Modeling and design of concrete hinges under general loading

Despite their successful applications in practice for over a century, the design of concrete hinges still entails considerable modeling uncertainty. Their dimensioning still relies on semi‐empirical recommendations originally proposed for admissible stress design, based on limited experimental data and generalized with engineering judgment. This presumably results in overly conservative designs in many cases, particularly when applied with modern design codes based on partial safety factors. This paper addresses these issues by revisiting the analytical modeling for one‐way Freyssinet concrete hinges under general loading based on approaches compatible with current design codes. A cross‐sectional analysis with confined concrete properties is proposed for the behavior under axial forces and bending moments, where the strength of the triaxially compressed concrete in the hinge throat is determined with a discontinuous stress field. The hinge resistance to shear forces is investigated with failure mechanisms inspired by failure modes observed in experiments. Based on the shear strength, a simple approach to estimate the torsional resistance of the throat is proposed. Remarks are also made on the analysis of combined actions. The predictions of the proposed models are validated against a wide range of test data, including a series of recently conducted own experiments. Overall, the proposed models agree better with the available experimental data than the existing approaches, potentially allowing for a more efficient design of new and assessment of existing concrete hinges.


| INTRODUCTION
Concrete hinges are monolithic articulation elements, providing flexibility and thus reducing restraints in Discussion on this paper must be submitted within two months of the print publication. The discussion will then be published in print, along with the authors' closure, if any, approximately nine months after the print publication. concrete structures by allowing large hinge rotations while transferring minor bending moments. The hinge rotation is accomplished by concentrated deformations in the hinge throat, which is formed with an abrupt and local reduction of the member cross-section, see Figure 1.
Concrete hinges have been successfully applied for over a century in buildings and especially bridges, 1,2 where they have proven to be a more durable and economical, practically maintenance-free alternative to mechanical bearings. 2 Nowadays, concrete hinges are primarily used in integral bridges, which are being promoted by bridge owners in several countries. 3 Despite the ample positive experiences, engineers are often confronted with considerable uncertainties when designing and assessing concrete hinges. This is mainly due to lacunae in the understanding of the fundamental mechanical behavior of concrete hinges, which impedes reliably quantifying the bearing capacity of concrete hinges under general loading. In fact, the design of concrete hinges is still based on old guidelines from the 1960s, [4][5][6][7][8][9] which are based on engineering judgment and limited empirical evidence and are only partially compatible with current design standards, as they were developed for admissible stress design. A thorough overview of the topic can be found in References 1, 10, and 11. Nonlinear finite element (NLFE) analyses can help understanding the complex three-dimensional behavior of concrete hinges and contribute to their adequate design. 12,13 However, such analyses require expertise and considerable modeling time, and their results can be very sensitive to parameters often unknown during the design phase. 14 Therefore, it is essential to provide reliable mechanical models for the behavior of concrete hinges that-even if NLFE analyses are carried out in detailed design-can be used as initial design (to be checked by NLFE) and as a plausibility check of the NLFE results.
The existing knowledge gaps regarding the structural behavior of concrete hinges hinder their efficient application and more widespread use. In this context, the authors started a broad research project analytically and experimentally investigating the structural behavior of one-way (or linear) Freyssinet concrete hinges. This most common type of concrete hinge permits the rotation about one axis while being moment-resistant in the perpendicular direction and contains moderate or no reinforcement crossing the hinge throat. Hence, the throat resists the loading primarily by the concrete (in contrast to Mesnager hinges, 15 which are conceived to transfer the load across the throat entirely with heavy reinforcement). The experimental part of the research project, consisting of seven large-scale tests on concrete hinges under general loading, has been published recently. 16 This paper focuses on the theoretical part of the project.
After some general geometrical and detailing recommendations, the paper proposes improvements to existing and presents new mechanical models for the loaddeformation behavior and load bearing capacity of concrete hinges. All stress resultants (internal actions of a hinge in a linear member) are investigated along with the hinge rotations, starting with the predominant ones of axial force and rotation, followed by a bending moment about the strong axis, shear forces in the transverse and longitudinal directions, and finally a torque. For each action combination, the corresponding section of the paper starts with a state-of-the-art summary of the available recommendations and models, introduces the new models, and finally compares the current and proposed new models with experimental data. The paper concludes with some remarks on the interaction between various concomitant actions and a summary of the main findings.
This work focuses on the behavior and resistance of the concrete hinge throat; the behavior of the adjacent blocks is only discussed when directly relevant to the throat behavior. Long-term and repeated loading effects on the hinge behavior are not treated. Moreover, only the mechanical modeling (with mean material values) is discussed, without considering safety concepts and the statistical distribution of material properties; naturally, in the design of concrete hinges according to current structural standards, characteristic values and pertinent safety factors on material properties and actions must be applied. Figure 1a-c shows the geometry of a typical one-way concrete hinge and introduce the coordinate system and nomenclature adopted in this paper. The six stress resultants that can act on a concrete hinge (in a linear element) are shown in Figure 1e: axial (normal) force N x (defined positive in compression), transverse and longitudinal shear forces V y and V z , respectively, torque M x , bending moment M y about the strong axis (perpendicular to the hinge rotation axis), and hinge resisting moment M z . Note that the latter, caused by a rotation r z about the longitudinal axis of the hinge, would vanish in an ideal one-way hinge.

| GENERAL CONSIDERATIONS AND RECOMMENDATIONS
In the following, some geometrical and detailing recommendations are given for designing efficient and wellfunctioning concrete hinges. These recommendations are based on fundamental mechanical reasonings-as substantiated by the models presented in the following sections-practical experience and past experimental campaigns, as well as guidelines of other researchers. 1,5,11,17,18 The recommendations are summarized here since the following sections of this paper assume that the concrete hinge has been detailed accordingly.
The main geometrical recommendations given by Leonhardt and Reimann 19 are summarized in Figure 1d. To be effective as a hinge and permit sufficient rotational capacity while opposing little moment resistance, the throat width d 1 should be kept as narrow as structurally possible-in any case significantly narrower than the width of the adjacent member-while considering the constructability. The throat height h t should be kept as short as practically feasible since increasing h t diminishes the confinement of the throat created by the deviation of the stress trajectories and by the impediment of lateral throat expansion by the adjacent blocks, 20,21 hence reducing its strength. The throat should not extend over the entire longitudinal length b of the blocks but be provided with recessed ends to reduce the risk of extensive spalling. 17 Moreover, the perimeter of the throat should be rounded off (filleted) to reduce stress concentrations and the extent of superficial spalling 11 ; note, however, that despite this measure, slight superficial spalling is likely to occur around the throat perimeter already at relatively small axial forces and rotations. 16,17,19,22 As mentioned before, the adjacent blocks provide confinement to the concrete in the throat region. In order to ensure this confinement, the hinge shoulders (recession) should not be more inclined than strictly required to facilitate the manufacturing of the hinge (e.g., extraction of the throat inserts during demolding), thereby allowing the placement of confinement reinforcement in the blocks as close as possible to the throat. This confinement reinforcement should consist of a fine grid of closed ties (or adequately anchored bars such as headed bars) and ensure sufficient reinforcement in the longitudinal direction (e.g., ρ z ≥ 0.5 ρ y ) 20,21,23 ; see, for instance, Figure 1f.
Large reinforcement contents across the throat should be avoided as they convey several disadvantages: they complicate the hinge manufacturing, impede concrete shrinkage and thus cause cracks through the throat, hinder beneficial creep processes (crack closure, stress peak reduction), increase the hinge resisting moment, and potentially compromise the triaxial compressive stress state in the throat. 15,19 However, a moderate reinforcement, that is, A s /(b 1 d 1 ) < 5%, 24 in the form of vertical bars crossing the hinge throat may be useful, primarily as a precaution against brittle shear failure or to cover accidental design situations (e.g., impact loading or earthquake). 16,17,[25][26][27][28] This throat reinforcement should cross the throat along the hinge rotation axis (y = 0) in order to minimize the impediment of the hinge rotation; the bars can be slightly inclined in the y-direction (crosswise) to improve shear transfer in the transverse direction (V y ). Some authors recommended not to use ribbed reinforcing bars through the throat to avoid the risk of localized (tensile) yielding in the throat, which would interfere with the closure of the throat in compression. 11,19 The authors of this paper deem this recommendation unnecessary as long as steel strains are checked in the serviceability state (e.g., via the procedure proposed in Section 3.2.5). In order to exclude corrosion problems due to the small concrete cover in narrow hinge throats, stainless steel reinforcement should be used in the throat; note that due to the small quantity required, this has no relevant influence on the overall cost.
The constructability of concrete hinges is a very challenging point that requires careful consideration. The concrete mix properties, including maximum aggregate size and rheology, should be carefully chosen to ensure good compaction in narrow throats and heavily reinforced blocks. Depending on the casting procedure used, the minimum allowable throat width might be limited by the constructability criteria and not by static requirements. In complex situations, it is advisable to cast a prototype hinge and then saw-cut to check concrete quality.

| AXIAL FORCE N x AND HINGE ROTATION r z
Most concrete hinges are provided primarily in order to accommodate high axial compressive forces N x while facilitating large hinge rotations r z about the longitudinal axis. Consequently, these are the predominant actions that concrete hinges experience in structural applications, and the main emphasis of past theoretical and experimental investigations was on studying the hinge behavior under these standard load cases. This combination is studied first, and in most detail, in this paper as well. Note that due to symmetry, only positive rotations are considered for the mathematical formulations described in this section. Moreover, likewise N x , also the axial stresses and strains are defined positive in compression.

| State of the art
As mentioned in Section 2 and thoroughly discussed in References 16,17,20, and 21, the concrete in the throat region is subjected to a triaxial compressive stress state. It can thus resist axial compressive stresses several times higher than the uniaxial concrete compressive strength f c0 and undergo large inelastic strains. Applying a rotation r z to an axially compressed hinge causes the axial stresses to increase on one side of the throat and decrease on the opposite side. With increasing rotation, a crack starts penetrating from the tensile side of the throat and reduces the throat width that effectively transfers the axial load. 17 The rotation gradually shifts the position of the axial stress resultant away from the section centroid, resulting in a hinge moment M z resisting the applied rotation; concrete hinges are thus inherently imperfect hinges. The flexural stiffness and moment resistance of the hinge are significantly smaller than those of the adjacent blocks-due to the narrow throat-and are strongly affected by the loading history as well as creep and relaxation phenomena. 16 Most existing modeling approaches for the concrete hinge behavior essentially establish a semi-empirical, experimentally calibrated relationship between the geometry of the hinge, the axial force, and the maximum admissible hinge rotation. Predictions of the hinge resisting moment M y as a function of the hinge rotation r z have also been studied intensely, though primarily using elastic approaches that cannot account for the inelastic and path-dependent behavior of the hinge throat. Despite their limited practical relevance, M y À r z relationships can be helpful to (i) calibrate and validate the underlying modeling assumption against monotonic loading curves from experiments and (ii) roughly estimate the range of hinge moments that can be expected during the service life of a concrete hinge.
The next subsections outline two existing modeling approaches, relevant to deriving a new improved model.

| Leonhardt's model
The design rules proposed in the 1960s by Mönnig and Netzel, 4 Leonhardt and Mönnig, 5 and Leonhardt and Reimann 19 are arguably the most prominent approach for designing and assessing concrete hinges. Recently, they have been updated from their original deterministic formulation to a semi-probabilistic design concept by Marx and Schacht. 1,29 This updated version is considered in the following, referred to as Leonhardt's model for brevity.
As modeling the stress and strain state in the throat region with reasonable effort and complexity 19 is highly challenging, Leonhardt et al. proposed a rather drastic simplification to estimate the hinge rotation and resisting moment, see Figure 2. They proposed modeling the throat region based on a cross-sectional analysis at the throat, adopting Bernoulli's hypothesis of plane sections remaining plane (i.e., linear strain distribution) and assuming linear elastic concrete behavior. Informed by experimental results, they further assumed that only the region immediately adjacent to the throat contributes to the rotation. The height h h of this region was set equal to the width d 1 of the hinge throat based on a model calibration with test results. 19 These assumptions essentially correspond to idealizing the throat region as a cuboid of height h h (= d 1 ) and width d 1 that experiences a constant curvature χ z , 30 see Figure 2b. The total hinge rotation is obtained by integrating the curvature over the idealized hinge height: r z = χ z h h . By formulating equilibrium on the cross-section, Figure 2c, the hinge resisting moment can be expressed as for 0 ≤ ξr z ≤ 0:009, with λ = 1.2-4 d 1 /d but 0 ≤ λ ≤ 0.8. Equation (3) effectively limits the compressed throat width to d 1,c > 0.4d, which is supposed to ensure position stability and safe operation of the hinge, according to Schlappal et al. 30 Indirectly, Equation (2) can be interpreted as a limitation of the peak concrete stress σ t,max at the throat edge (Figure 2c), becoming higher with decreasing ratio of throat width to block width, d 1 /d, and increasing hinge rotation. Whereas this trend can be substantiated by the fact that the geometrical confinement increases as the compressed throat width d 1,c decreases (relative to the block width), 21 Equation (2) cannot be further substantiated mechanically and may result in arguable results. For instance, for a hinge with a throat width d 1 = 0.2d subjected to a rotation causing a halfdecompressed throat, that is, d 1,c = 0.5d 1 = 0.1d, Equation (2) allows an edge concrete stress σ t,max roughly 2.5 times higher than in a throat of equal compressed width d 1,c = d 1 = 0.1d in a hinge with d 1 = 0.1d that remains compressed over the entire width (at a smaller rotation). Moreover, Equation (2) is independent of the reinforcement content of the adjacent blocks, which is a major limitation of Leonhardt's model, as experimental evidence indicates that the reinforcement ratio is decisive for the bearing capacity of such partially loaded blocks. 23

| Schlappal's model
Although adopting most of the assumptions of Leonhardt's model, Schlappal et al. 30,31 accounted for reinforcement crossing the throat, assuming strain compatibility between reinforcement and concrete (i.e., perfect bond), and adopted linear elasticperfectly plastic stress-strain relationships for concrete and steel. Moreover, they proposed determining the maximum and minimum admissible axial forces more systematically by defining stress and strain limits in the considered cross-section for the serviceability limit state (SLS) and ultimate limit state (ULS). They determined the confined concrete strength based on the square root equation contained in most structural standards, which results in very conservative capacity predictions for hinges with narrow throats and heavily confined blocks (see Section 3.3). Furthermore, by adopting different concrete elastic moduli E c for SLS and ULS, some inconsistencies arise when calculating the hinge resisting moment M z (as it is impossible to determine a unique value of M z for given r z and N x , see also Section 3.3.1). Nevertheless, Schlappal's model introduces some adaptions that make the analysis of concrete hinges more compatible with other design approaches for compression members subjected to bending and compression, such as columns and prestressed beams. Hence, it will be used as the basis for the improvements proposed in the following section.

| Proposed modeling
Building on the work of Mönnig and Netzel, 4 Leonhardt and Mönnig, 5 Leonhardt and Reimann, 19 and Schlappal et al., 30,31 an improved approach is proposed for modeling the behavior of one-way concrete hinges subjected to axial force N x and rotation r z . The main adjustments over Schlappal's model concern (i) the adoption of a more realistic stress-strain relationship and compressive strength for the confined concrete, (ii) the use of the same stress-strain relationship but different strain limits for the SLS and ULS, and (iii) the formulation of the model using a general value h h for the idealized hinge height, which allows its subsequent calibration on test results. Note that, in general, different values for the idealized hinge height h h should be adopted to accurately predict different types of hinge deformations, for example, rotations about the transverse and longitudinal axes and axial shortening. In the following, the material modeling properties and the cross-sectional analysis of the throat are outlined first, before developing the hinge moment M z À rotation r z characteristic and the normal force N x À rotation r z interaction diagram.

| Stress-strain relationship for steel and concrete
For the reinforcement crossing the throat, a linear elastic-perfectly plastic stress-strain relationship with yield stress ±f sy is adopted, as shown in Figure 3a. For the concrete, the polynomial ascending compressive branch proposed by Karthik 32 and Karthik and Mander 33 is adopted, whereas the descending (compression softening) and tensile branches are ignored, as illustrated in Figure 3b: where f cc is the confined concrete strength, ε cc is the corresponding strain, and n = E c ε cc /f cc . The bottom expression in Equation (4) closely resembles Mander's well-known original relationship 34 but has the key advantage of being analytically integrable. As experimentally observed by Mander et al., 34 confinement has a roughly five times greater effect on the peak strain increase than on the peak stress increase, that is: where ε c0 is the strain corresponding to the uniaxial concrete strength f c0 . With Equations (4) and (5) the stressstrain relationship of confined concrete can be described with the basic material properties of the unconfined concrete, namely f c0 , ε c0 , and E c , and the confined concrete strength f cc . Note that Equations (4) and (5) were proposed based on tests on confined columns, where the confinement stresses are one order of magnitude smaller than in concrete hinges. 31 For this reason, their extension to concrete hinges must be done with caution, and the resulting models must be validated with experimental data. Nevertheless, the adopted concrete constitutive models appear more realistic than the linear elastic and linear elastic-perfectly plastic characteristics adopted in the models of Leonhardt and Schlappal, respectively.
Cross-sectional analyses of the hinge throat under normal force N x and rotation r z .

| Confined concrete strength f cc
The top and bottom halves of a one-way concrete hinge under axial compression essentially represent reinforced concrete blocks subjected to strip loading. Their bearing capacity, which corresponds to the confined concrete strength in the throat region, can be reliably determined with the Dual-Wedge (DW) stress field (Figure 3c), a mechanically consistent and experimentally validated model developed by the authors in a previous study targeting partially loaded areas. 20,21,23,35,36 The reader is referred to References 21 and 23 for a detailed model description.
Here, only the main stress-field-related aspects and expressions are summarized. Where applicable, the general expressions reported by Markic et al. 21 are simplified for the conditions typical for concrete hinges.
In the DW stress field, the bearing capacity of the block is fully exploited when the confinement and splitting reinforcement yield in tension and the concrete below the loaded strip fails in triaxial compression. The main stress field parameters are the width d 1 of the loaded strip (i.e., the throat width), the width d 2 available for the load dispersion, the uniaxial concrete compressive strength f c0 , and the confining stress σ conf generated by the confinement and splitting reinforcement placed in the adjacent blocks (for the bearing capacity estimation of unreinforced specimens, σ conf might be taken roughly equal to the concrete tensile strength; in design, however, it is not advisable to rely on the tensile strength of concrete, and it is recommended to always provide a minimum transverse reinforcement content of at least ρ y ≥ 0.1f c0 /f sy 23 ). Following Markic et al., 21,23 d 2 is chosen equal to the width d cs of the confinement core, see Figure 1f. The confining stress σ conf , where 0 ≤ σ conf ≤ f c0 , is assumed to be constant along the height of the stress field and can be determined from expressions contained in design codes, typically depending on the content and layout of the confinement reinforcement. According to the DW stress fields, the maximum average axial stress across the throat width, that is, the confined concrete strength, is: The inclination α of the compressive strut in the stress field (see Figure 3c) is limited by the following condition: where and α min is the minimum admissible strut inclination given by geometrical constraints, for instance to prevent that the stress field extends beyond the block dimensions. The stress σ α in the compressive diagonal strut is limited MARKI C AND KAUFMANN to As Equations (6)-(10) are interdependent, an iterative (usually numerical) solution procedure is generally required to determine the confined concrete strength f cc . Every combination of α and σ α that fulfills Equations (7) and (10) corresponds to a lower-bound solution for the bearing capacity; its maximum value is obtained by choosing α as low and σ α as high as admissible. Figure 4 shows an exemplary axial strain and corresponding stress distribution across a hinge throat subjected to axial compressive force N x and rotation r z . Adopting Bernoulli's plane sections remain plane hypothesis, the axial strain distribution can be uniquely described by two parameters, for example, the maximum strain ε t,max at the throat edge and the curvature χ z = r z /h h :

| Cross-sectional analysis of the throat
The acting axial force N x is resisted by axial forces in the concrete and in the reinforcing bars crossing the throat: The total force F s in the reinforcing bars is readily obtained as where A s,eff = A s cos 2 α s is the effective area of the inclined bars crossing the throat at an angle α s to the xaxis. By limiting the reinforcement force to zero (right part of Equation 13), any reinforcement contribution in compression is neglected; this conservative assumption is partially compensated by using the gross (instead of the net) concrete cross-sectional area of the throat. The resulting concrete compressive force F c is obtained by integrating the concrete stresses over the throat crosssection. As the adopted constitutive relationship (Section 3.2.1) is analytically integrable, it is possible to derive parameters (η and λ) that facilitate the transformation of any stress distribution into a statically equivalent rectangular stress block. 32,33 Hence, the concrete compressive force can be expressed as where σ c is the concrete stress according to Equation (4), d 1,c is the width of the compressed throat section η and λ are the parameters defining the statically equivalent stress blocks (see Figure 4), whose product equals and the normalized strains at the edges of the compressed section are Note that for ε t,c,min ¼ ε t,c,max , that is, r z = 0, Equation (16) becomes ηλ ¼ 1 À 1 À ε t, max ð Þ n ≤ 1. The hinge resisting moment is obtained by integrating the product of the concrete stresses and the respective ycoordinate: with λ = parameter defining the width of the statically equivalent stress block, and ηλ according to Equation (16).

| Hinge resisting moment
M z À rotation r z characteristic As mentioned in Section 3.1 and observed in several experiments, 16,30 the actual M z À r z characteristic of concrete hinges is highly time and path-dependent. The high effort required to model such a complex behavior theoretically (e.g., with an incremental formulation over the load history) would be hardly justified and of minor relevance due to (i) the unknowns regarding the load path in real constructions and (ii) the in any case significantly lower bending stiffness and resistance of the hinge relative to the adjacent members. In practice, it is thus usually sufficient to approximate the M z À r z curve for a monotonically applied rotation, which can then be used to bound the range of expected hinge moments during the service life.
The resisting moment M z in a hinge subjected to constant axial force N x and a monotonically applied rotation r z can be iteratively computed as follows: (14)- (17). iv. Check if the resulting axial force N x according to Equation (12) is equal to the applied axial force. If not, iterate from (i) with an updated value of ε t,max . v. Determine M z from Equations (19) and (20). 3.2.5 | Axial force N x À rotation r z interaction diagram For the design and assessment of concrete hinges, N x À r z interaction diagrams describing the admissible combinations of N x and r z are most useful. Such diagrams can be defined by defining limits for the concrete compressive strains, the tensile strain of the reinforcing bars crossing the throat, and the crack width, based on experimental observations and engineering judgment. The maximum and minimum admissible axial forces for a given rotation r z , respecting the strain limit, follow directly from Equations (12)- (16). Repeating the procedure for all admissible rotations yields the N x À r z interaction diagram for the considered hinge.
Pertinent limits must be chosen based on specified requirements and the considered limit states. Possible values for the SLS and ULS are proposed in the following.
At the SLS, the following limits are proposed: i. The concrete compressive strains (at the compressed throat edge) shall not exceed an admissible strain ε c,SLS . A plausible value for ε c,SLS is the strain corresponding to the end of the elastic, fairly straight part of the stress-strain relationship of confined concrete, which roughly corresponds to ε c,SLS = f cc /E c (see Figure  3b). This limit should ensure that no large plastic deformations (excluding long-term effects), causing deterioration of the hinge, occur in service conditions. ii. The tensile strains in the throat reinforcing bars shall not exceed the admissible strain ε s,SLS (defined as negative). A reasonable value is given by the yield strain ε sy , hence preventing plastic deformations compromising bending crack closure during unloading. iii. The opening of the bending crack in the throat shall not exceed a notional value w SLS (defined as positive), which should ensure a proper crack opening and closing during rotation cycles without deteriorating the rough interface, which in turn could compromise the shear resistance of the throat. A value of w SLS = 0.2 mm appears reasonable to this end (as commonly stipulated by design code to ensure durability and appearance criteria). With Bernoulli's plane section hypothesis and assuming that the tensile deformation along the idealized hinge height h h is entirely localized in one crack, this corresponds to limiting the maximum tensile strain (at the tensile throat edge) to Àw SLS /h h .
Limiting the crack opening to prevent cyclic interface deterioration appears more pertinent to the authors than the argumentation of "position stability" used by Schlappal et al. 30 to support the crack length limitation proposed in their model (see Section 3.1.1). Presumably, the crack length limitation had been inspired by geotechnical engineering, where commonly the gaping joint under a foundation is limited to avoid overturning. Note that due to the complex behavior and the many modeling simplifications, the predicted crack opening and the crack length in the throat should be considered as notional values, which may differ considerably from the actual values (as applies to any crack width calculation in structural concrete).
Expressing the three limits proposed above in terms of ε t,max results in the following maximum strains for the upper limit of N x : and for the lower limit of N x : where and the maximum admissible rotation in SLS is Note that in cases r z,sy = r z,max,SLS and r z,sy = 0, the strain ε t,max is defined over the entire range of admissible rotations solely by the top and bottom expression of Equation (22), respectively. At the ULS, the following limits are proposed: i. The concrete compressive strain shall not exceed ε c,ULS , where ε c,ULS = ε cc is a plausible value. Considering the high deformation capacity of confined concrete and the relatively flat softening branch after the peak stress, a strain limit exceeding ε cc might also be used. However, as it will be evident from the resulting N x À r z diagrams, the proposed limit strain leads to abundant rotation capacities that rarely become governing in design. ii. The tensile steel strains shall not exceed the strain ε s,ULS (defined as negative), with ε s,ULS = ε su being a plausible value in order to avoid premature steel rupture.
Reformulating these two limits in terms of ε t,max results in the following maximum strains for the upper limit of N x : and for the lower limit of N x : where the maximum admissible rotation at ULS is Figure 5 shows N x À r z interaction diagrams obtained with the procedure outlined above for an exemplary concrete hinge with typical geometry and material properties (see figure caption) at the SLS and ULS. All combinations of N x and r z within the lower and upper limit of N x are admissible at the considered limit states. Characteristic states in the diagrams-such as decompression, maximum rotation, half compressed throat, completely decompressed throat, and reaching of the steel limit strain-are identified with markers and explained by means of strain planes in Figure 5a,b. Comparing these two diagrams, it is evident that significantly lower values of N x and r z are admissible at SLS than ULS; in particular, the rotations are an order of magnitude lower on common cases. Figure 5c illustrates the results of a sensitivity study on the SLS (top) and ULS (bottom) interaction diagrams, for the parameters indicated above the respective diagrams. Unsurprisingly, the throat width d 1 is the most relevant parameter. The variation of σ conf and A s primarily affects the upper and lower limit of N x , respectively. The crack width limit w SLS influences only the lower N x limit at SLS, while the variation of the model parameter h h essentially corresponds to scaling the diagram along the r z axis.
As an additional design criterion, it appears reasonable to limit the quasi-permanent compressive axial force in order to avoid excessive nonlinear creep in compression, for example, to 45% of the ultimate load carrying capacity of the hinge as proposed by Schlappal et al. 18 in line with EN1992-1-2 (regulating the design of concrete bridges 37 ). Depending on the share of permanent and transient loads, such a criterion may truncate the interaction diagram in Figure 5a horizontally at the top. This is, however, not further considered here as long-term effects are beyond the scope of the study.

| Model comparison and validation with test results
3.3.1 | Resisting moment M z À rotation r z curves Figure 6 shows experimentally observed M z À r z curves for hinges under constant axial force from literature and own tests [16][17][18]22,38 together with the model predictions. The experimental data used for the comparison are summarized in Appendix A. Although no explicit formulation of the M z À r z relationship was given by Schlappal et al., 30,31 this can readily be obtained by integrating the product of the concrete stresses and the respective y-coordinates over the throat. As Schlappal et al. adopted different stress-strain relationships for concrete at SLS and ULS, two different M z À r z curves are obtained. Only the ULS curves are plotted since the SLS curves coincide with the initial straight part of Leonhardt's curves.
All experimental M z À r z curves present an initial, straight part which progressively flattens with increasing rotation due to macro-cracking and plastic deformation in the throat. 16 In most tests, large cyclic rotations were imposed to the hinge, hence the corresponding M z À r z curves present hysteresis loops (note that for the tests illustrated in Figure 6l-n only discrete M z À r z points are available). As discussed in Section 3.2.4, the goal of the models is not to reproduce this hysteretic behavior but to give a reasonable approximation of the M z À r z curve for   a monotonically applied rotation, which can be used to validate the model assumptions and bound the range of expected hinge moments during service life. It appears however reasonable to assume-as common in seismic research-that the curve corresponding to a monotonically applied rotation is similar to the backbone of the hysteresis,   38 Fessler, 22 Base, 17 and Markic et al. 16 that is, the curve passing through the turning points of the hysteresis loops. The idealized hinge height h h in the proposed model was chosen equal to 0.5d 1 , which was found to adequately predict the initial bending stiffness (note that h h does not influence the moment capacity at large rotations).
For low axial forces (low values of q x /f c0 = N x /(b 1 d 1 f c0 ), indicated in the graphs) and moderate rotations, see Figure 6g-i, the predictions of the proposed and Leonhardt's models are similar (which is to be expected since the concrete remains in the elastic range) and satisfactorily match the backbone of the hysteresis loops. Schlappal's model also yields accurate moment predictions at large rotations; however, the initial stiffness is significantly underestimated due to the unrealistically low elastic modulus E c adopted at ULS. For higher axial forces, see Figure 6c-f,j,k, the difference between the models becomes more significant. Schlappal's model tends to significantly underpredict the bending moments because it underestimates the confined concrete strength. Note that in several cases, Schlappal's model could not be applied because the imposed normal force exceeds the predicted hinge capacity, even though in the tests the specimens could carry the load without particular signs of distress. On the contrary, Leonhardt's model tends to overestimate the hinge moment because it assumes linear concrete behavior even at very large stresses, several times higher than f c0 .
Overall, the proposed model performs satisfactorily and better than the existing models in almost all the compared experiments.
3.3.2 | Normal force N x À rotation r z interaction diagram Figure 7 shows the theoretical N x À r z interaction diagrams for various hinges from literature and own   [16][17][18]22 together with the load path applied to the specimens during testing. The data used for the comparison are summarized in Appendix A. The SLS diagrams are plotted primarily to provide an impression of their shape and proportion compared to the ULS curves, as they cannot be rigorously validated with the available data. Nonetheless, it can be concluded that they are safe, as overall no significant inelastic deformations or distress signs were reported at the loads corresponding to the SLS, apart from the expected minor effects in SLS, such as fine splitting and bending cracks in the adjacent blocks and throat, respectively, and superficial flaking of the cement skin at the fillet of the throat.
The ULS diagrams (computed with mean material properties, without safety factors) are supposed to indicate the most extreme N x À r z combinations that can be carried while still ensuring the structural safety of the hinge. Very few experiments are available where large axial forces and rotations were applied. 16 Moreover, there are merely three-highly valuable-tests where failure could be achieved (see Â markers in Figure 7a,h,k); in all other experiments, the capacity of the testing apparatus was insufficient in terms of axial force and/or rotation to cause failure of a large-scale concrete hinge in compression; note that once a part of the hinge throat has completely vanished due to the large inelastic deformations, the adjacent blocks enter in direct contact and can continue carrying large forces and rotations. Nevertheless, by achieving N x À r z combinations outside the ULS interaction diagram without causing a failure, the test data confirm that the corresponding region of the diagram is conservative.
Generally, Figure 7 shows that the ULS interaction diagrams obtained from the proposed model allow for a substantially wider range of N x À r z combinations than the two previously existing approaches, while still being conservative. The proposed approach is particularly superior for hinges with narrow throats and heavily reinforced adjacent blocks, see for example Figure 7a-c. This is primarily due to the refined stress field employed in the proposed model to compute the confined concrete strength. Schlappal's and the proposed model allow for significantly higher rotations at low compressive as well as tensile axial forces, as they account for reinforcement crossing the throat and generally do not restrict the crack length in the throat. At very low compressive forces, Schlappal's model allows for significantly larger rotations than the proposed model due to the larger height of the idealized throat adopted in Schlappal's model. For hinges with a wide throat relative to the block width (0.

| BENDING MOMENT M y ABOUT THE STRONG AXIS
One-way hinges can be subjected to significant bending moments about their strong axis (perpendicular to the hinge rotation axis) caused by lateral horizontal loads on the structure (such as wind, earthquake, and impact loads) and must be designed accordingly.

| State of the art
Analogously to the behavior in the transverse direction, the bending moment about the strong axis corresponds to an eccentricity of the axial force. As the bending moment increases, the throat is decompressed over part of its length (over the entire throat width d 1 ), causing a reduction of the contact area transferring the axial load. If the bending moment cannot be resisted by an axial force  eccentricity alone, an adequate throat reinforcement must be provided.
Current design recommendations provide little guidance regarding the bending resistance of concrete hinges about the strong axis. Leonhardt and Mönnig's 5 and the British 24 recommendations are among the few that contain explicit recommendations in this regard, by prescribing that the throat shall not decompress due to this bending moment. Assuming a linear-elastic behavior of concrete in compression, this corresponds to limiting the position of the resultant normal force within the kern of the cross-section, which for rectangular cross-sections equals M y = M y,dec = b 1 N x /6. Higher bending moments are permitted according to the Leonhardt's recommendations only if the hinge is heavily reinforced (literally "armored" 11 ) with large-diameter bars across the throat.
The limitation of the moment to the decompression moment is primarily based on engineering judgment and presumably overly conservative; furthermore, it does not consider the real stresses and strains in the throat. Jiang and Saiidi 27 proposed computing the hinge bending resistance with a simplified cross-sectional analysis adopting a rectangular stress block of increased concrete strength (f cc = 1.25 f c0 ) to account for the confinement of the throat. Apart from the simplistic account of concrete confinement, Jiang's approach appears promising, and is similar to the model proposed below.

| Proposed modeling
Essentially, the interaction between axial force and moment (or rotation) about the strong axis can be analyzed with a cross-sectional analysis which confined concrete properties (Figure 8), analogously to the procedure proposed for the transverse direction in Section 3.2. As the rotation and moment act about the y-axis instead of the z-axis, in Equations (13)- (20), the variables y, r z , d 1 , and b 1 must be replaced by -z, r y , b 1 , and d 1 , respectively. Moreover, because the reinforcement crossing the throat is usually distributed along the throat length, the total steel force and moment comprise the contribution of several bars, see Figure 8. For the computation of the steel force, Equation (13) must be adapted to where A s,eff,i and z s,i are the effective cross-sectional area and the z-coordinate of the ith bar crossing the throat.
The total steel force is then obtained by adding up the forces of all the bars F s ¼ P i F s,i . The moment M y about the strong axis is finally given, analogously to Equation (19), by For the computation of the admissible load combinations, strain and crack width limits can be chosen analogously to those in the transverse direction (Section 3.2.4).
If only an approximation of the ultimate bending resistance about the strong axis is of interest, as is often the case in preliminary design, one may assume ε t,max = ε cc and a rectangular stress block of intensity f cc over 85% of the compressive throat length, as common in crosssectional design for bending (which is conservative, as multiaxially compressed concrete has a significantly higher deformation capacity than uniaxially compressed concrete). The confined concrete stress f cc and strain ε cc at peak capacity can be determined according to Sections 3.2.1 and 3.2.2. Table 1 summarizes the specimen properties, the applied maximum axial compressive forces and bending moments, and the resistance predicted by the proposed model of the few experiments available on concrete hinges that failed under moments about the strong axis. Despite the many model simplifications, the proposed model leads to a good agreement with the test results, with an average deviation of 10% on the conservative side. Note that generally, the maximum bending moments were substantially higher than the decompression moment. A comparison of the predicted and actual moment M y À rotation r y curves of Specimen CH5 16 reveals that, in order to predict the initial stiffness accurately, a significantly higher idealized effective hinge height h h must be chosen (h h ≈ 3d 1 ) than the value obtained in Section 3.2.4 for predicting r z (h h = 0.5d 1 ). More test results are necessary to confirm and better quantify this finding.

| SHEAR IN THE TRANSVERSE AND LONGITUDINAL DIRECTIONS
The shear forces acting on concrete hinges are usually considerably smaller than the simultaneously acting axial force in persistent and transient design situations, and hence, the shear resistance is not critical. 17,28 However, significantly higher shear forces may occur in exceptional and accidental design situations such as earthquake and impact, requiring reliable predictions of the shear strength.
Similar to the shear strength of concrete structures without shear reinforcement and the shear transfer across interfaces, the shear resistance of concrete hinges a somewhat controversial topic. 28 Due to the hinge geometry, shear failures of concrete hinges are constrained to the throat region, hence the established shear design approaches for slabs and beams cannot be applied. On the other hand, the shear transfer across the throat differs from common interface shear cases due to the high axial compressive forces and hinge rotations.

| State of the art
Current design guidelines for concrete hinges 1,7,8,24,39,40 typically limit the shear resistance perpendicular to the hinge axis to a fraction of the axial load: where the ratio k = V y,u /N x is commonly independent of the axial force, hinge geometry, reinforcement content, and material properties. Its value diverges considerably among the different guidelines, with recommendations between 1/8 and 3/4. Some guidelines account for the beneficial effect of reinforcement crossing the throat by adding to Equation (30) a semi-empirical term depending on the reinforcement content. 1 The shear resistance V z,u parallel to the hinge axis (i.e., longitudinal direction) is not explicitly treated in any current guideline.
Recently, 28,41 it has been proposed to analyze the shear resistance of concrete hinges with approaches for concrete-to-concrete interfaces, considering all relevant contributions to the interface shear transfer, that is, adhesive bond, mechanical interlocking, friction, and dowel action of the reinforcement. As an example, consider the pertinent clause of Eurocode 2 42 (adapted to the nomenclature used in this paper): where A s and α s are the total cross-sectional area and inclination to the interface normal, respectively, of the reinforcing bars crossing the interface, and the parameters c 1 and μ are to be taken equal 0.45 and 0.7, respectively, for rough interfaces such as cracked concrete faces. 42 The upper limit in Equation (31) prevents failure of the diagonal concrete strut in the members adjacent to the interface. As the approach was proposed and calibrated for concrete interfaces with different geometries, subjected to (theoretically) uniform shear and axial stresses, the latter being substantially lower than those typical for concrete hinges, it is uncertain whether they are accurate for concrete hinges. Therefore, a different modeling approach is proposed in the next section.

| Proposed modeling
The shear resistance of concrete hinges is investigated employing the kinematic method of the theory of plasticity, inspired by a modeling approach proposed by Sørensen et al. 43 for shear keyed connections. After a At 900 kNm, the adjacent blocks entered in direct contact beyond the throat at the compressive throat edge; the moment could be further increased.
summary of the main assumptions and fundamentals of the underlying theory, the shear resistance of one-way concrete hinges subjected to axial and shear forces is predicted with two types of failure mechanisms observed in experiments.
Only the hinge resistance is considered; the shear resistance of the adjacent blocks can be determined using standard structural design methods for beams and columns and is not part of this paper.

| Basic assumptions and fundamental basis
The reader is referred to References 44 and 45 for a comprehensive introduction to the kinematic method of the theory of plasticity. Concrete and reinforcing steel are treated as rigid-perfectly plastic materials obeying the associated flow rule. In this study, a modified Coulomb failure criterion with zero tension cut-off is considered, with a uniaxial compressive strength f c0 and an internal angle of friction φ = arctan(3/4) ≈ 37 , hence a cohesion c = f c0 /4, as shown in Figure 9.
Upper-bound solutions for the ultimate load are obtained by considering kinematically admissible collapse mechanisms and applying the virtual work principle (equating the increments of external and internal work). The closest value to the real ultimate load is obtained by investigating different mechanisms and selecting the lowest solution, whereby for each mechanism, the ultimate load has to be minimized with regard to possible free geometrical parameters.
In this study, failure mechanisms containing discrete failure surfaces, that is, displacement discontinuities separating the specimen in volumes undergoing rigid body motions, are considered. The increment of external work is the scalar product of the external (applied) forces and the corresponding displacement increments. The increment of the internal work consists of dissipation in the yielding reinforcement and in the concrete displacement discontinuities. The former-assuming that the reinforcement only carries forces in its direction (i.e., no dowel action)-is the scalar product of the yield force of the reinforcing bars and the corresponding displacement increment. The reinforcement must be adequately anchored to be accounted for. The dissipation in the concrete displacement discontinuities is given by the scalar product of the stress and the plastic strain increments vectors. The stress vector is determined by the assumed yield criterion for the concrete, and the strain increments vector is assumed orthogonal to the yield surface, 44,46 see Figure 9. As derived by Chen and Drucker, 47 the dissipation in a failure surface in a modified Coulomb material with small tension cut-off is obtained with where δ is the relative displacement increment between the two rigid parts separated by the yield line, ψ is the angle between the yield line and the displacement vector δ, and A c is the failure surface area. In the case of a plane strain state, deformations with ψ < φ are not admissible as no corresponding state of stress satisfying the associated flow rule can be found. Considering the brittle nature of concrete in tension and the possibility of preexisting cracks, the tensile strength of concrete is often neglected in design, f ct = 0.

| Failure Mechanism A
In most experiments, the shear failure of the hinge occurred due to the formation of an inclined crack running diagonally through the hinge throat. This failure mechanism, referred to as Mechanism A, is schematically shown in Figure 10a for a reinforced concrete hinge subjected to axial compression N x and a shear force V y in the F I G U R E 9 Modified Coulomb failure criterion for concrete and plastic strain increment vectors. The cohesionless material idealization used in the simplified design method is shown in purple.
transverse direction. The external work done by the applied loads is with δ x = δ sin(ψ À θ) and δ y = δ cos(ψ À θ) being the displacement increments in the x-and y-directions, respectively. The dissipation in the yielding reinforcement is where A s is the total cross-sectional area of the X-shaped reinforcement crossing the throat at an angle α s /2 to the x-axis, and δ k,1 and δ k,2 are the relative displacements in the direction parallel to the reinforcing bar axes (see Figure 10a). According to Equation (32) and disregarding the concrete tensile strength, the energy dissipated in the concrete is where is the area of the concrete failure surface inclined at angle θ to the horizontal axis. Formulating the work equation, W E = W I,c + W I,s , and solving for V y , an upper-bound for the shear force that can be carried for a given normal force N x is obtained: The lowest upper-bound solution is obtained by minimizing the expression above for the parameters ψ and θ; in this case, this corresponds to choosing the angle θ as large as geometrically admissible (i.e., crossing opposite corners of the throat): The angle ψ opt that minimizes the upper bound solution of Mechanism A is then found by differentiating Equation (37) for ψ and setting it equal to zero. This results in F I G U R E 1 0 Failure mechanisms of a concrete hinge subjected to axial force N x and transverse shear V y : (a) Mechanism A with a failure plane running diagonally through the throat; (b) Mechanism B with a failure plane developing steeply from the throat edge and continuing horizontally beneath the first reinforcement layer. with The maximum admissible shear force V y,u,A that can be carried for a given normal force N x is finally obtained by inserting ψ opt and θ opt into Equation (37). The shear resistance V z,u,A of the hinge in longitudinal direction can be determined with an analogous procedure, with adapted values of the parameters α s and θ opt = arctan(h t /b 1 ).

| Failure Mechanism B
It was experimentally observed by Markic and Kaufmann 16 that a different failure mechanism might become governing for hinges with large throat widths d 1 relative to the width d of the adjacent blocks. The observed failure surface developed steeply from the edge of the throat and penetrated into the adjacent member until it intersected the first reinforcement layer, continuing beneath the latter until the edge of the member. Such a failure mode, referred to as Mechanism B, is schematically shown in Figure 10b for a shear failure in the transverse direction.
This type of failure mechanism might yield lower ultimate loads than the failure mechanism A if the failure surface A c2 between concrete and the first reinforcement layer is attributed a reduced contribution to the internal work, as caused by the reduced net concrete area and the low cohesion and friction between concrete and reinforcement. These effects are indirectly accounted for in the following, by lowering the dissipation in this weak interface A c2 by a factor ν c2 . As a default value, ν c2 = 0.2 is proposed.
The external work done by the applied loads and the dissipation in the yielding reinforcement can be expressed as in Mechanism A with Equations (33) and (34), respectively. The energy dissipated in the concrete is where the areas of the inclined and horizontal concrete failure surfaces are and h f is the distance of the first reinforcement layer from the crack base. The direction of the displacement vector is limited to ψ ≥ φ + θ by surface A c2 . Formulating the work equation and solving for V y yields: The lowest upper bound solution is given by minimizing the expression above for the parameters ψ and θ.
Analogously, failure Mechanism B can be formulated for a shear failure in longitudinal direction by adapting the expressions for the concrete areas A c1 and A c2 .

| Simplified model
The determination of the throat shear resistance using the two failure mechanisms described above is not always straightforward, particularly for Mechanism B, where an iterative procedure is needed and the factor ν c2 of the weak interface A c2 needs to be estimated. At least for preliminary design purposes, a simple expression that directly yields a conservative approximation of the shear resistance of the hinge is thus desirable. This can be achieved by conservatively assuming a cohesionless material, corresponding to the purple yield criterion shown in Figure 9. Consequently, the dissipation of the concrete vanishes, and Equations (37) and (44) reduce to Assuming that the failure surface must be geometrically contained within the throat, the lowest upper bound solution is given by ψ opt,simpl = φ, with the largest geometrically admissible angle: for failure in transverse y-direction for failure in longitudinal z-direction Note the similarity of Equation (45) and the first part of Equation (31), indicating an analogy between the failure mechanisms and shear friction theory. Figure 11 illustrates the sensitivity of the shear resistance predicted by the proposed failure mechanisms to the variation of the input parameters. Each mechanism was minimized with respect to the parameters ψ and θ to obtained the lowest ultimate load. It can be seen that with increasing throat cross-sectional area compared to the block area, the governing mechanism changes from A to B. The simplified model consistently yields the lowest ultimate load. The figure also shows that the axial load N x is the parameter with the most pronounced influence on the shear resistance. The concrete strength f c0 and steel cross-section also play an important role, whereas the bar inclination has a negligible effect (for a symmetric X-shaped placement of the bars). With respect to the parameters h f and ν c2 of Mechanisms B, the predicted shear resistance is much more sensitive to a variation of the empirical parameter ν c2 , which entails a large uncertainty. Figure 12 compares the predictions of various models to failure loads of concrete hinges tested in shear. The experimental data used for the comparison are summarized in Appendix A. The shear resistance prediction error relative to the failure load recorded in the tests is

| Model comparison and validation with test results
Influence of a 50% variation of various parameters on the shear resistance predicted by the proposed failure mechanisms.  28 Base, 17 Marki c et al., 16 and Reintjes. 48 plotted against the specimen names ( Figure 12a) and the normalized average axial stress across the throat ( Figure 12). The factor ν c2 for surface A c2 in the proposed model was chosen equal to 0.2 for all specimens. Only the results of the governing failure mechanism are plotted in Figure 12, which is Mechanism A for all specimens but CH7, agreeing well with the failure mode descriptions from the experiments where available. The approach for concrete interfaces was applied disregarding the upper limit in Equation (31). Note that Specimens B3 and B4 of Base 17 and V2 and V3 of Reintjes 48 were additionally subjected to a rotation that significantly reduced the compressed throat width; for this reason, d 1,c is used instead of d 1 in the computation of the shear resistance (see Section 7 for combined action). Figure 12 shows that the proposed failure mechanisms as well as the expression for concrete interfaces predict the observed failure loads well on average. However, both approaches exhibit a relatively large scatter and overpredict the shear strength in some cases, particularly for low axial stresses, N x /(b 1 d 1 ) < 0.15f c0 . A plausible reason for the scatter at low axial load is that adhesive bond and mechanical interlock largely contribute to the shear resistance at low normal stresses, and these transfer mechanisms exhibit a large scatter and are very sensitive to the presence of cracks and the concrete compaction quality (particularly challenging in the throat region of the hinges). The overpredictions at low axial load may be due to unintentional hinge rotations r z occurring during the application of the shear load: at low axial forces, even small hinge rotations can cause a significant reduction of the compressed throat width and, consequently, of the shear resistance predicted by the failure mechanisms. At higher axial stress level, the proposed failure mechanisms match significantly better with the test results and the predictions are slightly on the conservative side. The simplified model proposed in Section 5.2.4 yields safe predictions for all tests (even when disregarding the reduced compressed throat width in Specimens B3, B4, V2, and V3). Despite being significantly on the conservative side (À40% error on average), this simple approach still allows for a significantly more efficient design compared to Leonhardt's recommendations 5 (À89% error on average).

| TORQUE
Little attention has been given to the torsional resistance of concrete hinges in previous studies. To the authors' knowledge, torques are mentioned in no guideline for concrete hinges. While they often are of minor importance in design indeed, significant torques can result in concrete hinges in bridge piers, particularly in bridges with a curved horizontal alignment. Unless their torsional resistance is known, pairs of concrete hingesresisting the torque by shear forces with a horizontal lever arm-would have to be provided at each pier, as is common for bearings, requiring larger pier dimensions to carry the same axial load.

| State of the art
The only work known to the authors on the torque resistance on one-way concrete hinges is a theoretical study by Tue and Jankowiak, 41 who proposed a simple and pragmatic approach assuming that the torque is carried by a shear force couple in the transverse direction. The shear stress resistance was assumed as a fixed ratio of the acting axial compressive load (see current shear resistance approach in Section 5.1).

| Proposed modeling
The torsional resistance of a concrete hinge could be modeled by employing failure mechanisms analogously to the resistance to direct shear forces (Section 5.2). However, in the case of torsion, complex failure surfaces must be considered to obtain kinematically admissible mechanisms that yield realistic resistances. 49 In the following, a much simpler approach is proposed instead.
Similarly to Tue and Jankowiak, 41 it is assumed that the torque is resisted by shear stresses τ yx in y-direction alone. A triangular shear stresses distribution is assumed, as shown in Figure 13. Note that the assumed distribution is a crude, presumably conservative simplification, differing strongly from the classic shear stress distribution in a rectangular cross-section (as obtained, e.g., using Prandtl's elastic membrane analogy or the plastic sand hill analogy 50 ), where effectively half of the torque is resisted by shear stresses τ zx in z-direction that are completely neglected here. Nonetheless, it appears justified by the very narrow throat width (sensitive to the hinge rotation), the associated uncertainty in the lever arm of the shear stresses τ zx and most importantly the fact that the classic torsion models predict resistances an order of magnitude smaller than the experimentally observed torques, indicating that significant load redistributions occur after cracking.
Computing the moment about the x-axis, corresponding to the assumed shear stresses τ yx ≤ τ yx,u , results in the following hinge resistance to torque: where τ yx,u = maximum admissible shear stress. The latter can be derived from the hinge resistance to direct shear force (Section 5) as τ yx,u = V y,u /(b 1 d 1 ).
Alternatively to the triangular shear stress distribution, a fully plastic distribution with constant shear stress blocks τ yx = τ yx,u on each side of the neutral axis might also be considered and would result in a ratio 1/4 instead of 1/6 in the right part of Equation (47).

| Model validation with test results
Specimen CH4 by Markic and Kaufmann 16 is the only experiment known to the authors of a one-way concrete hinge tested in torsion. The specimen failed under a torque M x = 361 kNm while subjected to a normal force of 3 MN. With the failure mechanisms presented in Section 5.2, the predicted transverse shear resistance of the specimen amounts to V y,u = 1773 kN. With Equation (47), the predicted torsional resistance is M x,u = 310 kNm, that is, 14% lower than the experimentally observed strength. Assuming a rectangular shear stress distribution would result in M x,u = 465 kNm, 29% higher than the actual resistance. The single available experimental result thus indicated that the simple approach proposed, with a triangular shear stress distribution, leads to a conservative estimate of the torsional resistance. Moreover, the pronounced softening behavior of unreinforced concrete hinges after reaching the peak shear resistance appears to prevent a further redistribution of shear stresses.
More experimental data would, however, be required to consolidate these findings.

| INTERACTION BETWEEN VARIOUS CONCOMITANT ACTIONS
In the previous sections, the resistance of concrete hinges subjected to axial force and one concomitant action was investigated. In reality, an arbitrary combination of actions can simultaneously act on concrete hinges, which must be designed accordingly. The design for combined actions is, however, highly challenging based on the semiempirical models underlying the current design guidelines, which usually do not even mention the interaction between concomitant actions. The models proposed in this paper for axial force, rotations, bending moments and shear forces, by having a sound mechanical basis, can be more easily adapted and combined to account for the effect of various concomitant actions; note that this is less evident for the strongly simplified model for torques, validated on merely one test. This section discusses preliminary design proposals for combined actions based on the models for the individual load-cases developed in the paper, requiring further analytical and experimental validation which is envisaged for a future study.
As discussed in the previous sections, a hinge rotation r z and a moment M y about the strong axis produce flexural cracking and reduce the compression throat width d 1,c and length b 1,c , respectively. This has a detrimental effect on the shear resistance, as shown by tests, 17 and presumably on the torsional resistance of the hinge. Analytically, the effect of r z and M y on the shear and torsional resistances can be accounted for by using the compressed throat width d 1,c and length b 1,c instead of d 1 and b 1 , respectively, in the shear equations of Section 5.2.
The rigorous analysis of axial force and biaxial bending, that is, concurrence of rotation r z and moment M y , would imply a lengthy iterative process for obtaining the depth and inclination of the neutral axis and integration of the concrete stresses over polygonal areas. Alternatively, commercial software for the cross-sectional analysis of reinforced concrete columns can be used by adapting the concrete properties to resemble those of confined concrete (Sections 3.2.1 and 3.2.2). If a simple capacity verification at ULS is sufficient, the following expression can be used: where M z,u and M y,u are the moment capacities under the simultaneously acting axial force N x computed according to Sections 3.2.4-3.2.5 and 4.2, respectively. A conservative verification is obtained with the exponent p = 1, whereas p = 2 might be slightly unconservative for high axial force. Note that unlike in perfectly plastic materials where convexity is granted by assuming an associated flow rule, 50 the interaction diagram of (N x , M y , M z ) may exhibit concave parts due to the underlying strain limitations (see Sections 3.2.5 and 3.3.2).
Analogously, the capacity to biaxial shear can be verified with suitably oriented failure surfaces or with the simple interaction relationship: where V y,u and V z,u are the unidirectional shear capacities under the simultaneously acting axial force N x , determined according to Section 5.2. A conservative verification is obtained with the exponent p = 1. Preliminary calculations indicate that a less conservative exponent of p = 2 is admissible; however, a rigorous mathematical verification is still missing.

| SUMMARY AND CONCLUSIONS
For the last half century, the design of concrete hinges has relied on semi-empirical design recommendations proposed based on limited experimental data and engineering judgment. While these recommendations were pertinent and highly valuable at the time, their limitations present a major obstacle to the efficient application and a more widespread use of concrete hinges. This paper aims to alleviate the current uncertainties in the design and assessment of concrete hinges by revisiting their analytical modeling approaches based on models consistent with the philosophy of modern design codes. In the first part, this paper summarized existing geometrical and detailing recommendations for concrete hinges, concluding that these are pertinent and can be mechanically justified. In particular, (i) concrete hinges should have narrow and short throats with flat shoulder inclination on order to be efficient and satisfactorily functioning, (ii) the confinement and splitting reinforcement in the adjacent blocks should start immediately after the hinge recession and contain sufficient bars in the transverse and longitudinal directions, and (iii) in order to improve the hinge robustness against accidental loadings, a small amount of reinforcement can be placed through the throat in the form of vertical bars of stainless steel.
In the subsequent chapters, the paper investigated the mechanical behavior of concrete hinges. A cross-sectional analysis with confined concrete properties is proposed for the behavior of concrete hinges under axial forces N x and bending moments (about the hinge axis or orthogonal to it). A polynomial, analytically integrable stress-strain relationship is adopted for the confined concrete in the hinge throat. Its triaxial stress state and confined compressive strength are determined by means of a discontinuous stress field that consistently accounts for the beneficial effect of reinforcement in the adjacent blocks. By defining pertinent limits for strains and crack widths, expressions are developed for N x À r z interaction diagrams for the SLS and ULS. The hinge resistance to shear forces is investigated with failure mechanisms based on the kinematic method of limit analysis, inspired by the failure modes observed in experiments. Through conservative simplifications of the failure mechanisms, a simple expression is proposed for conservative estimations of the shear resistance. The hinge resistance to torque is estimated based on the direct shear resistance. Finally, some remarks are made on the analysis of combined actions.
Overall, the proposed models are mechanically better justified and agree better with the available experimental data than the existing analytical modeling approaches, potentially allowing for a more efficient design and assessment of concrete hinges. The proposed model can be used as an alternative to demanding NLFE analyses or as a plausibility check of their results. More test data for a broader range of parameters and load combinations would be welcome to systematically validate the increased resistances predicted by the proposed models over the existing guidelines. Further theoretical and experimental work is especially needed for torque and combined actions.

NOMENCLATURE
Normal forces, stresses, and strains are classed as positive when they are compression.