Beams, Deflection, Columns, Mohr's Circle, and Combined Loading

Beam model before formulas

A beam question starts with support conditions, loads, and the section where stress or deflection is requested. Draw reactions first, then shear and moment relationships. For simple cases, the FE Reference Handbook tables can replace a full derivation, but only when the support condition and loading pattern match exactly.

Bending stress is sigma = Mc/I, or sigma = M/S when section modulus S is available. Use the moment at the section of interest, not automatically the maximum moment unless the question asks for maximum stress. The c distance is measured from the neutral axis to the point where stress is requested. The moment of inertia must be about the bending axis.

Shear, moment, and deflection

Shear and moment diagrams are sign-convention tools. A consistent convention matters more than matching a textbook's diagram orientation. Concentrated forces jump the shear diagram. Distributed loads change shear slope. Concentrated couples jump the moment diagram. The maximum bending moment commonly occurs where shear crosses zero or at a boundary for cantilevers.

Deflection formulas are especially boundary-condition sensitive. A cantilever with an end load, a cantilever with a uniform load, and a simply supported beam with a midspan load have different constants and different maximum locations. If a superposition problem combines loads, keep signs and reference directions consistent.

Columns and buckling

Column questions test stability rather than material yielding alone. Euler buckling uses P_cr = pi^2 E I/(K L)^2 for long, slender columns in the elastic range. The effective length factor K depends on end conditions. Use the weak-axis I because the column buckles about the axis with least resistance.

Do not compare axial stress to yield and stop if the member is slender. A column can buckle at a load below the yield load. Conversely, Euler buckling may not govern for short columns; use the model indicated by the problem and handbook guidance.

Mohr's circle and stress transformation

Mohr's circle organizes plane stress transformation. Inputs are sigma_x, sigma_y, and tau_xy with a sign convention. The circle center is average normal stress, and the radius is sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2). Principal stresses are center plus or minus radius. Maximum in-plane shear stress is the radius.

For the FE exam, the biggest risk is sign and angle. The physical plane rotation angle is half the angle measured on Mohr's circle. If only principal stresses or maximum shear are requested, you may not need to draw the full circle, but you must still use the correct center and radius.

Combined loading check

Combined loading means add stress components at the same point. Axial force creates uniform normal stress. Bending adds tension on one side and compression on the other. Torsion adds shear. Transverse shear may matter near the neutral axis but is often zero at the extreme outer fiber of a rectangular section.

A practical FE routine is: find internal loads, compute section properties, calculate normal and shear stresses at the requested point, transform stresses if needed, then compare the governing stress or factor of safety to the failure criterion stated in the problem.