Calculation of Shear Area in SHAPE-THIN

Technical Article

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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 the cross-section area. This value allows you to consider the shear deformation when determining the internal forces. In contrast to the effective shear area of DIN EN 1993-1-1, the shear area calculated here is only used for the determination of internal forces. Therefore, the effective shear area of DIN EN 1993-1-1 is used for a stress calculation. The reduction of the cross-section area results from the different distribution of the material law and the equilibrium in the cross-section, which leads to a contradiction. The reason for this contradiction is the hypothesis of keeping the cross-sections flat, because the cross-section would actually warp if a shear force effect occurs. For this reason, the shear area is introduced in the strength theory. The derivation of this shear area is described below.

Equating the deformation energy II * for a member element dx

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

When calculating a rectangle, the shear correction factor κ is obtained. This factor indicates how much the cross-section area has to be reduced.

Example 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, the shear surface can be determined directly. The following are some shear correction factors:
Rectangle: 0.833
I-beam: ~ AWeb

The comparison of the numerical values shows that, if the shear deformation is taken into account, it is necessary to consider exactly which profile shape is available. The shear correction factors vary within a wide range, depending on whether they are solid cross-sections, thin-walled open or thin-walled closed cross-sections.

Example of a T-section

The shear areas for simple cross-sections can thus be very easily calculated. For example, if you have a T-section, SHAPE-THIN automatically determines the shear area for it.

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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