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Force In 2D Elements

The Timoshenko convention is used for forces in 2D elements. This means that a moment MxM_{x} is based on the stress in the xx direction. With the Timoshenko convention if a slab is in compression on the top face in both the xx and yy directions the moments are both negative. Consequently starting from the assumption that tensile stress is positive, we have the following relationships for the forces and moments

Nx=t2t2σxxdzNy=t2t2σyydzNxy=t2t2σxydzQx=t2t2σxzdzQy=t2t2σyzdzMx=t2t2σxxzdzMy=t2t2σyyzdzMxy=t2t2σxyzdz\begin{aligned}N_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xx}dz}& N_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yy}dz}& N_{xy} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xy}dz}\\ Q_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xz}dz} & Q_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yz}dz} \\ M_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xx}zdz} & M_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yy}zdz} & M_{xy} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xy}zdz}\end{aligned}

Following from this a plate that has a positive in-plane stress in xx/yy will have a positive force resultant and a positive bending stress in xx/yy (i.e. positive stress at the top surface relative to the bottom surface) will have a positive moment.

When the structure is linear these simplify to:

Nx=σp,xxtNy=σp,yytNxy=σp,xytQx=σxztQy=σyztMx=σb,xxt26My=σb,yyt26Mxy=σb,xyt26\begin{aligned}N_{x} &= \sigma_{p,xx}t&N_{y} &= \sigma_{p,yy}t&N_{xy} &= \sigma_{p,xy}t\\ Q_{x} &= \sigma_{xz}t&Q_{y} &= \sigma_{yz}t\\ M_{x} &= \sigma_{b,xx}\frac{t^{2}}{6}&M_{y} &= \sigma_{b,yy}\frac{t^{2}}{6}&M_{xy} &= \sigma_{b,xy}\frac{t^{2}}{6}\end{aligned}

where the superscripts pp and bb refer to in-plane and bending stress terms.

When in-plane and bending thickness modifiers are user the in-plane forces are based on the in-plane thickness and the moments and shear forces are based on the bending thickness. Stresses are always based on the actual thickness of the element.