Thermal stress 02: quasi-rigid link and two trusses

Determine the thermal stresses that develop when the temperature of one member is increased.

Case Description

A quasi-rigid link pivots about a hole at its right end. The link rests atop a cylindrical brass rod. A cylindrical steel rod anchors down the left end of the link. The temperature of the brass rod is increased from 20° C (the stress-free temperature of the assembly) to 50° C. The resulting thermal expansion induces a tensile stress in the steel hold-down rod. Since the brass rod is not free to expand unimpeded, an axial compressive stress develops in it. We will compare this axial stress to the theoretical solution.

Rigid elements are not available in the subject analysis type. Therefore, quasi-rigid behavior is achieved by making the link relatively large and setting the material stiffness to the maximum supported value. In the diagram that follows, all dimensions are in millimeters.

thermal stress diagram

Dimensions (mm)

Link: 1100 x 300 x 200 overall
Brass Rod: 30 diameter x 300 long
Steel Rod: 22 diameter x 900 long (excluding head)

Study Type and Parameters

Study Type: Thermal Stress
Stress-Free Reference Temperature: 20° C

Mesh Parameters

Mesh Type = Solid, Tetrahedral
Mesh Size = 30 mm, absolute
Element Order = Parabolic
Adaptive Mesh Refinement: None
Local Mesh Controls:
- Pivot hole in Link: 15 mm Mesh Size
- Steel Rod and associated hole faces in Link: 5 mm Mesh Size
- Brass Rod and associated hole faces in Link: 7mm Mesh Size

Material Properties

Property	Link	Brass Rod	Steel Rod
Modulus of Elasticity (MPa)	1.3 x 10⁶	105,000	200,000
Poisson's Ratio	0.3	0.31	0.3
Thermal Expansion Coefficient (/ °C)	1.2 x 10^-5	1.88 x 10^-5	1.2 x 10^-5

Note: The thermal expansion coefficients of the Steel Rod and Link do not really matter, since these parts will not experience a temperature change.

Constraints

Pivot hole in Link: Pin Constraint (Radial and Axial directions)
Bottom end of Steel Rod: Fixed
Bottom end of Brass Rod: Y Constraint only
Cylindrical face of Brass Rod: Pin Constraint (Tangential direction only)

Note: This constraint prevents the rod from rotating about any axis but allows it to grow radially due to thermal expansion. Therefore, the rod is statically stable, and artificial thermal stresses are not caused by the constraint scheme.

Thermal Loads

Link and Steel Rod (entire body of each): Applied Temperature of 20° C
Brass Rod (entire body): Applied Temperature of 50° C

Contact Parameters

Holes in the link for both rods are sized for clearance. Therefore, contact only occurs in two locations:

Between the top end of the Brass Rod and the end of the associated blind hole in the Link.
Between the bottom of the head of the Steel Rod and the bottom of the associated counter-bore in the Link.

The settings for both contact sets are as follows:

Contact Type: Separation
Penetration Type: Symmetric
Thermal Conductance: 1 x 10-6 W / (m2·K)

Note: All three bodies receive an applied temperature, to make the Brass Rod hotter and to ensure that the temperatures of the other two parts remain at their stress-free values. To minimize heat flow where the parts meet, zero thermal conductance is desirable. However, a zero conductance value causes the solution to fail. Therefore, we specify an infinitesimally small conductance value instead.

Theoretical Solution

thermal stress theory diagram

Considering the preceding diagram, we have the following relationship from the Beer and Johnston reference:

ΣME = 0 RA * (0.75 m) - RB * (0.3 m) = 0 RA = 0.4 RB

Deformations

The method of superposition is used. With RB removed, the temperature rise of the cylinder causes point B to move down a distance δT. The reaction RB causes a displacement δ1 of the same magnitude as δT so that the final deflection of point B is zero.

Due to a temperature rise of 30° C (50° C - 20° C), the length of the Brass Rod, when unconstrained, increases by δT according to the following equation:

δT = L(ΔT)α = (0.3 m)(30° C)(1.88 x 10-5/° C) = 0.0001692 m

We note that δD = 0.4 δC and that δ1 = δD + δB-D, where δB-D is the change in length of the Brass Rod.

δC = RAL / (AE)

Given that, for the Steel Rod, L = 0.9 m, A = πD2/4, D = 0.03 m, and E = 105 x 109 Pa:

δC = 4 RA(0.9 m) / [π(0.03 m)2(105 x 109 Pa)] = 11.83797097 x 10-9 RA δD = 0.4 δC = 0.4 (11.83797097 x 10-9 RA) = 4.73518839 x 10-9 RA δB-D = RBL / (AE)

Given that, for the Brass Rod, L = 0.3 m, A = πD2/4, D = 0.022 m, and E = 105 x 109 Pa:

δB-D = 4 RB(0.9 m) / [π(0.03 m)2(105 x 109 Pa)] = 4.04203030 x 10-9 RB

Given that RA = 0.4 RB:

δ1 = δD + δB-D = [4.73518839 x 10-9 (0.4 RB) + 4.04203030 x 10-9 RB] = 5.936105657 x 10-9 RB

But, δT = δ1:

0.0001692 m = 5.936105657 x 10-9 RB RB = 28,503.536 N

Stress in the Brass Rod

σB = RB/A = 4 (28,503.536 N) / [π (0.03 m)2] = 40,324,254 Pa

Therefore, σB = 40.324254 MPa

Comparison of Results

The stress in the Brass Rod is essentially axial. However, the solid representation of the rod, and the way that it is constrained, make it possible for bending deflection and bending stresses to occur. This behavior is not experienced in a true truss, which is the basis of the theoretical solution. To ensure that we do not include bending effects in our comparison, we select a point along the neutral axis of the rod (where the bending stress component is zero). In addition, we do not want to check stresses close to a constraint or contact area, where the results may be skewed by local effects. Therefore, we compare the theoretical axial stress of the truss (Brass Rod) to the Fusion Normal YY Stress result at the centroid of the Brass Rod (-300, -150, 0). The minus sign of the result indicates that the stress is compressive, as expected (whereas positive stress components are tensile):

thermal stress y-stress diagram

Location	Theoretical Result (Beer and Johnston)	Fusion Result	% Difference
Centroid of Brass Rod	-40.324254 MPa	-39.805 MPa	-1.288 %

Note: You can attribute the difference between the thermal stress study results and the theoretical solution to the following factors:

The contact and constrained ends of the rods do not behave like the endpoints of true trusses, which act like ball joints. Moments or eccentric reactions can occur at the ends of the solid representations.
The cross-sectional area of the solid rods is affected by the mesh and likely does not exactly match the area of a truly circular cross-section.
The quasi-rigid link is not truly rigid. The slight deformation of the link (due to bending along its length and local compression at the rod contact areas) partially unloads the two rods. This effect reduces the calculated stress magnitudes in the Brass Rod.

Despite these known differences between the theoretical and solid finite element models, the results are still quite close.

Reference

Beer, Ferdinand P. and Johnston, Jr., E. Russell, Mechanics of Materials , McGraw-Hill, Inc., 1981, sample problem 2.4, page 58.