SM 143 Statically indeterminate & Compatibility equations

Statically indeterminate objects

Object is “restrained”

More unknowns than equil equations, which means compatibility conditions need to be considered.

Superposition requirements

• The load must be linearly related to the stress and displacements

• No significant displacement or changes in geometry are allowed

“Compatibility conditions”

Logical equations developed by using equil equations & ΔL (δ) equations in context, then solved simultaneously.

ΔL is written as δ, because change is small.

Make sure δ > gap in stress questions

Examples:

1. Global equil to relate forces to each other, then a comp equation that sets total δ = 0. Simultaneously solve to find values of forces.

2. Global equil to relate forces, then comp equation that sets δ of both objects of interest equal to each other. Solve simultaneously.

Increase in temperature of restrained object

1. ↑ in temp results in displacement

2. Displacement is retrained so thermal stress occurs

3. ΔL_T = 0

ΔL due to ΔT

α is coefficient of thermal expansion which is given.

Thermal stress

Force exerted back onto object due to elongation

Find total δ, then subtract gap size from it. Use difference as δ in δ = PL/AE to find P exerted by wall.

If >1 segment on object, account for it by adding more terms to δ equation for each segment. eg. δ_diff = δ_AB + δ_BC, where P are equal in each term