My title https://doi.org/10.14311/APP.2022.36.0244 Acta Polytechnica CTU Proceedings 36:244–252, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague INFLUENCE OF PLASTIC TENSION CHORD DEFORMATION ON THE SHEAR CAPACITY OF PRESTRESSED BEAM ELEMENTS Sebastian Thoma∗, Oliver Fischer Technical University of Munich, Chair of Concrete and Masonry Structures, Theresienstraße 90, N6, 80333 München, Germany ∗ corresponding author: sebastian.thoma@tum.de Abstract. The shear strength of prestressed beam elements is of great interest with regard to the evaluation of existing concrete bridges. Accompanying extensive experimental test series, the adaptation and modification of existing models for the evaluation of the load-bearing capacity also represents an important step. Against this background, considerations are made for the isolation of critical influencing factors on the effective concrete compressive strength and quantified on the basis of a numerical framework. Subsequently, a sensitivity analysis is performed on this basis to determine the influence of individual parameters and their interaction. Taking into account the non-linear relationships at the cross-section level and in the cracked compressive stress field in the web, the shear reinforcement ratio and the strain in the longitudinal reinforcement are of major importance. Keywords: Plastic tension chord, prestressed concrete, sensitivity analysis, shear strength, stress fields. 1. Introduction For the assessment of existing bridge structures and with respect to special questions related to prestressed concrete bridges, theoretical and experimental efforts to evaluate the shear capacity have been increased in the recent past [1–3]. In particular, characteristic bearing mechanisms of prestressed continuous girders with a low amount of shear reinforcement are con- tinuously under discussion. From own experimental investigations, which are briefly presented schemati- cally in section 2, the question for the evaluation of possible influencing factors on the effective concrete compressive strength of the cracked compressive stress field is developed and described by means of a numeri- cal model, cf. section 3. Subsequently, the proportion of the variance of the individual input variables in the total variance of the response spectrum is evalu- ated using a global sensitivity analysis [4] in section 4. From this, dominant parameters and model-inherent interactions can be derived. Section 5 discusses main results and some influences of the numerical model are evaluated. Existing approaches to consider the phenomenon of compression softening together with the presented results may be viewed from a new angle. This procedure provides a valuable basis for possible model modifications based on it. 2. Experimental Evidence The aim of the experimental research is the systematic illustration of the influence of a gradually reduced de- gree of longitudinal reinforcement and the associated effects on internal equilibrium. For this purpose, nine prestressed beam elements are tested using the sub- structure technique. Figure 1 illustrates the concept. This method has already been deployed at the insti- tute to study various aspects in regard to the shear strength of prestressed beam elements [1, 5], while the basic idea for testing detached subsystems has already been used successfully before [6]. Including digital image correlation and fibre-optic sensors along longitudinal reinforcement and tendon axis, the exper- imental results show that the force flow in prestressed beams is subject to a highly statically indeterminate interaction that adopts a static equilibrium depending on communicating damage processes. In this way, the shear capacity is not considered in an exposed manner, but is to be classified with conformity of the stresses and strains under all acting internal forces, whereby in particular the mixed reinforced tension chord of re- inforcing steel and tendon controls the cross-sectional strain and dominance of individual load-bearing com- ponents. The analysis shows that even with yielding longi- tudinal reinforcement under high bending moments in field and support regions, the ultimate system ca- pacity is determined by a shear failure as long as the prestressing steel can provide the corresponding in- crease in strain. The evaluation of the tension chord deformation and the evolution of the cracked stress field in the web prove the great importance of arching actions in the description of the bearing behaviour of prestressed beam elements. A comprehensive analysis and discussion of experimental results will be pre- sented in other publications and are not the subject of this paper. The brief outline of the experiments carried out is intended solely as a background and forms the basis for the considerations set out below, which can assist in evaluating the results and further refining the model concepts. 244 https://doi.org/10.14311/APP.2022.36.0244 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 36/2022 Influence of plastic chord deformation on shear capacity M V 𝜀sl a) b) c) fce = f(𝜀1) Figure 1. Experimental setup and focus of presented studies: a) Substructure technique for investigating the shear strength of prestressed beam elements, based on the shear field of a continuous beam between the point load in the span and the inner support. Reinforcement and tendons are anchored in tension in the cut edges/load plates; b) Deformed system and internal forces acting on it. The push cam profiling enables the transmission of shear forces in analogy to bridges in segmental construction. The fans of the cracked compressive stress field are indicated in the field and support areas, a deflection takes place at tendon axis. The effective web compressive strength, controlled by the reduction factor kc, is the focus of the presented investigations. Relevant details related to this factor are explained in the following section 3; c) Qualitative profile of bending moment and shear force in the tested beam element. The load is successively increased until a shear failure occurs. Under increasing load and a considerably reduced degree of longitudinal reinforcement, plastic strains result in the longitudinal reinforcement, which require a corresponding increase in strain in the tendons in order to guarantee the equilibrium of the internal forces. 245 Sebastian Thoma, Oliver Fischer Acta Polytechnica CTU Proceedings 3. Compression Softening Approach 3.1. Background Taking into account the test results, which show that the system load-bearing behaviour of prestressed cross- sections is still limited by the shear capacity despite plastic tension chord deformation, and based on as- sumptions of the plasticity theory for the prediction of the shear strength, the effective concrete compres- sive strength of the cracked stress field is of crucial importance. As stated in [7], the assumption of a shear force bearing capacity derived from shear reinforcement is always limited by the bearing capacity of the concrete compression struts, which in turn depends on the ef- fective concrete compressive strength and thus on the general state of strain. As a result, in addition to the shear reinforcement ratio, the longitudinal reinforce- ment ratio and the longitudinal strain - coupled by the load distribution and system slenderness - may have a major influence on the shear force bearing ca- pacity. The factor kc, see equations 4 and 5, thus decides on the extent to which plasticity-theoretical approaches can be used to describe the shear strength of reinforced concrete beams. For design purposes, the effective concrete compres- sive strength is usually determined using constant reduction factors, which is a simple, conservative esti- mate. In the course of one’s own considerations, and especially in the case of plastic chord deformation, it is reasonable to explicitly take into account the actual state of strain. Concrete girders, and in particular prestressed systems, develop a varying strain over beam height. The development of arching actions and the deviation of the stress field at the tendon axis cannot be represented in panel tests, which have contributed significantly to the description of various compression softening approaches [8–10]. Still, the relations within the membrane element can be trans- ferred to the beam web if an equivalent measure of the longitudinal distortion can be formulated. Fol- lowing the convention in the fib Model Code 2010 [11] or respectively the Modified Compression Field Theory (MCFT) [9, 12], the measure of effective lon- gitudinal strain is defined at the mid-depth of the member. However, the member is not defined by the cross-sectional contour, but must be understood as a web (following a classical plasticity-theoretical cross- sectional division) between compression and tension chord. The longitudinal strain at half the height of the inner lever arm of the chord forces may serve as a clearer formulation here. 3.2. Numerical Model For the weighted evaluation of dominant influencing factors on the reduction factor kc and consequently on the effective concrete compressive strength, various assumptions are linked to a numerical model, the solution of which is solved as an optimisation problem, using sequential least squares [13, 14]. By iterating the strain plane for a cross-section under an acting bending moment and normal force, the lever arm of the internal forces is given after integrating the concrete compressive stresses, cf. figure 2. At the level of zm/2, the average longitudinal strain state is obtained, which is further supplemented by components caused by shear, cf. equation 1, adopted from [7]. 𝜎c Δ𝜎p 𝜎s 𝜀c zm/2 zm/2 Fc Ft 𝜀x,MN800 250 Figure 2. Investigated cross section [mm], strain, stress, inner forces and decisive strain εx,MN due to bending and axial forces 𝜀z 𝜀x 𝜀2 𝜀1 𝜀 𝛾/2 𝜃 kc 𝜀1 a) b) Figure 3. Core components of the optimisation prob- lem: a) Mohr’s circle of strains in pure shear b) Re- duction factor kc, introduced equation according to [10, 15] Since the proportion of longitudinal strain due to shear εx,V is already formulated as a function of the compressive strut angle θ, equation 1 is already part of the constrained nonlinear optimisation. Based on equilibrium conditions and an explicit consideration 246 vol. 36/2022 Influence of plastic chord deformation on shear capacity of the strains in longitudinal and shear reinforcement, which limit the range of possible strut angles (for a more detailed discussion, please refer to [7, 15]), the correlated values of the compression strut inclination and reduction factor kc, respectively the effective com- pressive strength fce, can be determined, cf. equations 4 and 5. Assuming θ can evolve freely, taking into ac- count its boundary conditions (derived from the yield strength and tensile strength of the reinforcement), the angle can be determined according to equation 2. The iterative routine is obvious in linking equation 2 and figure 3. After convergence the further steps are straight- forward, and the principal strain ε1 as well as the reduction factor kc are determined, cf. equations 3 and 4. εx,V = V /2 · cot θ 2 (zs/zm · EsAs + zp/zm · EpAp) (1) tan θ = √ ρswfy fce − ρswfy (2) ε1 = εx,MNV + (εx,MNV − εc2) · cot2 θ (3) kc = 1 1.2 + 55 · ε1 ≤ 0.65 (4) fce = kc · ( 30 fc )1/3 · fc (5) The model uses additional, generally accepted sim- plifications, which are essentially due to the basic outlines of plasticity theory and will not be the focus of further attention here. Related to the proposed model structure, some questions with respect to model sensitivity arise: • The principal strain εc2 is assumed to be constant at -0.002. Is this, essentially empirically based, assump- tion [9] valid or do implicitly considered influencing factors falsify a generally valid statement? • What are the consequences of an increased strain level in the longitudinal reinforcement for estab- lished models for predicting the shear strength in prestressed beam elements? Is εx a sensitive param- eter for the response surface Y? • Are there significant interaction effects between dif- ferent input parameters of the model? 4. Global Sensitivity Analysis To address the above questions, the numerical model from section 3.2 is subjected to a sensitivity analysis. A global sensitivity analysis (GSA) distils the influence of possible uncertainty of selected n input parameters on the result of a model. Possible interactions of the input values (so-called second order effects) can also be determined by variance-based methods. In the following Sobol’s Method [16, 17] enables a quantified evaluation of these issues. Essential information on the characteristics of the GSA approaches used in this research is based on the considerations in [4], unless otherwise referenced. The method does not require an analytical model, but forms its correlation ratio on the basis of a problem spectrum to be defined initially, which contains all input parameters (which may be subject to the GSA), their distribution function and possible bounds. This data set is mapped into n dimensions by means of low-discrepancy sampling and then passed individually to the numerical model. Critical aspects and details of the implementation are described below. 4.1. Sobol’s sequence Aiming for minimized discrepancy, generated se- quences of parameter values should avoid holes and clusters in the hypercube. This task is adequately accomplished by Sobol’s sequences [16, 17]. In addi- tion, Sobol’s sequence ensures order independence for different input parameter axes. Without going into further detail, this is enabled by its direction numbers which are derived from distinct primitive polynomials for every required parameter vector [18]. Sobol’ se- quences are a quadrature rule and become unstable if samples of a size that is not a power of 2 are used, or if the first point is omitted, or if a sequence becomes sparse [19]. This method thus offers a tried and tested approach for evenly covering possible parameter con- stellations, which essentially eliminates the danger of unfavourable density of a parameter constellation. Figure 4 shows an example of uniformly distributed random coordinates and the spread using a Sobol’ sequence. The difference in the uniform coverage of the area is obvious. 0 1 0 1 0 1 0 1 0 1 Figure 4. Random samples and Sobol’ sequence in two dimensions (27 = 128 samples) The model study accounts for seven parameters, which are listed in table 1 together with their assumed 247 Sebastian Thoma, Oliver Fischer Acta Polytechnica CTU Proceedings uniform distribution. Uniform distributions are as- sumed for the parameter space because no specific configuration with known uncertainty is to be exam- ined, but an overall impression and overview of the model sensitivity is to be given. The limit values of the distributions are based on the test programme referred to at the beginning or assumed as realistic limit values when considering existing reinforced concrete bridges. All parameters are part of the resistance side and are included in the iteration of the initial strain plane and/or subsequent iteration to determine the stress and strain state in the web. In addition to the range of the longitudinal reinforcement ratio, the lower limit of which leads to plastic chord deformations in the cross- section analysis, the shear reinforcement ratio is also chosen relatively low with regard to the conditions in existing prestressed concrete bridges. Taking into ac- count the applied variation of the concrete compressive strength, a bandwidth of about 0.7 to 4.0 times the minimum shear reinforcement ratio ρw,min results for the general case according to ρw,min = 0.16ḟctm/fyk [20]. Only λ represents a component of the load side to a certain extent, since the ratio of acting moment, which is increased independently of the GSA for all combinations of possible input values, and shear force, which also has a share in εx (see equation 1), is con- trolled via a variation of the shear slenderness. In addition, the variation of the tendon slope influences the effective shear force (after subtracting the ver- tical component Vp = Pmt · sin αp) in the web and ultimately the proportion εx,V. Parameter Uniform distr. range ρsl [%] [0.8, 2.0] ρsw [%] [0.09, 0.36] λ [-] [2.5 - 5.0] εc2 [-] [-0.002, -0.001] fc [MPa] [30.0, 50.0] σp [MPa] [550.0, 600.0] αp [deg] [1.0, 7.0] Table 1. Input parameters for sensitivity analysis. Variance-based methods are powerful in quantifying the relative importance of input factors or groups. The main drawback of variance-based methods is the cost of the analysis, which, in the case of computationally intensive models, can become prohibitive even when using the approach described above. With Saltellis extension of the Sobol’s sequence [17], which is used within this scope, the resulting matrix has N (̇2n + 2) rows, where n is the number of parameters. For a full set of Sobol’ indices (S1, S2 & ST, cf. section 4.2) a model with 7 factors requires to execute the model at least 16000 times, taking N = 1000. Whether the assumption for N was chosen sufficiently large can be judged from the confidence intervals. This results in over 200.000 model evaluations in total due to a relatively finely chosen incremental bending moment rise. In a general case, the validity and robustness of a composite indicator, in this case the reduction factor kc may depend on a number of factors. • The model chosen for estimating the measurement error in the data • The mechanism for including or excluding indicators in the index; The choice of factors fed into a GSA is subjective. Particularly in the case of comparatively complex, non-linear numerical models, the model character itself has a considerable influence on the response surface Y. Theoretically, one could provide the GSA with as many input parameters as pos- sible, but this seems unnecessarily CPU-intensive. Not pursued further here, but possible in the first place, seems to be the gradual reduction of the in- put parameters with subsequent evaluation of the correlation coefficient to the results of a fully packed analysis without static default values. Alternatively, qualitative screening using the Morris method [21] can be performed as the first step in model survey. • The indicators preliminary treatment; In the in- vestigations presented here, the limiting value 0.65 according to equation 4 is used in the framework of the iteration. The GSA evaluation deliberately excludes this criterion in order to take into account possible influences caused by the asymptotic charac- ter of the equation at low principal tensile strains. • The type of normalization scheme applied to the indicators to remove scale effects • The amount of missing data and the imputation algorithm; Under high bending moment, it is the- oretically possible that the iteration of the strain plane does not reach a stable equilibrium according to the defined stress-strain relationships and limit strains. This leads to a lack of the input value εx,MN in the further iterative consideration of the condi- tions for the web element, and to a non-response bias depending on the error handling. This scenario was prevented by previous comparative calculations so to avoid inconvergent result data falsifying the GSA. 4.2. Sobol’s Method After running the numerical model for Sobol’s se- quence, one gets an array that forms the so-called response surface. Sobol’s method evaluates the part of the total variance of the response that can be at- tributed to input parameter Xn. The numerical imple- mentation is based on the SALib package [22]. Three measures can be obtained for each parameter: • first order index S1: contribution (without interac- tion) of a parameter to the response variance • second order index S2: interaction of input param- eters to the response variance 248 vol. 36/2022 Influence of plastic chord deformation on shear capacity • total effect index ST: total contribution (including interaction) of a parameter to the response variance A measure of sensitivity is to calculate the variance of the conditional expectation V ar [E(Y |Xn)] referred to the total variance of Y, V ar(Y ), cf. equation 6. In other words, the total variance of the response Y that can be attributed to input parameter Xn defines its weight within the numerical model and, in a further step, enables a better understanding of model-inherent features beyond qualitative screening and, ultimately, opens up initial starting points if a model modification appears desirable. S1n = V ar [E(Y |Xn = xn)] V ar(Y ) (6) Any numerical model is some sort of mapping input parameters to output results. The following equation 7 shows the added parts that influence the result y, namely scalar values, the sum of functions evaluated for each parameter (modelling the effect of each indi- vidual parameter) and the interaction of parameters. y = f (x) =f0 + Np∑ n=1p fn (xn) + ∑ 1≤Np fn,n′ (xn, xn′ ) + . . . + f1,2,...,Np ( x1, . . . , xNp ) (7) The total variance of the response Y (the total of all y-evaluations) can be decomposed into partial vari- ances, attributing variability of the response Y to each input parameter, including interactions. Different con- ditional variances: Dn = V ar [E (Y |Xn)] (8) Variance on two conditions: Dn,n′ = V ar [E (Y |Xn, Xn′ )] − Dn − Dn′ (9) The first order Sobol’ index S1n calculates the impact of the input parameter Xn by estimating the partial variance of Y explained by this parameter. It estimates by how much the variance of the response is reduced, on average, when the parameter Xn is fixed, i.e. it measures the contribution of the parameter Xn to the total variance of the response. The total effect index for a parameter Xn is defined: STn = Sn + ∑ n≤n′ Sn,n′ + . . . (10) The total effect index represents the total contribu- tion (including interactions) of a parameter Xn to the response variance; it is obtained by summing all first-order and higher-order effects involving the pa- rameter Xn. Further input variables of the model that are not included in the GSA, but are neverthe- less to explicitly assume a variable character, require a recalculation of the sensitivity indices for all Sobol’ samples. In the present case, this is the case for the applied bending moment from external load. This procedure is comparatively computationally intensive, but it allows a view on a potentially variable prioriti- sation of the parameters according to Table 1, which should be quite common, especially for non-linear model considerations. 5. Results and Discussion The GSA shows which of the selected input parame- ters have an influence on kc and which values would have been sufficiently taken into account within the scope of the investigations with the assumption of a constant value, since no significant contribution to the variance of the result is evident. The major fac- tor is the transverse reinforcement ratio. It is the most important influencing variable over the complete variation of the moment acting on the cross-section. This seems to be plausible with regard to equation 2. Described from a phenomenological point of view, the stirrup strain imposes a proportional transverse strain on the concrete compression struts, which reduces the effective concrete compressive strength. For low shear reinforcement levels, this issue is particularly critical because the small cross-sectional area mobilises high strains early on for comparatively small forces after shear cracking. Depending on the crack opening and ductility of the reinforcement, rupture of the rein- forcement may occur. Further questions arise in this regard, which will not be discussed in detail here. A model without parameter interaction leads to∑ ST = 1, which is obviously not the case here. Under external bending moment of about 0.45 to 0.5 MNm, interesting dependencies between longitu- dinal and shear reinforcement ratio are revealed, cf. figure 5. The concrete compressive strength also ap- pears to gain in importance in the meantime before ST settles back down to the initial level. Figure 6 shows also a remarkable interaction between ρsw and εc2 develop in this range. This intermediate effect is due to the limiting conditions of the possible band- width of permitted compression strut angles. After exceeding the decompression stresses, small longitu- dinal strains of the same order of magnitude as εc2 initially enter into the equations. As expected, the in- fluence of the longitudinal strain rises under increasing moment, while the shear reinforcement ratio, which is still important, decreases significantly. A moment of 0.9 MNm for the considered example cross-section (figure 2) results in the longitudinal reinforcement reaching its yield stress. The high longitudinal strain εx,MNV becomes more important for the iterative determination of the com- pression strut angle and the principal strain ε1 under increasing load. Against the background of high uti- lization rates for bending and shear of many existing 249 Sebastian Thoma, Oliver Fischer Acta Polytechnica CTU Proceedings My [MNm] asl asw asl asl asw Figure 5. Individual parameter importance S1 and total prioritisation ST affecting kc concrete bridges, the question arises to what extend the mostly constant factors in standards provided rep- resent a reasonable assumption. Directly linked to this is, of course, the question of an adequate formulation of the effective concrete compressive strength or, at a distance, the question of the fundamental capacity of the model conception for shear strength discussed here. The generally constant assumption of the principal strain εc2 even proves to be a definite parameter to be taken into account, largely independent of the acting moment, if one considers modifying the numerical model with respect to kc. The importance of a rea- sonable assumption or parameterized formulation for εc2 becomes clear when looking at figure 3. Selected principal compressive strain and longitudinal strain εx are of the same order of magnitude in absolute terms and have an influence on the compression strut angle according to their difference. 6. Conclusions The paper presents a numerical model that allows a closed-loop determination of the effective concrete compressive strength in a beam web under shear load- ing. Of further interest is the reduction factor kc used, which is influenced to different degrees by different parameters. In order to determine the individual influ- ence of the parameters and possible model-determined interaction effects, a GSA was carried out, which was able to identify the shear reinforcement ratio and the assumption of the principal strain εc2 in general, and My [MNm] S2 Figure 6. S2 Indices: Input Parameter interaction affecting kc the longitudinal reinforcement ratio and correspond- ing, ultimately plastic chord deformation under high moment loading in particular as the decisive adjusting screws. In light of the experimental investigations, which achieved high shear capacities despite plastic chord deformation, a few questions arise: • Do the empirically derived formulations for deter- mining the reduction factor kc sufficiently approxi- mate the conditions in a prestressed beam web or are the model ideas of plasticity theory described here (especially for cross-sectional segmentation) not applicable? • Which possibilities for a more refined estimation of the main strain εc2 seem practicable and reason- able? These questions are part of further considerations, which should be evaluated in the context of the duc- tility of the reinforcement, which was already briefly addressed in the course of the discussion of results. List of symbols kc Reduction factor for concrete compressive strength due to transverse tensile strain [–] εc Concrete strain [–] εx,MN Axial concrete strain due to normal forces and bending moment at half the height of the inner lever arm [–] εx,V Axial concrete strain due to shear [–] ε1 Concrete principal tensile strain [–] 250 vol. 36/2022 Influence of plastic chord deformation on shear capacity ε2 Concrete principal compressive strain [–] θ Compression strut angle [deg] fce Effective concrete compressive strength [MPa] fc Concrete compressive strength, derived from cylinder tests [MPa] ρsl Longitudinal reinforcement ratio, defined as ρsl = Asl/(bw · d) [–] ρsw Geometric shear reinforcement ratio, defined as ρsw = asw/bw [–] λ Shear slenderness, defines as λ = M/(V · d) [–] zm Lever arm of inner forces [mm] zs Lever arm between Fc and Fs [mm] zp Lever arm between Fc and Fp [mm] σc Concrete (compressive) stress [MPa] σs Steel stress due to normal forces and bending moment [MPa] σp Prestress in tendons [MPa] ∆σp Increased tendon stress due to normal forces and bending moment [MPa] αp Tendon slope [deg] F c Resultant inner compressive force [MN] F t Resultant inner tension force as weighted sum of Fs and Fp [MN] V Shear force [MN] Es Modulus of elasticity of longitudinal reinforcement [MPa] Ep Modulus of elasticity of prestress tendon strands [MPA] As Cross-sectional area of longitudinal reinforcement in tension [m2] Ap Cross-sectional area of prestressed tendons [m2] S1 Sobol’s first order index - individual parameter con- tribution to total response variance [–] S2 Sobol’s second order index - contribution of parame- ter interactions (higher-order effects) to total response variance [–] ST Sobol’s total index combines margins of S1 and S2 for each parameter [–] References [1] N. 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The Journal of Open Source Software 2(9), 2017. https://doi.org/10.21105/joss.00097. 252 https://doi.org/10.1080/00401706.1991.10484804 https://doi.org/10.21105/joss.00097 Acta Polytechnica CTU Proceedings 36:244–252, 2022 1 Introduction 2 Experimental Evidence 3 Compression Softening Approach 3.1 Background 3.2 Numerical Model 4 Global Sensitivity Analysis 4.1 Sobol's sequence 4.2 Sobol's Method 5 Results and Discussion 6 Conclusions List of symbols References