Beam Theory — A Short, Exam■Oriented
Essay (SI)
Scope: linear elastic beams, small deflection, prismatic members. Sign convention: sagging M positive.
1) What Beam Theory Is For
Beam theory predicts internal forces (shear V, moment M), stresses (σ, τ), and deflections v(x) for
slender members under transverse loading. It is the backbone of machine/structural design: quick,
closed■form, and accurate when the underlying assumptions hold.
2) Modeling Assumptions & Regimes
Euler–Bernoulli (EB) model: cross■sections remain plane and perpendicular to the deflected centerline;
shear deformation is neglected. Good when the span■to■depth ratio L/h ■ 10–15, material obeys
Hooke’s law, and rotations are small.
Timoshenko (TB) model: relaxes the perpendicularity assumption and includes shear deformation and
rotary inertia (important in dynamics). Prefer TB for deep beams, sandwich cores, or short spans where
shear strains are not negligible.
3) Equilibrium & Differential Relations
For distributed load w(x) (N/m) acting upward positive: dV/dx = −w(x), dM/dx = V(x). The elastic curve
satisfies EI v''(x) = −M(x) in EB theory. Slope θ = v'(x), curvature κ = v''(x) (for small rotations). A point
load P causes a negative jump of V by P; an applied couple M■ causes a positive jump of M by M■.
4) Shear and Moment Diagrams (SFD/BMD) — Workflow
1 Draw the global free■body diagram and compute support reactions by ∑F=0, ∑M=0.
2 Cut sections and write piecewise w(x), V(x), M(x). Apply continuity and jumps at loads/couples.
3 Sketch using slopes: slope of M is V; area under w equals −∆V; area under V equals ∆M.
4 Where V = 0 you may have an extremum of M. Corners in M occur at concentrated couples.
5) Stresses in Beams
Normal (bending) stress: σ(x,y) = M(x) y / I, where y is measured from the neutral axis and I is the
second moment of area. Extreme■fiber stress: σ_max = M c / I with c the distance to the farthest fiber.
Transverse shear (EB): τ = V Q / (I t), with Q the first area moment of the portion above/below the point,
t the local thickness. For a rectangle, τ_max = 1.5 V/A at the neutral axis; for a solid circle, τ_max =
(4/3) V/A.
Shear flow in built■up or thin■walled open sections: q = V Q / I; the average web shear stress is
q/t_web; flange shear■flow guides connector sizing.
6) Deflection of Beams
Direct integration: integrate EI v''(x) = −M(x) twice and use boundary conditions (BCs) to find constants.
Moment■area: (I) change in slope between A and B equals area under M/(EI) from A to B; (II) deflection
of B relative to the tangent at A equals the first moment of that area about B. Superposition: sum
responses of simpler load cases (linear regime required).
7) Energy Methods and Castigliano
Strain energy in bending (EB): U = ∫ M(x)^2 / (2 E I) dx. Deflection δ at the application point of a load P is
δ = ∂U/∂P (treating P as a variable). For indeterminate structures, enforce compatibility by setting the
deflection at a released support to zero to solve for the redundant.
8) Indeterminate Beams (One Redundancy) — Quick Recipe
1 Choose a reaction R as redundant and remove it → primary determinate structure.
2 Compute deflection at the redundant’s location due to actual loads (δ_L).
3 Apply a unit load in the direction of R and compute flexibility f = ∫ (M_L M_1)/(E I) dx.
4 Compatibility: δ_L + f R = 0 (plus settlements/thermal terms if any). Solve for R and recover V, M, v.
9) Canonical Closed■Form Results (Prismatic EB, Max Values)
Case (span L)
M_max
Slope at support / tip
Deflection v_max
Simply supported, central P
PL/4
θ_support = P L^2 / (16 E I)
P L^3 / (48 E I) at midspan
Simply supported, uniform w
w L^2 / 8
θ_support = w L^3 / (24 E I)
5 w L^4 / (384 E I)
Cantilever, end P
P L (at fixed end)
θ_tip = P L^2 / (2 E I)
P L^3 / (3 E I)
Cantilever, uniform w
w L^2 / 2
θ_tip = w L^3 / (6 E I)
w L^4 / (8 E I)
Fixed–fixed, uniform w
M_end = − w L^2 / 12 θ_end = 0
w L^4 / (384 E I)
10) Section Properties & Practical Notes
Second moment of area about the centroidal neutral axis: rectangle I = b h^3 / 12; circle I = π d^4 / 64;
hollow circle I = π (D^4 − d^4) / 64; use the parallel■axis theorem I = I_c + A d^2. For I■sections,
compute flange/web parts about the centroid and include parallel■axis terms. In practice: keep a
consistent unit system (N■mm or N■m), track sign conventions, and verify results by area/slope checks
on V/M diagrams.
11) When EB Fails & Notes on Timoshenko Beams
If L/h is small (deep beam), shear deformation causes EB to under■predict deflections. In Timoshenko
theory the rotation ϕ(x) of a cross■section is independent of slope θ(x) = v'(x). Kinematics: γ_xz = ϕ − θ;
constitutive τ = κ G γ_xz (κ is the shear■correction factor). Governing equations couple bending and
shear; closed■form solutions exist for simple cases; otherwise use finite elements.
12) Micro■Example: Simply Supported Beam, Central Load P
Reactions: R_A = R_B = P/2. For 0 ≤ x ≤ L/2: V = P/2, M = (P/2) x. By symmetry the other half mirrors.
Max bending stress at midspan: σ_max = (M_max c)/I = (P L / 4) (c/I). Deflection curve from EI v'' = −M:
integrate twice with BCs v(0)=0, v(L)=0 and symmetry v'(L/2)=0 → v_max = P L^3 / (48 E I).
13) Exam Checklist (Fast)
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Draw SFD/BMD correctly (use area/slope checks).
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Compute I about the correct axis; include holes via subtraction.
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Use σ = M c / I and τ = V Q / (I t); don’t mix units.
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Pick a deflection method and stick to it; enforce BCs cleanly.
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For 1■DOF indeterminates, use unit■load/Castigliano compatibility.
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If geometry is deep/short, mention EB limits or TB applicability.
References (for deeper study)
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R. C. Hibbeler, Mechanics of Materials, 7th ed. (or later), Pearson.
S. P. Timoshenko & D. H. Young, Elements of Strength of Materials; S. Timoshenko, Strength of Materials.
S. P. Timoshenko & J. N. Goodier, Theory of Elasticity.
W. C. Young & R. G. Budynas (eds.), Roark’s Formulas for Stress and Strain.
A. Ugural & S. Fenster, Advanced Mechanics of Materials and Applied Elasticity.
© 2025. This handout is for study use; equations are given in SI units unless noted.