Longitudinal Steel Stresses in Beams Due to Shear and Torsion in AASHTO-LRFD Specifications

Rahal, Khaldoun N

The alternative shear and torsion design method (the general method) in the current AASHTO-LRFD specifications and the Canadian A23.3 CSA building code requires a check on the adequacy of the longitudinal steel to resist the stresses not only from bending and axial loads, but also from shear and torsion. This paper presents an evaluation of the general method equation for the calculation of the stresses in the longitudinal steel in members with adequate transverse reinforcement. Longitudinal steel strain readings from tests are compared with the values calculated based on the general method equation for longitudinal force. The study covers the cases of nonprestressed members subjected to combined shear and/or torsion and to prestressed and nonprestressed members subjected to combined torsion and bending. The general method equation is found to give satisfactory results.

Keywords: beams; bond; longitudinal reinforcement; prestress; shear; strain; stress.

(ProQuest Information and Learning: … denotes formulae omitted.)

INTRODUCTION

Shear and torsion cause diagonal cracks in concrete beams. These inclined cracks cross the orthogonal reinforcement and hence cause tension not only in the transverse steel, but also in the longitudinal steel.

Accounting for the effects of shear and torsion on the longitudinal steel is necessary to ensure a safe design and to avoid the possibility of brittle failures. For example, exterior support regions in edge beams and supports regions of simple beams are prone to brittle bond-shear and/or torsion failure. Figure 1 shows a simply supported beam after the development of diagonal cracks. The diagonal crack crosses both the transverse and tension longitudinal reinforcement at an inclined angle, and causes stresses in both. One of the cracks may originate from the inner face of support plate. Despite the relatively low bending moment across this crack, the stresses in the longitudinal steel due to shear at this location are significant and therefore cannot be neglected. Proper development of both the transverse reinforcement and longitudinal reinforcement into the concrete should be provided.

The stirrups can be easily detailed for proper development into the concrete. The longitudinal steel, however, is cut or hooked at the edge of the beam as shown in Fig. 1, leaving a relatively short distance as development length to transfer the force to the concrete via bond stresses. This is especially critical in deep beams where the stresses in the support region are larger in comparison with shallow beams and in prestressed girders where the shear forces resisted are relatively large and the length of development remaining across the transfer length is relatively small. Beams subjected to high torsional stresses suffer from the same risk.

The effect of longitudinal steel is well recognized, and is accounted for in the detailed equation for the concrete contribution V^sub c^ to the shear resistance in the current ACI code.1 The code, however, disregards the effect of shear on the longitudinal stresses. Moreover, it permits the designer to use the simplified equation for V^sub c^, and hence disregards the mutual effect between shear and the stresses in the longitudinal reinforcement. Torsion is treated in a more rational manner where the transverse as well as the longitudinal reinforcement are proportioned to resist the torsional moment.

In 1994, the AASHTO-LRFD Specifications2 and the Canadian CSA-A23.3 building code3 adopted an alternative shear and torsion design method. The original method is the traditional method, based on the modified 45-degree truss model. The alternative method, named the general method, is a simplification of the modified compression field theory.4-6 This method recognizes the effect of shear and torsion on the longitudinal reinforcement and requires a check on the adequacy of the longitudinal steel to resist the forces not only from bending and axial loads, but also from shear and torsion. As discussed previously, the contribution of the shear and/or torsion to the longitudinal force is significant. Hence, the accuracy in calculating the force and the stresses in the longitudinal steel due to shear and torsion is critical to ensure this steel is safely proportioned and to avoid the possibility of brittle bond-shear (or torsion) failures.

In this paper, the experimental results available in the literature from 17 transversely-reinforced nonprestressed and partially prestressed concrete beams are used to check the accuracy of the general method equation for adequacy of longitudinal reinforcement. The strain measurements in the test specimens are compared to the strain values calculated from the AASHTO equation for the longitudinal forces.

RESEARCH SIGNIFICANCE

The longitudinal reinforcement in beams should be proportioned to resist the forces not only from bending, axial load, and the prestressing operation, but also from shear and torsion. This could be critical in many cases such as near the support regions of beams, where the longitudinal stresses due to shear and torsion are significant while the development length past the inner face of the support is limited. The current AASHTO-LRFD and CSA A23.3 alternative shear and torsion design provisions include an equation to check the adequacy of the longitudinal prestressed and nonprestressed steel to resist the combined effects. This paper evaluates the accuracy of this equation by comparing its results with experimental results from prestressed and nonprestressed beams subjected to combinations of shear, torsion, and bending.

GENERAL METHOD

This section gives basic details of the shear and torsion design requirements in the general method as given in the current 1998 AASHTO-LRFD.7 More details about the background of the development and the evaluation of these provisions can be found elsewhere.6,8,9 The general method in the Canadian CSA-A23.3 code3 is similar except that the AASHTO-LRFD Specifications2,7 were adopted to suit bridge girders and prestressed concrete.

The general method assumes that concrete, steel, and inclined prestressing contribute to the shear resistance. The nominal shear strength V^sub n^ at a section is given by

… (1)

where V^sub c^ is the concrete contribution provided by the residual tensile stresses in the cracked concrete, V^sub s^ is the shear contribution provided by the stirrups when ample longitudinal reinforcement is provided, and V^sub p^ is the component of the effective prestressing force in the direction of the applied shear. The term β is a factor that depends on the ability of concrete to transmit tensile stresses, f’^sub c^ is the 28-day compressive strength of concrete cylinder, b^sub v^ is the effective shear width, d^sub v^ is the effective shear depth, A^sub v^ is the area of stirrups within a spacing s, f^sub yv^ is the yield strength of the stirrups, and θ is the angle that the principal compressive stresses and strains make with the longitudinal axis of the beam. Diagonal shear cracks are assumed to be oriented at the angle θ.

In torsion, the concrete contribution is neglected and the torsional strength is assumed to come solely from the steel contribution according to the following equation

… (2)

where A^sub t^ is the area of one leg of the torsional reinforcement, and A^sub 0^ is the area enclosed in the shear flow path. AASHTO permits A^sub 0^ to be taken as 0.85A^sub 0h^, where A^sub 0h^ is the area enclosed in the outermost transverse torsional reinforcement.

For combined shear and torsion, the total area of one leg of the transverse reinforcement is taken as

… (3)

The values of β and θ used to calculate V^sub c^, V^sub s^, and T^sub s^ in Eq. (1) and (2) are given in two separate tables in the CSA-A23.3 code3 and AASHTO-LRFD Specifications.7 AASHTO’s table for members with adequate transverse reinforcement is reproduced in Table 1 whereas that for other members can be found elsewhere.2,7 The factors β and θ in Table 1 depend on the level of longitudinal strain in the section ε^sub x^ and on the normalized nominal shear stress v/f’^sub c^. The term ε^sub x^, which is a measure of the level of longitudinal strain in the section, is conveniently and conservatively taken at the level of the centroid of the flexural tension reinforcement. This term can be calculated as

… (4)

where A^sub s^ and E^sub s^ are, respectively, the area and modulus of elasticity of the non-prestressed reinforcement in flexural tension zone of section; A^sub ps^ and E^sub p^ are, respectively, the area and modulus of elasticity of the prestressed reinforcement in flexural tension zone of section; N^sub u^ is the factored applied axial load (positive if tensile); M^sub u^ is the factored bending moment; and f^sub po^ can be taken as the effective prestress after losses. The flexural tension side is taken as the half-depth containing the tension zone. The term ε^sub x^ in Eq. (4) is obtained by dividing the estimated force by the estimated tensile stiffness of the longitudinal reinforcement. Recent work by Collins and his colleagues10-11 suggested that for members with adequate transverse reinforcement, Eq. (4) gives relatively conservative results, and recommended calculating the longitudinal strain at the center of the section. This strain is approximated to be about half the value calculated in Eq. (4).

The adequacy of the longitudinal reinforcement for resisting the stress that can be developed is checked by ensuring that

… (5a)

or

Q^sub n^ ≥ Q (5b)

where f^sub yl^ is the yield strength of nonprestressed reinforcement in tension zone, and f^sub ps^ is the average stress in the prestressing reinforcement at the time for which the resistance is required.

The second part of Eq. (5a), referred to in this paper by Q, represents the maximum factored force that can develop in this steel due to the combined effects of flexure, axial load, shear, and torsion. The first part of the equation, referred to in this paper by Q^sub n^, represents the nominal tensile force that the longitudinal steel can resist. In accordance with the modified compression field theory, it is assumed that at ultimate conditions, the prestressing steel “does not remember” that it was prestressed, and the maximum attainable strength is assumed to be the ultimate strength irrespective of the prestrain. It is also to be noted that the contribution of combined shear and torsion to the force in the longitudinal steel is taken as the square root of the sum of the squares of the individual contributions from torsion and shear. This value is an intermediate value between the larger force that takes place on the side of the section where the shearing stresses due to shear and torsion are additive, and the smaller force on the opposite side where these stresses are subtractive.

According to MCFT,4-6 the longitudinal force in the tension zone due to the shear force is given by (0.5V^sub n^cotθ – 0.5f^sub 1^b^sub w^d^sub v^) where f^sub 1^ is the concrete tensile stresses averaged across many cracks. In Eq. (4), the second term of the expression is neglected and hence the longitudinal strain indicator is conservatively calculated.

The longitudinal force due to shear in Eq. (5) is based on the calculation at the location of the crack because this ultimate strength equation is applicable at the location of the largest stresses. The contribution of shear to the longitudinal force is given by the MCFT6 as (V^sub n^ – 0.5V^sub s^) cotθ, which can be rearranged to give (V^sub c^ + 0.5V^sub s^)cotθ. It is to be noted from the latter expression that the shear taken by the concrete causes twice as much tension as the shear taken by the stirrups. The code uses the shear contribution expression after replacing V^sub n^ with V^sub u^/φ and excluding the contribution of the inclined prestressing V^sub p^, as shown in Eq. (5). Hence, for the case of shear, Eq. (4) gives a conservative estimate of the contribution of V to the longitudinal force or strain.

The shearing stress v due to shear and torsion in hollow sections is given by

… (6)

In solid sections, redistribution of shearing stresses is possible.12 For this case, the combined shearing stress in the general method is given by

… (7)

To ensure that the provided transverse reinforcement yields before crushing in the concrete, Table 1 sets an upper limit to the normalized shear stress represented by

v/f’^sub c^ ≤ 0.25 (8)

Rahal and Collins8 showed that the upper limit ranges between 0.25 and 0.3 depending on the longitudinal strain ε^sub x^, and that the 0.25 limit is a conservative practical value suitable for the whole range of strain values.

EVALUATION OF LONGITUDINAL FORCE EQUATION

The accuracy of Eq. (5) is checked against the test results from 17 transversely reinforced beam specimens available in the literature. These specimens included partially prestressed and nonprestressed concrete beams subjected to combined torsion and/or bending, and nonprestressed concrete beams subjected to combined shear and/or torsion. A list of these beams and their observed and calculated capacities are shown in Table 2.

Prestressed and nonprestressed beams under pure torsion

While the case of pure torsion is not a practical one, longitudinal strains in symmetrically reinforced beams subjected to pure torsion are uniform and can be easily measured, and hence comparison with calculated values is straightforward. In nonprestressed members subjected to pure torsion, the longitudinal strain ε^sub x^ from Eq. (4) and the longitudinal stress calculated based on the left-hand side of Eq. (5a) are equivalent because the concrete contribution to the torsional strength is neglected. This is not the case in partially or fully prestressed sections where, for example, the longitudinal strain ε^sub x^ can be compressive due to a large prestressing force while the stressed calculated based on Eq. (5a) is always tensile.

Transversely reinforced specimens, PT4 and P1, tested by Mitchell and Collins13 are used for the evaluation, and their cross-sectional details are shown in Fig. 2(a). Specimen PT4 was not prestressed, while Specimen P1 was partially prestressed. Targets were placed on the longitudinal reinforcement before loading, and the longitudinal steel strains were measured using 200 mm mechanical gauges at numerous load stages during the tests.

Figure 2(b) shows the observed torque versus the reported average and maximum longitudinal strains in PT4. This specimen resisted a maximum torque of 70.1 kN·m and failed after the transverse reinforcement showed significant yielding. The calculated ultimate torque was 63.4 kN·m. (T^sub exp^/T^sub calc^ = 1.11). Refer also to Table 2. The calculated longitudinal force Q in half the reinforcement is approximately 263 kN. At a crack location in this section subjected to tensile longitudinal strains, this force corresponds to a strain ε^sub L^ of 1.65 × 10^sup -3^ (which is the same value calculated using Eq. (4), as expected). The ratio of observed-to-calculated longitudinal force is therefore 1.053. The calculated strain values are plotted in Fig. 2(b), showing that these results are accurate and conservative.

Figure 2(c) shows the observed torque-longitudinal strains relationship for the partially prestressed beam, P1. This specimen resisted a maximum torque of 86.1 kN·m and failed after the transverse reinforcement showed significant yielding. The calculated ultimate torque was 77.8 kN·m, giving a ratio of T^sub exp^/T^sub calc^ = 1.11. The values of Q and Q^sub n^ are 346 and 483 kN, respectively, and hence this steel is adequate. The strain calculated from Eq. (4) is 0.713 × 10^sub -3^ and is adequate to estimate the state of strain in the beam, as shown in Fig. 2(c). The strain required to achieve a 346 kN force in the longitudinal steel is 0.0055. Designing the development of the longitudinal steel for Q = 346 kN yields a conservative value because Eq. (5) neglects the effect of prestressing and is suitable for ultimate conditions in the bars. Hence, the use of the general method for this partially prestressed member is shown to be conservative and adequate.

Combined torsion and bending in nonprestressed members

Five underreinforced hollow beam specimens were tested in Series TBU14 under combined torsion and bending. These specimens had unsymmetrical nonprestressed longitudinal reinforcement, with significantly larger amounts of steel in the bottom flange. Figure 3 gives the details of the cross sections of the specimens and the observed torque-bending moment interaction diagram. The diagram shows the effect of unsymmetrical longitudinal reinforcement. In pure torsion, the relatively weak top longitudinal reinforcement can be critical in determining the ultimate capacity. Adding a relatively small bending moment introduces compression in the top flange and increases the torsional strength. It is to be noted that the top reinforcement is relatively smaller than the typical minimum torsional reinforcement, and was selected in Onsongo’s experimental program14 possibly to maximize the effect of unsymmetrical reinforcement and to show its effect on the shape of the interaction diagram.

Grids of targets were attached to the hoop steel to measure the strains in all four faces of the beams, as shown in Fig. 3(a). The targets were 152 mm apart longitudinally and 76 mm apart transversely. Fifteen gauge readings were measured in the longitudinal direction at different levels of loading in each of the bottom and top faces. The averages from these readings are used to compare the measured strains and the results of Eq. (5).

In applying the AASHTO equations to the TBU Series, Eq. (4) and (5) need to be checked for the top and bottom flanges. For the top flange, the sign of the M^sub u^/d^sub v^ factor in Eq. (4) and (5) should be reversed from positive to negative. In accordance with AASHTO’s definition of the tension zone, the tension reinforcement when the top steel is checked includes the three No. 4 top bars and the four upper No. 3 skin reinforcing bars.

Figure 3(b) shows the calculated torque-bending moment interaction diagram. At relatively large bending moments, the adequacy of the bottom longitudinal reinforcement checked using Eq. (5) was critical, and very good agreement is observed between the observed and calculated strength. At intermediate values of M, the upper limit set on the torsional stress in Eq. (8) was critical, and the comparison showed that the calculated results are conservative and adequate. At relatively low bending moments, the AASHTO method correctly predicts lower torsional capacity with lower bending moment. Because the amount of top longitudinal reinforcement was relatively low, the results were conservative.

Figure 4(a) and (b) compares the calculated and measured strains in the bottom longitudinal steel in Specimens TBU2 and TBU3. The top steel was in compression. Figure 4(c) shows the comparison in TBU4 for both the top and bottom steel, both of which were in tension, with the bottom steel being more critical. In all three cases, AASHTO’s general method calculated very accurately the stresses and strains in the longitudinal reinforcement. Better accuracy was obtained where the amount of the steel is practically large.

Figure 4(d) compares the calculated and measured strains in the bottom and in the top longitudinal steel in Specimen TBU5. Fig. 3-Cross section details and T-M interaction diagrams The method correctly calculated larger strains in the top for nonprestressed TBU14 beams. reinforcement, which was observed in the experimental tests as reverse curvature. The method gives very conservative results, possibly because of the inadequately small amount of top longitudinal reinforcement.

Hence, comparing the results of AASHTO’s general method with the observed results of Series TBU specimens shows that the method is accurate and conservative in calculating the ultimate capacity and the force in the longitudinal reinforcement. Developing the longitudinal steel based on the results of Eq. (5) gives conservative results, especially in the more practical case where the bending moment is significant and where the amount of steel is sufficient.

Combined torsion and bending in prestressed members

Five hollow partially prestressed beam specimens in Series TB15 were tested under combined torsion and bending. Figure 5 shows the details of the cross section of the specimens and the observed interaction diagram. These specimens had symmetrical longitudinal reinforcement, which is reflected in the shape of the interaction diagram in that the presence of bending moment decreased the torsional strength.

At relatively large bending moments, the adequacy of the longitudinal reinforcement (Eq. (5)) was critical in determining the ultimate strength, and very good agreement between the experimental and calculated results is observed. At intermediate and low bending moment values, the strength is dictated by the amount of transverse steel (Eq. (3)), and the calculated results are conservative. The effect of the intermediate layer of skin reinforcement is not significant and was hence neglected in the calculations.

The steel strains were measured by a Zurich gauge. Each of the four longitudinal corner bars had six strain readings, while the middle bars had three readings. The gauge readings length ranged from 150 to 200 mm. The averages from these readings are used to compare the measured strains and the results of Eq. (5).

Figure 6 shows the torque versus the average and maximum longitudinal steel strains for the four beams subjected to torsion. In TB3, tested under predominant bending, Eq. (5) requires full development of the longitudinal steel, whichis consistent with the relatively large strains measured in the nonprestressed reinforcement shown in Fig. 6(a). The response of Specimen TB2 tested at intermediate T/M is shown in Fig. 6(b). According to Eq. (5), the steel in this specimen is to be developed for approximately 80% of the full capacity, and that compares well with the measured strains at the level of the calculated maximum torque. The results of Eq. (4), which accounts for the effects of prestressing, also compare well with the measured strains.

The response of Specimen TB1 tested under a relatively large T/M is shown in Fig. 6(c). At the level of calculated load, Eq. (5) requires the steel to be developed for at least 65% of its capacity. This appears to be adequately conservative when compared to the response in the figure. The strain calculated using Eq. (4) also compares well with the state of strain in the longitudinal bar.

The response of Specimen TB4 tested under pure torsion is shown in Fig. 6(d). The longitudinal force Q is approximately 55% of the maximum longitudinal force Q^sub n^, and this is shown to be adequate and conservative. The results of Eq. (4) slightly underestimate the strains at the level of calculated capacity.

Hence, the equations of the general method of AASHTO have been shown to be conservative and adequate to calculate the ultimate bending and torsional capacity and the force in the longitudinal steel. Development of this steel based on Eq. (5) leads to conservative results.

Shear in nonprestressed member

Specimen A60-95 was tested by Rahal and El-Shaleh16 in shear and flexure in a four-point loading arrangement at a shear-to-depth ratio of 2.77 (Fig. 7(a)). One of the strain gauges was attached to the longitudinal bar at 45 mm from the inner face of support. Figure 7(a) gives details of the specimen, which failed across a diagonal crack originating from the load location. It is to be noted that the hook at the end of the beam was a standard hook according to the ACI Code.1

Figure 7(b) shows the observed longitudinal strains near the face of support, which are significant in this case. The analysis by the general method is based on a failure crack originating from the inside face of the support and extending a distance d^sub v^cotθ. The moment used in Eq. (4) is the average value across this crack. For simplicity, it can be taken at d^sub v^ away from the face of the support, as shown in Fig. 7(a), Section A-A. Checking the longitudinal force at the location of the strain gauge near the face of the support should, however, be based on the relatively small moment at that location (which is 0.125 × V), or simply on neglecting this small moment. Figure 7(b) shows the strains based on Q = 153 kN (when M is neglected) and Q = 213 kN (when M = 0.125V). The latter value agrees well with the observed results. The observed and calculated results show that a 144 kN shear force causes a 153 kN longitudinal force that should be resisted by the steel and developed within a relatively short length.

Combined shear and torsion in nonprestressed members

In solid sections, the general method allows redistribution of the shearing stresses by taking the total shearing stress as the square root of the sum of the squares of the individual stresses from shear and torsion (Eq. (6) and (7)). The method uses a similar combination of the effects of V and T in calculating the longitudinal force and strains, as shown in Eq. (5) and (4). This overall effect can be considerably smaller than the maximum value at the critical side where the stresses are additive, and slightly less than the average value at the center of the section assuming a linear variation of longitudinal strains.

The four tests for combined shear and torsion by Rahal and Collins17 are used in the evaluation of Eq. (5). Figure 8 shows the details of the specimens’ cross section near the point of contraflexure in the test region and the observed shear-torsion interaction diagram. The test setup of the four specimens was designed17 to minimize the effect of bending moment. The torque-to-shear ratio at the point of contraflexure ranged from pure shear to predominant torsion.

Grids of strain targets were glued to the surface of the concrete at 200 mm spacing and used to measure the strains at different levels of load. Assuming perfect bond between concrete and steel, the average concrete longitudinal strains can be considered adequate measures of the steel strains for the purpose of evaluating Eq. (5).

Figure 8 compares the observed and calculated interaction diagrams. An average concrete compressive strength of 45.6 MPa was assumed in the calculations. Torsional moments below one quarter of the cracking moment can be neglected, which affects the area of predominant shear. On the other side of the curve, adding a relatively small shear force to the pure torque case can be resisted by the concrete contribution Vc and hence does not increase the demand on the stirrups. This explains the horizontal part of the curve in the predominant torsion part of the diagram. Figure 8 shows a good agreement between the calculated and observed ultimate strength in combined shear and torsion.

Figure 9 compares the longitudinal steel strains calculated using Eq. (5) with the observed concrete surface strains. The reported average strains are the mean values from readings from eight square grids: three on each of the vertical faces, and one on each of the horizontal faces. These grids were 200 × 200 mm in size and were centered at the point of contraflexure. The reported maximum values are the mean values from the three grids on the critical face of the specimens where the shearing stresses are additive.

Specimen RC2-2 was tested in shear and its response is shown in Fig. 9(a). The calculated maximum shear force is 720 kN, which gives V^sub exp^/V^sub calc^ = 1.11. At V^sub calc^ = 720 kN, Eq. (5) calculates a longitudinal force of Q = 454 kN, which causes a 182 MPa stress and a 0.91 × 10^sup -3^ strain in the longitudinal reinforcement. The available capacity of the tensile reinforcement is 1200 kN. Assuming a perfect bond between steel and concrete, the strain value is plotted in Fig. 9(a) and agrees well with the observed strains. Equation (4) calculates a longitudinal strain indicator ε^sub x^ of 1 × 10^sup -3^. It is to be noted that in spite of the reasonable amount of longitudinal reinforcement in the section (1.29% in the bottom layer), the stresses in the longitudinal steel exceeded 180 MPa at the level of calculated shear load.

Specimen RC2-4 was tested at a torque-to-shear ratio T/V = 76 mm. Figure 9(b) shows the average observed longitudinal strains, and the maximum ones observed at the critical face of the beam where the shear and torsional stresses were additive. The calculated ultimate shear force and torsional moment were 569 kN and 43.4 kN·m, respectively, which were about 80% of the observed capacity. Equation (5) calculates a longitudinal force of 423 kN, which causes a 170 MPa stress and a 0.85 × 10^sup -3^ strain in the longitudinal reinforcement. The figure shows a good agreement between the calculations of Eq. (5) and the observed strains. Equation (4) calculates a longitudinal strain indicator ε^sub x^ of 1.0 × 10^sup -3^.

Instead of calculating the square root of the sums of squares, an addition of the contributions of T and V to the longitudinal force in Eq. (5) gives a larger force in the steel as per the following equation

… (9)

For Specimen RC2-4, Eq. (9) gives 29% larger force than the actual value from Eq. (5) and, as expected, seems to be more suitable for calculating the maximum strains at the critical face of the section (Fig. 9(b)).

Specimen RC2-1 was tested at a torque-to-shear ratio T/V = 156 mm. Figure 9(c) shows the observed and calculated strains. The calculated ultimate shear force and torsional moment were 453 kN and 70.5 kN·m, respectively, which were approximately 85% of the observed capacity (refer also to Table 2). Equation (5) calculates a longitudinal force of 416 kN, which causes a 166 MPa stress and a 0.83 × 10^sup -3^ strain in the longitudinal reinforcement. The calculated longitudinal force is slightly unconservative, and is increased by approximately 40% if Eq. (9) is used as shown in the figure. Equation (4) calculates a longitudinal strain indicator ε^sub x^ of 0.99 × 10^sup -3^.

Specimen RC2-3 was tested under predominant torsion at a torque-to-shear ratio T/V = 1216 mm. Figure 9(d) shows the longitudinal strains at different torsional moments. The calculated ultimate shear force and torsional moment were 97 kN and 118 kN·m, respectively, which were approximately 87% of the observed capacity (refer also to Table 2). Equation (5) calculates a longitudinal force of 430 kN, which causes a 172 MPa stress and a 0.86 × 10^sup -3^ strain in the longitudinal reinforcement. Using Eq. (9) instead of Eq. (5) increases the longitudinal force by 25%. The longitudinal strain indicator εx calculated using Eq. (4) is 1.0 × 10^sup -3^.

Hence, comparing the results of AASHTO’s method with the observed results from Series RC2 specimens shows that the method is accurate and conservative in calculating the ultimate capacity and the force in the longitudinal reinforcement.

CONCLUSIONS

This paper presented an evaluation of the AASHTOLRFD equation used for checking the adequacy of the longitudinal reinforcement in beams subjected to shear, torsion, axial load, and bending. The calculations of the equation were compared with steel and concrete strain readings from 17 partially prestressed and nonprestressed beams with transverse reinforcement subjected to combinations of shear, torsion, and bending.

The comparison showed that AASHTO’s equation provided accurate estimates of the longitudinal force in nonprestressed beams. Similar direct comparisons were not possible in partially prestressed beams, but it was shown that the equation yielded conservative results in these beams.

The experimental and calculated results showed that the longitudinal steel stresses due to shear can be considerable. In one of the simple beams, for example, the stresses at the inside face of support exceeded 150 MPa despite the relatively large tension reinforcement (1.29%) provided in the section.

The contributions of shear and torsion to the longitudinal force in the equation are combined based on the square root of the sum of the squares. Modifying this to a linear combination increased the longitudinal force by 25 to 40% in the beams studied. The larger values correlated better with the observed longitudinal strains on the critical side of the section where the shearing stresses are additive.

It is believed that the check of the adequacy of the longitudinal steel for the effects of shear and torsion is one of the major features and contributions of AASHTO-LRFD’s general method.

ACKNOWLEDGMENTS

The research reported in this paper was made possible by a grant from Kuwait University, Grant No. EV 122. This support is gratefully acknowledged.

REFERENCES

1. ACI Committee 318, “Building Code Requirements for Structural Concrete (ACI 318-02) and Commentary (318R-02),” American Concrete Institute, Farmington Hills, Mich., 2002, 443 pp.

2. American Association of State Highway and Transportation Officials, “AASHTO-LRFD Bridge Design Specifications and Commentary,” First Edition, 1994, 1098 pp.

3. CSA A23.3-94, “Design of Concrete Structures (A23.3-94),” Canadian Standards Association, Rexdale, Ontario, Canada, 1994, 199 pp.

4. Vecchio, F. J., and Collins, M. P., “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear,” ACI JOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

5. Vecchio, F. J., and Collins, M. P., “Predicting the Response of Reinforced Concrete Beams Subjected to Shear Using the Modified Compression Field Theory,” ACI Structural Journal, V. 85, No. 3, May-June 1988, pp. 258-268.

6. Collins M. P., and Mitchell, D., Prestressed Concrete Structures, Prentice Hall, Englewood Cliffs, N.J., 1991, 766 pp.

7. American Association of State Highway and Transportation Officials, “AASHTO-LRFD Bridge Design Specifications and Commentary,” Second Edition, 1998.

8. Rahal, K. N., and Collins, M. P., “Background of the 1994 CSA-A23.3 General Method of Shear Design,” Canadian Journal of Civil Engineering, V. 26, No. 6, Dec. 1999, pp. 827-839.

9. Collins, M. P.; Mitchell, D.; Adebar, P.; and Vecchio, F. J., “A General Shear Design Method,” ACI Structural Journal, V. 93, No. 1, Jan.-Feb. 1996, pp. 36-45.

10. Angelakos, D.; Bentz, E. C., and Collins, M. P., “Effect of Concrete Strength and Minimum Stirrups on Shear Strength of Large Members,” ACI Structural Journal, V. 98, No. 3, May-June 2001, pp. 290-301.

11. Rahal, K. N., and Collins, M. P., “Experimental Evaluation of ACI and AASHTO-LRFD Design Provisions for Combined Shear and Torsion,” ACI Structural Journal, V. 100, No. 3, May-June 2003, pp. 277-282.

12. Rahal, K. N., and Collins, M.P., “Analysis of Sections Subjected to Combined Shear and Torsion-A Theoretical Model,” ACI Structural Journal, V. 92, No. 4, July-Aug. 1995, pp. 459-469.

13. Mitchell, D., and Collins, M. P., “Behavior of Structural Concrete in Pure Torsion,” Publication No. 74-06, Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada, Mar. 1974, 88 pp.

14. Onsongo, W. M., “The Diagonal Compression Field Theory for Reinforced Concrete Beams Subjected to Combined Torsion, Flexure, and Axial Load,” PhD thesis, Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada, 1978, 246 pp.

15. Mardukhi, J. “The Behaviour of Uniformly Prestressed Concrete Box Beams in Combined Torsion and Bending,” MASc thesis, University of Toronto, Toronto, Ontario, Canada, 1974, 73 pp.

16. Rahal, K. N., and Al-Shaleh, K., “Minimum Transverse Reinforcement in 65 MPa Concrete Beams,” ACI Structural Journal, V. 101, No. 6, Nov.-Dec. 2004, pp. 872-878.

17. Rahal, K. N., and Collins, M. P., “Effect of Cover Thickness on Shear and Torsion Interaction-An Experimental Investigation,” ACI Structural Journal, V. 92, No. 3, May-June 1995, pp. 334-342.

ACI member Khaldoun N. Rahal is an associate professor in the Department of Civil Engineering at Kuwait University, Kuwait. He is a member of Joint ACI-ASCE Committee 445, Shear and Torsion, and is Past President of the ACI Kuwait Chapter.

Copyright American Concrete Institute Sep/Oct 2005

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