Calculation of Shear Area in SHAPE-THIN

Technical Article

Design of cross-sections usually requires many different cross-section properties. In RFEM and RSTAB, all required properties of standardized cross-sections are available in the cross-section library and can be used directly for the calculation. If the cross-sections are not standardized, SHAPE-THIN allows you to use these cross-sections, too. You can simply enter the geometry to determine all required cross-section properties. The following example shows the calculation of a shear area on a practical example.

Theoretical Background of Shear Area Calculation

The shear area is a calculated reduction of a cross-sectional area. By using this value, you can consider the shear deformation when determining the internal forces. In contrast to the effective shear area of EN 1993-1-1, the shear areas calculated here will only be used to determine the internal forces. Therefore, the shear area of EN 1993-1-1 applies for the stress calculation. The reduction of the cross-sectional area results from different distribution of the material law and the cross-section equilibrium, which leads to a contradiction. This contradiction is due to the hypothesis that the cross-sections remain the same, although the cross-section would actually be subjected to warping when the shear force effect occurs. Therefore, the shear area is introduced into the strength of materials. The derivation of this shear area is described below.

Equalization of Strain Energy II* for Member Element dx

$$\begin{array}{l}\int_\mathrm A\frac{\mathrm\tau^2(\mathrm z)}{2\;\cdot\;\mathrm G}\mathrm{dA}\;=\;\int_{{\mathrm A}_\mathrm s}\frac{\mathrm\tau_\mathrm m^2}{2\;\cdot\;\mathrm G}{\mathrm{dA}}_\mathrm s\;=\;\frac{\mathrm Q^2}{2\;\cdot\;\mathrm G\;\cdot\;{\mathrm A}_\mathrm s}\\\frac1{2\;\cdot\;\mathrm G}\;\int_\mathrm A\left[\frac{\mathrm Q\;\cdot\;{\mathrm S}_\mathrm y(\mathrm z)}{{\mathrm I}_\mathrm y\;\cdot\;\mathrm b(\mathrm z)}\right]^2\mathrm{dA}\;=\;\frac1{2\;\cdot\;\mathrm G}\;\cdot\;\frac{\mathrm Q^2}{{\mathrm A}_\mathrm s}\\\\\mathrm{dA}\;=\;\mathrm b(\mathrm z)\mathrm{dz}\\{\mathrm A}_\mathrm s\;=\;{\mathrm A}_\mathrm{sz}\;=\;\frac{\mathrm I_\mathrm y^2}{\int_{{\mathrm z}_\mathrm o}^{{\mathrm z}_\mathrm u}{\displaystyle\frac{\mathrm S_\mathrm y^2(\mathrm z)}{\mathrm b(\mathrm z)}}{\displaystyle\mathrm d}{\displaystyle\mathrm z}}\\\mathrm Q\;=\;{\mathrm Q}_\mathrm z\;\rightarrow\;{\mathrm A}_\mathrm{sz}\\\mathrm Q\;=\;{\mathrm Q}_\mathrm y\;\rightarrow\;{\mathrm A}_\mathrm{sy}\end{array}$$

When calculating a rectangle, the result is the shear correction factor κ. This factor indicates to what extent the cross-sectional area must be reduced.

Example for Rectangle:
$$\begin{array}{l}{\mathrm I}_\mathrm y\;=\;\frac{\mathrm b\;\cdot\;\mathrm h^3}{12}\\\mathrm b(\mathrm z)\;=\;\mathrm b\\{\mathrm S}_\mathrm y(\mathrm z)\;=\;\mathrm b\;\int_{-\frac{\mathrm h}{\mathrm s}}^\mathrm z\overline{\mathrm z}\mathrm d\overline{\mathrm z}\;=\;-\frac{\mathrm b}2\;\cdot\;\left(\frac{\mathrm h^2}4\;-\;\mathrm z^2\right)\\-\frac{\mathrm h}2\;\leq\;\mathrm z\;\leq\;\frac{\mathrm h}2\\\int_{{\mathrm z}_\mathrm o}^{{\mathrm z}_\mathrm u}\mathrm S_\mathrm y^2(\mathrm z)\mathrm{dz}\;=\;\int_{-\frac{\mathrm h}{\mathrm s}}^\frac{\mathrm h}2\frac{\mathrm b^2}4\;\cdot\;\left(\frac{\mathrm h^4}{16}\;-\;\frac12\;\cdot\;\mathrm h^2\;\cdot\;\mathrm z^2\;+\;\mathrm z^4\right)\mathrm{dz}\;=\;\frac{\mathrm b^2\;\cdot\;\mathrm h^5}{120}\\{\mathrm A}_\mathrm s\;=\;\frac{120\;\cdot\;\mathrm b^2\;\cdot\;\mathrm h^6\;\cdot\;\mathrm b}{144\;\cdot\;\mathrm b^2\;\cdot\;\mathrm h^5}\;=\;\frac56\;\cdot\;\mathrm b\;\cdot\;\mathrm h\;=\;\frac56\;\cdot\;\mathrm A\\\mathrm\kappa\;=\;\frac56\end{array}$$

For simple cross-section types, it is possible to directly conclude the shear area. Some of the shear correction factors are:
Rectangle: 0.833
I-beam: ~ Aweb

The comparison of the numerical values shows that you always have to pay attention to the cross-section type when considering the shear deformation. The shear correction factors vary within a wide range, depending on whether there are solid cross-sections, thin-walled open cross-sections, or thin-walled closed cross-sections.

Example of a T-Section

The calculation of shear areas for simple cross-sections is thus very easy. For example, if there is only a T-section, SHAPE-THIN determines the shear area for this cross-section automatically.

Figure 01 - Input in SHAPE-THIN

Analytical solution for shear area calculation:
$$\begin{array}{l}{\mathrm I}_\mathrm y\;=\;13,304\;\mathrm{cm}^2\\{\mathrm z}_\mathrm m\;=\;8.786\;\mathrm{cm}\\\mathrm b(\mathrm z)\;=\;1\;\mathrm{cm}\\\mathrm h\;=\;40\;\mathrm{cm}\\\mathrm d\;=\;45\;\mathrm{cm}\\{\mathrm S}_{\mathrm y1}\;=\;\mathrm b(\mathrm z)\;\cdot\;(\;\mathrm h\;-\;{\mathrm z}_\mathrm m\;-\;\mathrm z)\;\cdot\;\left(\frac{\mathrm h\;-\;{\mathrm z}_\mathrm m\;-\;\mathrm z}2\;+\;\mathrm z\right)\\{\mathrm S}_{\mathrm y2}\;=\;\mathrm b(\mathrm z)\;\cdot\;\mathrm d\;\cdot\;-({\mathrm z}_\mathrm m\;-\;\mathrm b(\mathrm z))\\{\mathrm A}_\mathrm{sz}\;=\;\frac{\mathrm I_\mathrm y^2}{\int_{-30.124}^{8.786}{\displaystyle\frac{{\mathrm S}_{\mathrm y1}(\mathrm z)^2}{\mathrm b(\mathrm z)}}\mathrm{dz}\;+\;\int_{9.286}^{8.786}{\displaystyle\frac{{\mathrm S}_{\mathrm y2}(\mathrm z)^2}{\mathrm b(\mathrm z)}}\mathrm{dz}}\;=\;30.17\;\mathrm{cm}^2\end{array}$$


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