Beam Deflection and Indeterminate Beams

Introduction

Strength analysis based only on bending stress is incomplete. A beam may satisfy its stress limits yet deform enough to damage finishes, misalign machinery, impair drainage, or produce uncomfortable vibration. Beam-deflection analysis determines the slope and vertical displacement caused by loading and provides the compatibility equations needed for statically indeterminate beams.

This chapter develops four classical methods: double integration, moment-area, conjugate-beam, and superposition. It also introduces compatibility-based analysis of selected indeterminate beams. The discussion assumes slender, prismatic beams undergoing small elastic deflections.

Learning Objectives

By the end of this chapter, the reader should be able to:

• distinguish slope, deflection, and curvature;
• derive and apply the elastic-curve equation;
• determine slopes and deflections by integration, moment-area, conjugate-beam, and superposition methods;
• select suitable boundary and continuity conditions;
• solve selected statically indeterminate beams using compatibility;
• check signs, units, boundary conditions, and physical reasonableness;
• recognize the limitations of Euler–Bernoulli beam theory.

1. Fundamental Concepts

The elastic curve is the deflected centroidal axis of a beam. The slope θ at a section is the rotation of the tangent to this curve, measured in radians. The deflection v is the transverse displacement of the section, measured in metres or millimetres. The curvature κ describes how rapidly the beam’s slope changes along its length.

For a beam with constant flexural rigidity, the Euler–Bernoulli relation is

where E is Young’s modulus (Pa), I is the second moment of area about the neutral axis (m⁴), and M(x) is the internal bending moment (N·m). The product EI is the flexural rigidity (N·m²). Depending on the chosen sign convention, a negative sign may appear; consistency is more important than the particular convention.

The slope is

and therefore

For variable rigidity, the more general equation is

These equations assume small rotations, linear elastic material behaviour, plane cross-sections remaining plane, and negligible shear deformation. They are most accurate for slender beams with length-to-depth ratios commonly greater than about 10, although the required limit depends on geometry and loading.

Boundary and continuity conditions

Integration produces two constants for each continuous region. They are found from physical conditions such as:

• at a fixed support: v = 0 and θ = 0;
• at a simple support: v = 0;
• at a free end with no applied couple: M = 0, hence d²v/dx² = 0;
• at a free end with no transverse force: V = 0, hence d³v/dx³ = 0 when EI is constant;
• at an internal continuity point: deflection and slope are continuous unless a hinge, release, or concentrated deformation is present.

A concentrated force changes the shear force, while a concentrated couple causes a jump in the bending moment. The deflection and slope themselves normally remain continuous at an ordinary load point.

2. Double-Integration Method

The double-integration method begins by writing M(x) in terms of x, substituting it into the elastic-curve equation, and integrating twice. For a constant-EI beam,

The method is systematic and particularly useful when the moment function is simple. If the loading or cross-section changes, the beam is divided into regions and separate equations are written.

Worked Example 1: Cantilever with an end load

A cantilever of length L carries a downward point load P at its free end. Let x be measured from the fixed support. Using a sign convention in which downward deflection is negative, the bending moment is

Thus,

Integrating once,

At the fixed end, θ(0) = 0, so C₁ = 0. Integrating again,

Since v(0) = 0, C₂ = 0. Therefore,

At the free end x = L,

The negative signs indicate downward displacement and clockwise rotation under the selected convention. The dimensions are consistent: PL²/EI is dimensionless and PL³/EI has units of length.

Singularity functions

Macaulay or singularity notation can represent several loads in one equation. A bracket such as <x−a>ⁿ is zero for x < a and equals (x − a)^n for x ≥ a. A concentrated force, applied at a, can therefore be included without dividing the beam into many regions. The notation is powerful, but its sign convention and treatment of point moments must be stated before use.

3. Moment-Area Method

The moment-area method converts the M/EI diagram into geometric relationships.

Theorem 1: Change in slope. The change in slope between two points A and B equals the algebraic area under the M/EI diagram between them:

Theorem 2: Tangential deviation. The deviation of point B from the tangent drawn at A equals the first moment of the M/EI area between A and B about B:

The phrase “deviation of B from the tangent at A” is essential. It is not automatically the vertical deflection of B from the original, undeformed axis. The correct reference tangent must be identified.

The method is efficient when the M/EI diagram has simple rectangular, triangular, or parabolic areas. It is also useful for locating maximum deflection, because maximum deflection occurs where θ = 0.

Example 2: Cantilever with an end load by moment-area

For the cantilever in Example 1, the M/EI diagram is triangular, with maximum magnitude PL/EI at the fixed end and zero at the free end. Its area is

Since the fixed-end slope is zero, the free-end slope magnitude is A, giving θ_L = PL²/(2EI). The first moment of this triangular area about the free end is its area multiplied by the distance from the free end to the triangle centroid, 2L/3:

This agrees with double integration.

4. Conjugate-Beam Method

The conjugate-beam method replaces the real beam by an imaginary beam of the same length loaded by the M/EI diagram. The loading on the conjugate beam is therefore

The key relationships are:

• shear force in the conjugate beam corresponds to slope in the real beam;
• bending moment in the conjugate beam corresponds to deflection in the real beam.

Support transformations are required:

The method follows from equilibrium: the derivative of conjugate-beam moment is conjugate shear, and the derivative of conjugate shear is the applied M/EI loading. It is especially convenient when the M/EI loading can be replaced by simple resultant forces.

For a simply supported beam, the conjugate beam is also simply supported. For a cantilever, the conjugate beam is fixed at the real beam’s free end and free at the real beam’s fixed end. The result must still be interpreted with the original beam’s sign convention.

5. Superposition

For a linear elastic beam, the response to several loads equals the algebraic sum of the responses to the individual loads:

Superposition requires linear material behaviour, small deflections, unchanged supports, and load-independent E and I. It is practical when standard cases are available in beam tables.

Common formulas for a simply supported beam of span L and constant EI include:

The first two formulas are standard for the stated loading and support conditions. The couple case is sensitive to the exact loading and sign convention and should not be used without verifying the corresponding moment diagram.

Worked Example 3: Central load on a simply supported beam

A simply supported steel beam has L = 4 m, a central load P = 10 kN, E = 200 GPa, and I = 8.0 × 10⁻⁶ m⁴. The maximum deflection is

Substitution gives

The numerator is 640,000 N·m³ and the denominator is 76,800,000 N·m², so

This value should be compared with the project’s serviceability limit, not merely with the stress limit.

6. Statically Indeterminate Beams

A beam is statically indeterminate when equilibrium equations alone cannot determine all reactions. Additional equations come from deformation compatibility and constitutive relations.

The general force-method procedure is:

1. determine the degree of static indeterminacy;
2. select redundant reaction(s);
3. remove the redundant(s) to form a statically determinate primary structure;
4. calculate displacement at the released coordinate caused by real loads;
5. calculate displacement caused by a unit redundant or by the redundant itself;
6. impose the support compatibility condition;
7. solve the redundant reaction(s), then recover all other reactions;
8. compute slopes, deflections, and stresses as required.

For one redundant R, compatibility commonly has the form

where Δ_load is displacement at the redundant coordinate due to applied loads, f_RR is flexibility at that coordinate per unit redundant, and Δ_prescribed is the permitted displacement. For a fixed support, Δ_prescribed = 0.

The flexibility coefficient may be found by virtual work:

and the load displacement is

Here M is the bending moment from the applied loads on the released structure, and m_R is the bending moment caused by a unit force in the redundant direction. These equations assume linear elasticity and that bending deformation dominates. If shear or axial deformation matters, their corresponding virtual-work terms must be added.

Worked Example 4: Propped cantilever with a uniformly distributed load

Consider a prismatic cantilever of length L fixed at A and supported by a vertical prop at B. A uniform load w acts downward over the full length. The vertical reaction at B, R_B, is the single redundant. Remove the prop, leaving a cantilever.

At B, the downward free-end deflection caused by w is

The upward redundant R_B produces an upward free-end deflection

Compatibility requires zero deflection at B:

After cancelling L³/EI,

The total downward load is wL, so vertical equilibrium gives the fixed-end vertical reaction

The fixed-end moment follows from moment equilibrium about A. Taking counterclockwise moments as positive,

Thus the algebraic fixed-end moment is

The negative sign indicates a hogging internal moment at the fixed support under the stated convention. This example illustrates that compatibility, not equilibrium alone, determines the prop reaction.

7. A Practical Compatibility Example

Illustrative case: machinery-support beam

Background. A maintenance platform uses a steel beam to support a motor. The beam is simply supported and has a concentrated load at midspan. The design team initially checks bending stress but observes excessive movement at the motor shaft.

Problem. The beam has L = 4 m, P = 10 kN, E = 200 GPa, and I = 8.0 × 10⁻⁶ m⁴. Determine the midspan deflection and assess the serviceability issue.

Analysis and solution. The central-load formula gives δ_max = 8.33 mm, as calculated earlier. If the project limit is L/500, the allowable deflection is 4,000/500 = 8.0 mm. The beam therefore exceeds the stated limit by approximately 0.33 mm, even though the difference is small.

Engineering response. Increasing I is often more efficient than increasing yield strength because deflection varies inversely with EI. Increasing I by at least the ratio 8.33/8.00 ≈ 1.042 would meet the numerical limit in an idealized calculation; a practical design would include uncertainty, connection flexibility, dynamic effects, and load combinations. A deeper section, a shorter span, an intermediate support, or a stiffer connection could be considered.

Learning points. Serviceability is a separate design requirement. Deflection formulas are valid only for their stated loading and support conditions. Real design also requires checks for strength, lateral stability, fatigue, vibration, connections, corrosion, construction tolerances, and applicable building or machinery standards.

8. Choosing a Method

A robust solution often uses one method for calculation and another for checking. For example, a cantilever result obtained by integration can be checked using moment-area or a standard table.

9. Limitations and Advanced Considerations

Euler–Bernoulli theory neglects transverse shear deformation. For short, deep beams, sandwich beams, timber members with low shear stiffness, or beams made from soft materials, Timoshenko beam theory may be more appropriate. In that theory, total rotation includes bending and shear contributions.

The classical equations also become inadequate when deflections are large enough to change the geometry significantly, when material behaviour is plastic or viscoelastic, or when cracking, contact, yielding, or support settlement changes the structural model. Variable E or I must be included explicitly. Temperature gradients and imposed settlements can create deflections and reactions even without external transverse loads.

In professional practice, finite-element software is useful for complex geometry, variable sections, elastic supports, and load combinations. However, software does not replace modelling judgement. Boundary conditions, units, mesh quality, releases, and sign conventions must be verified using hand calculations and equilibrium checks. Digital twins and sensor-based structural monitoring can compare measured displacement with analytical predictions, but measurement uncertainty and calibration remain important.

Best Practices

• Draw the free-body diagram before writing a moment function.
• State the x-coordinate, sign convention, E, and I.
• Use consistent units; N–m–Pa or N–mm–MPa are both acceptable.
• Apply every boundary and continuity condition explicitly.
• Check dimensions of each final expression.
• Confirm that deflection is zero where a support requires it.
• Locate maximum deflection by setting θ = 0, not automatically at midspan.
• Compare deflection with serviceability limits and consider vibration where machinery or people are involved.
• Verify indeterminate results with equilibrium and physical reasoning.
• Do not report more significant figures than justified by the input data.

Chapter Summary

Beam deflection is governed, under Euler–Bernoulli assumptions, by the relationship between bending moment and curvature, EIv'' = M. Integration gives slope and deflection directly, while the moment-area and conjugate-beam methods provide geometric and statics-based alternatives. Superposition permits standard responses to be added when the structure remains linear elastic. Statically indeterminate beams require compatibility in addition to equilibrium; the force method releases redundants, calculates flexibility, and restores the displacement constraint. Reliable solutions require correct boundary conditions, consistent signs and units, independent checks, and recognition of shear, large-deflection, material, and support limitations.

Key Terms

Elastic curve: Deflected centroidal axis of a beam.

Slope: Rotation of the beam tangent, θ = dv/dx.

Deflection: Transverse displacement of a beam section.

Curvature: Rate of change of slope along the beam.

Flexural rigidity: Product EI governing bending stiffness.

Euler–Bernoulli theory: Beam theory neglecting transverse shear deformation.

Double integration: Method integrating the moment-curvature equation twice.

Moment-area method: Method relating M/EI areas to slope changes and tangential deviations.

Conjugate beam: Imaginary beam loaded by the real beam’s M/EI diagram.

Superposition: Addition of individual linear responses.

Compatibility: Geometric requirement that connected components deform consistently.

Redundant: Reaction or force not obtainable from equilibrium alone.

Primary structure: Determinate structure formed after removing redundants.

Flexibility coefficient: Displacement produced per unit redundant force.

Serviceability: Performance concerned with deflection, vibration, appearance, and function.

Prismatic beam: Beam with constant cross-sectional properties along its length.

Singularity function: Bracket notation used to express piecewise loading compactly.

Virtual work: Energy-based method for calculating displacement through load–moment products.

Review Questions

Short Answer Questions

1. Define beam slope and deflection, and state their usual units.
2. What physical conditions are imposed at a fixed support?
3. State the two moment-area theorems.
4. Why can equilibrium alone not solve a statically indeterminate beam?
5. List two assumptions of Euler–Bernoulli beam theory.

Long Answer Questions

1. Derive the free-end slope and deflection of a cantilever carrying a point load at its free end using double integration.
2. Explain the conjugate-beam method, including the support transformations.
3. Describe how the moment-area method determines the deflection of a point relative to a tangent.
4. Explain the force method for a beam with one redundant reaction.
5. Discuss the differences among double integration, moment-area, conjugate-beam, and superposition methods.

Analytical / Application Questions

1. A simply supported beam of span 5 m carries a central 12 kN load. If E = 200 GPa and I = 1.2 × 10⁻⁵ m⁴, determine the maximum deflection and assess it against an L/360 limit.
2. A propped cantilever carries a uniform load. Explain how removing the prop and imposing zero displacement leads to the prop reaction; identify the assumptions behind the calculation.
3. A short, deep beam supports a heavy machine and shows deflection larger than predicted by Euler–Bernoulli theory. Identify likely causes and recommend an improved analysis strategy.

Multiple Choice Questions with Answers

Question 1

For a prismatic Euler–Bernoulli beam, which equation relates bending moment to curvature?

A. EI d²v/dx² = M

B. EI dv/dx = M

C. EIv = M

D. EI d³v/dx³ = M

Correct Answer: A. EI d²v/dx² = M

Question 2

A cantilever of length L carrying an end load P has free-end deflection magnitude equal to:

A. PL/(EI)

B. PL²/(2EI)

C. PL³/(3EI)

D. PL⁴/(8EI)

Correct Answer: C. PL³/(3EI)

Question 3

The change in slope between two points equals the:

A. Area under the shear-force diagram

B. Area under the M/EI diagram

C. First moment of the load diagram

D. Maximum bending moment divided by EI

Correct Answer: B. Area under the M/EI diagram

Question 4

In the conjugate-beam method, the applied loading is:

A. The shear-force diagram

B. The bending-moment diagram multiplied by EI

C. The M/EI diagram

D. The deflection diagram

Correct Answer: C. The M/EI diagram

Question 5

Superposition is valid for beam deflection when the response is:

A. Linear elastic with unchanged boundary conditions

B. Fully plastic

C. Independent of stiffness

D. Always associated with large rotations

Correct Answer: A. Linear elastic with unchanged boundary conditions

Question 6

A fixed support in the real beam corresponds to what in the conjugate beam?

A. A fixed support

B. A simple support

C. An internal hinge

D. A free end

Correct Answer: D. A free end

Question 7

For a simply supported beam with a central point load P, the maximum deflection is:

A. PL³/(24EI)

B. PL³/(48EI)

C. 5PL³/(384EI)

D. PL²/(48EI)

Correct Answer: B. PL³/(48EI)

Question 8

The additional equation required for a statically indeterminate beam usually comes from:

A. Compatibility of deformation

B. The density of the material

C. The beam’s colour

D. The support reaction being zero

Correct Answer: A. Compatibility of deformation

Question 9

A simply supported beam has L = 4 m, P = 10 kN at midspan, E = 200 GPa, and I = 8 × 10⁻⁶ m⁴. Its maximum deflection is approximately:

A. 0.833 mm

B. 4.17 mm

C. 8.33 mm

D. 16.7 mm

Correct Answer: C. 8.33 mm

Question 10

Which beam is most likely to require a theory including shear deformation?

A. A very slender steel beam

B. A short, deep beam

C. A long beam with small deflection

D. A prismatic beam under a small central load

Correct Answer: B. A short, deep beam