2D element force and moment results

When dealing with elements with no thickness (such as fabrics) or composite materials (such as reinforced concrete) it is more useful to work with stress resultants or forces, than with stresses. For concrete the stress values are based on the properties of an equivalent isotropic material. The checks used for stress results noted above can also be applied to force results. For fabric elements the force resultants are calculated directly but for elements with thickness they are calculated from the in-plane and bending stresses and the element thickness. Details of these are given in the GSA Theory.

The moment per unit length results are a tensor

$M_{x}$, $M_{y}$, $M_{y}$

and the force results are a tensor (in-plane forces per unit length)

$N_{x}$, $N_{y}$, $N_{y}$

and a vector (through-thickness shears per unit length) $Q_{x}$, $Q_{y}$

The Wood-Armer results are derived from the moment terms as follows

$$\begin{aligned} M^*_x &= M_x + \text{sgn}(M_x)\left|M_{xy}\right| \text{ and}\\ M^*_y &= M_y + \text{sgn}(M_y)\left|M_{xy}\right| \end{aligned}$$

Note that the moment terms follow the Timoshenko convention in which the 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.

Sign convention

• Positive in-plane forces result from positive in-plane stresses.
• Positive bending moments result from positive top surface stresses relative to the bottom surface stress. (This corresponds to the Timoshenko (1964) sign convention.)

This is illustrated as follows.

Note that moments are based on the bending stress (i.e. they are not defined as being about any axis).

Derived forces and moments

In-plane principal forces and principal moments can be calculated in a similar manner to principal stresses:

• Principal forces: No in-plane shear force
• Principal moments: No in-plane twisting moment