Development of Design Guidelines for Cantilevered Portions of Bridge Decks

Report of Finite Element Study to be used as Calibration Model for Simplified Design Procedure

By

Nathan Loewen, doctoral candidate supervised by Dr. Stiemer

April 2004

Table of Content

1 Model Description.......................................................................................................32 Bilinear Material Model..............................................................................................6

2.1 Results Under Applied Shear Load.....................................................................62.1.1 Load-Deflection Response...........................................................................62.1.2 Element Stresses..........................................................................................8

2.1.2.1 Linear Response.......................................................................................82.1.2.2 Nonlinear Response...............................................................................11

2.2 Results Under Applied Bending Moment..........................................................132.2.1 Load-Deflection Response.........................................................................132.2.2 Element Stresses........................................................................................15

2.2.2.1 Linear Response.....................................................................................152.2.2.2 Nonlinear Response...............................................................................16

3 Concrete Material Model...........................................................................................173.1 Concrete Model Results.....................................................................................18

4 Conclusions................................................................................................................21APPENDED FILES...........................................................................................................22

IntroductionThe BC Ministry of Transportation (MoT) is currently seeking to develop design guidelines for cantilevered portions of bridgedeck subjected to impact loading of the guardrail/barrier. The cantilevered portion of the bridgedeck considered extends from the outermost bridge girder to the edge of the bridge. MoT has requested an analysis to be carried out using rational methods such as yield-line method, and verify the results with nonlinear finite element analysis. These results will then be used to develop a simplified design procedure based on hand calculations.

The purpose of this subproject is to investigate different methods of modelling the nonlinear material properties of the reinforced concrete slab using ANSYS software. The results of intersest are how the modelling of the nonlinearities affects the distribution of forces within the bridge deck, and the failure modes. Two non-linear material models were investigated to model the reinforced concrete. The first uses a bilinear isotropic material to model the composite behaviour of the reinforced concrete slab. The second uses a brittle material to model the solid concrete elements, and reinforcement is modelled discretely by using bilinear spar elements.

1 Model DescriptionFigure 1 shows a plan view of region of the bridge deck under consideration. An impact load is applied to the guardrail, and this is transferred into the bridge slab, which in turn carries the forces back to the outermost bridge girder running parallel to the bridge. The figure also shows the region of the finite element mesh and the coordinate system used. The coordinate system is oriented such that the x-direction is perpendicular to the flow of traffic, and the y-direction is parallel to the flow of traffic. This coordinate system will be referred to in discussion of applied forces, resultant stresses and displacements. The region of finite element mesh takes advantage of the x-axis symmetry of the problem when the applied forces are perpendicular to the guardrail. The dimensions of the model used in this analysis were as follows: the cantilever length was set to 2000mm, the model width (width of meshed region) was 6000mm, and the slab depth was 300mm.

Figure 2 shows an isotropic view of the finite element mesh, applied loads, and boundary conditions. The finite element analysis was carried out using ANSYS 8.0 software. The elements used in the analysis have 3 DOF at each node. All DOF were fixed at the boundary with the bridge girder (i.e. a rigid girder was assumed), and at the edge of the model away from the applied load. The length of the model was chosen to be sufficiently long so that the boundary condition at the edge of the model would have minimal effect on the results. Nodes along the x-axis were restrained from translation in the y-direction in order to model the symmetry condition. Loads were applied to multiple nodes at the region of applied load in order to avoid high stress concentrations at single nodes. Bending moments were applied as force couples over multiple nodes.

The analysis input file is written in the ANSYS Parametric Design Language (APDL), which allows complete parameterization of the model by the user. Input parameters include:

Material properties Geometric properties (slab depth, cantilever length, width of model) Rebar properties (spacing and area in x- and y-directions) Applied loads (x- and z-direction forces, moment about y-axis) Mesh size

Due to the limited scope of the subproject, only the applied loads and mesh sizes were varied, and the remaining properties were assigned values for typical bridge slab construction.

Region of Applied Load(Fx, Fz, or My)

X

Y

FEM Coordinate System

Region of FE mesh

Bridge Girder

Cantilevered Portion of Bridge Deck

Line of Model Symmetry

CENTERLINE OF BRIDGE GIRDER

PLAN VIEW OF BRIDGE DECK SHOWING REGION OF FINITE ELEMENT MODEL

Figure 1: Overview of bridge deck model

Figure 2: Finite element mesh, showing applied load and boundary conditions

2 Bilinear Material ModelThe first nonlinear material model implemented was a bilinear isotropic model. This was used to model the behaviour of the reinforced slab as a whole, and not the stresses in the individual components of the concrete and rebar. A single element type (SOLID45, 3D structural solid with plasticity capabilities) was used to model the combined effect of the concrete and rebar. The user defines the yield stress, and the post yield tangent modulus, which in this case was set to zero (elastic-perfectly-plastic condition).

The input was parameterized so that two parameters as used to describe the composite behaviour of the reinforced concrete slab. The first parameter is the initial stiffness of the reinforced concrete slab, which is the concrete elastic modulus multiplied by the moment of inertia of the transformed-to-concrete composite slab section (Ec*Ic). The model uses an equivalent initial stiffness (Emodel*Imodel) where Imodel and Emodel are calculated by equations 1 and 2:

(1) Imodel=tslab3/12

(2) Emodel*Imodel=Ec*Ic ----> Emodel=Ec*Ic/Imodel

The second parameter is the plastic moment of the composite slab. The user defines a fully plastic moment (Mp), and the program calculates an effective yield stress Fymodel based on the plastic modulus of the modelled slab, Zmodel:

(3) Zmodel=tslab2/4

(4) Fymodel=Mp/Zmodel

Since the bilinear model only allows one yield stress to be input, the x- and y-direction plastic moments are equal. It is however possible to vary initial stiffess in x- and y-direction by using orthotropic elastic material properties, Ex and Ey. An additional limitation of the bilinear model is that it is not capable of modelling brittle failure due to crushing of the concrete, and the true stress distribution through the reinforced concrete slab is not known.

Results for two load cases are given below: an applied vertical load Fz, and an applied bending moment My.

2.1 Results Under Applied Shear Load

2.1.1 Load-Deflection ResponseFigure 3 plots the load-deflection response for the bilinear material model under vertical load for various mesh sizes. In all cases the mesh in the x-y plane remains constant, but the number of element layers through the slab depth changes. Results are plotted for 1, 3, and 6 element layers. It is seen that the results for 3 and 6 layers are very similar, indicating that a sufficiently converged solution is achieved with 6 layers. The results for a single layer of elements predict a higher yield load, and a higher plastic collapse load. This is attributed to the location of the Gauss points being further away from the extreme slab fiber compared to the 3 and 6 layer models. Contour plots of the nodal

displacements are given in figures 4 and 5 for load cases near first yield, and near plastic collapse.

Figure 3: Load-displacement plot for bilinear material model under vertical load

Figure 4: Displacement plot under vertical load of 48.0 kN (near first yield)

Figure 5: Displacement plot under vertical load of 108.0 kN (near plastic collapse)

2.1.2 Element StressesFigures 6 to 11 plot the element stresses, first in the linear range, and then in the nonlinear range. With the bilinear model, the stress output does not represent the true stresses in the concrete or rebar since effective properties are used to model the composite behaviour of the slab. The stresses plotted are indicative of the magnitude of the total forces in the composite slab for a given region, and are therefore used in determining the regions of first yield, and plastic collapse mechanisms.

2.1.2.1 Linear ResponseFigure 6 plots the von Mises stresses in the slab near the point of first yield. It is seen that the maximum stresses occur along the x-axis at the point where the slab connects to the bridge girder. Note that the contour plot is scaled so that the maximum stress plotted is the effective yield stress (Fymodel as described above), which is 1.1Mpa in this case. Since ANSYS extrapolates Gauss point stresses to nodal stresses, it may occur that the plotted nodal stress exceeds the yield stress (which of course is not possible in an elastic-perfectly-plastic material). The regions where nodal stresses are greater than the yield stress appear grey in the plots.

Figures 7 and 8 plot the x- and y-direction stresses, respectively. For the x-direction plot maximum stresses occur in a similar region to Figure 6, with the maximum tensile forces appearing near the top surface, and the maximum compression force at the bottom surface. Figure 8 illustrates a secondary bending effect, where the applied vertical load is

being redistributed laterally (in the y-direction) through bending to adjacent portions of the cantilevered slab. This results in compression in the top surface near the point of applied load, and tension in the bottom surface.

Figure 6: Von Mises stress under vertical load of 48.0 kN (near first yield)

Figure 7: x-direction stress under vertical load of 48.0 kN (near first yield)

Figure 8: y-direction stress under vertical load of 48.0 kN (near first yield)

2.1.2.2 Nonlinear ResponseRefer to the animation file “Animation of Equivalent Stress Plot under Vertical Force.avi” which shows the von Mises stresses plotted with increasing load to the point of plastic collapse. Displacements are scaled by a factor of 10 in the animation. Figures 9 through 11 plot the stress components near the point of plastic collapse. It is observed that as the cantilevered slab yields near the point of applied load, the additional load is redistributed laterally to adjacent portions of the slab. This redistribution continues to the point of collapse. It is difficult to determine the exact collapse mechanism. It appears that a point is reached where the load can no longer be redistributed laterally, and rotation occurs along a line extending diagonally outward from the point where the x-axis intersects the bridge girder.

Figure 9: Von Mises stress under vertical load of 108.0 kN (near plastic collapse)

Figure 10: x-direction stress under vertical load of 108.0 kN (near plastic collapse)

Figure 11: y-direction stress under vertical load of 108.0 kN (near plastic collapse)

2.2 Results Under Applied Bending Moment

2.2.1 Load-Deflection ResponseFigure 12 plots the load-displacement curve for the bilinear model subject to an applied moment about the y-axis. Figures 13 and 14 plot the nodal displacements near first yield and plastic collapse loads. It can be seen that in this case the plastic collapse mechanism is much more isolated than in the case of the applied vertical load, and that the applied moment is not able to redistribute itself to a greater portion of the slab.

Figure 12: Load-displacement plot for bilinear material model under subject to end moment

Figure 13: Displacement plot under moment of 20e+3 kN-mm (near first yield)

Figure 14: Displacement plot under moment of 34e+3 kN-mm (near plastic collapse)

2.2.2 Element StressesRefer to the animation file “Animation of Equivalent Stress Plot under Bending Moment.avi” which again shows the von Mises stresses plotted with increasing load to the point of plastic collapse. Figures 15 and 16 plot the von Mises stresses near first yield and plastic collapse. Again, both the first yield and plastic collapse mechanisms are very localized, resulting in a simple bending failure of the slab at the point of the applied moment.

In order to avoid this collapse mechanism, the applied moment could be distributed furthur into the slab by reinforcing the model near the point of applied load (and reinforcing the slab near the connection to the guardrail in reality). This could allow some redistribution of bending forces laterally (in the y-direction), similar to that seen for the vertical load case above, except in this instance the applied bending force about the y-axis would be transferred laterally through torsion in the slab about the x-axis.

2.2.2.1 Linear Response

Figure 15: Von Mises stress under moment of 20e+3 kN-mm (near first yield)

2.2.2.2 Nonlinear Response

Figure 16: Von Mises stress under moment of 34e+3 kN-mm (near plastic collapse)

3 Concrete Material ModelA secondary concrete model was also used to model the reinforced concrete slab using the SOLID65 element, which is compatible with the concrete material model. This is a perfectly brittle material offering cracking capabilities in tension, and crushing capabilities in compression. The nonlinear material properties that are defined are as follows:

fc: uniaxial crushing stressft: uniaxial cracking stressCclosed: shear transfer coefficient through a closed crack, 0.0-1.0Copen: shear transfer coefficient through an open crack, 0.0-1.0

In order to limit the number of parameters in the current bridge deck model, the cracking stress was set to 0.0 Mpa , and the crushing stress set to 50.0 Mpa. Typically the rebar in the bridge deck would be sized such that yielding occurs in the rebar well before crushing occurs in the concrete in order to ensure a ductile response. Shear transfer coefficients were set to 1.0 for both open and closed cracks.

The SOLID65 element allows either discrete or smeared reinforcement. Discrete reinforcement is achieved by adding spar elements where rebar is required. Smeared reinforcement can be defined in 3 directions, with the volume ratio, material properties, and rebar orientation defined for each direction. Discrete reinforcement was used in the current model since the rebar is more realistically isolated to the top and bottom of the slab, instead of being distributed throughout the slab. The rebar was modelled with 3D spars elements. These elements were defined with binlinear, elastic-perfectly plastic material properties, with a yield stress of 300 Mpa and a tangent modulus of 0.0 Mpa.

The nonlinearities within the SOLID65 element are handled as follows. The failure criteria for concrete subjected to a multiaxial stress state is defined as follows:

F is a function of the princical stresses s1, s2 and s3. S defines the failure surface, and is a function of the principal stresses and input parameters fc and ft. Concrete failure has the following 4 domains:

(comp-comp-comp) (tens-comp-comp) (tens-tens-comp) (tens-tens-tens)

For each of the 4 domains, there is a unique function for F and S. Cracking is permitted in three orthogonal directions at an integration point. When a crack has been detected at an integration point, the material matrix is modified to a cracked material matrix, Dck.

The cracked material matrix depends on whether cracking has occurred in 1, 2, or 3 planes.

3.1 Concrete Model ResultsThe concrete model was run using 6 equally spaced layers of elements. Two layers of rebar elements were modelled: one between the 1st and 2nd layer, and one between the 5th and 6th layer. Vertical rebar elements were also modelled. The rebar grid can be seen in Figure 19.

Some convergence problems were encountered running the concrete model, which limited the scope of this portion of the analysis. Unfortunately the model parameters were not scaled to match the bilinear model, and therefore direct comparison of results is not possible. However, failure modes and trends are compared.

Figure 17 plots the load displacement relationship under an applied vertical load. Although some softening occurs, one can see that the concrete model in this case does not achieve nearly the same ductility as the bilinear model, and the concrete model predicts a brittle failure mode. This is assuming that the model no longer converged due to model instability (collapse) as opposed to some other convergence problem. Figure 18 plots the von Mises stress in the concrete at the last converged step. It is seen that the compressive stresses near the point of applied load are at the crushing stress of 50 Mpa, and this appears to be the failure mode. Note that high compressive stresses were also noted in this region in the bilinear model under an applied vertical load. Figure 19 plots the stresses in the rebar elements (fy=300 Mpa). Rebar elements plotted in red show elements that have yielded in tension, and blue elements have yielded in compression.

The rebar in this model was proportioned so that rebar yielding would occur long before concrete crushing, and therefore a more ductile response was expected. Additional analysis was run with the concrete crushing capability removed (i.e. concrete is linear in compression), and the same linear behaviour shown in figure 17 continued indefinitely (no failure mode was predicted up to a very high load). This is somewhat puzzling to the author since eventually all of the compressive force will be in the extreme Gauss points, and this compressive force will be equal to the sum of the tensile yield forces in the rebar, and plastic rotation will occur. Future analysis would involve investigating how a ductile response can be achieved using the concrete material model with the bilieanear rebar model.

Figure 17: : Load-displacement plot for concrete material model under vertical

load

Figure 18: Von Mises stress at last converged point (fc=50Mpa)

Last converged point

Figure 19: Rebar element stress at last converged point (fy=300Mpa)

4 ConclusionsAs expected, the bilinear model exhibits a plastic failure mode under both an applied vertical force and an applied bending moment. In the case of the applied vertical force, the forces in the slab were redistributed over a large region of the slab prior to plastic collapse. In the case of the applied moment, the failure mode was highly localized to the region of the applied load. This load could possibly be further redistributed by locally reinforcing the region of applied bending moment.

The concrete model exhibited a brittle failure mode, despite the fact that the rebar was proportioned to create a ductile composite slab. The stress distribution in the concrete at the point of failure roughly matched the stress distribution in the bilinear model at the point of first yield. When the crushing capability of the concrete element was removed, the model still did not exhibit plastic deformations. Further modeling needs to be done to achieve a ductile response using the concrete material model with discrete reinforcement elements.

APPENDED FILES

1.) CIVL539_TP_Input_BilinMat.dat: ANSYS input file for bilinear material model.

2.) CIVL539_TP_Input_ConcMat.dat: ANSYS input file for concrete material model.

3.) CIVL539_TP_PostPro.dat: ANSYS postprocessing file.

4.) Animation of Equivalent Stress Plot under Vertical Force.avi: animation of bilinear model results under applied vertical force.

5.) Animation of Equivalent Stress Plot under Bending Moment.avi: animation of bilinear model results under applied bending moment.