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Structures under Combined Loading

Key Concepts: Structures commonly support multiple loads such as axial, torsion,
shear, bending, and distributed loads as well as internal pressures. The structural

response of structures under the conditions of combined loads is found by superposition

of the structural response for each separate type of loading.

In a Nut Shell: Combine the standard formulas for normal stress, shearing stress, and

bending stress by superposition to determine the total stresses developed in the structure.

The figures below illustrate structures under combined loads. The top one shows a circular bar
supporting both an axial force, P, and a torque, T_o. The middle figure depicts a beam with shear loading, V, which causes bending, and an axial load, P. The bottom figure shows a pressure vessel subjected to internal pressure P_i and axial torque, T_o.

Strategy: Combined loading of structures results in elements on the surface and within

that undergo normal and shear stresses acting on the faces of the element. Then use Mohr’s Circle to find the principal stresses at the location of the element.

Start with a Free Body Diagram of the structure at the location where you want to find the

stresses. Calculate the forces and moments at that section. Then use them to calculate the stresses associated with each individual load such as σ = Mc/I for bending stress, σ = P/a for axial stress, τ = Tc/J for torsional stress, and τ = VQ/It for shearing stress. Note that the

total normal stress at the section typically combines both bending stress and axial stress.

The same is true for combining shear stresses due to torsion and transverse shear.

Click here for examples.

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