Overview

PE exam study guide for structural mechanics: beam analysis, shear/moment diagrams, deflection, column buckling, truss analysis, and combined loading.

This topic accounts for 6 out of 40 questions on the PE Civil Civil Breadth (AM) exam.

Key Concepts

Beam Reactions and Shear/Moment Diagrams

For static equilibrium: sum of forces = 0, sum of moments = 0. Simply supported beam with uniform load w: R = wL/2, M_max = wL^2/8 at midspan, V_max = wL/2 at supports. Cantilever with uniform load: R = wL, M_max = wL^2/2 at fixed end. Key relationships: dV/dx = -w(x), dM/dx = V(x). Moment is maximum where shear = 0.

Simple Beam Maximum Moment Uniform Load Simple Beam Maximum Shear Uniform Load Cantilever Maximum Moment Uniform Load Bending Moment Bending Stress Shear Stress In Beams

Beam Deflection

Simply supported beam, uniform load: delta_max = 5wL^4 / (384EI) at midspan. Cantilever, point load at tip: delta = PL^3/(3EI). Cantilever, uniform load: delta = wL^4/(8EI). Moment of inertia for rectangles: I = bh^3/12. For common steel shapes, use the AISC manual tables.

Beam Deflection Uniform Load Simple Span Simple Beam Maximum Deflection Uniform Load Maximum Deflection In Simple Beam

Column Buckling (Euler)

Euler critical load: Pcr = pi^2 EI / (KL)^2. K = effective length factor: pinned-pinned K=1.0, fixed-free K=2.0, fixed-pinned K=0.7, fixed-fixed K=0.5. Slenderness ratio = KL/r where r = sqrt(I/A). Use the weak axis (minimum I) for buckling. Critical stress: Fcr = pi^2 E / (KL/r)^2. Valid only for elastic buckling (below proportional limit).

Euler Buckling Load Euler Elastic Buckling Stress Fe Slenderness Ratio Limit

Truss Analysis

Method of joints: sum of forces at each joint = 0. Method of sections: cut through max 3 unknowns and apply equilibrium. Zero-force members: at a joint with only 2 non-collinear members and no external load, both are zero-force. Common types: Pratt, Warren, Howe trusses. All members carry only axial force (tension or compression).

Combined Loading and Stress Transformation

Normal stress: sigma = P/A +/- Mc/I. Shear stress: tau = VQ/Ib (beam shear) or Tr/J (torsion). Principal stresses from Mohr's circle: sigma_1,2 = (sigma_x+sigma_y)/2 +/- sqrt[((sigma_x-sigma_y)/2)^2 + tau_xy^2]. Maximum shear stress = radius of Mohr's circle. Von Mises: sigma_vm = sqrt(sigma_x^2 - sigma_x*sigma_y + sigma_y^2 + 3*tau_xy^2).

Principal Stresses From Mohr S Circle Axial Stress Shear Stress Bending Stress

Common Exam Question Types

Calculate beam reactions for simple and cantilever beams
Draw shear/moment diagrams and find maximum values
Compute beam deflection under various loading
Determine Euler buckling load with effective length factor
Identify zero-force members in trusses
Calculate principal stresses using Mohr's circle

Exam Tips & Strategies

For beam problems: M_max = wL^2/8 (simply supported uniform), delta_max = 5wL^4/384EI

Euler buckling: always use the weak axis (smaller I) and correct K factor

Zero-force members: at unloaded joints with only 2 non-collinear members, both are zero-force

Mohr's circle center = (sigma_x + sigma_y)/2, radius involves both normal and shear stress

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Structural Mechanics Study Guide