1. CTC / MTC 322 Strength of Materials
Chapter 3
Axial Deformation and Thermal Stress

2. Chapter Objectives
Compute the elastic deformation of a member due to an axial tensile or compressive load
Design axially loaded members to limit their deformation to a given value
Define coefficient of thermal expansion
Compute the amount of thermal deformation due to temperature change if a member is unrestrained
Compute the thermal stress in a member due to temperature change if the member is restrained
Compute the stress in components of a composite structure made of more than one material

3. Elastic Deformation
Stress – the internal resistance to an external force offered by a unit area of the material from which a member is made, or, more simply, force per unit area
Stress = force / area = F / A
Strain – unit deformation, calculated by dividing the total deformation by the original length
Strain = ε = total deformation / original length

4. Axial deformation, δ For an axially loaded member,
ε = δ / L , where δ = total deformation, and L = original length
Modulus of elasticity = normal stress / normal strain, or E = σ / ε
Solving for strain: ε = σ / E = δ / L
Solving for deformation: δ = σ L / E
But, σ = F / A
Therefore, δ = F L / A E

5. Axial deformation, δ
For an axially loaded member, δ = F L / A E if the following conditions apply:
Member is straight
Uniform cross section over length considered
Material is homogeneous
Load applies along centroidal axis (no bending)
Stress is below the proportional limit
Also, the equation applies only when all factors, F, L, A, and E are constant over the section being analyzed
In some cases, member can be divided into segments where factors are constant and the principle of superposition can be used to calculate total deformation (Example 3 – 8)

6. Thermal Expansion When heated a metal part tends to expand
If unrestrained, the part will expand, but no stress will be developed
If restrained, the expansion may be prevented, in which case stress will be developed
Different materials change dimensions at different rates
Coefficient of thermal expansion, α = unit change in dimension for a unit change in temperature
U.S. Units – in / (in - ˚F ), or 1 / ˚F
S.I. Units – mm / (mm - ˚C ), m / (m - ˚C ), or 1 / ˚C
Change in length due to thermal expansion
δ = α x L x ∆T
Where, L = original length of member and ∆T = change in temperature

7. Thermal Stress
If restrained, deformation due to temperature change will be prevented, and stress will be developed
ε = δ / L = α x L x ∆T / L = α (∆T)
But, E = σ / ε , or σ = E ε
Therefore, σ = E α (∆T)
This stress is independent of (in addition to) stress from external forces
In some cases, member is initially free to expand, but after some initial deformation, further deformation is prevented
Determine ∆T1 which will bring member to the point of restraint
Calculate ∆T2 = ∆T - ∆T1 , the remainder of the temperature change
Calculate the stress due to this temperature change (∆T2 )

8. Stress in Members Made of More Than One Material
When two or more materials in a member share the load, the elastic properties of the materials must be considered
The same situation applies when the load is shared by two or more members made from different materials
It can be shown that, provided all members undergo equal strains:
σ2 = FE2 / (A1E1 + A2E2 )
σ1 = σ2 E1 / E2
The subscripts 1 and 2 refer to materials 1 and 2, respectively
See Example 3-15