Deflection of Beams and Shafts
The elastic curve, four methods for slopes and deflections, and the solution of statically indeterminate beams.
Learning objectives
- Relate internal bending moment, curvature, slope, and transverse displacement using a declared sign convention.
- Select integration, discontinuity functions, moment-area, or superposition for a linear-elastic beam response.
- Apply boundary and compatibility conditions to calculate slope, displacement, or a redundant reaction.
- Account for changing flexural rigidity using a piecewise \(M/EI\) relation.
- Audit a deflection prediction against units, support conditions, small-slope applicability, and serviceability intent.
12.1The Elastic Curve
The Elastic Curve
The deflected shape of a beam's centroidal axis. Deflections must often be limited — for stability, to protect brittle finishes (plaster, concrete), and most importantly because slopes and displacements supply the extra equations needed for statically indeterminate beams. Sketching rules: supports that resist a force restrict displacement; supports that resist a moment (fixed ends) restrict slope and displacement.
Reading the Curve from the Moment Diagram
By the beam sign convention, positive moment bends the beam concave up; negative moment concave down; a point of zero moment is an inflection point where the curvature reverses. The maximum deflection occurs either where the slope is zero or at an unsupported end — drawing the moment diagram first makes the elastic curve easy to sketch.
- \(\rho\) = radius of curvature at the point; \(EI\) = the flexural rigidity (always positive), the beam's resistance to bending deformation.
Physical meaning
Curvature is the rate at which a beam tangent turns with distance. A larger \(EI\) makes a given moment produce less turning; with \(v\) positive upward and sagging moment positive, this page uses \(EI v''=M\).
12.2Slope and Displacement by Integration
The exact curvature \(1/\rho = v''/[1 + (v')^2]^{3/2}\) makes the governing equation nonlinear, but design codes keep beam slopes small, so \((dv/dx)^2 \ll 1\) and the curvature reduces to \(v''\). With constant EI, three equivalent forms follow:
- Starting from \(M(x)\) needs only two integrations (two constants); starting from \(w(x)\) needs four.
- Boundary conditions: pin or roller → \(v = 0\); fixed end → \(v = 0\) and \(\theta = dv/dx = 0\).
- Continuity conditions: where separate moment functions meet, slope and deflection from adjacent regions must match: \(\theta_1(a) = \theta_2(a)\), \(v_1(a) = v_2(a)\).
Problem-solving strategy
- Draw the free-body diagram and decide whether the target is a curve, slope, displacement, or compatibility condition.
- Declare \(x\), \(v\), and moment signs; obtain \(M(x)\) before choosing integration, moment-area, discontinuity functions, or superposition.
- List support and interface conditions before integrating.
- Use \(M/[E(x)I(x)]\) piecewise where stiffness changes.
- Audit endpoints, signs, units, small rotations, and the applicable serviceability limit.
Recognition strategy
A single-load cantilever has a simple moment function and two known wall conditions, making direct elastic-curve integration the most transparent method.
Given information
- \(L=1.20\ \mathrm{m}\), \(P=2500\ \mathrm{N}\), and constant \(EI\).
- Set \(x=0\) at the wall; upward \(v\) and sagging \(M\) are positive.
Model and interpretation
The prismatic bracket is linear elastic with small rotation. Its moment is \(M(x)=-P(L-x)\), and the wall imposes \(v(0)=v'(0)=0\).
Verification
Units: \(PL^3/EI\) is \(\mathrm{m}\), and \(PL^2/EI\) is dimensionless radians.
Signs and direction: Negative values mean downward displacement and clockwise rotation under the declared convention.
Plausibility: \(0.480\ \mathrm{mm}\ll1.20\ \mathrm{m}\) and \(\lvert\theta_L\rvert\ll1\), consistent with small-slope theory.
Engineering interpretation
The ideal beam prediction is modest, but wall connection flexibility and local deformation can add movement; they matter if the bracket locates a precision component.
12.3Discontinuity Functions
One Expression for the Whole Beam
With several loads, region-by-region integration multiplies the constants and continuity conditions. Discontinuity functions write the load or moment as a single expression valid over the whole beam, so only the boundary conditions remain. Macaulay functions handle distributed loads:
\[ \langle x - a\rangle^n = \begin{cases} 0 & x < a \\ (x - a)^n & x \ge a \end{cases} \quad (n \ge 0), \qquad \int \langle x - a\rangle^n dx = \frac{\langle x - a\rangle^{n+1}}{n+1} + C \]Singularity functions (\(n = -1, -2\)) represent a concentrated force \(P\langle x - a\rangle^{-1}\) and couple \(M_0\langle x - a\rangle^{-2}\); they integrate by simply raising the exponent. In a moment expression, a force P applied at \(a\) contributes \(P\langle x - a\rangle^1\) and a couple contributes \(M_0\langle x - a\rangle^0\).
12.4Slope and Displacement by the Moment-Area Method
A semigraphical technique — ideal when the M/EI diagram is built of simple shapes. It uses the same assumptions as integration (small slopes, bending only).
The angle between the tangents at any two points of the elastic curve equals the area under the M/EI diagram between them — counterclockwise from tangent A to tangent B if the area is positive.
The vertical deviation of point A from the tangent drawn at B equals the moment of the M/EI area between A and B, taken about A (\(\bar{x}_A\) = distance from A to the area's centroid). Note \(t_{A/B} \ne t_{B/A}\) in general. Deviations are measured perpendicular to the original axis; positive areas place A above the tangent.
12.5Method of Superposition
Adding Tabulated Deflections
The governing equation \(EI\,v'''' = w(x)\) is linear and assumes small deformations — the two requirements of superposition — so the deflection from several loads is the algebraic sum of the deflections from each load acting alone. Engineering handbooks tabulate slopes and deflections for standard cases; combining table entries solves most practical problems in a few lines. Frequently used entries:
\[ \text{Cantilever, end load: } v = \frac{PL^3}{3EI} \qquad \text{Simply supported, center load: } v = \frac{PL^3}{48EI} \qquad \text{Simply supported, uniform load: } v = \frac{5wL^4}{384EI} \]Recognition strategy
Both standard loads act on the same linear, constant-stiffness support case and the response is wanted at their shared symmetry point, so tabulated midspan responses can be added.
Given information
- \(L=4.00\ \mathrm{m}\), \(w=3.00\ \mathrm{kN/m}\), \(P=10.0\ \mathrm{kN}\), \(EI=5000\ \mathrm{kN\,m^2}\).
- Downward displacement is reported negative.
Model and interpretation
Assume linear elasticity, small displacement, and no change in contact load as the beam moves. These assumptions make superposition valid.
Verification
Units: Each numerator is \(\mathrm{kN\,m^3}\), so division by \(\mathrm{kN\,m^2}\) gives meters.
Signs and direction: Both loads bend the beam downward, so two negative contributions must add.
Plausibility: \(4.67\ \mathrm{mm}/4000\ \mathrm{mm}=1/857\), a small displacement relative to span.
Engineering interpretation
The point load contributes more than the distributed load. A serviceability decision must compare the calculated total with the project limit and include dynamic machine effects if they are relevant.
12.6Statically Indeterminate Beams and Shafts
Redundants and Degree of Indeterminacy
A beam is statically indeterminate when its unknown reactions outnumber the equilibrium equations. The extra reactions — those not needed for stable equilibrium — are redundants, and their count is the degree of indeterminacy: a propped cantilever (4 reactions, 3 equations) is indeterminate to the first degree; a two-span continuous beam to the second. Any reaction whose removal leaves the beam stable may be chosen as the redundant. The solution must satisfy equilibrium and compatibility simultaneously — by integration (12.7), moment-area (12.8), or superposition (12.9).
12.7Statically Indeterminate — Method of Integration
Extra Unknowns, Extra Conditions
Write \(M(x)\) including the unknown redundant reaction, integrate \(EI\,v'' = M\) twice, and note that the indeterminate beam has more boundary conditions than integration constants — exactly enough surplus to solve for the redundants too. A propped cantilever, for example, supplies three conditions (\(v = 0\) at both supports, \(\theta = 0\) at the wall) against two constants: the third condition determines the redundant.
12.8Statically Indeterminate — Moment-Area Method
Geometric Compatibility from M/EI Diagrams
Draw the M/EI diagram in terms of the unknown redundant — most conveniently as separate diagrams for each load and for the redundant (the superposition of diagrams keeps the shapes simple). Then enforce the support geometry through the theorems: e.g. for a propped cantilever, the deviation of the roller from the wall tangent must vanish, \(t_{B/A} = 0\), which yields the redundant directly.
12.9Statically Indeterminate — Method of Superposition
The Force Method for Beams
Remove the redundant to create the statically determinate, stable primary beam; let the applied loads deflect it by \(v_B\) at the redundant's location; then apply the redundant alone, deflecting the point back by \(v'_B\). Compatibility at the support requires the net displacement to vanish — an equation solved directly for the redundant (hence "force method"), with the remaining reactions following from equilibrium.
Recognition strategy
The requested output is a serviceability decision at midspan for a standard simply supported center load. The tabulated elastic response is appropriate only after stating shaft, bearing, and load idealizations.
Given information
- \(L=1.50\ \mathrm{m}\), \(P=1800\ \mathrm{N}\), \(E=200\times10^9\ \mathrm{Pa}\), \(I=8.00\times10^{-6}\ \mathrm{m^4}\).
- Model bearings as simple supports and take downward deflection as negative.
Model and interpretation
The shaft is straight, prismatic, linear elastic, and lightly deflected. Bearing compliance, pulley overhang, torque-induced twist, and rotating-load dynamics are excluded from this bending-only screen.
Verification
Units: \(PL^3/EI\) reduces to meters; the comparison is made in millimeters on both sides.
Signs and direction: The negative result is downward, consistent with the pulley load; the limit uses magnitude.
Plausibility: \(0.0791\ \mathrm{mm}/1500\ \mathrm{mm}\approx5.3\times10^{-5}\), so small-slope bending theory is self-consistent.
Engineering interpretation
The static beam model clears the alignment limit, not a complete shaft design. A professional release still checks bearing stiffness, actual load positions, fatigue, critical speed, torsional twist, tolerances, and the governing equipment specification.
Chapter recap
- Use moment curvature to connect loading to shape, and impose geometric conditions at every support.
- Select integration for custom functions, moment-area for simple \(M/EI\) geometry, and superposition only for linear compatible cases.
- Before moving on: sketch the curve, state signs, apply boundary and compatibility conditions, treat variable \(EI\) explicitly, and compare a checked result with serviceability intent.