# 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

In, for example, [2] it is often recommended to define a girder grill. A grid can also be used to model the biaxial structural behavior of the concrete slab of a composite beam. However, the modeling effort for this is greater and at local discrete points, the grid is inaccurate. In the following, the modeling of a girder grate is compared with that of an orthotropic plate.

Figure 02 - "Edit Surface Stiffness - Orthotropic" in RFEM

According to the system, the definition of the grating is described by means of a simple system, then the orthotropic plate. Finally, the results and deviations are explained.

#### System

- Cross-section steel: HE-A 200
- 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-MASSIVE and imported into RFEM with the defined eccentricity of the cross-section to the concrete slab. The effective width of the cross-section is assumed to be 60 cm. The centroid of the cross-section is shifted upwards by 0.8 cm compared to the joint between concrete and steel. Therefore, the joint is calculated for the support. The supports are shifted downwards by 5 cm.

Figure 05 - Support Positioning

The support scheme itself was selected in such a way that no restraints due to restrained deformation occur.

The load is applied identically for both systems.

- LC1 = 5 kN/m²
- LC2 = 10 kN (x-direction = center of field, y-direction = outer edge)

#### Girder grating system

Prerequisites for Support Grating (from [1] ):

- constant height
- straight beam bridge
- simply symmetrical cross-section
- Both main beams are supported in each support axis, with the support axis running perpendicular to the longitudinal axis of the bridge.
- almost rigid transverse stiffeners in the support axes
- unobstructed warping in the support axes
- The structural system used must be able to calculate shear-resistant member elements.

Calculated value of bending stiffness (from [2]):

$$(\mathrm{EI})^\mathrm I\;=\;{\mathrm E}_\mathrm c\mathrm I^\mathrm{Platte}\;=\;{\mathrm E}_\mathrm c\;\cdot\;\frac{\mathrm b\;\cdot\;\mathrm d³}{12\;\cdot\;(1\;-\;\mathrm\mu²)}\;=\;3.300\;\cdot\;\frac{120\;\mathrm{cm}\;\cdot\;(10\;\mathrm{cm})³}{12\;\cdot\;0,8}\;=\;20,6\;\cdot\;\mathrm E^{06}\;\mathrm{kNcm}²$$

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
- A
_{y}= 833 cm²

The input is made in the program by means of effective cross-section values. The shear stiffness of the members is taken into account.

#### Orthotropic plate system

In the orthotropic plate system, the main beams are modeled identically to the grid. These beams are subsequently integrated into the concrete slab. The stiffness in the longitudinal direction is completely taken over by the main beams and in the transverse direction by the concrete slab. The FE mesh size is defined identically to the distance of the cross beams as 50 cm.

The stiffness matrix of the orthotropic plate is symmetrical and occupied only on the main diagonal. The stiffnesses for bending in the longitudinal direction of the plate and torsion were defined identically to the transverse members of the grating 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 input is made by user-defined stiffnesses.

Figure 07 - Stiffness Matrix of Slab Plane

#### Summary

Figure 08 - Comparison of Results

Figure 09 - Deformations in Load Case 2

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