Understanding Axial Deformation and Thermal Stress in Materials

1. CTC / MTC 322Strength 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 = σL / E • But, σ = F / A • Therefore, δ = FL / A E

5. Axial deformation, δ • For an axially loaded member, δ = FL / 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 = Eα (∆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