https://doi.org/10.14311/APP.2022.33.0284 Acta Polytechnica CTU Proceedings 33:284–289, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague LOWER BOUND ANALYSIS FOR SHEAR ASSESSMENT OF FULL-SCALE RC GIRDERS SUBJECTED TO AXIAL TENSION Takeru Kanazawa Hokkai Gakuen University, 1-1, Minami 26, Nishi 11, Tyuo-ku, Sapppro, Hokkaido, Japan correspondence: t-kanazawa@hgu.jp Abstract. Axial tension force exerted as a result of a temperature change or shrinkage can cause the collapse of RC structural members. Design code provisions and analytical models such as Modified Compres- sion Field Theory (MCFT) yield reasonable estimates of shear strength of RC beams subjected to axial tension. Nevertheless, their semi-empirical nature is not necessarily appropriate for shear assessment of existing RC structural members. The extra conservativeness and empirically determined parameters might require unnecessary maintenance work. A generalised model with rigorous formulation must be developed. This paper presents a purely theoretical model to predict the shear strength of RC beams under axial tension based on limit analysis. Without regressive functions and empirical functions, lower bound analysis enables shear strength derivation when the force equilibrium and strain compat- ibility are satisfied. Accuracy of the analysis was verified by comparison of its predictions with three experimental shear strengths of full-scale RC girders. An equal level of accuracy was observed between the analytical solutions and MCFT-based predictions. Keywords: Axial tension, full-scale RC girders, lower bound analysis, shear assessment. 1. Introduction The sustainability of RC structures is an important issue worldwide because of their exposure to a vari- ety of loading with environmental and mechanical ac- tions. The partial collapse of the Wilkins Air Force Depot in 1955 [1] spurred the refinement of design practices for shear under axial tension caused by a temperature change or shrinkage. The brittle col- lapse was reproduced in laboratory investigation [1]. Results demonstrated that tensile stress of approxi- mately 1.4 N/mm2 can reduce shear strength by 50% [2]. Design code provisions have difficulty in predict- ing such detrimental effects of axial tension because of empirically determined parameters with safety fac- tors. In this respect, several well-established theories for shear [3–5] are based not only on the force equi- librium, but on the failure kinematics. The former and latter respectively correspond to lower and up- per bound solutions. The lower bound analysis is the basis of the shear design expression in most in- ternational codes. This lower bound is the reason underlying their inherent conservativeness, which is appropriate for designing new structures. For struc- tural assessment of existing RC members, however, such conservativeness is of less importance than de- termining the actual bearing capacity [6]. Among the currently available theories, the Modi- fied Compression Field Theory (MCFT) has provided a consistent framework to consider the influence of axial force and to facilitate its incorporation into var- ious international codes and specifications. Never- theless, the MCFT predictions might elicit some dis- crepancy with several test results of RC beams under axial tension [2]. Current understanding should be supplemented theoretically for rational shear assess- ment because the limitations of existing test results include laboratory-scale specimens without web rein- forcement [7], despite recent progress [8, 9]. This paper presents a rational analysis based on the lower bound theorem. The analytical structure is based on the combined upper and lower bound anal- ysis [10], which predicted the shear strengths of RC beams accurately with no axial force and no web re- inforcement. The present work extends their origi- nal theory to consider the effectiveness of web rein- forcement and the negative effects of axial tension. Rigorous formulation for shear failure is presented, including shear resisting mechanisms such as aggre- gate interlock and dowel action of longitudinal re- inforcement under both force equilibrium and strain compatibility. The developed analysis is validated by comparison of its solutions with MCFT-based pre- dictions and test results of three full-scale RC girders subjected to axial tension [7]. The notation used for this study is summarised in the Appendix. 2. Model Formulation 2.1. Analytical framework In the theoretical framework of limit analysis [11], the exact solution lies between lower and upper bound solutions, for which the former and latter respec- tively satisfy force equilibrium and compatible pat- terns of failure, in addition to yield criteria. One therefore finds the exact solution by maximising the lower bound solutions. To derive shear strength with 284 https://doi.org/10.14311/APP.2022.33.0284 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 33/2022 Analysis for Shear Assessment of Full-Scale RC Girders Figure 1. Constitutive law of concrete. Figure 2. Constitutive law of steel reinforcement. reasonable simplification, the following assumptions are introduced. • Concrete and steel reinforcement are in a plane stress state. • Perfect bonding exists between steel reinforcement and concrete. • Concrete strength equal to over a compression zone; concrete behaves as a rigid - perfectly plastic material, as presented in Figure 2. • Steel reinforcement behave as an elastic - perfectly plastic material, as shown in Figure 2. It is noteworthy that the physical quantities such as forces, stresses and strains in compression are neg- ative. Those denoted with a prime have positive com- pression. 2.2. Force equilibrium Figure 3 shows the free-body diagram with the shear resisting components against the external shear and axial tension force, respectively denoted as V and N. This equilibrium condition yields the following ex- pressions, respectively, for vertical force, horizontal force, and moment. V − Vc − Fd − Fw − b ! t f sin θ dt = 0 (1) Figure 3. Free-body diagram. Figure 4. Critical crack shape. ! h αh bf ′c dy − Fs1 − F ′ s2 + N + b ! t f cos θ dt = 0 (2) [γa + cot β (h − d1)] Fd + (h − d1) Fs1 − (h − d2) F ′s2 + a (1 + γ) 2 Fw + aVc − h 2 N − h (1 + γ) 2 ! h αh bf ′c dy + b "! t f sin θ x dt − ! t f cos θ y dt # = 0 (3) Therein, $ t dt denotes the path integral from point A to B in Figure 3. One can replace these integrations with the following integrations with respect to x and y. ! t cos θ dt = ! a γa dx, ! t sin θ dt = ! αh 0 dy (4) Mathematical treatment of the path integral re- quires an expression for the critical crack shape shown in figure 4 on the x − y coordinates. y2 = α2h2 (x − aγ) a (1 − γ) (5) Dividing equations 1, 2 and 3 respectively by bd1f ′c and abd1f ′c, one obtains non-dimensionalised expres- sions as presented below. 285 Takeru Kanazawa Acta Polytechnica CTU Proceedings Figure 5. Strain compatibility. v = (1 − α) τc + φs1τs + ωφswσw + α µ f (6) 1 − α µ − φs1σs1 + φs2σs2 + n + λ (1 − γ) f = 0 (7) % γ + cot β " ζ − 1 λ #& τsφs1 + " ζ − 1 λ # φs1σs1− " ζ − δ λ # φs2σs2 + 1 + γ 2 ωφwσw + (1 − α) τc − ζ 2 n − ζ 2µ ' 1 − α2 ( − α 3µ (1 − 4γ) f = 0 (8) In those equations, the non-dimensional static vari- ables are defined as shown below. ) **********+ **********, v = V bd1f ′c , σs1 = Fs1 As1fs1 , σs2 = F ′s2 As2fs2 , σw = Fw Aswfsw , τc = Vc bd1f ′c (1 − α) , τs = Fd As1fs1 , n = N bd1f ′c , f = f f ′c . (9) Equation 6 is the objective function in this study. The non-dimensional shear strength of v is maximised with respect to the static variables. 2.3. Compatibility conditions Though lower bound solutions do not require compat- ibility conditions, an inequality constraint for aggre- gate interlock ' f ( can be derived from strain com- patibility. Figure 5 presents the strain compatibility on the cracked section. The assumption of Bernoulli- Euler beam (plane sections remain planar) yields the linear distribution of longitudinal strain. Those of bottom and top reinforcement are obtained respec- tively as shown below. εs1 = d1 − (1 − α) h (1 − α) h ε′cu = 0.0035 - µ 1 − α − 1 . (10) εs2 = (1 − α) h − d2 (1 − α) h ε′cu = 0.0035 - δµ 1 − α − 1 . (11) The neutral axis depth under the effect of axial tension is calculable by arranging Equation 6 with re- spect to k = (1 − α)h/d1 as presented in the equation below. k = ε′cuEs f ′c / (ρs1 + δρs2) µ 1 − α − (ρs1 + ρs2) 0 − f λ (1 − γ) − n (12) Equation 12 gives an upper limit of the aggregate interlock in maximisation because k must be positive according to its definition. 2.4. Yield criterion and solution technique The following von Mises yield criterion is introduced between τs and σs1. 3τ 2s + σ 2 s1 = 1 (13) Although Hweé et al. [10] used another yield criterion for concrete, it need not be addressed in this study because the rigid - perfectly plastic be- haviour is assumed. The complete set of equations enables the maximisation of the objective function of Equation 6 with respect to the six static vari- ables ' σs1, σs2, σw, τc, τs, and f ( . The possible ranges of those variables are presented as the follow- ing. σs1, σs2, σw ∈ 〈0, 1〉 τc, τs, f ∈ 〈0, ∞〉 (14) All of those expressions must be positive. The up- per limits of σs1, σs2, and σw and correspond with yielding of reinforcement. Although no upper limit is imposed on the others, Equation 7, 8, 12, and 13 render them physically admissible values. Maximisation is performed using the optimisation algorithm of MATLAB [12], under the equality and inequality constraints listed in Table 1. All possible 286 vol. 33/2022 Analysis for Shear Assessment of Full-Scale RC Girders Objective function Equation 6 Equality constraints (force equilibrium) Equation 7 and 8 Equality constraints (yield criterion) Equation 13 Inequality constraints (neutral axis depth) Equation 12 ≥ 0 Inequality constraints (yield criterion) 3τ 2s + σ2s1 ≤ 1 Table 1. List of constraints for optimisation. Figure 6. Cross-sectional details of inverted-T beams (unit: mm). Figure 7. Shear strength dependence on applied ax- ial force. values of v are calculated within the following ranges of kinematic variables. 1 − µ ≤ α ≤ 1 αh a (1 − γ) ≤ β ≤ ∞ 0 ≤ γ ≤ 1 (15) The lower limits of and stem respectively from the assumption that the longitudinal strain of bottom re- inforcement in Equation 10 must be positive, and that the critical crack is directed upwards at y = 0 (Fig- ure 3). To obtain more optimal solution, the maximised so- lutions are then minimised with kinematic variables. The partial derivative of equation (6) with respect to determines whether the objective function is mono- tonically decreasing or increasing. The minimised shear strengths are calculable as presented below. ∂v ∂α = − τc + f µ (16) ) *******************+ *******************, vana = φs1τs + ωφswσw + f µ " if ∂v ∂α < 0, αmax = 1 gives vana # vana = µτc + φs1τs + ωφswσw + 1 − µ µ f " if ∂v ∂α > 0, αmax = 1 − µ gives vana # (17) 3. Results and Discussion 3.1. Experiment results The analytical solutions have been compared with test results obtained for full-scale RC girders [7]. The studied girders had the inverted-T configuration (Fig- ure 6) to support the deck in flexural tension. Fig- ure 7 presents the detrimental effect of axial tension on shear strength. The axial force of 900 kN decreases the shear strength by 14%. The axial force magni- tudes of 3-200-P and 4-300-P were found respectively to represent shrinkage-induced stress only, and both the shrinkage and temperature-induced stress. It is noteworthy that Smith et al. [7] tested seven speci- mens in all. The three specimens shown in figure 7 were extracted for model validation because the oth- ers included the use of cut-off bars and epoxy injected specimen, of which the contributions were not formu- lated. 3.2. Model validation To calculate the shear strength by the developed anal- ysis, static and kinematic constants were determined as listed respectively in Tables 2 and 3. These val- ues were calculable using information referred from reports of the related literature [7]. The rectangular area of bd1 in Figure 6 was used for calculation be- cause the contributions of the flanges under flexural tension are negligible. In addition, the values of ε′cu 287 Takeru Kanazawa Acta Polytechnica CTU Proceedings n = N bd1f ′c φs1 = ρs1fs1 f ′c φs2 = ρs2fs2 f ′c φw = ρswfsw f ′c 1-0-P − 0.221 0.110 0.064 3-200-P 0.083 0.269 0.135 0.079 4-300-P 0.128 0.267 0.133 0.081 Table 2. Non-dimensional static constants. λ = a d1 ζ = h a µ = d1 h σ = d1 d2 ω = s d1 1-0-P 3-200-P 2.60 0.407 0.943 0.066 0.265 4-300-P Table 3. Non-dimensional kinematic constants. Vexp vexp vana VAASHT O vana vexp VAASHT O Vexp + Vdl (kN) (−) (−) (kN) (−) (−) 1-0-P 902 0.0773 0.0809 810 1.05 0.88 3-200-P 780 0.0814 0.0879 743 1.08 0.93 4-300-P 783 0.0835 0.0847 712 1.01 0.89 Table 4. Accuracy of the developed analysis. and Es in equation (12) are assumed respectively as 0.0035 and 200 kN/mm2 because the original data were not found in the literature [7]. Table 4 presents comparisons among test results (Vexp), present analysis (vana), and MCFT-based pre- diction by AASHTO LRFD bridge design specifica- tion [13]. The analytical solutions show adequate ac- curacy (vana/vexp) with specimens under axial ten- sion (3-200-P and 4-300-P) and without axial tension (1-0-P). In all cases, the static variable of σw was equal to one when the optimisation terminated. This value represents the yielding of web reinforce- ment, which was observed in the experiments as well. Furthermore, results demonstrate better agreement than the MCFT-based predictions, except for those for specimen 3-200-P. Although the analytical pre- dictions of 3-200-P and 3-400-P show a consistent decrease of shear strength with the increase of ap- plied axial force, no such tendency was observed in the experiment as presented in Figure 7. Further in- vestigation about this discrepancy will be reported in another paper with a broad range of experimental data. 4. Conclusions Shear strengths under the effect of axial tension load- ing were derived analytically based on the lower bound theorem. The theoretical basis treated the shear strength derivation as an optimisation prob- lem under equality and inequality constraints ob- tained from force equilibrium, strain compatibility and yield criteria. Model validation showed good correspondence between experimental and analytical shear strengths without any regression functions and empirically determined parameters. The accuracy was an equal level with MCFT-based predictions. The generality of this formulation enables the consis- tent treatment of axial compression, and gives quan- titative results of each shear component such as ag- gregate interlock of cracked concrete and dowel action of longitudinal reinforcement. Although further veri- fication by comparison with a number of test results is necessary, the developed analysis might contribute to the provision of comprehensive understanding of shear behaviour under axial force, together with the MCFT [2] and other well-established theories [3–5]. Research is currently underway to extend this theory to axially compressed and tensioned beams without web reinforcement. List of symbols f ′c Concrete compressive strength V External shear force N External force of axial tension Vc Shear component of uncracked concrete Fw Shear component of web reinforcement f Aggregate interlocking force per unit area θ Inclination angle of critical crack to x-axis Fd Dowel force of bottom reinforcement Fs1, Fs2 Longitudinal forces of bottom and top rein- forcement, respectively α Critical crack height divided by total depth 288 vol. 33/2022 Analysis for Shear Assessment of Full-Scale RC Girders β Slope of critical crack at point B γ Longitudinal distance between support and point B divided by shear span a Shear span b Web width d1 Effective depth h Total depth d2 Distance between top reinforcement and upper perimeter of beams s Spacing between adjacent web reinforcement t Coordinate axis along critical crack ω Spacing between adjacent web reinforcement divided by effective depth µ Effective depth divided by total depth λ Slenderness (shear span-to-depth) ratio ζ Total depth divided by shear span δ d1 divided by d2 k Neutral axis depth divided by effective depth ε′cu Ultimate compressive strain of concrete (0.0035) Es Modulus of elasticity of longitudinal reinforcement (200 GPa) ρs1, ρs2, ρsw Reinforcement ratios of bottom, top and web reinforcement, respectively As1, As2, Asw Areas of bottom, top and web reinforce- ment, respectively fs1, fs2, fsw Yield strengths of bottom, top and web reinforcement, respectively εs1, εs2, εsw Longitudinal strains of bottom and top re- inforcement, respectively φs1, φs2, φsw Reinforcement degrees of bottom, top and web reinforcement, respectively Acknowledgements Financial support from a Grant-in-Aid for Scientific Re- search from the Japan Society of Promotion of Science (19K15062) is greatly appreciated. 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