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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=dA
Area Moments of Inertia
Iyy=z2dA
Izz=y2dA
Area Products of Inertia
Izz=y2dA
Radius of Gyration
Rg=IminA
Elastic Section Modulus
Ey=Iyyzmax
Ez=Izzymax
Max Coordinate Extension
ymax=max|y|
zmax=max|z|
Plastic Section Modulus
Py|z|dA
Pz|y|dA
Torsional Constant
Solid
It=Iyy+Izz+(zωyyωz)dA
ω - Warping function
(see below for warping function)
Shell open
It=13t3ds
t - Shell thickness
Shell closed
It=2AmiFsi
Ami - Area enclosed by cell i
Fsi - Shear flow in cell i
Elastic Torsion Modulus
Solid
Et=Itmax(y2+z2+zωyyωz)
Shell open
Et=Itmaxt
Shell closed
Et=Itmax(Fsit)
Shear Center
ys=IyzIyωIzzIzωIyyIzzI2yzIyω=yωdA,Izω=zωdA
zs=IyzIyωIyzIzωIyyIzzI2yz
Warping Constant (normalized to the shear center)
Iωω=ω2dA
Shear deformation coefficients
αzz=1Q2y(τ2xy|Qz=0+τ2xz|Qz=0)dA
αzy=1QyQz(τxy|Qy=0τxy|Qz=0+τxz|Qy=0τxy|Qz=0)dA
αzz=1Q2z(τ2xy|Qy=0+τ2xz|Qy=0)dA
Shear stiffness factors
kyy=1αzz
kyz=1αyz
kzz=1αyy
Shear stiffness
Sii=kiiGA
Warping Function
2ω=0
(ωyz)ny+(ωz+y)nz=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=Izz

/2=Iyy

/12=Iyz

K1=Kyy

K2=Kzz