsxx −= θρ (A-19a) 1tan fvfsyv −= θρ (A-19b) 12 )cot(tan fvf −+= θθ (A-19c) Conditions are expressed in terms of average stresses. The average principal tensile stress after cracking, 1f , was suggested by Collins and Mitchell (1991) to be as follows: 1 1 5001 ε+= crff (psi) (A-20) where crf is taken as '4 cf . It can be seen, as shown in Fig. A-17(j), that the average principal stress decreases as the average principal strain increases. On a local level, however, stresses vary and differ from the values calculated from Eqs. A-19 and A-20. The failure of the member can be governed by either average stresses or local stresses at a crack. Therefore, the conditions at a crack also need to be checked. At a crack, the concrete tensile stresses decrease to zero and thus the tensile stresses in the reinforcement increase significantly. Assuming that cracks are parallel, the equilibrium conditions at a crack can be found from Fig. A-17 (c and d): Longitudinal direction: 0)cos()cos()sin( =−− θθθρ crcrcicrxsxcr AvAvAf (A-21a) Transverse direction: 0)sin()sin()cos( =−+ θθθρ crcrcicrvsycr AvAvAf (A-21b) where crA = area of crack plane, and civ = interface shear stress at a crack (see Eq. A-23). Therefore, Eqs. A-21(a) and (b) become as follows: θθρ cotcot cisxcrx vvf += (A-22a) θθρ tantan cisycrv vvf −= (A-22b) From Eqs. A-22(a) and (b) it is apparent that as civ at a crack increases, the stress in the longitudinal reinforcement increases but the stress in the transverse reinforcement decreases. The shear stresses that can be transmitted across a crack, civ , can be expressed as a function of crack

A-14 width. A limit on civ was proposed by Vecchio and Collins (1986), based on the experimental data of Walraven (1981), and simplified by Bhide and Collins (1989), as follows: )( 63.0 243.0 16.2 ' inandpsi a w f v cci ++ ≤ (A-23) Fig. A-17(k) shows the normalized magnitude of the allowable shear stress on a crack in the case that the maximum aggregate size is 0.75 inch. In the Eq. (A-23), the crack width, w , can be obtained as the average crack spacing multiplied by the principal tensile strain, as shown in Fig. A-17(g). Thus, θε msw 1= (A-24) where the average crack spacing, θms is taken as: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ += mymx m ss s θθθ cossin1 (A-25) In Eq. (A-25), mxs and mys are the estimated crack spacings in the longitudinal and vertical directions, respectively. The expression for these estimated crack spacings are taken from the CEB-FIP Code (1978) and, for uniform tensile strain, these expressions become: x bxx xmx dkscs ρ125.0102 +⎟⎠ ⎞⎜⎝ ⎛ += (A-26a) y byy ymy d k s cs ρ125.0102 +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ += (A-26b) where xc , yc : distance between midsection and longitudinal and transverse reinforcement, respectively. xs , ys : spacing of longitudinal and transverse reinforcement, respectively 1k : coefficient for bond characteristics of bars (0.4 for deformed bars, 0.8 for plain bars) bxd , byd : bar diameter of longitudinal and transverse reinforcement, respectively. In a typical member, the average strain in the stirrups exceeds the yield strain under high load and, then, Eqs. A-19(b) and A-22(b) must be equivalent. Thus,

A-15 )(tan 63.0 243.0 16.2 tan ' 1 inandpsi a w f vf cci θθ ++ =≤ (A-27) Eq. (A-27) limits the average principal tensile stress in cracked concrete so that possible failure of the aggregate interlock mechanism can be taken into account in the MCFT. Both the CFT and MCFT can predict the shear behavior of members with shear reinforcement for all loading histories. However, the CFT predicts no shear strength for those members without shear reinforcement as it neglects the contribution of the tensile stress carried by cracked concrete. The MCFT can predict shear behavior even for those members without shear reinforcement because it accounts for the tensile stresses carried by cracked concrete. A.2.6 Rotating-Angle Softened-Truss Model and Fixed-Angle Softened-Truss Model Two models, the Rotating-Angle Softened-Truss Model (RA-STM) and the Fixed-Angle Softened-Truss Model, have been developed by Hsu and his researchers (Hsu 1988, 1993; Belarbi and Hsu 1991, 1994, 1995; Pang and Hsu 1992, 1995; 1996). These methods utilize equilibrium, compatibility and stress-strain relationships for softened concrete. A.2.6.1 Rotating-Angle Softened-Truss Model (RA-STM) This model assumes that the crack angle in post-cracking concrete coincides with the principal stress and strain directions. Since the crack angle in typical elements decreases as the shear increases, this model is called the Rotating-Angle Softened-Truss Model. The average stresses in the reinforced concrete element shown in Fig. A-18(a) can be expressed as sum of the stresses in the concrete (Fig. A-18b, c) and the stresses in the steel (Fig. A-18d). Two coordinate systems are used as shown in Fig. A-18(e); the l and t axes represent the longitudinal and transverse directions, respectively, while d and r represent the principal compressive and tensile directions with an angle, θ , from the l axis, respectively. Equilibrium equations can be derived by superposing the stresses in the concrete and the stresses in the steel bars and assuming that the steel bars resist axial stresses only. The resultant equations are: lll frd ρθσθσσ ++= 22 sincos (A-28a) ttrdt fρθσθσσ ++= 22 cossin (A-28b) θθσστ cossin)( rdtt −= (A-28c) where lσ , tσ , tlτ = normal and shear stresses in each direction in l - t coordinate, dσ , rσ = principal stresses in the d and r directions, respectively, lρ , tρ = reinforcement ratios in the l and t directions, respectively, and lf , tf = steel stresses in the l and t directions, respectively.

A-16 Compatibility relationships can be derived from Mohr’s strain circle. These relationships can simply be obtained from the Eqs. A-28(a), (b), and (c) by replacing lσ by lε , tσ by tε , dσ by dε , rσ by rε , and tlτ by 2/ttγ . Then, the compatibility equations become: θεθεε 22 sincos rd +=l (A-29a) θεθεε 22 cossin rdt += (A-29b) θθεεγ cossin)(2 rdtt −= (A-29c) Belarbi and Hsu (1995) suggested stress-strain relationships to account for concrete softening that are similar to those of MCFT but somewhat more complex: ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= 2 '' ' 2 co d co d cod f εζ ε εζ εζσ εε σ if 1' ≤ co d εζ ε ε (A-30a) ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − −−= 2' ' 1/2 1/ 1 o cod cod f ε ε σ ζ εζεζσ if 1' > co d εζ ε ε (A-30b) where for “proportional loading” r o εζ σ 4001 9.0 += and ro ε ζ ε 5001 1 += (A-30c) for “sequential loading” r o εζ σ 2501 9.0 += and 1=oεζ (A-30d) The predictions from Eq. (A-30) give very similar results to those from Eq. (A-18) in the Compression Field Theory (CFT). Belarbi and Hsu (1994) also suggested that the average stress-strain relationship of concrete in tension after cracking was as follows: 4.0)500,12( r cr d f εσ = (psi) when 00008.0>rε (A-31) where crf , the cracking stress of concrete, is taken as '75.3 cf (in psi).

A-17 The principal tensile stress, dσ , predicted by Eq. (A-31), decreases more rapidly with increasing principal tensile strain, rε , than the stress predicted by Eq. (A-20) as used in the MCFT. Unlike the MCFT, where local stress conditions are checked to account for transmissibility of stresses at a crack, the RA-STM accounts for local yielding of steel bars by using an adjusted average stress-strain curve for mild steel bars embedded in concrete. As shown in Fig. A-18(l), average yield stresses for embedded bars are smaller than the average yield stresses of bare bars. The relationships are: sss Ef ε= if ns εε ≤ (A-32a) ⎥⎦ ⎤⎢⎣ ⎡ −−⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++−= ρ φε 000,1 45/21)25.002.0()291.0( s y s ys f EBBff if ns εε > (A-32b) where ⎥⎦ ⎤⎢⎣ ⎡ −−−= ρ φε 000,1 45/21)293.0( B E f s y n , ρ 5.1 ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= y cr f fB , and φ = the initial crack angle. The shear strength predictions by the RA-STM are similar to those predicted by the MCFT for members with low amounts of reinforcement but are somewhat lower than those of the MCFT for members with large amounts of reinforcement. The RA-STM model does not account for the concrete contribution to shear due to the assumption that the crack angle coincides with the principal stress direction in cracked concrete, resulting in a vanishing of shear stresses along cracks. To account for the concrete contribution to shear, the Fixed Angle Softened Truss Model (FA-STM) was developed by Hsu and his researchers. A.2.6.2 Fixed-Angle Softened-Truss Model (FA-STM) Unlike the RA-STM which considers the reorientation of the crack angle in post-cracking behavior, the FA-STM assumes that the initial crack angle remains constant. For this reason the model is called the Fixed-Angle Softened-Truss Model (FA-STM). Pang and Hsu (1995) recommended that the FA-STM, rather than the RA-STM, be used for cases where the crack angle changes by more than 12 degrees after initial cracking. The two coordinate systems used in this model are shown in Fig. A-18(g); The 1-2 coordinate system represents the principal tensile and compressive directions for an initial crack angle, φ , from the l axis, with the direction of the latter dependent on the principal concrete stress direction just prior to cracking (Fig. A-18h). The equilibrium and compatibility equations for the FA-STM can be derived from the transformation of stresses and strains into the two coordinate systems as follows: Equilibrium equations: lll f ccc ρφφτφσφσσ +++= cossin2sincos 212122 (A-33a)

A-18 tt ccc t fρφφτφσφσσ +−+= cossin2cossin 212122 (A-33b) )sin(coscossin)( 222112 φφτφφσστ −++−= ccctl (A-33c) where lσ , tσ , tlτ are the normal and shear stresses in each direction in thel - t coordinate system (positive for tension), c2σ , c1σ are the average normal stresses of the concrete in the 2 and 1 directions, respectively, c21τ is the average shear stress in the concrete in the 2-1 coordinate system and φ is the angle of the initial inclined cracks. Compatibility relationships are: φφγφεφεε cossinsincos 212122 ++=l (A-34a) φφγφεφεε cossincossin 212122 −+=t (A-34a) )sin(coscossin)(2 222112 φφγφφεεγ −++−=tl (A-34c) where the 2ε , 1ε = average normal strain in the directions 2 and 1, respectively, and 21γ = average shear strain in the 2-1 coordinate. Note that the principal compressive direction in the concrete after cracking does not coincide with the crack angle, and thus, φθ ≠ . The shear resistance of an element subjected to pure shear has been derived by Pang and Hsu (1996) as: tt tt c tt cc t ffff ff ρρρρ σρρσττ ll ll ll l ++++= 2 )()()( 1 2 1 2 21 (A-35) The first term is the “concrete contribution”, cV , and the second term is the “steel contribution”, sV . When the steel stresses reach their local yield stresses, '' yf , at cracks, the local tensile strength of the concrete, c1σ , must be zero and Eq. (A-35) becomes: '''' '''' 2 21 2 )( tyty tyty c t ff ff ρρρρ ττ ll ll l += (A-36) where ⎥⎦ ⎤⎢⎣ ⎡ −−= ρ φ 000,1 deg45/21'' yy ff and '' yf becomes '' yfl or '' tyf for the longitudinal reinforcement and transverse reinforcement, respectively.

A-19 If the crack angle coincides with the d -r coordinates, the concrete shear stress term, c21τ , vanishes and Eq. (A-36) gives: '''' tytyt ff ρρτ lll = (A-37) which does not involve a “concrete contribution” term, cV . It should be noted that the first terms in both Eqs. A-35 and A-36 become infinite when no transverse reinforcement exists and thus those expressions cannot predict a concrete contribution to shear resistance for such members. Further, the assumption that the crack angle remains parallel to the principal direction of the applied stresses is true only when the amounts of longitudinal and transverse reinforcement are equal. In most practical cases, however, those amounts are different and the shear crack angles depend on the principal strain direction. A.2.7 Truss Model with Crack Friction A.2.7.1 Introduction The failure of concrete members or structures in shear is mostly determined by the formation of inclined cracks, one of which opens widely due to excessive strains in the reinforcements crossing one of those cracks. The opening of that crack eventually leads to crushing of the concrete on the compression face of the member and sometimes to non-ductile behavior because yielding of the reinforcement does not occur before failure of the member. Therefore it is understandable that designers have always first looked for the weakest sections along such cracks and determined the amount of reinforcement required there. For shear design, an intensive observation of the shear cracking and failure of beams led Mörsch (1909, 1922) to regard the concrete between the inclined cracks as struts of a truss (Fig. A-19). Mörsch looked at the equilibrium along the failure surface in the B-region, (beam region), initially by means of graphic statics. The same approach was later used by Lessig (1959), who proposed skew, spatial failure surfaces for reinforced concrete beams subjected to torsion, and to torsion combined with shear forces and bending moments. The ASCE-ACI Committee 426 in their State-of-the-Art Report on Shear (1973) was also guided by the concept of looking at failure mechanisms. They extensively reviewed the different shear failure modes and the possible actions and mechanisms for shear transfer at cracks. Generally, "failure mechanism approaches" are not restricted to the kinematic theory of plasticity, but are approaches characterized by the modeling of the actual failure surface in a member, or the critical crack and the localized crushing of the concrete compression zone. In this sense, failure mechanism approaches have a common feature with fracture mechanics approaches, where the localization of the failure zone, either in tension or compression, plays the major role in the failure concept. For shear it is important to note that rarely are all the shear transfer actions modeled in the theories presented in the foregoing sections. Despite the fact that a failure mechanism approach is used, it may well be that a lower bound estimate of the failure load is attained, as for any other non- linear analysis, if safe assumptions for the material characteristics and shear transfer actions are made. Figure 2 2-7: Description of RA-STM and

A-20 The “truss model with crack friction” for members with transverse reinforcement is such a failure mechanism approach and is a “discrete” approach with respect to cracking. It is based on the research of dei Poli et al. (1987, 1990), Gambarova (1979), Kirmair (1985/87), Kupfer and his coworkers Mang, Bulicek, Moosecker and Karavesyrouglou (1979, 1983, 1991), Bulicek (1993) and Reineck (1990, 1991, 1995). The shape and the geometry (inclination) of the crack and the spacing of the cracks are modeled. Therefore, this approach is, in principle, different to smeared approaches based on truss analogies or compression field theories. The shear design method “truss model with crack friction” was implemented as a shear design procedure in the 1999 FIP Recommendations “Practical Design of Structural Concrete” and is explained in this section. The method is also used for shear design in the new German code DIN 1045-1 (2001), which is presented in Section B.1.8. The background for this method is explained more extensively by Reineck (1995, 1998) and also described in Chapter 3.4 of the ASCE-ACI 445 report (1998). A.2.7.2 Equilibrium The shear design method “truss model with crack friction” starts from the fundamental free-body diagram shown in Fig. A-20. That diagram is obtained by separating the member along an inclined crack in the B-region of a structural concrete member with transverse reinforcement. The approach is similar to that of Mörsch and shown in Fig. A-19. The forces acting on the free body are the forces at the end support, the chord forces, the forces in the stirrups and the friction forces along the crack, as shown in Fig. A-20b. In Fig. A-20 the dowel force in the longitudinal reinforcement is neglected, even though that force plays a role in members without transverse reinforcement. Furthermore, for simplicity, as shown in Fig. A-20 the chords are assumed to be parallel to the axis of the member so that there are no vertical components of an inclined compression chord or an inclined tension chord of the truss. That condition means that a different model is necessary in the “disturbed region” (D-region) of the beam near the support because the compression chord must descend towards the support if that support is located beneath the beam as shown in Fig. A-20. The basic requirement for shear design is: VRd ≥ VSd (A-38) where: VSd = shear force at about a distance z cotθ from the face of the support. The basic equation for the shear resistance follows directly from vertical equilibrium: VRd = Vswd + Vfd + Vpd + Vccd (A-39) where: Vswd = shear force carried by the stirrups across the crack Vfd = vertical component of the friction forces at the crack (Fig. A-20b) Vpd = vertical component of the force in the prestressing tendon, and Vccd = vertical component of the force in an inclined compression chord. From Eqs. A-38 and A-39, the design shear force for the web of a structural concrete member is defined as: VSd,web = VSd - Vpd - Vccd (A-40)

A-21 Therefore, the web must provide the following resisting shear force: VRd,web = Vswd + Vfd ≥ VSd,web (A-41) The shear force component Vswd carried by the vertical stirrups across the inclined crack at the ultimate capacity is given by: Vswd = (Asw /sw) fywd z cotβr (A-42) where: Asw = area of transverse reinforcement sw = stirrup spacing in the longitudinal direction fywd = design yield strength of transverse reinforcement z = internal lever arm, and βr = crack angle The shear force component Vsw is known at any load level, (and not at the ultimate capacity only), if the shear force component Vf due to friction is known, in addition to the amount of transverse reinforcement and the crack angle. The force Vf is the vertical component of the combined friction forces Tf and Nf across the inclined crack in the web, as shown in Fig. A-20(b). Normally only a part of the force Tf combines with Nf to provide an inclined compressive force, but additionally a component without axial stresses exists on the crack surface. The shear force component Vf due to friction represents the "concrete contribution" appearing in many codes such as ACI 318, as explained later. A.2.7.3 Inclination and Spacing of Inclined Cracks The crack inclination, as well as the crack spacing, must be assumed based either on tests or determination by a non-linear analysis. The angle of the inclined cracks is normally assumed at 45° for a reinforced concrete member. However, Kupfer and Moosecker (1979) have also pointed out, that the angle could be up to 5° flatter, due to a reduced modulus of elasticity caused by micro- cracking. Flatter angles occur for prestressed concrete members or for members with axial compression, and steeper angles occur for members with axial tension. For such members, the angle of the principal compressive stress at the neutral axis for the uncracked state is commonly assumed as the crack angle. The spacing of the inclined cracks is mainly determined by the amount of reinforcement and relevant formulas have been proposed by Gambarova et al. (1979, 1991), dei Poli et al. (1987, 1990) and Kirmair (1985/87), amongst others. A.2.7.4 Constitutive Laws for Crack Friction Any approach, like the "truss model with crack friction", relies on constitutive laws for the transfer of forces across cracks by friction or interface shear. This shear transfer mechanism is clearly defined in the works of Fenwick and Paulay (1968), Taylor (1972, 1974) and others. However, only a few tests and no theories were available initially for formulating reliable constitutive laws. This situation has changed considerably through the research of the last 20 years by Hamadi (1976), Walraven (1980), Walraven and Reinhardt (1981), Gambarova (1979), Daschner (1980), Nissen (1987), and Tassios and Vintzeleou (1987) in Europe, White and Gergeley and Mattock and Hawkins in the USA (Section 11.7 of ACI 318-02), and Okamura in Japan(JSCE 1986). An extensive state-of-

A-22 the-art report on interface shear has been provided by Gambarova and di Prisco (1991). The constitutive law proposed by Walraven (1980) is often used by others because it describes not only the shear stress-slip relationship for different crack widths but also the associated normal stresses. A.2.7.5 Shear Force Component Vf due to Crack Friction The shear force component Vf in Eqs. A-39 and A-40, that is transferred by friction across the crack, depends on the available slip and the crack width. These displacements have to be calculated in order to determine the strains in the chords and in the web. In return, the displacements and the strains must be compatible with the forces in the model according to the constitutive laws for the shear force components. The variation in the force Vf is plotted in Fig. A-21. Its value depends on the magnitude of the shear, on the strain conditions in the member, on the longitudinal strain εx in the middle of the web, and on the crack spacing, in addition to the assumed friction law. However, for simplicity a constant value Vf may be assumed for code purposes, as indicated in Fig. A-21(a). Because the web area is reinforced by the stirrups, the value for Vf is influenced to a minor extent only by size effects and the longitudinal reinforcement ratio. The practical result for shear design is a constant value for the shear Vf,. In the 1999 FIP Recommendations the following value was specified for Vf , along with the crack angle, for reinforced concrete members without axial forces: Vfd = 0.070 (bw z fcwd) (A-43a) cotβr = 1.20 i.e. βr ≈ 40° (A-43b) The results of Eqs. A-41, A-42 and A-43 are plotted in the simple, non-dimensional design diagram, Fig. A-22, which is well known and used in many codes. The crack friction governs the design for low and medium shear stress values and only for a small range of very high shears, is the strength of the compression struts fcwd reached. That value is characterized by the quarter circle in Fig. A-22. For low shear stresses or low reinforcement ratios, ρw, the strength is limited by the minimum reinforcement ratios ρw,min specified in codes, and the corresponding values ωw,min therefore represent the lower limit for applying Eq. (A-43). For members without transverse reinforcement that capacity is far lower than the value for Vf. Clearly this result means that Vfi is completely different to the ultimate shear force Vct for members without transverse reinforcement. The approach “truss model with crack friction” can consider the influence of axial forces as well as prestress. Flatter cracks occur and Vf is reduced as consequence of the negative longitudinal strain, εx. The influence of the crack inclination is very pronounced as shown in Figs. A-20(b) and A- 21(b). For cracks flatter than about 30°, the shear force component Vf no longer plays any role and the struts are parallel to the cracks. In the 1996 FIP Recommendations, the following relationships are proposed for members with axial compression or prestress, and the practical results are shown in Fig. A-23. cotβr = 1.20 - 0.2 σxd / fctm (A-44a) Vfd = 0.10 (1 - cotβr /4) (bw z fcwd) ≥ 0 (A-44b) where: σxd = NSd /Ac = axial stress [( -) in compression] fcwd = 0.80 f1cd = compressive strength of inclined struts

A-23 In the case of axial tension, the cracks may be steeper than 45° and the strain εx is positive. In the FIP Recommendations the following relations are given for members with axial tension: cotβr = 1.20 - 0.9 σxd /fctm ≥ 0 (A-45a) Vfd = 0.10 (1 - 0.36/ cotβr) (bw z fcwd) ≥ 0 (A-45b) The influence of axial tension is quite noticeable, because the value Vfd = 0 is already reached at a value of σxd = 0.933 fctm and a crack angle of about 70°, corresponding to a value cotβr = 0.36. This result leads to high stirrup amounts. The approach may be on the safe side because the crack angle is evaluated on basis of the uncracked state, rather than the cracked state. Unfortunately, there are too few tests with which to compare predictions and propose a better relationship. The shear force component Vf in DIN 1045-1 was made dependent on the concrete strength and its value decreases with increasing strength. Values are given in Section B.1.8. A.2.7.6 The Truss - Model When the shear force component Vf is known, all the forces are defined at the crack or failure surface, so that the state of stress in the struts between the cracks is also known. Obviously the solid concrete between the cracks is the strut of a truss formed together with the stirrups (Fig. A-24a). This was the action that Mörsch utilized in his truss analogy (see Fig. A-19). The frictional forces cause a biaxial state of stress as shown in the Fig. A-24(b) with a principal compression field at a flatter inclination than that of the cracks, and a tension field perpendicular to it. For high shear forces the minor principal stress turns into compression because the normal stresses due to friction prevail. However, these compressive stresses are so small, that they are usually neglected. Then, only the truss of Fig. A-24(a) remains with an uniaxial compression field. That result is the model normally used for theory of plasticity analyses. The minor principal stress is tension for small shear forces, so that the state of stress can be represented by the two truss models shown in Fig. A-25. The usual truss model, with uniaxial compression struts inclined at the angle θ in Fig. A-25(a), is superimposed on a truss with compression struts in the same direction and concrete ties perpendicular to the struts (Fig. A-25b). Then there are two load paths for shear transfer, as defined by Schlaich, Schäfer and Jennewein (1987) and as also shown earlier by Reineck (1982), and with different explanations by Lipski (1971, 1972) and by Vecchio and Collins (1986) in their "modified compression field theory". The model in Fig. A-25(b) is the same as that proposed by Reineck (1990, 1991) for members without transverse reinforcement, so that the transition from members with, to members without, transverse reinforcement is consistently covered. For simplicity only the truss with a uniaxial compression field is modeled, because it is applicable for both the intermediate and high shear stress ranges, υd = τRd / fcwd. With the shear force components Vsw and Vf defined by the equations given above, the angle θ of the inclined struts in the web may be derived as follows: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − = webSd fd r V V , 1 cotcot βθ (A-46a)

A-24 or zf s A V ywd w webSw webSd ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛= , ,cotθ (A-46b) with VRd,web = Vswd + Vfd ≥ VSd,web This means that the angle θ varies with the magnitude of the applied shear force, i.e. the angle θ increases with increasing applied shear force VSd,web. The upper limit of the resistant shear force may then be derived from the truss analogy as follows: VRd,web,max = bw z fcwd sinθ cosθ = bw z fcwd / (cotθ + tanθ) (A-47a) For θ = 45° and fcwd = 0.80 f1cd (ν = 0.80) the highest value is reached with VRd,web,max = 0.5 bw z fcwd = 0.4 bw z f1cd (A-47b) However, this check is rarely required, because it is only relevant in a small range of very high shear stresses, as can be seen from Figs. A-22 and A-23. For example, for reinforced concrete beams web compression failures occur only for values of υd > 0.472. A.2.7.7 Strength of Concrete between Cracks The concrete between the cracks is uncracked and forms the strut. However, apart from the compressive stresses due to the truss action there are also transverse tensions in the struts due to the friction stresses and due to the forces induced by the bonded stirrups. This action leads to a reduction in strength compared to the uniaxial compressive strength allowed in compression chords, which is f1c = 0.95 f´c (not considering any reduction for a rectangular stress block). Additional reasons for such a reduction in strength of the inclined struts are the smaller effective width of the strut, (rough crack surfaces), and the disturbances caused by the crossing stirrups. Consequently it was found by Schlaich and Schäfer (1983), Schäfer, Schelling and Kuchler (1990), Eibl and Neuroth (1988) and Kollegger and Mehlhorn (1990) that the following value may be assumed for the concrete strength of the struts: fcwu = 0.80 f1c (A-48) where f1c = uniaxial compressive strength of concrete, (strength of slender prism). This strength has a relatively high value compared to the lower values proposed elsewhere as so- called “effective strengths” e.g. ν f1c = 0.60 fc or 0.50 fc as used in the theory of plasticity. This result illustrates that these lower effective strengths are meant also to cover insufficient transfer of forces across the cracks by friction. The practical outcome of the foregoing higher value for the compressive strength is that, for high ratios of transverse reinforcement, the capacities are far higher for the variable truss angle method in the EC 2 than for the theory of plasticity, Reineck (1991, 1999). A.2.7.8 Concluding Remarks on Truss-Model with Crack Friction The "truss-model with crack friction” is a failure mechanism approach, where the expected failure cracks are considered discretely. The possible transfer of forces over the crack due to friction

A-25 or interface shear is modeled explicitly. The approach gives directly the shear capacity and the required amount of transverse reinforcement. The contribution of friction across the cracks provides a physical explanation of the “concrete term Vc” as used in US Codes as the vertical component Vf of the forces transferred across the crack. For practical design it can be assumed that this shear force component Vf is independent of the applied shear force Vu. This assumption yields, for a reinforced concrete beam, the diagram of Fig. A-22, which is similar to proposals that use a concrete contribution Vc , like the approach proposed by Leonhardt (1965, 1977), the approach in the EC 2, part 1 (1991) and the approach in ACI 318. Similar diagrams (Fig. A-23) may be derived for members subjected to shear and axial compressive forces or prestress, the influence of which is covered consistently by using the “truss-model with crack friction”. The shear force component Vf for members with shear reinforcement is different to the ultimate shear force Vct for members without transverse reinforcement. Due to the fact that the web is reinforced by the stirrups, the value for Vf is higher than that of Vct, and the size effect, as well as the influence of the longitudinal reinforcement ratio, are far less pronounced. The state of stress is defined at the crack, and from this definition the principal stresses between cracks can be evaluated. The resulting biaxial stress field is represented by the superposition of the two trusses in Fig. A-25. These trusses are statically admissible stress-fields for the web in the B- region of a beam, so that the “truss model with crack friction” fulfils the requirements of a lower- bound method, provided safe strength limits are defined in the constitutive laws for the shear transfer mechanisms at the crack. These models demonstrate that both approaches, discrete and smeared ones, result in similar design models, and only the failure criteria are different. In discrete shear transfer approaches, friction values are explicitly checked, whereas in smeared approaches limits for the strut angle or the strength of the struts are empirically derived. It is important to note that any controversy over the “standard method” versus the “variable strut angle method” is futile and unproductive. More important is whether the behavior and the test results are realistically and economically covered as is the case for the “truss-model with crack friction”. The “truss-model with crack friction” also fulfils an indispensable requirement for a modern design code, namely that the resulting shear design procedure is consistent with the design rules for strut-and-tie models, which is important for practicing engineers. The resultant truss models are clearly defined so that transition from the B-region, where the “truss-model with crack friction” applies, to the D-region, where the strut and tie model applies is seamless. This result is also relevant for future ACI 318 and AASHTO-LRFD Codes, because the design format for the “truss-model with crack friction” is very similar to that for the present shear design rules of ACI 318, since both use "concrete terms". A. 3.1 Concluding Remarks The smeared models of both the Modified Compression Field Theory (MCFT) and the Fixed-Angle Softened –Softened Truss Model (FA-STM) have been adjusted so that they provide excellent agreement with test data for a wide range of test variables. However, both models are sufficiently complicated that computer programming of the models is almost a necessity for their practical use. Further, the models cannot be readily used to check the appropriateness of an existing design. The Truss-Model with Crack Friction provides a failure mechanism approach that is easy to understand and readily useable to check an existing design. Further, the approach provides a rational explanation for why the Vc term of the AASHTO Standard Specification and ACI Code 318 is appropriate.

A-26 However, the basic difference is that while the Vc term in the AASHTO Standard Specification and ACI 318 is assumed to be the shear force at diagonal cracking, the Vc term for the “truss-model with crack friction” is a function of the shear that can be transferred by friction across the inclined crack in a B-region. While the two Vc terms may have similar values the basic concepts associated with each are fundamentally different with the Vc term of the “truss-model with crack friction usually being less than the shear force at diagonal cracking. The appropriate angle of the truss associated with the crack for the “truss-model with crack friction” will vary with the type of cracking, “flexure-shear” or “web-shear”, anticipated for the B-region being checked. Thus it is rational to relate the effectiveness of the transverse reinforcement to the type of diagonal cracking predicted. A simplified failure mechanism with Vc related to the “truss-model with crack friction” concept and the contribution of the transverse reinforcement determined using a variable angle truss whose angle is dependent on the type of diagonal cracking is a simplified and rational approach justified by theory, experience, and the economy of the result.

A-27 Figure A-1 Shear Transfer/Actions Contributing to Shear Resistance Figure A-2 Distribution of Internal Shear Resistance (ASCE-ACI Committee 426, 1973) d cot θ V : Shear in Compression Zonecc V : Dowel Action V : Aggregate Interlock V : Residual Tensile Stress in Concrete V : Steel Contribution V : Vertical Component of Prestressing Steel d ca cr s p Vsupport d V = A fs s v Vcr Vd Vca Vp θ Vd Vcc

A-28 Figure A-3 Response of Concrete in Uniaxial Tension (Gopalaratnam and Shah, 1985) Figure A-4 Influence of Concrete Compressive Strength on Shear Strength (Kuchma and Kim, 2002) 1Specimen 0 Displacement, δ µ( in) 600 500 400 300 200 100 4 3 2 1 300 600 900 1200 1500 1800 2100 2400 0 10 20 30 40 50 60 ( m) µ (MPa) Av er ag e N et A xi al S tre ss , f ct (p si ) 2 3 4 496fc| 559 531 512 Average values of 4 specimens 0 0 50 100 150 200 250 0 5000 10000 15000 f'c (psi) vu te st (p si ) Moody, Viest, Elstner,and Hognestad : h=305 mm, rh = 1.89 %, a/d=3.41 Yoon and Cook : h=750 mm, rho=2.80 %, a/d=3.23 Collins and Kuchma : h=1000 mm, rho=1.01 %, a/d=2.92 Collins and Kuchma : h=1000 mm, rho=0.76 %, a/d=2.92 Angelakos and Collins : h=1000 mm, rho=1.01 %, a/d=2.92 ACI 318-02 ρ l ρ l ρ l ρ l ρ l

A-29 3.2 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0 0.25 0.20 0.15 0.10 0.05 0 (MPa)(psi) max. aggregate size a = 25 mm (1 in.) a = 2.5 mm (0.1 in.) ρ A s b wd = = = = 0.4% 3500 psi (24.1 MPa) 52 ksi (386 MPa) 6” 8” d = 4 in. 12” 24” 20” 39” b = 39” d = 79” b = 59” d = 118” d 12d d 0 1000 2000 3000 d (mm) 0 20 40 60 80 100 120 d (inches) V calculated here Air Force Warehouse Beams PCA Tests of Model Air Force Warehouse Beams fc f y a=10 mm (0.39 in.) 25.4 mm = 1” ' cf ν , ' cf ν Figure A-5 Size Effect in Shear (Kuchma and Collins, 1998) Web-Shear Crack Flexure-Shear Crack Figure A-6 Types of Inclined Cracks

A-30 (a) Beam Deep Very Short Slender Very slender Short Flexural capacity Failure Inclined cracking and failure 1.0 2.5 6.5 a/d (b) Moments at cracking and failure Shear vailure Flexural capacity Inclined cracking and failure 1.0 2.5 6.5 (c) Shear at cracking and failure M om en t a t l oa d po in t Mn Sh ea r a a d V V VV Inclined cracking a/d Inclined cracking Figure A-7 Effect of /a d on Shear Strength of Beams without Stirrups (McGregor, 1988)

A-31 Figure A-8 Diagonal Tension Failure (ASCE-ACI 426, 1973) Loss of bond due to splitting crack a) Shear-tension failure b) Shear-compression failure 1 Anchorage failure 2 Bearing failure 3 flexure failure 4 Arch-rib tension failure 5 Compression strut failure 1 2 3 3 4 5 Tension tie Compression strut Figure A-9 Modes of Shear Failures in Short Beams (ASCE-ACI 426, 1973) Figure A-10 Modes of Failures in Deep Beams (ASCE-ACI 426, 1973) Crushing Inclined cracks Crushing T T+ T∆ Figure A-11 Modes of Shear Failures of I-Beams (ASCE-ACI 426, 1973)

A-32 (a) Cross section V M M = 0 s 0.5V 0.5V jd / 2 A fv v f2 f2 s/ 2 f2 V 45o jd bw Av s (b) Diagonal stresses and longitudinal equilibrium (c) Stirrup force V Figure A-12 Equilibrium Conditions for 45o Truss Model (Collins and Mitchell, 1991) (a) Cross section V M M = 0 s 0.5Nv jd cos θ A fv v f2 f2 f2 D jd bw s (b) Diagonal stresses and longitudinal equilibrium (c) Stirrup force θ s sin θ Av 0.5Nv Figure A-13 Equilibrium Conditions for Variable-Angle Truss Model (Collins and Mitchell, 1991)

A-33 fsy fcy fsx fcxv Es fc| L |c 1 ρ fsxx f2 y v x ρ fsyv 12 2θ y x 0.5γm 12 2θ ‰ L 1 L 2 L x L y L x L 1 L 2 L y v γ/2 Initial cracks Later cracks fs f2 L s fy L 2 f2max (a) Free Body Diagram (b) Concrete Stress Circle (c) Strains in Cracked Element (d) Strain Circle (e) Reinforcement (f) Cracked Concrete Equilibrium Conditions Compatibility Relationships Stress-Strain Relationships σ L ρ θx fsx v cot  tan cotv ( )θ θ ρ θy fsy v tan f2 tan 2θ ( )L  Ly 2 ( )L  Lx 2 fs fy L Es s } f2 L |c L |c L 2 L 22  2 f2max f2max fc| 0.8 170 L 1 where f2 L 1 L 2 L 2 f2 L L ‰  L  ‰x tan 1 tan( ) ( )1 2 2 2 / L L  L ‰  ‰y tan 1 tan( ) ( )1 2 2 2 / N L  L ‰xy x 2 tan( ) 2 / Figure A-14 Basic Relationships for Compression Field Theory (Mitchell and Collins, 1974) Figure A-15 Shear Panel Test (Vecchio and Collins, 1982) Figure A-16 Compressive Stress-Strain Relationship for Cracked Concrete (Vecchio and Collins, 1986)

A-34 v f2 y v x 12 2θcr v (c) Free Body Diagram (d) Local Concrete Stresses σv vci 2θ vci ρ θ x sxcr cotf v + vci cot θ ρ θv sycr tanf v θ - vci tan Compatibility Conditions Stress-Strain Relationships L L ‰  L  ‰x tan 1 tan( ) ( )1 2 2 2 / L L  L ‰  ‰y tan 1 tan( ) ( )1 2 2 2 / N L  L ‰xy x 2 tan( ) 2 / Average Strains: w s L m‰ 1 s m‰ Crack Widths: where sin‰ cos‰ smx smy / 1 θ ‰ L 1 L 2 L x L y Initial cracks Later cracks (c) Strains in Cracked Element ‰ L 1 L 2 L x L y Initial cracks (g) Crack Width s m‰ w s L m‰ 1 smx, smy : see equations 2-29(a) and (b) L 1 fcr w s L m‰ 1 L cr L 1 L 1 Eq.(2.23) Eq.(2.30) Average Stresses: Stresses at Cracks: fc| f1 fcr 1 500  L 1 vci } 0.18 0.31  24 w a  16 fsy fcy fsx fcxv L |c y v x 12 2θ y x 0.5γxy 12 2θ L 1 L x L y L x L 1 L 2 L y v γ/2 L 2 (a) Free Body Diagram (b) Average Concrete Stresses e (f) Average Strains Equilibrium Conditions σ L ρ θ x sx cotf v - f1 ρ θv sy tanf v - f1  tan cotv ( )θ θ - f1 f2 f2 L 1 L 2 Es fc| 1 L s L 2 f2 fs fy L Es s } f2 L |c L |c L 2 L 22  2 f2max f2max fc| 0.8 170 L 1 where tan 2 θ ( )L  Ly 2 ( )L  Lx 2 (h) Reinforcement (i) Cracked Concrete in Compression (j) Cracked Concrete in Tension w a 0 0 1 2 0.2 0.4 0.6 a = 0.75 in vci fc| Concrete in Compression: Concrete in tension:Reinforcement: (at crack slip) f2 ρx sx f ρv sy f ρx sxcr f ρv sycr f fsycr fsxcr tan‰ f1 vci where fcr fc|4Average Tensile Stresses : Allowable Local Tensile Stresses : fs fy f2max f2 f1 f1 f1 (k) Allowable Local Shear Stress on Crack where f1 Figure A-17 Description of Modified Compression Field Theory (Vecchio and Collins, 1986) e equations -26( ) and (b) q.(2.20) Eq.(2.27)

A-35 fcr 1 M Equilibrium Equations Es L s fs L Es s } (l) Average Stress-Strain Curve of Steel Bars in Concrete Reinforcement in Concrete: fs fy Œ r (a) Reinforced Concrete  lt  lt Œ l Œ t Œ l fl Œ t t f Œ d += Œ Œl l l f= + σ θ + sinr 2σ θ d cos2 σ = + σ θ + t r t t cos 2 Œ f σ θ d sin2  lt r − σ θ θ sin cos (σ )d l t 2 l t dr  lt  lt cŒ t Œ l c  lt  lt cŒ t Œ l c θ (b) Concrete (d) Reinforcement l t dr θ 1 M l t 2 Œ 2Œ 1 Œ 2 Œ 1 Œ d Œ r (g) Principal axes 2-1 for Applied Stresses (e) Principal axes d-r for Stresses on Concrete (h) Assumed Crack Direction in FA-STM (f) Assumed Crack Direction in RA-STM Stress-Strain Relationships L 1 fcr L cr L 1 L |c L 2 f2 L 1 L 2 fc| L 2 f2 ζ L |c ζ L |c L 2 L 2 2  2 (i) Softened Stress-Strain Curve of Concrete in Compression w Concrete in Compression: Concrete in tension: f1 f1  L |c Œ p L p = ζ L |c Œ 2 Œ 1 c Œ 1 c Shear Strain, N 21 S he ar S tre ss o f C on cr et e,  21 Ep L n L y (k) Average Stress-Strain Curve of Concrete in Shear fn fo 1 cŒ 2 ζfc| L 2 ζ L |cif } ζ L 2 1-  2 cŒ 2 ζfc| L 2 ζ L |cif ! / ζ L |c 1 2/  1 1 00   L 1 ζ 0.9where L 1 0.00008 0.4 L 1 0.00008! Ec whenŒ 1 c L 1 L 1 0.00008} where Ec fc|47,000 when fcr fc|3.75 (psi) (psi)  21 N 21 c  21m c 1 6  1 N 210 where N 210 0.85 L 10 1   ltmcos 2φ ρ − l l lf y Œ ρ − t ty t f Œ  21m c 1/ 2 ' ' sin2φ Concrete in shear: fs fy 1.5 fy| (0.91 - 2B) + 0.02 + 0.25B L s L y 1 1000 ρ 1 φ / (45 deg) Bear Steel Bars L s L n when !L s L n when where L n L y (0.93 - 2B) 1 1000 ρ 1 φ / (45 deg) B ρ 1 f cr fy fcr fc|3.75 (psi), Compatibility Conditions L L Ll = + θ sinr 2d cos2θ L L Lt r cos= + θ2d sin2θ N  L Llt r − θ θ sin cos ( )d Equilibrium Equations Œ  Œl l l f= + σ φ + 2 φ φ + sin sin cos1 21 2σ φ 2 cos2 σ = φ + σ φ + 2 φ φ + t 1 21 t t cos sin cos2  Œ f σ 2 sin2  lt 1 21 + σ φ φ + τ ( φ − φ)sin cos cos sin2 2(−σ )2 L L L Nl = + φ + φ φ sin sin cos1 21 22 cos2φ L L L Nt 1 21 cos sin cos= φ + φ − φ φ22 sin2 N  L Llt 1 21 + φ φ + γ ( φ − φ)sin cos cos sin2 2(− )2 Compatibility Conditions Rotating Angle Softened Truss Model (RA-STM) Rixed Angle Softened Truss Model (FA-STM) (c) Concrete Stresses 2θ σ Œ r Œ d  Œ l  lt  Œ t  lt  Reinforced Concrete Memberane Elements c c c c c c c c c (i) Average Stress-Strain Curve of Concrete in Tension cŒ t  lt Œ l c  lt Steel Bars in Concrete Figure A-18 Description of RA-STM and FA-STM (Hsu, 1988, Pang and Hsu, 1996)

A-36 Figure A-19 The Approach by Mörsch (1909,1922) for Shear Design a) free-body with forces acting on it b) forces due to friction Figure A-20 Free-Body of an End Support of a Beam with Applied Forces

A-37 a) reinforced concrete members b) prestressed concrete members Figure A-21 Results of Parameter Studies for the Shear Force Component Vf (Kirmair, 1985 and 1987) Figure A-22 Dimensioning Diagram for the Design of Vertical Stirrups for RC Members without Axial Forces

A-38 Figure A-23 Dimensioning Diagram for Vertical Stirrups for Prestressed Concrete Members and Members with Axial Compression a) truss action b) biaxial stresses due to friction Figure A-24 Forces and Stresses in the Discrete Concrete Struts between the Inclined Cracks

A-39 a) truss model with uniaxial compression field in B-region b) biaxial tension-compression field in concrete for low shear Figure A-25 Models for B-regions with Shear Force