Rotational springs for localized crack modeling in structural concrete

This paper presents a mechanical and numerical approach to model localized deformations occurring in concrete structures with less than minimum reinforcement. The presented models are based on, and validated against large‐scale experiments conducted in a previous study to investigate the behavior of a less than minimally reinforced slab in the framework of the structural safety assessment of an existing cut and cover tunnel. The model deploys rotational springs embedded in a nonlinear finite element analysis to capture deformations concentrating in an inclined crack. An iterative solution procedure ensures equilibrium and compatibility in the inclined crack section by adjusting the spring stiffnesses, which depend on the acting loads. A case study shows that the localized crack has a minor effect on the global structural behavior. In contrast, localization strongly influences deformations and stress resultants in the crack, and compressive membrane action significantly impacts the load–deformation behavior of the localized crack, which is relevant particularly for the shear strength.


| INTRODUCTION
Modern design standards commonly specify a minimum bending reinforcement, assuring that the bending resistance (conservatively assumed to correspond to the yield moment m sy ) 1 is higher than the cracking moment m r , thereby avoiding brittle failure upon crack formation and contributing to ductile behavior.However, existing engineering structures, whose analysis is becoming increasingly important as they approach the end of their service life, often do not comply with these minimum reinforcement requirements locally.Design models and codes based on the Theory of Plasticity 2 are not a priori applicable to these structures since the required deformation capacity cannot be deemed to be sufficient.Instead, refined analyses are required, which need to account for the potentially localized deformations in the parts of the structure lacking appropriate minimum reinforcement.
This paper presents a mechanical approach, which describes the load-deformation behavior of partially less than minimally reinforced concrete structures and its implementation into a numerical procedure.The study originates from the structural safety assessment of an existing cut and cover tunnel, whose geometry with the outer wall, intermediate columns, and symmetry lines is shown in Figure 1.The lap splice above the haunch marks an abrupt change from the sufficiently reinforced frame corner ðm À r < m À sy Þ to the slab with less than minimum reinforcement ðm À r > m À sy Þ.To describe the load-deformation behavior of this structure, it is of particular interest to understand the phenomena related to cracking of the cross-section in the vicinity of the lap splice, that is, at the transition from sufficiently to insufficiently reinforced areas: Assume that external loads are applied to the uncracked structure, and increased until the (closing) bending moment in the frame corner equals the cracking moment m r , that is, the tensile stresses σ c in the top fibers of the cross-section reach the concrete tensile strength f ct , and a bending crack forms.If the structure is provided with adequate minimum reinforcement in the cracked sections, the applied loads and the corresponding bending moment can be further increased until the yield moment m sy is reached.During this load increase, the cracked region extends to adjacent parts of the structure experiencing a lower bending moment, that is, a stabilized crack pattern 3 forms.If a cross-section is provided with less than minimum reinforcement (m r > m sy ), however, such as at the lap splice in the structure illustrated in Figure 1, the cracking moment cannot be sustained.Instead, the bending moment drops to the yield moment m sy upon crack formation; at this point, an isostatic structure would fail in a brittle manner.In a hyperstatic system, load redistributions may prevent collapse upon crack formation, but the associated behavior is highly complex: since the cracking moment cannot be reached anymore due to the lacking resistance, (i) any sections that had cracked previously (closer to the frame corner) are unloaded, and (ii) the cracked region will not extend further, that is, beyond the lap splice in Figure 1, hence no stabilized crack pattern develops. 4Instead, deformations concentrate in a localized crack, marking a discontinuity. 5Note that due to the weak section caused by the lap splice and its vicinity to the frame corner (Figure 1), initial cracking is likely to occur at this location, hence a single crack is expected there.
The behavior of a structure with localized deformations in cracks cannot be adequately analyzed by standard FE models based on continuum mechanics.Furthermore, the effect of transverse shear and its interaction with bending moments needs to be considered, accounting for the lacking transverse reinforcement. 6In the following, the behavior of the frame corner of the cut and cover tunnel illustrated in Figure 1-which is affected by all issues mentioned above-is thus discussed along with the results of a large-scale frame corner experiment conducted at ETH Zurich, 1 helping to understand the governing effects.Based on the findings, a mechanical model for a discontinuity-hereafter referred to as localized crack-in the deformation field of a structure with less than minimum reinforcement is presented.The model is implemented into a nonlinear finite element framework using rotational springs.Compared with traditional interface elements, springs have the advantage of allowing for the implementation of a customized bending moment-rotation behavior.The influence of the discontinuity on the behavior on a structural level is subsequently analyzed by means of the example of the cut and cover tunnel as a case study.Particular attention is given to the impact of compressive membrane action, 7 which has not been investigated in the experiments but is expected to significantly influence the behavior of the localized crack.

| EXPERIMENTAL CAMPAIGN
The main assumptions concerning the mechanical behavior of the localized crack are backed by two largescale frame corner experiments (FC1 and FC2) conducted by Beck et al. 1 at ETH Zurich, where the analysis carried out in this study is limited to specimen FC2. Figure 2a,b illustrates the specimen dimensions and the reinforcement layout, which correspond to those of the frame corner of the cut and cover tunnel illustrated in Figure 1. Figure 2c shows the specimen after the experiment and illustrates the applied loads; the reinforcement resisting hogging moments is indicated for reference as well.The position of the force F 1 was chosen such that upon initial loading, with F 2 ¼ 0, the combination of bending moment and shear force corresponded to that expected in the cut and cover tunnel.After cracking, the two forces F 1 and F 2 were controlled such that the shear force 2d), such that the shear resistance could be determined for this crack width and compared with that predicted by corresponding models (e.g., References  5,8,9).Note that in a real structure, where moment redistributions occur and compressive membrane action is activated upon crack formation (see Sections 4 and 5), a smaller crack width is expected and hence, a lower bound for the shear strength was obtained in the test.
The experimental results confirmed the theoretical considerations on localized cracking (Section 1): A single crack formed in the slab adjacent to the frame corner, starting at the end of the lap splice at the top and running to the edge of the haunch, that is, the crack was inclined (see Figure 2c).No further bending cracks on the slab top occurred in its vicinity.Note that the behavior of a sufficiently reinforced cross-section can be studied in Figure 2c as well: in contrast to the localized crack zone, the load introduction zone of F 1 was provided with more than the minimum reinforcement for sagging moments, and a stabilized crack pattern with multiple cracks developed.

| Assumptions and idealizations
The model for the load-deformation behavior of the localized crack is based on the following assumptions: i.All employed elements (shell and rotational spring elements) allow only for unidirectional loading.Thus, the back bone of the load history is calculated without unloading or load (i.e., bending moments for the rotational springs) reversals.ii.The localized crack develops with a constant inclination from the edge of the haunch to the end of the lap splice (i.e., neglecting the slight curvature observed in the experiment).Its geometry is thus defined by the geometry of the structure and the reinforcement layout.iii.In plan view, the crack runs orthogonal to the main (hogging moment) reinforcement direction.iv.The principal directions of shear forces, bending moments, and membrane forces are orthogonal to the crack in plan view.
While conditions (i)-(iii) are incorporated in the assembly of the model before the calculations, the validity of condition (iv) must be checked in the postprocessing of the results.

| Modeling concept
The localized crack is implemented in a 3D FEA software by deploying rotational springs at the crack location.Their stiffnesses are adjusted iteratively by conducting cross-sectional analyses in the inclined crack until equilibrium and compatibility according to the deployed material models are reached in each spring element.The three levels of modeling presented in this section describe (i) the modeling of the structure and the integration of the localized crack, (ii) the behavior of the localized crack on a cross-section level, and (iii) the deployed concrete, steel and bond models.

| Structural modeling
The overall structure is discretized utilizing layered Shell181 elements with three translational and three rotational degrees of freedom (DOFs) in the finite element (FE) software Ansys Mechanical APDL. 10 The nonlinear behavior of the reinforced concrete shell elements is modeled with a user-defined material based on the Cracked Membrane Model 11 and the Tension Chord Model (TCM), 3 implemented into ANSYS Mechanical APDL by Thoma et al. 12,13 Rotational springs are introduced at the predefined position of the localized crack (see Section 2), according to Figure 3.They allow for modeling a concentrated rotation at the crack location.Assuming orthogonality of principal force and bending moment directions to the crack (see Section 3.1), the behavior of the localized crack is governed by three stress resultants (bending moment m y , normal force n y , and transverse shear force v yz in the coordinate system defined in Figure 3), just like a line beam subjected to uniaxial bending and normal force.The nonlinear bending moment-rotation characteristics of the spring elements (schematically illustrated in Figure 3) depend on the sectional forces and require an iterative solution as outlined in the following sections.The following sections focus on the modeling of the localized crack.Regarding the constitutive model of the shell elements, the reader is referred to References 12 and 13.

| Localized crack modeling
The model for the mechanical behavior of the localized crack, shown in Figure 4, is based on concepts established by Marti et al. 14 As described in Section 2, a single crack is assumed to propagate from the end of the lap splice to the bottom of the slab at the haunch.The crack has an inclination β c , defined by the position of the lap splice (Figure 2c) and the distance (d À x) from the negative bending reinforcement, where d ¼ static depth, and x ¼ depth of the compression zone.
The crack width w r can be determined in a mechanically consistent way by integrating the steel strains ε s over the embedment length l b : where l b is the length required to activate the force in the reinforcing bar at the crack by bond stresses; note that concrete tensile strains are neglected as they are much smaller than the steel strains.Using Equation ( 1), the crack opening can be related to the force in the reinforcing bar at the crack by solving the second order differential equation of bond using any bond shear stress-slip relationship. 3,15ssuming small crack rotations, that is, tan θ CS ð Þ≈ θ CS , the localized crack rotation θ CS follows from geometry: Hence, the kinematics of the localized crack are uniquely determined by two primary unknowns: the steel stress at the crack σ sr and the compression zone depth x.Using the hypothesis of Navier-Bernoulli, these unknowns can be determined from a cross-sectional analysis of a slab segment including the inclined crack plane, as illustrated in Figure 5, assuming that the shear force v yz is transferred by aggregate interlock acting in the inclined crack, that is, the vertical component of the force v yz,R. 5To obtain the sectional forces by integrating the stresses, the cross-section is divided into n layers with a small depth δz, similar as proposed by References 16,17 or 18. Equilibrium of the free body in Figure 5a yields the normal force n y and bending moment m y per unit width: Þare the concrete and steel stresses, respectively, as functions of the generalized strains (ε m = strain at mid-depth of the section and χ = curvature), a s,l = reinforcement area per unit length, and z l = depth of layer l.
The nonlinear equations (3), combined with appropriate stress-strain relationships of concrete and reinforcing steel-such as the nonlinear constitutive laws used in this study, illustrated in Figure 6a,b and further discussed in the following subsection-yield the sectional forces (n y , m y ) for any meaningful combination of the generalized strains (ε m , χ).On the other hand, by iteratively solving these equations, one gets the strains (ε m , χ) for any combination of applied sectional forces (n y ,m y ).The steel stress σ sr at the crack, the corresponding strain ε sr , as well as the depth of the compression zone x, follow directly.The crack width w r and the localized crack rotation θ CS are determined considering Equations ( 1) and (2).

| Material modeling
For concrete in compression, the constitutive model of Sargin 19 is adopted, with no post-peak softening 12 for numerical stability (Figure 6a). Figure 6b shows the stress-strain behavior of the bare reinforcement, which is assumed bilinear.
Since less than minimum reinforcement is provided at the lap slice, a pull-out type behavior results, as illustrated in Figure 4.The response is modeled using the pull-out model (POM), which has been developed and validated specifically for such cases 4 and adopts the same stepped, rigid-perfectly plastic bond stress-slip relationship as the TCM.The bond stresses are normalized using the concrete tensile strength f ct , which is determined on the lines of Raphael. 20Setting the parameters ξ 0 ¼ 2 and ξ 1 ¼ 1 12 before and after yielding, respectively: see Figure 6c, the stresses and strains along the reinforcing bar shown in Figure 4 are readily obtained, and integrating the steel strains one gets the relationship between crack opening and steel stresses at the crack illustrated in Figure 6d.(d) relationship between the crack opening and the steel stress at the crack according to the POM. 4 F I G U R E 7 Numerical algorithm of spring stiffness updating.After assembling the FE model, the fixed node connections at the crack location (solid dots in Figure 3) are replaced by linear spring elements, which are infinitely stiff for all degrees of freedom (DOF) except the crack rotation (θ y ).For the corresponding spring elements, a secant stiffness m y =θ y ¼ k 0 is estimated, and an initial FE analysis is carried out, resulting in sectional forces m y,i,j , n y,i,j ,v yz,i,j and rotations θ FEA,i,j of each spring element representing the localized crack, where the loop variables i and j denominate the spring element and the iteration step, respectively.A cross-sectional analysis at each spring element, according to Section 3.2, yields the theoretical crack rotations θ CS,i,j , which are used to update the secant spring stiffnesses after each solution step of the global FE analysis.For numerical stability, the theoretical updating factors r θ,i,j ¼ θ FEA,i,j =θ CS,i,j are attenuated by an exponent 1=η ≤ 1, that is, the secant stiffness is adjusted using the relationship k θy,i,jþ1 ¼ r 1=η θ,i,j Á k θy,i,j .This process is iterated until θ FEA,i,j ¼ θ CS,i,j in every spring is achieved.

| Solution strategy
The default FE solver of Ansys Mechanical APDL is used in this paper with Newton-Raphson iteration and full (non-symmetric) stiffness matrices.The mechanical and numerical models described in this chapter are used in two different cases: the experimental validation (Chapter 4) and the cut and cover tunnel case study (Chapter 5).Table 1 summarizes the most relevant parameters of the solution strategy 21 regarding kinematic compatibility and equilibrium, consisting of (i) parameters concerning the global FE solution with iterations over several solution steps until global equilibrium is reached, and (ii) local iterations in the spring elements for each solution-and substep.The former comprises the average mesh sizes, the number of substeps n SUBS (number of load increments in nonlinear FE solution) in each solution step and the acceptable range for r θ,i,j , denoted r θ,lim .The latter includes the tolerances for the cross-sectional analysis in spring element i and solution step j α ε,i,j,k ¼ Δε m,i,j,k =ε m,i,j,kÀ1 , α χ,i,j,k ¼ Δχ i,j,k =χ i,j,kÀ1 as well as the initial value (k 0 ) and the attenuation factor (η) for the rotational spring stiffness.
The number of solution steps needed to reach convergence according to the implemented solution strategy is n SOL ¼ 3 in the experimental validation (fast convergence in the isostatic system without moment redistributions) and n SOL ¼ 4…7 in the case study. ) ) )

| FE model
The numerical model presented in Section 3 is validated using the experiments conducted by Beck et al. 1 Figure 8a,b shows the FE model of the cantilever.The haunch is modeled with an offset, that is, the reference plane of the shell element is located at the top of the slab.All material properties in Figure 8c were taken from Reference 1, except for ε c0 (denominating the concrete strain at f c ), which was estimated.The concrete strength f c was determined from a core sampled from the specimen after the test, to account for irregularities during casting.The steel properties indicated are dynamic values obtained from material tensile tests, carried out on samples taken from the same production batch as the specimen reinforcement.

| Results
Figure 9a shows the load steps at which specimen FC2 1 is analyzed with the FE model presented in Section 4.1.The analysis starts after the formation of the localized crack and continues to the peak load; the unloading observed in the experiment thereafter (Figure 2c) cannot be captured by the model (see Section 3.1).
The load-deformation behavior and the crack width are shown in Figure 9b,c, comparing the experimental results to the numerical predictions.A good agreement between experimental and numerical values can be observed, with a slight tendency toward overestimating the stiffness.
Figure 9d compares the different modeling approaches used to quantify the influence of (i) the constitutive model of the shell elements (linear or nonlinear) and (ii) localized crack modeling, on the loaddeformation curve at the position of the load F 1 .Nonlinear shell element model results with and without the localized crack (NLFEA + C and NLFEA, respectively) are compared with linear elastic shell element modeling, again with and without the (nonlinear) localized crack (LFEA + C and LFEA).The different approaches are summarized in Table 2; note that NLFEA + C corresponds to the approach introduced in Section 3.
The comparison of the different predictions to the experimental results in Figure 9d highlights the necessity of considering both nonlinear components (localized crack and nonlinear shell elements) for a satisfactory prediction of the load-deformation response: linear elastic modeling strongly underestimates the deformations, even considering the localized crack.NLFEA still underpredicts the deformations, with an approximately constant offset with respect to the experiment, roughly corresponding to the deformations caused by the localized crack.Interestingly, the latter are fairly independent of the shell element model adopted (compare differences between NLFEA and NLFEA + C, and between LFEA and LFEA + C, respectively).Note that this observation may not apply to other systems, particularly if they are statically indeterminate and compressive membrane action is activated upon cracking.

| Geometry
The case study highlighting the effect of the localized crack on the structural level follows the example introduced in Section 1 and the experiments from Section 2. Figure 10a,b illustrate the geometry of the analyzed cut and cover tunnel section with its kinematic boundary conditions.The concrete slab with a thickness of 1.2 m is supported by a concrete wall, a circular steel column and a concrete column; all assumed to be clamped in the tunnel base plate.Adjacent structural members are considered by introducing corresponding kinematic boundary conditions at the level of the slab.The reinforcement layout depicted in Figure 10c corresponds to the situation investigated in the experiments, that is, a localized crack can occur at the transition between the frame corner and slab reinforcement (lap splice at y = 2.06 m, less than minimally reinforced slab for 2.06 m < y < 4.81 m). Figure 10d shows the reinforcement contents defined in the FE model.Reinforcement for sagging moments (subscript inf), as well as the reinforcement in the wall and the column, are constant throughout the entire model.In contrast, the top reinforcement in the slab (sup) is doubled over the columns and reduced to half in the transition area between frame corner and slab.

| Analytical models
Four different analytical models, M1-M4, are used to study the structural behavior in this case study.All have the same geometry and reinforcement layout and are to a vertical load q z ¼ 160 kN=m 2 .The only difference is the modeling of the tunnel wall at y = 0.In Models M1-M3, the wall is substituted by a uniaxially movable, clamped support in the slab axis (free translational DOF in the y-direction), with a lateral force of varying magnitude, that is, q y ¼ 0, 250, and 500 kN=m 2 in M1, M2, and M3, respectively.Model M4 includes the tunnel wall, and (instead of a lateral force at the top) is subjected to a triangularly distributed lateral pressure with a maximum value of 164 kN=m 2 at its foot.Figure 11 summarizes the static and kinematic boundary conditions of all models in section A-A (see Figure 10a).
These four models allow isolating the impact of the normal force, while all other parameters are kept constant.It also allows studying the effect of a common simplification in engineering practice, that is, to replace a wall with a clamped support in the slab axis, often neglecting the effect of horizontal forces, and hardly ever quantifying the influence of these simplifications.

| Discussion of structural behavior
This section analyses the global structural behavior of Models M1-M4 using the modeling approaches introduced in Table 2.The applicability of the localized crack model is discussed along with the results in terms of deformations and stress resultants, globally in the entire structure and in the localized crack.

| Applicability of localized crack model
As introduced in Section 3.1, the localized crack model used in this study presumes orthogonality of the principal directions of the stress resultants to the crack.Figure 12 shows the deviation of the directions of the principal bending moments φ m ð Þ, normal forces φ n ð Þ, and shear forces φ v ð Þ from the direction orthogonal to the crack for Model M4 in section B-B (see Figure 10a), determined with NLFEA + C. The deviations are insignificant (max.9 for the direction of the principal shear forces φ v ), and localized crack modeling is thus applicable.

| Deformations
Figure 13a-d show the vertical deformations and stress resultants in section A-A (see Figure 10a) for Models M1-M4 and the modeling approaches from Table 2. To understand the impact of the localized crack, it is important to realize that the rotational spring representing the crack introduces a local stiffness reduction to the FE model.The relative stiffness decrease is slightly more pronounced in LFEA + C than NLFEA + C due to the higher stiffness of the shell elements in the linear elastic model.
The vertical deflections are mainly governed by the constitutive model of the shell element (LFEA or NLFEA), and much less affected by the localized crack.The impact of the latter is more significant for linear elastic than nonlinear modeling due to the mentioned higher local stiffness reduction, and is most influential in Model M1 with q y ¼ 0. In Model M4, including the tunnel wall, the influence of the localized crack on the deflections can largely be neglected.

| Stress resultants
The localized crack significantly influences the stress resultants m y , n y , and v yz near the frame corner (y = 0).Toward the opposite end of the slab, the influence declines, and the constitutive model of the shell becomes the governing factor.The effect can exemplarily be studied based on the profile of the bending moments resulting from LFEA and LFEA + C: in the vicinity of the nonlinearity introduced by the localized crack, the bending moments for LFEA + C are approximately equal to those obtained with full nonlinear modeling (NLFEA +C), while with increasing distance from the crack, they approach the LFEA and NLFEA solution, respectively.Similar observations hold for the shear forces.
Principal directions in section B-B according to Figure 10a.
The normal force distribution in section A-A exhibits a pronounced dependency on the constitutive model of the shell elements.NLFEA leads to a distinct redistribution of normal forces n y between section A-A (mid-span section without intermediate steel column) and the column section around x = 5.97.This effect is most obvious in Model M1, where the sum of the normal forces n y in any entire section in x-direction (e.g., section B-B) must vanish by equilibrium as no lateral forces are applied, but significant tensile forces n y are obtained in the nonlinear analyses along section A-A.This is due to the yielding of the top reinforcement over the intermediate steel column, whichif unrestrainedwould cause large elongations in the column section.However, these elongations are restrained by the adjoining slab parts (such as section A-A) which tend to extend much less.Hence, compressive and tensile normal forces result in the column section and in section A-A, respectively.The related effects are captured with the NLFEA models, with a minor influence of the localized crack, but not by the linear analyses.

| Stress resultants in models M1-M3
Table 3 summarizes the stress resultants in section A-A at the position of the localized crack.It confirms the observations from Section 5.3 that while the effect of the F I G U R E 1 3 Stress resultants in section A-A (Figure 10) for different approaches and boundary conditions.(a) M1, q y = 0; (b) M2, q y = 250 kN/m; (c) M3, q y = 500 kN/m; (d) M4.
localized crack on the global behavior of the structure is limited, its local impact on the stress resultants is significant.The largest differences introduced by the crack are observed for LFEA + C in Model M1, where the bending moment m y reduces by a factor of 4.7 compared with the purely linear elastic model (LFEA).This ratio reduces in M2 and M3, where the higher compressive normal force (see Figure 11) increases the rotational stiffness of the localized crack, attracting more load.NLFEA exhibits a considerably smaller, yet still significant sensitivity on localized crack modeling (NLFEA + C), with an approximately constant ratio m y,NLFEA =m y,NLFEAþC of roughly 1.3 in Models M1-M3.The influence of the normal force (M3 compared with M1) on the bending moment in the localized crack is, again, more pronounced for linear elastic (m y,LFEAþC,M3 =m y,LFEAþC,M1 ¼ 1:8) than for nonlinear (m y,NLFEAþC,M3 =m y,NLFEAþC,M1 ¼ 1:4) shell element modeling.
The shear forces vary less pronouncedly than the bending moments, and in a slightly different manner.The maximum relative impact of the localized crack to v yz,LFEA,M1 =v 1:3, experiencing only a minor reduction for increasing compressive normal force.In NLFEA, the localized crack causes a reduction of shear force in the crack by roughly 5%, irrespective of the normal force.
The discussion of stress resultants in Models M1-M3 leads to the following key findings: i. Introducing a nonlinearity (through a localized crack in LFEA or full NLFEA with or without localized crack) leads to a significant change of stress resultants at the localized crack.The nonlinear approaches differ much less among each other than they do from the linear elastic analysis.ii.The higher the relative local stiffness decrease in the localized crack, the higher is its impact.Hence, it is highest for LFEA+C (compared with LFEA) in this case study.iii.As soon as a nonlinearity is introduced, the normal forces play a crucial role, as they significantly influence the bending stiffness.10a) for the different models and FE modeling approaches, excluding purely linear elastic analysis which does not allow for a pertinent calculation of steel stresses nor crack widths.Interpolated integration point results 12 are shown for the nonlinear modeling without localized crack.The distribution of the steel stresses and crack openings along Section B-B reveals-for all modeling approaches-that the outer parts without intermediate column experience larger crack openings and hence, higher steel stresses than the column section in the center.This is in line with the higher compressive normal force in the column section discussed above: Crack openings and steel stresses diminish with increasing compressive force, which outweighs the higher bending moments which it also causes (Figure 13a-c).
Including the effect of the localized crack significantly increases the steel stresses and crack widths (compare NLFEA + C and LFEA + C with NLFEA), which in the purely elastic regime are proportional to the squared steel The bending moment-rotation relationship is characterized by a decreasing slope at lower rotations, followed by a fairly linear branch until close to the last load step, where a significant stiffness increase is observed for both springs.The initial stiffness decrease is attributed to the increasing depth of the concrete compression zone and a corresponding decrease of the inner lever arm.At high vertical loads, with yielding of the slab reinforcement, compressive membrane forces rapidly increase, which leads to a stiffer response of the spring elements.The steel stresses do not reflect the stiffness increase at high rotations, as the stiffness of the reinforcement decreases after the onset of yielding.The steel stresses thus exhibit a gradually softening behavior, becoming fairly linearthat is, proportional to the crack rotation-at higher load.Finally, Figure 15 verifies the assumption of unidirectional loading, which is a prerequisite for applying the developed model, see Section 3.1.

| CONCLUSIONS
This paper presents a model consisting of nonlinear rotational springs to account for localized cracks in NLFE analyses of concrete structures provided with less than minimum reinforcement, that is, whose bending resistance is smaller than the cracking moment locally.The localized crack model and the NLFE analyses are validated against large-scale experiments and applied in a case study.
Based on this study, the following conclusions can be made regarding the mechanical behavior of less than minimally reinforced concrete structures and their numerical modeling with finite elements: When applying the presented model to a real-life concrete structure, the following points need to be considered: i.Whether a localized crack will occur, and where this crack will form, cannot be predicted with certainty.It depends on the applied loads, material properties (particularly f ct and its variation) and self-equilibrated stress states, induced for example, by restrained shrinkage, creep, prestressing or differential settlements.ii.When analyzing a structure with partially less than minimally reinforced areas, further potential behaviors, such as the formation of a stabilized crack pattern, need to be considered in addition to the deformation concentration discussed in this study.iii.When applying localized crack modeling, a thorough parameter study is required regarding the indicated parameters, and particularly the position of the localized crack: while the latter was predefined in this study by the strong discontinuities of the lap splice and the haunch end, it is typically not precisely known in existing structures.
(a) iv.In the case study, rather than the localized crack, the constitutive model of the shell elements governed the global structural behavior, that is, deformations and stress resultants throughout the entire structure.Presumably, this observation holds for most applications on a structural level.v.However, the local behavior near a deformation concentration in a less than minimally reinforced area (and thus pertinent structural verifications such as shear strength or fatigue) is significantly affected by localized crack modeling.Hence, even an otherwise linear elastic analysis may yield reasonable local results if the localized crack is modeled appropriately, for example, using the rotational springs introduced in this study.vi.Whether or not localized crack modeling is required, and how it is applied, depends on the structure and problem under consideration (global or local structural verification) and requires engineering judgment.

ACKNOWLEDGMENTS
The authors would like to express their gratitude to Prof.

1
Cut and cover tunnel.(a) Overview of structure; (b) detail of frame corner.

F I G U R E 2
Frame corner experiment FC2.(a) Geometry and negative bending reinforcement; (b) Reinforcement layout in section A-A; (c) Load introduction, bending moment and localized crack; (d) Loading history with loads and resulting bending moment at the edge of the haunch; (e) Crack width over time.Dimensions in [m] if not specified differently.(Adapted from Reference 1).

F I G U R E 3
Schematic overview of deployed elements.F I G U R E 4 Mechanical model for analyzing the localized crack and the corresponding distribution of the steel stresses and strains as well as the slip.

F I G U R E 5
Cross-sectional analysis at the inclined localized crack; (a) segment bounded by inclined crack and vertical reference section; (b) strains; (c) stresses; (d) forces per layer; (e) cross-section with layers.Note that strains and stresses in (b-d) refer to the inclined crack section.

F
I G U R E 6 Material models.(a) Stress-strain diagram of concrete; (b) stress-strain diagram of bare reinforcement; (c) stepped, rigid-perfectly plastic bond stress-slip relationship;

Figure 7
Figure 7 outlines the numerical workflow of integrating the cross-sectional analysis into the NLFEA framework.After assembling the FE model, the fixed node connections at the crack location (solid dots in Figure3) are replaced by linear spring elements, which are infinitely stiff for all degrees of freedom (DOF) except the crack rotation (θ y ).For the corresponding spring elements, a secant stiffness m y =θ y ¼ k 0 is estimated, and an initial FE analysis is carried out, resulting in sectional forces m y,i,j , n y,i,j ,v yz,i,j and rotations θ FEA,i,j of each spring element representing the localized crack, where the loop variables i and j denominate the spring element and the iteration step, respectively.A cross-sectional analysis at each spring element, according to Section 3.2, yields the theoretical crack rotations θ CS,i,j , which are used to update the secant spring stiffnesses after each solution step of the global FE analysis.For numerical stability, the theoretical updating factors r θ,i,j ¼ θ FEA,i,j =θ CS,i,j are attenuated by an exponent 1=η ≤ 1, that is, the secant stiffness is adjusted using the relationship k θy,i,jþ1 ¼ r Model of frame corner experiment.(a) Side view; (b) top view of FE model; (c) material properties.Dimensions in [m].

1 0
Cut and cover tunnel.(a) Geometry and kinematic boundary conditions; (b) dimensions; (c) reinforcement layout of the slab; (d) slab reinforcement content in FE model.Dimensions in [m].

5. 4 . 2 | 5 . 4 . 3 |
Figure14a-d compares the steel stresses and the crack widths along the localized crack (section B-B in Figure10a) for the different models and FE modeling approaches, excluding purely linear elastic analysis which does not allow for a pertinent calculation of steel stresses nor crack widths.Interpolated integration point results12 are shown for the nonlinear modeling without localized crack.The distribution of the steel stresses and crack openings along Section B-B reveals-for all modeling approaches-that the outer parts without intermediate column experience larger crack openings and hence, higher steel stresses than the column section in the center.This is in line with the higher compressive normal force in the column section discussed above: Crack openings and steel stresses diminish with increasing compressive force, which outweighs the higher bending moments which it also causes (Figure13a-c).Including the effect of the localized crack significantly increases the steel stresses and crack widths (compare NLFEA + C and LFEA + C with NLFEA), which in the purely elastic regime are proportional to the squared steel

F I G U R E 1 5
Load-deformation behavior in spring elements.(a) Mid-span section; (b) column section.
Solution strategy parameters.
T A B L E 1 T A B L E 3 Stress resultants in section A-A at the localized crack.
em.Dr. Peter Marti for the inspiration and the support in the development of the localized crack model.The authors would like to thank the Swiss Federal Railways for partially funding the experiments and the nonlinear structural analysis.Open access funding provided by Eidgenossische Technische Hochschule Zurich.