# Creative Use of Temperature in Stress Analysis  April 3, 2015 By: Peter Barrett

In over 30 years of implementing the Finite Element Method, I have found that using temperature distributions in non-traditional ways has often been an effective tool for saving time and for obtaining convergence of difficult nonlinear problems. Three of my favorite examples are presented in this post:

• Pre-load and Initial Strain Modeling
• Torsional Shear Stress Simulation
• Variable Stiffness/Break-Away Supports

Efficient convergence of structures with assembly pre-loads in finite element analyses can often be achieved by using “dummy” thermal loading.  This use of temperature loads was the most common method of modeling bolt pre-tension prior to the development of custom bolt pre-tension elements in recent years.  The bolt loads are induced though tensile strains in the pre-loaded member activated by the thermal expansion induced strain that is resisted by the bolt flanges or other supports. For a rigid connection, the required strain can be calculated directly from the thermal strain induced based on Coefficient of thermal Expansion multiplied by the delta temperature load.  However, in most cases, a trial and error process of adjusting either the coefficient of expansion or bolt temperature is needed to induce the correct bolt force, since the stiffness of the bolt flanges will reduce the net strain where the amount is typically unknown.  There is still an advantage to this approach in that users can control how the temperatures (hence bolt pre-load forces) are ramped on in a non-uniform nonlinear sequence, which can often enhance convergence speed. I have used this method in the past on models that require special tuning to converge.  Loose frictional contact connections and complex nonlinear material models, such as hyperelastic seals inserted between the connecting bodies, are two examples that come to mind.

Thermally induced initial strains can also be useful in a variety of other applications such as pre- or post-tensioning concrete reinforcement, cable stayed bridges, soil springs etc.  These initial strains are created through the combined input of a dummy coefficient of thermal expansion in conjunction with delta temperature loads (element temperature minus the reference temperature) imposed on the bolt, spring, wire, cable, etc. A series of load steps can be used to slowly ramp the temperature (i.e. model the tightening of the bolt or cable for example) at different rates.

Torsional Shear Stress Simulation
For a shaft under pure torsion loading, the only nonzero stresses are shear. It can be shown that the maximum shear stress under torsion occurs tangent to the free surface.  Closed form solutions, for example Saint Venant’s theory of torsion, utilize the Prandtl Stress Function which can represent the shear stress in non-circular shafts using spatial differential equations. The maximum shear stress is the largest slope of the stress function at the boundary. One can use this thermal analogy with heat conduction to solve an equivalent set of equations, where the heat flux represents the shear stress.  Instead of modeling a complex 3-d shaft and struggling with boundary conditions to simulate pure torsion. One can run a 2-d heat transfer solution using a unit thermal conductivity, zero temperatures on the boundaries, and a heat generation rate that is proportional to the cross-section torsional constant in order to derive accurate shaft shear stresses.  Figure 1 illustrates the procedure with a matching 2-d heat flux plot (top images) to 3d shear stress (bottom images) in a triangular shaft under pure torsion.

Figure 1- 2-d Thermal Flux Model vs. 3-d Shaft Maximum Shear Stress Comparison

## thermal flux.png Variable Stiffness/Break-Away Supports
A third novel way to use thermal loading is to leverage temperature-dependent material properties in order to provide a variable stiffness support. One example would be to define the modulus of elasticity as a function of temperature in a series of temporary support members.

Figure 2: Bridge Form Supports that one could model using variable stiffness/break away supports

## thermal flux2.png One could vary the temperature, and hence stiffness, of the modeled truss or beam elements to convert a stiff connection (high modulus) to a released support (near zero modulus) by changing the element temperature. A loading sequence of this type could mimic the removal of construction forms, for example.  A zero coefficient of thermal expansion should be used to eliminate any fictitious thermal strains in the simulation caused by the changes in temperature.

Have you used thermal loads in non-traditional ways in your FEA models in the past? I would love to hear how others have incorporated these techniques into their finite element modeling bag of tricks!