Beam stresses

Beam stresses are calculated from the forces, moments and the shape of the section. The assumption is that the stresses are linear through the thickness, so the stresses are incorrect for elements in which yield has taken place.

Where beam section properties have been modified from the base values the properties used in the stress calculation depend on the Stress calc. basis setting for the property. The Stress calc. basis determines whether the modified or unmodified properties are used and can also be set to result in the stresses not being calculated.

The beam stresses are as follows:

• A: Axial stress
• B: Bending stress
• C: Combined stress
• S: Shear stress

Only axial stresses are output for sections that have properties explicitly defined.

The axial stress is:

$$A=\frac{F_{x}}{\text { area }}$$

For symmetric sections the bending stresses are:

$$B_{y}=\frac{M_{y y}}{I_{y y}} D_{z}, \qquad B_{z}=-\frac{M_{z z}}{I_{z z}} D_{y}$$

For non-symmetric sections there is an interaction between the two directions. So a bending stress about one axis will depend on the moments about both axes. In this case the bending stresses are:

$$\begin{aligned} &B_{y}=\frac{1}{I_{y y} I_{z z}-I_{y z}{ }^{2}}\left(I_{z z} M_{y y}+I_{y z} M_{z z}\right) D_{z}, \\ &B_{z}=\frac{-1}{I_{y y} I_{z z}-I_{y z}{ }^{2}}\left(I_{y z} M_{y y}+I_{y y} M_{z z}\right) D_{y} \end{aligned}$$

The combined stresses are:

• C1 is the maximum extreme fibre longitudinal stress due to axial forces and transverse bending
• C2 is the minimum extreme fibre longitudinal stress due to axial forces and transverse bending

The shear stresses are:

$$S_{y}=\frac{F_{y}}{\text { area }_{y}}, \qquad S_{z}=\frac{F_{z}}{\text { area }_z}$$

The shear stresses are of most use for steel structures so the shear stresses are calculated based on the shear area ($area_y$ and $area_z$) in BS 5950-1:2000 cl.4.2.3, and they depend on the section shape and if the section is rolled or welded. The section dimensions are shown below:

Area is the area of the section. The shear areas are calculated as follows:

	Concrete		Steel: Welded		Steel: Rolled, formed, etc.
	z	y	z	y	z	y
I section	$0.8 × D × t$	$1.6 × B × T$	$d × t$	$1.8 × B × T$	$D × t$	$1.8 × B × T$
Channel	$0.8 × D × t$	$1.6 × B × T$	$d × t$	$1.8 × B × T$	$D × t$	$1.8 × B × T$
Rectangular hollow	$1.6 × D × t$	$1.6 × B × T$	$2 × d × t$	$1.8 × B × T$	$\frac{\text{area}\times D}{(D+B)}$	$\frac{\text{area}\times B}{(D+B)}$
Rectangular solid	$0.8 × \text{area}$	$0.8 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$
Angle	$0.8 × D × t$	$0.8 × B × T$	$0.9 × d × t$	$0.9 × B × T$	$0.9 × D × t$	$0.9 × B × T$
Tee	$0.8 × D × t$	$0.8 × B × T$	$d × t$	$0.9 × B × T$	$D × t$	$0.9 × B × T$
Circular hollow	-	-	$0.6 × \text{area}$	$0.6 × \text{area}$	$0.6 × \text{area}$	$0.6 × \text{area}$
Circular solid	$0.7 × \text{area}$	$0.7 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$	$0.9 × \text{area}$

Welded sections are assumed to have weld positions as shown above.

Sections not specified as steel or concrete

For sections not specified as steel or concrete the areas are

$$\text{area}_y = \text{area} \times k_y, \qquad \text{area}_z= \text{area} \times k_z$$

In the case of simple beams where the shear area factor is set to zero the actual area is used giving an average shear stress value.

Stress measures in beams

The derived stress values output from GSA for beam elements are:

• Maximum torsional stress
• Maximum shear stress (y and z)
• von Mises stress

For the following section shapes:

• Rectangular
• RHS
• Circular
• CHS
• I
• Tee
• Channel
• Angle
• Oval

A new property – $C_T$ (or $W_T$) the torsion modulus – is loaded or calculated for these sections.

Torsional stress

Torsional stress is calculated ignoring warping of the section by using the torsion modulus $C_T$ (or $W_T$). We then get the maximum torsional stress from

$$S_T = \frac{M_{XX}}{C_T}$$

The values for CT are calculated from Bautabellen für Ingenieure, Klaus-Jürgen Schneider, (pp 4.30 & 4.31) and Roark's Formulas for Stress and Strain (Table 10.7)

Shear stress

The calculation of the shear stresses $S_{Ey}$ and $S_{Ez}$ are calculated as described in the Technical Note: Calculation of shear areas with the exception of ovals which are calculated from the equation:

$$K = \frac{2}{3+(\frac a b)^2}$$

Where $K$ is the shear area factor, $n = 1.75$ for $a<b$ and $1.5$ for $a>b$, with load being applied in the $b$ direction. The maximum shear stress = shear force / [K(stress) × total area]. The expression gives a that tends to 2⁄3 (as for rectangles) for tall, narrow ellipses where $b\gg a$.

von Mises stress

A precise calculation of the von Mises stress is difficult so a number of simplifications have been made. The von Mises stress depends on the axial forces, the bending moments, the through thickness shear forces and the torsional moment.

The axial force and bending moments combine to give σxx. The shear stresses are a combination of the through thickness shear + torsion. To combine the shear stresses the through thickness shear stresses are rotated into components parallel and perpendicular to the surface of the section and the torsional shear stress is added to the component parallel to the surface.

$$\left\{\begin{array}{l} \tilde{\tau}_{x y} \\ \tilde{\tau}_{x z} \end{array}\right\}=\mathbf{T}\left\{\begin{array}{l} \tau_{x y} \\ \tau_{x z} \end{array}\right\}+\left\{\begin{array}{c} \tau_{T} \\ 0 \end{array}\right\}$$

The von Mises stress is then calculated from

$$\sigma_{v M}=\sqrt{\sigma_{x x}^{2}+3 \tilde{\tau}_{x y}^{2}+3 \tilde{\tau}_{x z}^{2}}$$

Note: In most cases this is an over estimate of the von Mises stress.