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.

Property modifiers

There are three property modifiers which affect the stiffness of the element.

• The in-plane stiffness modifier
• The bending stiffness modifier
• The through stiffness shear modifier

The modified stiffness will lead to differences in the displacements, which may be reflected in the stress and force calculations which are based on these displacements. However there is no further implication of the property modifiers on the stress and force results.

Taking, as an example, a simple cantilever model with a bending stiffness factor of 0.5. This would result in a halving of the bending stiffness and thus a doubling of the displacements. The stresses and moments are based on the strains (derived from the displacements) which are doubled. However the modifier reduces the bending stiffness by the same amount, meaning that the stress and moment are the same as if no property modifier had been applied.

In the general case the distribution of displacement, and not just the magnitude is changed, so unlike the simple case of a cantilever the stresses and moments will be different from a model with no property modifier.

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

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

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

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

These can then be integrated across the element

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