Chapter 8. Thermal Analysis

Chapter 8. Thermal Analysis

Two objectives:

•Determine the temperature field (steady or unsteady state)

•Stresses due to the temperature changes

I.Temperature Field

Fourier Heat Conduction Equation:

1-D Case:

where,

fx = heat flux per unit area, k = thermal conductivity, x

T = T(x) = temperature.

where, fx, fy, fz = heat flux in x, y and z direction, respectively, and in case of isotropy,

The Equation of Heat Flow is:

in which,

qv = rate of internal heat generation per unit volume, c = specific heat,

ρ = mass density.

For steady state (∂T∂t = 0) and isotropic materials, we can obtain:

This a Poisson equation.

Boundary Conditions (BC’s):

n

Finite Element Formulation for Heat Conduction:

This is obtained in a similar way as for the structural analysis, e.g., by starting with the interpolation T = NTe (N is the

shape function matrix, Te the nodal temperature).

Note that there is only one DOF at each node for the thermal problems.

Thermal Transient Analysis:

∂∂Tt ≠ 0.

Apply FDM (use time steps and integrate in time), as in the transient structural analysis, to obtain the transient temperature fields.

II.Thermal Stress Analysis

•Solve Eq. (7) first to obtain the temperature (change) fields.

•Apply the temperature change ∆T as initial strains (or initial stresses) to the structure.

1-D Case:

At temperature T1

At temperature T2

εo

Thermal Strain (Initial Strain):

with εe being the elastic strain due to mechanical load.

Example: The above shown bar under thermal load ∆T .

(a)If no constraint on the right-hand side, that is, the bar is free to expand to the right, then

ε = εo , εe = 0, σ = 0 , from Eq. (12). No thermal stress!

(b)If there is a constraint on the right-hand side, that is, the bar can not expand to the right, then

ε = 0, εe = −εo = −α∆T , σ = −Eα∆T ,

from Eqs. (10) and (12). Thus, thermal stress exists!

2-D Cases:

Plane Stress,

Here, ν is the Poisson’s ratio.

Observation: Temperature changes do not yield shear strains.

Total Strain:

Thermal Stress Analysis Using the FEM:

•Need to specify α for the structure and ∆T on the related elements (which experience the temperature change).

•Note that for linear thermoelasticity, same temperature change will yield same stresses, even if the structure is at two different temperatures.

•Differences in the temperatures during the manufacturing and working environment are the main cause of thermal (residual) stresses.