# 2-node Frame FEM element thermal formulation

1. Jul 11, 2012

### Ronankeating

$\int$hi all,

I need just little bit of confirmation about what I'm doing.

Normally I know how the stiffness matrix is composed for 2 node frame element in 3D space. which creates 12x12 matrix for that element, if there is also thermal loading on frame element additional coefficients(Q12x1 matrix) will contribute to the RHS of KxU=F equation.

Q=TRANSPOSE(\begin{bmatrix}Px1&0&0&0&-My1&Mz1&-Px2&0&0&0&My2&-Mz2\end{bmatrix})
where
Px = E*A*$\alpha$*ΔT
My = $\int \alpha$*E*ΔT*z*dA
Mz = $\int \alpha$*E*ΔT*y*dA

Is this the correct formulation of 3D frame elements thermal coefficients matrix?

Frame stiffness matrix works flawlessly, but if I introduce the Q on the right side of linear equations I always get the wrong solutions. The process itself is so simple but couldn't spot what I'm doing wrong.

Regards,

2. Jul 11, 2012

### AlephZero

A uniform temperature rise in a beam element will not create any bending. To bend the beam you would need a thermal gradient from one side to the other. That is sometimes included in thermal loading of plate elements, with heat flowing through the thickness of the plate, but it doesn't sound very relevant for a building frame element.

The "simple" way to derive the loads is consider what happens to an element without any external loads or restraints. It will expand by an amount $\alpha L \Delta T$ where $L$ is the length of the element. To force it back to its original length, you need to apply axial loads $F$ and $-F$ at the two ends, where $F = K \alpha L \Delta T$.

The axial stiffness $K = EA/L$, so $F = EA \alpha \Delta T$.

The load vector, in the element coordinate system, is
$Q = \begin{bmatrix} F & 0 & 0 & 0 & 0 & 0 & -F & 0 & 0 & 0 & 0 & 0\end{bmatrix}$.
(Warning, Q might be minus the above, but that's easy enough to check with a test problem.)

Remember you also need to subtract the thermal strains before you calculate the element stresses!