Redistribution of internal forces from the span to the support

This paper presents the results of evaluations on the redistribution of internal forces from the span to the support area using nonlinear finite element calculations. The motivation is the internalization of bridges, particularly the reconstruction of single‐span beams arranged in a row to a continuous beam system. When integralizing a bridge, it can be assumed that the existing bridge is designed according to standards, which are no longer valid. If the integralized bridge is loaded and designed according to the new standard, the problem may be that the existing structure has insufficient reinforcement. For single‐span bridges, the flexural reinforcement in the span may be considered critical. The degree of reinforcement in the support area can then be adjusted individually, which offers the possibility of transferring internal forces from the span to the support area if they cannot be fully absorbed by the existing reinforcement.


| INTRODUCTION
Integral bridges have advantages compared to conventional bridges with bearings and expansion joints in terms of the load-bearing behavior, driving comfort, and maintenance. They are characterized by low life cycle costs, which significantly increase the popularity of this construction method. 1,2 Many countries have an infrastructure network that is already aging, which is why more work must often be put into renovation rather than into new construction. [3][4][5] The reconstruction of existing bridges to integral bridges offers the opportunity to reduce life cycle costs in the future. When integralizing a bridge, it can be assumed that the existing bridge was designed according to standards that are no longer valid. In new load standards for bridges, for example, there are significantly heavier traffic load models applied. If the bridge to be integrated is designed according to the new standard, the problem may be that the existing structure does not have sufficient reinforcement. For single-span bridges, among other things, flexural reinforcement in the span can be considered critical. In case of problems with the shear force, measures can be taken, for example, through special additional calculations [6][7][8][9] or reinforcement measures. [10][11][12] When an integralization is done, the dead weight remains in the single span girder and the live loads are taken from the continuous girder system.
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.
The load case dead weight results in high stress in the span reinforcement. The action of the live load according to the new standard results in lower stress in the span due to the continuous girder system than in a single span girder. In the case of integralization, the degree of reinforcement at the support area can be adjusted individually, which offers the possibility of transferring internal forces from the span area to the support area if they cannot be completely absorbed by the existing reinforcement. Regarding the redistribution of internal forces by creep, it should be taken into account that the creep behavior has come to an end prior to the integralization. 13,14

| FUNDAMENTALS OF INTERNAL FORCES
In structural engineering, significant effort is made to design statically indeterminate load-bearing structures because in statically determined systems, no load-bearing reserves are available. The calculation of internal forces from equilibrium conditions only is not possible for statically indeterminate systems. The consideration of the compatibility conditions of the deformation of the structure allows the determination of the internal forces. Thus, the stiffness of the cross-section has a significant influence on the distribution of internal forces. The nonlinear material behavior, and subsequently, the cracking of concrete, changes the stiffness in the higher stressed zones of the structure. Accordingly, the stiffness and distribution of internal forces also depend on the amount of reinforcement. Linear elastic calculations do not consider the reduction of stiffnesses, which leads to moment redistributions. This results in partially high reinforcement concentrations in the areas of highest stress.

| Redistribution of internal forces
By locally changing the stiffnesses due to cracking, the internal forces can be shifted to areas of higher stiffness. A simplified method for determining the redistribution of moments in statically indeterminate systems can be performed using a linear elastic calculation with limited redistribution. The internal force distribution deviating from the elasticity theory can be understood as a redistribution of the internal forces of the linear-elastic system to the internal forces of the real system. The magnitude of the rearrangement can be described using the redistribution factor "δ" (Equation (1)) and redistribution ratio "1 À δ" (Equation (2)). 15 Here M is the real moment and M EL is the moment determined according to elasticity theory.
Limits are usually provided in standards for the redistributions considering the compression zone height of the highly stressed cross-section and plastic rotational capacity. [16][17][18] A more complex method for analyzing internal force redistributions is the nonlinear finite element analysis.

| Linear elastic analysis with limited redistribution
By changing the stiffnesses due to cracking of the structure in high stress areas, internal forces from the linear-elastic calculation of the structure in state I (uncracked) may be redistributed as shown in Figure 1. The redistribution can provide a more economical design of cross-sections and reveals any load-bearing reserves of the structure. The requirement for a redistribution of internal forces is a statically indeterminate structure because the stiffness has no influence on the internal forces for statically determinate support structures. The limitation of the redistribution depends, for example, according to Eurocode, 19 on the related compression height zone. The limit values given usually only apply to the redistribution of the internal forces from the support to the span. For the redistribution of internal forces from the span to the support area, verification of the rotational capability must generally be provided. 18 F I G U R E 1 Principle of linear-elastic internal force calculation with moment redistribution 2.1.2 | Nonlinear analysis design The use of nonlinear calculations enables the inclusion of realistic material behavior. The knowledge of the crosssection parameters and reinforcement layout must be given to perform nonlinear analysis. If this is not the case, a preliminary design of the structure can provide the input values. By considering the real stiffnesses of the structure according to the actions, the ultimate and serviceability limit states can be determined. 18 By including the material properties in the analysis, the verification of the plastic rotational capacity may be neglected.

| MOTIVATION: INTEGRALIZATION OF EXISTING BRIDGES
Expansion joints in bridges were and are used to keep the effects of restrained forces like imposed deformations or temperature as low as possible. But there is a desire to avoid expansion joints in bridge structures because they pose a durability problem in this area. The durability of the bearings is particularly affected because of damaged expansion joints and defect seals. In addition to corrosion on the steel components and cracks in the mortar bed of the bearing area, unintentional displacements can cause constraint forces that influence the load-bearing capacity of the structure. The high costs and effort that arise for the maintenance activities of bearings and expansion joints are addressed by the integralization of bridges. A second important point is the strengthening of the existing structure and the utilization of a system change to reduce or redistribute internal forces. The motivation for this is the increase in heavy traffic and associated change in the load assumptions for the structural analysis. The possibilities for integrating existing bridges are the formation of frame corners and the reconstruction of singlespan girders to a multi-span girder. 1

| Structural design
The connection or integralization of bridge structures can be conducted, for example, according to the "Integral Abutment Bridge" concept. 20 A distinction is made between fully integrated and semi-integrated constructions. Figure 2 shows the reconstruction concept for a rigid connection of the super-and substructure as a design variant for integralization.
F I G U R E 2 Schematic construction measures for integrating single span beams 20

| Case study: Integralization of a bridge
To determine realistic initial values for nonlinear analysis, the internal forces analysis and reinforced concrete design of a bridge were conducted according to the old Austrian Standard ÖN B4002 21 and ÖN B4200 22 (valid 50 years ago) for a single-span bridge with a length of 10 m. This was then integralized into a continuous beam of 2 Â 10 m and analyzed and designed according to the current Austrian Standards ÖN EN/B1991-2 23,24 and ÖN EN/B1992-2, 19,25 as shown in Figures 3 and 4. The slab thickness was 80 cm and the concrete strength class used was grade C30/37. The analysis and determination of the internal forces were conducted linearly elastically by Haller. 26 To consider the load-bearing behavior of the slab in the transverse direction, the analysis was conducted with shell elements.

| Required reinforcement quantities
The results of the analysis of the required reinforcement quantities for the two different systems according to the actions and design principles to old and new standards are presented in Table 1. According to the B4002 21 (old standard), a uniformly distributed load (UDL) of 5 kN/m 2 is applied to all lanes. In addition, two adjacent lanes are each subjected to a 250 kN truck. The width of a lane is 2.5 m. According to the EN/B 1991-2 23,24 (new standard), a UDL of 9 kN/m 2 is applied to the first lane and a UDL of 2.5 kN/m 2 to the remaining area. In addition, a double axle load with an axle load of Q ik = 300 kN was applied to the first lane and a double axle load with an axle load of Q ik = 200 kN for lane 2. The width of a lane is 3 m. The two standards are subject to a different safety concept. The design of the multispan girder shows that despite integralization, the available amount of reinforcement in the span is insufficient. Consequently, the design according to the current standard with the safety concept is more conservative than that according to the standard that was valid when the structure was built.
The linear elastic analysis results in a considerably lower moment at the support than in the span. The fact that the dead weight of the structure still acts on the system of the single span beam explains the low support moment. A redistribution of internal forces from the span to the support area seems appropriate because the design according to EC 19,25 requires 3.5% more reinforcement in the span of the existing structure. It is pointed out that in this analysis, only deadand traffic loads were considered in the preliminary design. 26

| PARAMETER STUDY FOR NONLINEAR NUMERICAL EVALUATIONS
In the context of this study, the redistribution of internal forces from the span to the support area in a reinforced concrete structure is evaluated after the integralization of the structure. The consideration of the influence of the stiffness distribution over the span length due to damage by previous and current traffic loads according to the standard should give a realistic estimation of this redistribution. In addition, it should also be determined whether plastic hinge formation in the span of concrete structures is possible or not and the subsequent load-bearing behavior of the structure is to be analyzed. Plastic hinge formation is characterized by the yielding of the reinforcement. The parameter study is intended to estimate how much reinforcement in the support area is required so that a sufficient redistribution of internal forces from the span to the support area can occur because of the stiffness distribution. The basis of the calculation is a single span beam with predefined dimensions. Based on the calculated reinforcement, as shown in Section 3.2, a nonlinear FE analysis is performed using a damage model for concrete.

| Test body properties and evaluated parameters
The concrete used in the NL FEA had a strength of class C30/37. 19 Longitudinal reinforcement was placed in the field and in the support area in the tension and compression zones respectively. The tensile reinforcement in the field area is continuous over both supports. The length of the tensile reinforcement in the support area was 2.5 m for the test specimens with 10 m length and 3.0 for the test specimens with 20 m length. The overlap length of the tensile reinforcement in the support area with the compression reinforcement in the field area was 1 m, taking into account that full bond was used in the analysis. The degrees of reinforcement can be found in Table 2.
The edge distance of the longitudinal reinforcement was 5 cm. Shear reinforcement was positioned as vertical bars in sufficient quantity. The parameter study was conducted on a beam with a width of 1.0 m using a two-dimensional model with plane strain elements. The reinforcement in the span was kept constant and the reinforcement at the support was varied during integralization. The reinforcement ratios ρ s = A s / (wÁd) and geometric parameters λ = l/d for the evaluated types are listed in Tables 2 and 3.

| NONLINEAR NUMERICAL EVALUATIONS
To effectively represent a realistic load-bearing behavior, a nonlinear analysis of the already presented bridge structure is conducted.
The following is assumed and considered in the modeling: • Nonlinear material behavior of concrete and reinforcing steel • Perfect bond • Damage due to dead weight and traffic loads • Construction and loading history The results of the nonlinear analysis (M NL ) are compared with those from the linear elastic analysis (M EL ), whereby the redistribution can be determined.

| Material model
The nonlinear material behavior of concrete and reinforcing steel are considered in the modeling in ABAQUS. [27][28][29]

| Reinforcing steel material behavior
For the analysis, the reinforcing steel is assigned an elastic-plastic stress-strain relationship. The range of plastic strain begins when the yield strength f y = 550 N/ mm 2 is exceeded and fails when the tensile strength f u = 600 N/mm 2 with an ε u ¼ 0:05 is reached. The plastic strain ε pl is obtained by subtracting the elastic strain ε el from the total strain ε tot (Equation (3)).
For the plastic range of reinforcing steel, the stress-strain relationship results are as presented in Table 4.  25 It assumes that the two main failure mechanisms are the tensile cracking and compressive crushing of the concrete material. The evolution of the yield (or failure) surface is controlled by two hardening variables,ε pl t andε pl c , linked to failure mechanisms under tension and compression loading, respectively.ε pl t andε pl c are referred to as tensile and compressive equivalent plastic strains, respectively. The associated damage variables and strains due to the tensile and compressive loading of concrete are determined for concrete grade C30/37. The required input parameters for the analysis with the Concrete Damage-Plasticity (CDP) model are chosen according to Simulia 27 and are listed in Table 5.

Concrete under compression stress-compressive crushing
According to EC, 19,25 a stress-strain relationship described by Equation (4) is recommended for internal force determination with nonlinear methods in the compression range.
In the analysis with ABAQUS, this relationship is separated into a linear and nonlinear part. The linear range is described by the elastic modulus E cm and is limited by 40% of the mean compressive strength f cm , as shown in Figure 5. The damage variable for concrete under compressive stress can take values from zero, representing the undamaged material, to one, which represents the total loss of strength. Abaqus automatically converts the inelastic strain values into plastic strain values using the relationshipε pl Concrete under tensile stress-cracking The post failure behavior for direct straining is modeled with tension stiffening, which allows the definition of the strain-softening behavior for cracked concrete. 27 This behavior also allows the effects of the reinforcement interaction with concrete to be simulated in a simple manner. Tension stiffening is required in the concrete damaged plasticity model. In reinforced concrete, the specification of the post failure behavior generally means giving the post failure stress as a function of the cracking strain, ε ck t . The cracking strain is defined as the total strain minus the elastic strain corresponding to the undamaged material; that is,ε ck t ¼ ε t À ε el 0t as illustrated in Figure 6. Tension stiffening data are given in terms of the cracking strain,ε ck t . When unloading data are available, the data are provided to Abaqus in terms of the tensile damage curves, d t Àε ck t , and Abaqus automatically converts the cracking strain values to plastic strain values using the relationship in Equation (8).
The Tension Stiffening Effect 29,33-35 (TSE) can be modeled indirectly by modifying the stress-strain curve of the concrete if the stress increase of the embedded rebar caused by the TSE for a given strain compared to a steel bar σ s,TSE , see Figure 7 is replaced by an equivalent concrete stress σ c,TSE . 37,38 This results in where ρ eff = A s /A c,eff denotes the effective reinforcement ratio. It represents the ratio of the cross-sectional area of the reinforcing bar to the tensile stressed part of the concrete cross-sectional area. The latter can be calculated according to Model Code. 36 The strains of the diagram are set as follows: Strain when the tensile strength of the concrete is reached: Strain at completed crack pattern: Strain at the beginning reduction of the tension stiffening effect: Yield strain of the reinforcing steel: Δε sr is the difference between the maximum steel strain at the location of the crack and the concrete or steel strain in the uncracked concrete. According to Model Code, 36 β t is chosen to be 0.4.

| Geometric model
The analysis was based on a two-dimensional model, as shown in Figure 8. The concrete body was modeled with plain strain shell elements-longitudinal and transverse reinforcement with truss elements. The element size was 50 mm for each of the shell elements (CPS4: 4-node bilinear plane stress quadrilateral) and truss elements (T2D2: 2-node linear 2D truss). Load distributing steel plates with elastic material behavior were modeled at the load F I G U R E 6 Response of concrete to uniaxial loading in tension F I G U R E 7 Idealized stress-strain relationship according to Model Code 36 considering the tension stiffening effect for (a) the reinforcing steel and (b) the concrete 37 introduction point and at the supports. Owing to the symmetry of the setup, only half of the body was modeled. The bond between the reinforcing steel and concrete body was rigid. The geometric dimensions and degrees of reinforcement are in Section 4.1. The load was applied at a distance of 0.44 l from the edge support. The element thickness in the transverse direction was 1.0 m.

| Analysis steps and load application
The distribution of the stiffness plays an essential role in the redistribution of internal forces in the nonlinear calculation. In the nonlinear material model of the concrete, damage was considered, which meant that preloading could be considered. The loads on the structures were divided into individual load steps in the analysis. Owing to the preload, the realistic effects from traffic load were well modeled. The loading history considering the reconstruction of the single span girder to a double span girder is presented in Table 6.

| Moment redistribution
Linear and nonlinear calculations were performed, and the internal forces were analyzed. The bending moments were determined based on the load and support reactions. The deviations of the bending moments calculated according to the nonlinear theory (M NL ) from the calculated values according to elasticity theory (M EL ) are referred to as moment redistribution. 15 The moment redistribution is described by the redistribution factor δ = M NL /M EL , as presented in Section 2.1. The results of the calculations are listed in Tables 7 and 8. The redistribution factor is observed at the time the reinforcement starts to yield in the span and when the ultimate load is reached. It can be observed that from the beginning of yielding, no significant redistribution takes place. Apart from the redistribution of the internal forces from the span to the support area, an increase in the reinforcement in the support area causes an increase in the bearing capacity. The Type 1 configuration corresponds to a concrete design based on the internal forces determined according to linear elasticity. Type 2 and 3 were variants where a significantly higher reinforcement at the support is considered than that required according to the elasticity theory. Because of the over-reinforcement of the support cross-section, a maximum load increase of 30% was possible. Figures 9-12 show the development of the redistribution factor δ for the span at the point of load application as a function of the load progress. It can be observed that the redistribution factor does not fall below the value δ Span = 0.87. Furthermore, there is no significant  Figure 13 shows the tensile damage variable d t of the concrete, which represents the cracked areas of the concrete. Figure 14 shows the plastic strain of the reinforcing steel; it can be observed that the reinforcement in the span and in the support area started yielding.   16 show the load-dependent span moment development for a reinforced concrete beam and a rolled steel profile. It can be observed that the development of internal forces already deviates from the linear theory from the first crack onwards and is a function of the stiffness distribution in the statically indeterminate system. The redistribution of internal forces occurs from the initial crack and not after the formation of the plastic hinge. There is a general trend that the plastic hinge formation has only a residual influence on the redistribution of internal forces owing to the previous redistribution.
For a rolled steel beam modeled with linear elastic as shown in Figure 16, there is internal force development according to the linear elasticity theory until the onset of yielding at one point. The redistribution of internal forces occurs because of the formation of a plastic hinge. It can be inferred that in a reinforced concrete beam, the redistribution of internal forces occurs predominantly before the formation of a plastic hinge owing to the formation of cracks in the concrete.

| SUMMARY
A parameter study was conducted on the redistribution of internal forces from the span to the support area using nonlinear finite element analysis. The motivation for this F I G U R E 1 3 FEA results-Tensile damage variable d t F I G U R E 1 4 FEA results-Plastic strain of the reinforcing steel F I G U R E 1 5 Load-related moment development for a reinforced concrete beam F I G U R E 1 6 Load-related moment development for a rolled steel profile is the integralization of existing bridges. The analyzed parameters were the length (l = 10 m and 20 m), slenderness (λ = 13.3 and 20), and degree of reinforcement (ρ Support = 0.21%, 0.34%, and 0.55%). The analysis used a concrete damage model and considered the construction and loading history. The calculations showed that the redistribution of internal forces from the span to the support area is strongly dependent on the reinforcement distribution (span and support) and the pre-damage of the concrete. In principle, a clear redistribution of the internal forces from the span occurred from a load level that was greater than the preload. The higher the degree of reinforcement in the support area, the greater the redistribution of the internal forces. The redistribution factor δ = M NL /M EL for the span was maximum δ = 0.89 at yielding and δ = 0.87 at the ultimate load. After the reinforcement started to yield, a residual amount of the redistribution of the internal forces could still be detected. The redistribution of internal forces from the span to the support area did not occur by plastic hinge formation but was due to the stiffness-dependent internal force distribution.