Force In 2D Elements

The Timoshenko convention is used for forces in 2D elements. This means that a moment $M_{x}$ is based on the stress in the $x$ direction. With the Timoshenko convention if a slab is in compression on the top face in both the $x$ and $y$ 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

$$\begin{aligned}N_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xx}dz}& N_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yy}dz}& N_{xy} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xy}dz}\\ Q_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xz}dz} & Q_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yz}dz} \\ M_{x} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xx}zdz} & M_{y} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{yy}zdz} & M_{xy} &= \int_{- \frac{t}{2}}^{\frac{t}{2}}{\sigma_{xy}zdz}\end{aligned}$$

Following from this a plate that has a positive in-plane stress in $x$/$y$ will have a positive force resultant and a positive bending stress in $x$/$y$ (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:

$$\begin{aligned}N_{x} &= \sigma_{p,xx}t&N_{y} &= \sigma_{p,yy}t&N_{xy} &= \sigma_{p,xy}t\\ Q_{x} &= \sigma_{xz}t&Q_{y} &= \sigma_{yz}t\\ M_{x} &= \sigma_{b,xx}\frac{t^{2}}{6}&M_{y} &= \sigma_{b,yy}\frac{t^{2}}{6}&M_{xy} &= \sigma_{b,xy}\frac{t^{2}}{6}\end{aligned}$$

where the superscripts $p$ and $b$ 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.

Force/moment in layered shell

Once the displacements are calculated the strains $\epsilon_i$ can be calculated for each layer

$$\epsilon_i = B_i u_i$$

Which relates back to the element displacements

$$\epsilon_i = B_i T_i u$$

Stresses can then be calculated from

\begin{align} \sigma _i &= C_i \epsilon_i \\ &= C_i B_i T_i u \\ \end{align}

The stresses can then be integrated to get the force and moment

\begin{align} N_i &= \sigma _{i,p} {t_i} \\ M_i &= \frac{\sigma_{i,b} t_i^2}{6} \\ Q_i &= \sigma _{i,q} t_i \\ \end{align}

These can then be integrated across the element

\begin{align} N &= \sum_i{N_i} \\ M &= \sum_i{(M_i + N_i z_i)} \\ Q &= \sum_i{Q_i} \\ \end{align}