# Comparison of Girder Grillage Calculation with Calculation Using Orthotropic Plates

### Technical Article

Composite beams in a three-dimensional 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

- Cross-Section Steel: HE-200 A
- Material Steel: S235
- Cross-Section Concrete: d = 100 mm
- Material Concrete: C30/37
- Load: 5 kN/m²

Figure 04 - Cross-Section Including Effective Widths

The composite cross-section is created in SHAPE-THIN and imported to RFEM with the defined eccentricity of the cross-section to the concrete flange. The effective width of the cross-section is set to 60 cm in this case. The centroid of the cross-section 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 (x-direction = mid-span, y=direction = outer edge)

#### Girder Grillage Structure

Requirements of the girder grillage (from [1]):

- Constant construction height
- Straight girder bridge
- Simple symmetric cross-section
- 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:

$$\begin{array}{l}({\mathrm{GI}}_\mathrm T)^\mathrm I\;=\;\mathrm k\;\cdot\;({\mathrm{GI}}_\mathrm T)\\{\mathrm G}_\mathrm c\;=\;\frac{{\mathrm E}_\mathrm c}{2\;\cdot\;(1\;+\;\mathrm\mu)}\;=\;\frac{3,300}{2\;\cdot\;(1\;+\;0,2)}\;=\;1,375\;\mathrm{kNcm}²\end{array}$$

Cross-Section 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 cross-section 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 user-defined 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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