Ch 1. Stress and Strain Multimedia Engineering Mechanics NormalStress Shear andBearing Stress NormalStrain Hooke'sLaw ThermalEffects IndeterminateStructures
 Chapter 1. Stress/Strain 2. Torsion 3. Beam Shr/Moment 4. Beam Stresses 5. Beam Deflections 6. Beam-Advanced 7. Stress Analysis 8. Strain Analysis 9. Columns Appendix Basic Math Units Basic Equations Sections Material Properties Structural Shapes Beam Equations Search eBooks Dynamics Fluids Math Mechanics Statics Thermodynamics Author(s): Kurt Gramoll ©Kurt Gramoll

MECHANICS - THEORY

Thermal Strain

 Material US (×10-6/oF) SI (×10-6/oC) Steel 6.5 12 Aluminum 13 23 PVC 30 55 Concrete 5.5 10
Coefficient of Thermal Expansion, α,
for Common Materials
(Expanded List in Appendix)

When physical materials are heated, they generally expand due to atomic-level changes. This expansion is proportional to the change in temperature. In terms of strain, this relationship is written as

 εT= α (ΔT)

where α is the coefficient of thermal expansion. This constant is different for all materials and is generally a positive number. There are a few materials that contract when heated, and then this constant would be negative.

The coefficient of thermal expansion represents a quantity (i.e. strain) per degree C or per degree F.

Thermal Expansion

Thermal Expansion where
ΔT = T2 - T1

The total deflection of a member that undergoes a temperature change, ΔT = T2 - T1, can also be written as

 δT= L α (ΔT)

The deflection is in the direction of the length L.

For a 3D homogenous object, thermal expansion will occur in all three directions. The total deflection in any given direction will be a function of the length, width or depth of the object.

There is no shear strain or shear deflection due to thermal expansion.

Combined Thermal and Mechanical
Strain or Deflection

Rod Between Two Fixed Walls -
Example of Stresses induced by
Thermal Expansion

Generally, thermal expansion is accompanied with mechanical deflection caused by a physical load. When both force and temperature changes are present, special care is needed when solving the problem. Both the thermal and mechanical deflections need to be compatible.

For example, if a simple rod is placed between two fixed walls and heated, the rod will try to expand. However, the walls will prohibit the normal thermal expansion and induce a compression force. This force is real and causes the rod to decrease in length. The final result is the thermal expansion is offset by the mechanical compression deflection (see diagram and animation at left).

When both mechanical and thermal conditions are present, a compatibility relationship needs to be introduced to solve the problem. Further discussion of compatibility conditions can be found in the following section on Indeterminate Axial Structures.

Other Material Properties

Stress-Cycle (σ-n) Curves
(also called s-n Curves)

As structural members experience multiple loading/unloading, their failure stress level will decrease. For example, car axles or even aircraft wings will experience millions of loadings over their service life. For aluminum, the failure stress can reduce by 70% of the original failure stress level.

To assist designers, charts called s-n curves are developed through experiments. The charts show the failure stress changes as the object is subjected to cycles. In some industries, such as aircraft, extensive testing on full scale models are required to insure the structure will not failure before a certain number of load cycles.

Typical Creep Curve

Viscoelasticity (Creep)

Another interesting property of many materials, especially plastics, is viscoelasticity or creep. This property models a material's tendency to flow over time when under stress. The most common effect of this is the deformation of an object over time when under a very low load. Even glass will creep over time. Plastics tend to exhibit higher creep characteristics then metals or ceramics.

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