Mechanics of Materials for the FE Exam
Mechanics of materials (also called strength of materials) extends statics by analyzing the internal forces and deformations within structural members. This topic is heavily tested on the FE Civil and FE Mechanical exams and appears on other disciplines as well. Understanding stress, strain, and deformation is essential for any practicing engineer.
Stress and Strain Fundamentals
Normal Stress
Normal stress (sigma) is the internal force per unit area acting perpendicular to a cross section:
sigma = P / A
Where P is the internal axial force and A is the cross-sectional area. Normal stress can be tensile (positive, pulling apart) or compressive (negative, pushing together). Units are pascals (Pa) or pounds per square inch (psi).
Shear Stress
Shear stress (tau) is the internal force per unit area acting parallel to a cross section:
tau = V / A
Where V is the internal shear force. Shear stress causes sliding deformation rather than stretching or compression.
Normal Strain
Normal strain (epsilon) is the change in length divided by the original length:
epsilon = delta_L / L
Strain is dimensionless. It can be expressed as a decimal, percentage, or in micro-strain units.
Hooke's Law
For linear elastic materials, stress is proportional to strain:
sigma = E * epsilon
Where E is the modulus of elasticity (Young's modulus). This relationship holds only within the elastic range of the material.
Axial Loading
For a member under axial load P, the deformation is:
delta = P * L / (A * E)
For members with varying cross sections or loads, break the member into segments and sum the deformations of each segment. Temperature changes also cause axial deformation: delta_T = alpha * delta_T * L, where alpha is the coefficient of thermal expansion.
Torsion
Torsion occurs when a member is twisted by applied torques. For circular shafts:
- Shear stress: tau = T * r / J, where T is the torque, r is the distance from the center, and J is the polar moment of inertia.
- Maximum shear stress: Occurs at the outer surface where r is maximum.
- Angle of twist: phi = T * L / (G * J), where G is the shear modulus and L is the shaft length.
- Polar moment of inertia: J = pi * d^4 / 32 for a solid circular shaft of diameter d.
Bending
Bending analysis is one of the most important topics in mechanics of materials. For a beam under bending:
- Flexural stress: sigma = M * y / I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia.
- Maximum flexural stress: Occurs at the point farthest from the neutral axis (top or bottom of the beam).
- Shear and moment diagrams: Graphical representations of internal shear force V(x) and bending moment M(x) along the beam length.
Drawing accurate shear and moment diagrams is a critical skill. Key relationships:
- dV/dx = -w(x), where w(x) is the distributed load
- dM/dx = V(x)
- The maximum moment occurs where V = 0 or changes sign
Combined Loading
Many real structures experience multiple types of loading simultaneously. The principle of superposition allows you to add the stresses from each load type independently (valid only for linear elastic behavior). Common combinations include:
- Axial plus bending
- Bending plus torsion
- Axial plus torsion
Mohr's Circle
Mohr's circle is a graphical method for determining principal stresses, maximum shear stress, and stresses on any inclined plane. Key results:
- Principal stresses occur on planes where shear stress is zero
- Maximum shear stress equals the radius of Mohr's circle
- The maximum shear stress plane is oriented 45 degrees from the principal stress planes
Practice constructing Mohr's circle from given stress states and extracting the required information.
Column Buckling
Euler's formula gives the critical buckling load for a slender column:
P_cr = pi^2 * E * I / (K * L)^2
Where K is the effective length factor that depends on the end conditions:
- Both ends pinned: K = 1.0
- One end fixed, one end free: K = 2.0
- Both ends fixed: K = 0.5
- One end fixed, one end pinned: K = 0.7
Euler's formula applies only to long, slender columns that buckle elastically. Short columns may fail by material yielding before buckling.
Beam Deflection
Common methods for calculating beam deflection include:
- Double integration method
- Using beam deflection tables (available in the reference handbook)
- Superposition of standard cases
The reference handbook contains deflection and slope formulas for many standard beam configurations. Learn to use these tables quickly.
Mechanics of materials is a substantial topic that rewards thorough preparation. Focus on understanding the concepts behind the formulas, and practice enough problems to recognize which approach applies to each question type.