Mechanics of Materials on the FE Exam
Mechanics of materials (also called strength of materials) is a core topic on the FE Civil, FE Mechanical, and FE Other Disciplines exams, typically accounting for 7 to 11% of the questions. This subject builds directly on statics and applies equilibrium principles to deformable bodies. Mastering these concepts is essential for both the exam and your engineering career.
Stress and Strain Fundamentals
Normal stress equals force divided by cross-sectional area, and it can be either tensile (positive) or compressive (negative). Shear stress equals the shear force divided by the area over which it acts. Strain is the deformation per unit length: normal strain for axial deformation and shear strain for angular deformation.
Hooke's Law relates stress to strain for linearly elastic materials: stress equals the modulus of elasticity (Young's modulus) times strain. For shear: shear stress equals the shear modulus times shear strain. Know the relationship between the elastic modulus, shear modulus, and Poisson's ratio.
Axial Loading
For members under axial loading, the deformation equals the force times the length divided by the product of the cross-sectional area and the elastic modulus. For members with varying cross sections, applied loads, or material properties along the length, break the member into segments and sum the individual deformations.
Thermal deformation problems require adding or subtracting thermal strain from mechanical strain. Thermal strain equals the coefficient of thermal expansion times the temperature change. Statically indeterminate axial problems require combining equilibrium equations with compatibility (deformation) equations.
Torsion
For circular shafts under torsion, the shear stress varies linearly from zero at the center to a maximum at the outer surface. The maximum shear stress equals the torque times the outer radius divided by the polar moment of inertia. The angle of twist equals the torque times the length divided by the product of the shear modulus and the polar moment of inertia.
Know the polar moment of inertia for solid and hollow circular cross sections. For solid shafts, J equals pi times the diameter to the fourth power divided by 32. For hollow shafts, use the difference between the outer and inner values.
Beam Bending
The flexure formula relates bending stress to the applied moment: bending stress equals the moment times the distance from the neutral axis divided by the moment of inertia. Maximum bending stress occurs at the point farthest from the neutral axis. For symmetric cross sections, this is at the top and bottom surfaces of the beam.
Shear stress distribution in beams follows the shear formula: shear stress equals VQ divided by Ib, where V is the shear force, Q is the first moment of area above (or below) the point of interest, I is the moment of inertia, and b is the width at the point of interest. For rectangular cross sections, the maximum shear stress occurs at the neutral axis and equals 1.5 times the average shear stress.
Beam Deflection
The FE exam may test beam deflection using the double integration method or by applying standard deflection formulas from the FE Reference Handbook. Common loading cases include simply supported beams with concentrated loads, uniformly distributed loads, and cantilever beams. Memorize or quickly locate the deflection formulas for these standard cases.
Column Buckling
Euler's formula gives the critical buckling load for long, slender columns: the critical load equals pi squared times the elastic modulus times the moment of inertia divided by the effective length squared. The effective length depends on the end conditions: pinned-pinned (K=1.0), fixed-free (K=2.0), fixed-pinned (K=0.7), and fixed-fixed (K=0.5).
Euler's formula applies only to long columns where buckling occurs in the elastic range. For shorter columns where yielding occurs before buckling, use the Johnson formula or other inelastic buckling criteria provided in the FE Reference Handbook.
Combined Loading
Real engineering members often experience combined axial, bending, shear, and torsional loads simultaneously. Use superposition to combine the stresses from each loading type. For combined normal stresses, add the axial and bending components. For combined states of stress, use Mohr's circle or the stress transformation equations to find principal stresses and maximum shear stress.
Failure theories (Maximum Shear Stress theory and von Mises theory) use principal stresses to predict whether a material will yield under combined loading. Know when to apply each theory: Maximum Shear Stress (Tresca) is more conservative, while von Mises is more accurate for ductile materials.