Comparison of Girder Grillage Calculation with Calculation Using Orthotropic Plates
Technical Article
Composite beams in a threedimensional analysis are usually connected with orthotropic plates. The longitudinal direction of the plate stiffness is defined by a main beam and the transverse direction by an orthotropic plate. The stiffness of the plate in the longitudinal direction is set almost to zero. This article explains the determination of stiffnesses in the orthotropic plate.
Figure 01  Road Bridge over L55 near Schwarzheide, Germany
For example, [2] often recommends defining a girder grillage. The grillage very well represents the biaxial structural behavior of the concrete flange of a composite beam. However, the modeling effort is greater in this case, and the grillage is inaccurate on local discrete points. Below, the modeling of a girder grillage is compared with the modelling of an orthotropic plate.
Figure 02  'Edit Surface Stiffness  Orthotropic' in RFEM
First, the definition of the girder grilage is described using a simple structure. Then the orthotropic plate is defined. Finally, the results and the differences are explained.
System
 CrossSection Steel: HE200 A
 Material Steel: S235
 CrossSection Concrete: d = 100 mm
 Material Concrete: C30/37
 Load: 5 kN/m²
Figure 04  CrossSection Including Effective Widths
The composite crosssection is created in SHAPETHIN and imported to RFEM with the defined eccentricity of the crosssection to the concrete flange. The effective width of the crosssection is set to 60 cm in this case. The centroid of the crosssection is shifted slightly upwards by 0.8 cm to the joint between the concrete and steel. Therefore, the joint is taken into account for the supports, which are shifted downwards by 5 cm.
Figure 05  Support Positioning
The support schema itself was selected in such a way so no restraints occur due to the restrained deformation.
The load is the same for both models.
 LC1 = 5 kN/m²
 LC2 = 10 kN (xdirection = midspan, y=direction = outer edge)
Girder Grillage Structure
Requirements of the girder grillage (from [1]):
 Constant construction height
 Straight girder bridge
 Simple symmetric crosssection
 Both main beams are supported on each support axis, which is perpendicular to the longitudinal axis of the bridge.
 Approximately rigid cross bracing in the support axes
 Unrestrained warping in the support axes
 The structural engineering software for truss analysis must be able to calculate member elements.
Calculated value of bending stiffness (from [2]):
$$(\mathrm{EI})^\mathrm I\;=\;{\mathrm E}_\mathrm c\mathrm I^\mathrm{Plate}\;=\;{\mathrm E}_\mathrm c\;\cdot\;\frac{\mathrm b\;\cdot\;\mathrm d³}{12\;\cdot\;(1\;\;\mathrm\mu²)}\;=\;3,300\;\cdot\;\frac{120\;\mathrm{cm}\;\cdot\;(10\;\mathrm{cm})³}{12\;\cdot\;0,8}\;=\;20.6\;\cdot\;\mathrm E^{06}\;\mathrm{kNcm}²$$
Calculated value of torsional stiffness:
$$\begin{array}{l}({\mathrm{GI}}_\mathrm T)^\mathrm I\;=\;\mathrm k\;\cdot\;({\mathrm{GI}}_\mathrm T)\\{\mathrm G}_\mathrm c\;=\;\frac{{\mathrm E}_\mathrm c}{2\;\cdot\;(1\;+\;\mathrm\mu)}\;=\;\frac{3,300}{2\;\cdot\;(1\;+\;0,2)}\;=\;1,375\;\mathrm{kNcm}²\end{array}$$
CrossSection Properties:
 I_{T} = 0 cm^{4}
 I_{y} = 6,250 cm^{4}
 A = 1,000 cm²
 Ay = 833 cm²
The entry is made in the program using the effective crosssection properties. The shear stiffness of the members is taken into account.
Orthotropic Plate Structure
In the orthotropic plate structure, the main beams are modeled in the same way as in the girder grillage. These girders are then integrated in the concrete flange. The stiffness is transferred completely by the main beams in the longitudinal direction and by the concrete flange in the transverse direction. The FE mesh size is defined identically to the distance of the secondary beam with 50 cm.
The stiffness matrix of the orthotropic plate is symmetrical and only applied to the main diagonals. The stiffnesses for bending in the longitudinal direction of the plate and torsion are defined identically to the transverse bars of the girder grillage with almost zero.
Calculated value of bending stiffness:
$$\mathrm D22\;=\;\frac{{\mathrm E}_\mathrm c\;\cdot\;\mathrm d³}{12\;\cdot\;(1\;\;\mathrm\mu²)}\;=\;206,000\;\mathrm{kNcm}/\mathrm{cm}$$
Calculated value of torsional stiffness:
$$\mathrm D33\;=\;{\mathrm G}_\mathrm{xy}\;\cdot\;\frac{\sqrt{\mathrm d_\mathrm x^3\;\cdot\;\mathrm d_\mathrm y^3}}{12}\;=\;13.8\;\mathrm{kNcm}/\mathrm{cm}$$
In the program, the userdefined stiffnesses are entered.
Figure 07  Stiffness Matrix of Slab Plane
Summary
Figure 08  Comparison of Results
Figure 09  Deformations in Load Case 2
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Figure 01  Road Bridge over L55 near Schwarzheide, Germany

Figure 02  "Edit Surface Stiffness  Orthotropic" in RFEM

Figure 03  Structural System

Figure 04  CrossSection Including Effective Widths

Figure 05  Support Positioning

Figure 06  Load Case 2

Figure 07  Stiffness Matrix of Slab Plane

Figure 08  Comparison of Results

Figure 09  Deformations in Load Case 2