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

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

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

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

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)

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

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 )

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