Cross Sectional Properties Calculated by HyperBeam

The beam cross section is always defined in a y,z plane.

The x-axis is defined along the beam axis. The coordinate system you define is called the local coordinate system; the system parallel to the local coordinate system with the origin in the centroid is called the centroidal coordinate system; the system referring to the principal bending axes is called the principal coordinate system.

For shell sections, only the theory of thin walled bars is used. This means that for the calculation of the moments and product of inertia, terms of higher order of the shell thickness t are neglected. Thickness warping is also neglected.

Area

A = \int d A

$A = \int d A$

Area Moments of Inertia

I_{y y} = \int z^{2} d A

$I_{y y} = \int z^{2} d A$

I_{z z} = \int y^{2} d A

$I_{z z} = \int y^{2} d A$

Area Products of Inertia

I_{z z} = \int y^{2} d A

$I_{z z} = \int y^{2} d A$

Radius of Gyration

R_{g} = \sqrt{\frac{I_{min}}{A}}

$R_{g} = \sqrt{\frac{I_{\min}}{A}}$

Elastic Section Modulus

E_{y} = \frac{I_{y y}}{z_{max}}

$E_{y} = \frac{I_{y y}}{z_{\max}}$

E_{z} = \frac{I_{z z}}{y_{max}}

$E_{z} = \frac{I_{z z}}{y_{\max}}$

Max Coordinate Extension

y_{max} = max | y |

$y_{\max} = \max | y |$

z_{max} = max | z |

$z_{\max} = \max | z |$

Plastic Section Modulus

P_{y} \int | z | d A

$P_{y} \int | z | d A$

P_{z} \int | y | d A

$P_{z} \int | y | d A$

Torsional Constant

Solid: $I_{t} = I_{y y} + I_{z z} + \int (z \frac{\partial ω}{\partial y} - y \frac{\partial ω}{\partial z}) d A$ $I_{t} = I_{y y} + I_{z z} + \int (z \frac{\partial ω}{\partial y} - y \frac{\partial ω}{\partial z}) d A$; ω - Warping function; (see below for warping function)
Shell open: $I_{t} = \frac{1}{3} \int t^{3} d s$ $I_{t} = \frac{1}{3} \int t^{3} d s$; t - Shell thickness
Shell closed: $I_{t} = 2 \sum A_{m i} F_{s i}$ $I_{t} = 2 \sum A_{m i} F_{s i}$; $A_{m i}$ $A_{m i}$ - Area enclosed by cell $i$ $i$; $F_{s i}$ $F_{s i}$ - Shear flow in cell $i$ $i$

Elastic Torsion Modulus

Solid: $E_{t} = \frac{I_{t}}{max} (y^{2} + z^{2} + z \frac{\partial ω}{\partial y} - y \frac{\partial ω}{\partial z})$ $E_{t} = \frac{I_{t}}{\max} (y^{2} + z^{2} + z \frac{\partial ω}{\partial y} - y \frac{\partial ω}{\partial z})$
Shell open: $E_{t} = \frac{I_{t}}{max t}$ $E_{t} = \frac{I_{t}}{\max t}$
Shell closed: $E_{t} = \frac{I_{t}}{max (\frac{F_{s i}}{t})}$ $E_{t} = \frac{I_{t}}{\max (\frac{F_{s i}}{t})}$

Shear Center

y_{s} = \frac{I_{y z} I_{y ω} - I_{z z} I_{z ω}}{I_{y y} I_{z z} - I_{y z}^{2}} I_{y ω =} \int y ω d A, I_{z ω} = \int z ω d A

$y_{s} = \frac{I_{y z} I_{y ω} - I_{z z} I_{z ω}}{I_{y y} I_{z z} - I_{y z}^{2}} I_{y ω =} \int y ω d A, I_{z ω} = \int z ω d A$

z_{s} = \frac{I_{y z} I_{y ω} - I_{y z} I_{z ω}}{I_{y y} I_{z z} - I_{y z}^{2}}

$z_{s} = \frac{I_{y z} I_{y ω} - I_{y z} I_{z ω}}{I_{y y} I_{z z} - I_{y z}^{2}}$

Warping Constant (normalized to the shear center)

I_{ω ω} = \int ω^{2} d A

$I_{ω ω} = \int ω^{2} d A$

Shear deformation coefficients

α_{z z} = \frac{1}{Q_{y}^{2}} \int (τ_{x y}^{2} ∣ ∣_{_{Q_{z} = 0}} + τ_{x z}^{2} ∣ ∣_{_{Q_{z} = 0}}) d A

$α_{z z} = \frac{1}{Q_{y}^{2}} \int (τ_{x y}^{2} |_{_{Q_{z} = 0}} + τ_{x z}^{2} |_{_{Q_{z} = 0}}) d A$

α_{z y} = \frac{1}{Q_{y} Q_{z}} \int (τ_{x y} ∣ ∣_{_{Q_{y} = 0}} τ_{x y} ∣ ∣_{_{Q_{z} = 0}} + τ_{x z} ∣ ∣_{_{Q_{y} = 0}} τ_{x y} ∣ ∣_{_{Q z = 0}}) d A

$α_{z y} = \frac{1}{Q_{y} Q_{z}} \int (τ_{x y}^{} |_{_{Q_{y} = 0}} τ_{x y}^{} |_{_{Q_{z} = 0}} + τ_{x z}^{} |_{_{Q_{y} = 0}} τ_{x y}^{} |_{_{Q z = 0}}) d A$

α_{z z} = \frac{1}{Q_{z}^{2}} \int (τ_{x y}^{2} ∣ ∣_{_{Q_{y} = 0}} + τ_{x z}^{2} ∣ ∣_{_{Q_{y} = 0}}) d A

$α_{z z} = \frac{1}{Q_{z}^{2}} \int (τ_{x y}^{2} |_{_{Q_{y} = 0}} + τ_{x z}^{2} |_{_{Q_{y} = 0}}) d A$

Shear stiffness factors

k_{y y} = \frac{1}{α_{z z}}

$k_{y y} = \frac{1}{α_{z z}}$

k_{y z} = \frac{- 1}{α_{y z}}

$k_{y z} = \frac{- 1}{α_{y z}}$

k_{z z} = \frac{1}{α_{y y}}

$k_{z z} = \frac{1}{α_{y y}}$

Shear stiffness

S_{i i} = k_{i i} G A

$S_{i i} = k_{i i} G A$

Warping Function

\nabla^{2} ω = 0

$\nabla^{2} ω = 0$

(\frac{\partial ω}{\partial y} - z) n_{y} + (\frac{\partial ω}{\partial z} + y) n_{z} = 0

$(\frac{\partial ω}{\partial y} - z) n_{y} + (\frac{\partial ω}{\partial z} + y) n_{z} = 0$

For solid sections, the warping function is computed using a finite element formulation. This may lead to un-physically high stresses in geometric singularities (sharp corners) that get worse with mesh refinement. This may cause problems computing the elastic torsion modulus.

Nastran Type Notation

$/ 1 = I_{z z}$ $/ 1 = I_{z z}$

$/ 2 = I_{y y}$ $/ 2 = I_{y y}$

$/ 12 = I_{y z}$ $/ 12 = I_{y z}$

$K 1 = K_{y y}$ $K 1 = K_{y y}$

$K 2 = K_{z z}$ $K 2 = K_{z z}$