Flexure, Shear Stress and Beam Deflection

Prerequisite knowledge: force equilibrium, SI dimensional analysis, algebra and first-year calculus.

Learning objectives: state a model and its validity range, calculate physically meaningful engineering quantities, reconcile equations with a free-body diagram or deformation compatibility, and identify at least two failure modes excluded by the idealization.

Unit context and engineering interpretation

Beam stress and beam deflection are both governed by section stiffness, yet they answer different design questions. Flexural stress uses the section modulus I/c; deflection uses the full flexural rigidity EI. Changing beam depth alters I approximately with depth cubed in a rectangular section. The outer fiber stress therefore scales with depth squared under constant bending moment and constant width, whereas deflection changes more sharply. This explains why a modest depth increase can be valuable for long spans.

The famous elastic beam formulas rest on assumptions. Plane sections are assumed to remain plane and perpendicular to a small-deflection neutral axis, and shear deformation is omitted. These assumptions work best for long slender beams where bending is dominant. A deep timber beam, a sandwich panel with a soft core or a short composite member may require Timoshenko beam theory or a continuum model. Likewise, near a point load, a bearing plate creates three-dimensional contact stresses outside one-dimensional beam theory.

Use an independent approximate check whenever possible. For example, under a central point load, a simply supported beam has a triangular moment diagram. Integrating the curvature function twice with zero end deflections yields the familiar P L³/(48EI) midspan result. Under a uniform full-span load, the maximum becomes 5 w L⁴/(384EI). Both carry deflection units; both grow rapidly with span. A simulation must reproduce zero displacement at its ideal pin and roller, even when the line is graphically exaggerated for visibility.

Lesson 6.1 — Bending stress and neutral axis

Physical mechanism and governing idea

Under classical elastic bending of a prismatic homogeneous beam, plane sections remain plane and longitudinal stress varies linearly through depth. For bending about a principal centroidal axis, sigma=-M y/I. The most highly stressed fibers lie far from the neutral axis. This is a local beam-section approximation, not a full stress analysis at a notch.

Governing relationship

sigma_x = -M y/I; S=I/c; sigma_max=|M|/S

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A section with twice the elastic section modulus carries half the bending stress under the same moment.

What the simplified formula misses

At large curvature, near abrupt supports, or in cracked composite sections, the elastic transformed-section assumptions need re-evaluation.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 6.1 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 6.2 — Transverse shear through beam sections

Physical mechanism and governing idea

Shear stress is generally nonuniform across a beam section. The engineering VQ/(I b) formula connects transverse shear resultant to first area moment Q and local width b. In a rectangular section, shear peaks at the neutral axis and reaches zero at its top and bottom surfaces in the textbook idealization.

Governing relationship

tau = VQ/(I b); rectangular tau_max=1.5 V/A

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A rectangular beam carrying 10 kN shear over 5000 mm² area has average 2 MPa and ideal maximum about 3 MPa.

What the simplified formula misses

This approximation is not appropriate at all corners and supports, or when transverse shear deformation strongly affects the member.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 6.2 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 6.3 — Deflection from curvature

Physical mechanism and governing idea

Small-deflection Euler-Bernoulli theory approximates curvature by y double prime and relates it to M/EI with an adopted sign convention. Integrating curvature twice introduces constants set by support displacement and rotation conditions. The shape of the bending moment function determines the deflection profile.

Governing relationship

E I y(x)=M(x) with a consistent signed coordinate convention

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

For a midspan point load on a simple beam, maximum downward deflection magnitude is P L³/(48EI).

What the simplified formula misses

The formula neglects shear deflection and geometric nonlinearity, so it becomes less accurate for deep, soft-shear, or strongly deflected beams.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 6.3 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 6.4 — Serviceability and stiffness selection

Physical mechanism and governing idea

Strength and serviceability are independent questions. A beam may meet bending stress limits but deflect excessively for partitions, finishes or equipment. Span has a cubic influence on point-load deflection and a fourth-power influence on uniform-load deflection. Support conditions and composite action also matter significantly.

Governing relationship

delta_max(point central)=P L³/(48EI); delta_max(UDL)=5wL⁴/(384EI)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

Doubling the span while preserving EI and central load increases elastic deflection eightfold.

What the simplified formula misses

Allowable deflection depends on application and design standard. A single span/360 limit is not universal and should not be imposed without context.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 6.4 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Worked numerical casebook

The following 18 worked cases are intended for dimensional fluency and model interpretation, not as a substitute for unique field measurements. Cases change materials, loads and geometry and deliberately restate their validity conditions.

When reproducing a result, document a complete free body or section cut, record the input units, and calculate at least one quantity without looking at the worked result.

Case 6.01: Midspan deflection in a service gantry

A simply supported beam, span 5 m, has E=210 GPa, I=90×10⁶ mm⁴, a central point load 8 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 1.102 mm from the point load and 0.000 mm from the distributed load, totaling 1.102 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.01: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.02: Midspan deflection in a construction prop

A simply supported beam, span 5 m, has E=210 GPa, I=50×10⁶ mm⁴, a central point load 3 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.744 mm from the point load and 0.000 mm from the distributed load, totaling 0.744 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.02: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.03: Midspan deflection in a testing rig

A simply supported beam, span 3 m, has E=69 GPa, I=75×10⁶ mm⁴, a central point load 5 kN and uniform load 3 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.543 mm from the point load and 0.611 mm from the distributed load, totaling 1.155 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.03: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.04: Midspan deflection in a industrial platform

A simply supported beam, span 6 m, has E=69 GPa, I=50×10⁶ mm⁴, a central point load 8 kN and uniform load 3 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 10.435 mm from the point load and 14.674 mm from the distributed load, totaling 25.109 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.04: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.05: Midspan deflection in a equipment mount

A simply supported beam, span 2 m, has E=210 GPa, I=75×10⁶ mm⁴, a central point load 12 kN and uniform load 2 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.127 mm from the point load and 0.026 mm from the distributed load, totaling 0.153 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.05: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.06: Midspan deflection in a maintenance bracket

A simply supported beam, span 4 m, has E=210 GPa, I=50×10⁶ mm⁴, a central point load 10 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 1.270 mm from the point load and 0.000 mm from the distributed load, totaling 1.270 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.06: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.07: Midspan deflection in a temporary support

A simply supported beam, span 2 m, has E=210 GPa, I=50×10⁶ mm⁴, a central point load 10 kN and uniform load 2 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.159 mm from the point load and 0.040 mm from the distributed load, totaling 0.198 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.07: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.08: Midspan deflection in a pipe rack

A simply supported beam, span 4 m, has E=200 GPa, I=25×10⁶ mm⁴, a central point load 5 kN and uniform load 1 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 1.333 mm from the point load and 0.667 mm from the distributed load, totaling 2.000 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.08: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.09: Midspan deflection in a solar panel rail

A simply supported beam, span 3 m, has E=200 GPa, I=25×10⁶ mm⁴, a central point load 10 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 1.125 mm from the point load and 0.000 mm from the distributed load, totaling 1.125 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.09: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.10: Midspan deflection in a transit shelter

A simply supported beam, span 3 m, has E=200 GPa, I=25×10⁶ mm⁴, a central point load 8 kN and uniform load 1 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.900 mm from the point load and 0.211 mm from the distributed load, totaling 1.111 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.10: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.11: Midspan deflection in a lightweight crane hook

A simply supported beam, span 4 m, has E=200 GPa, I=25×10⁶ mm⁴, a central point load 3 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.800 mm from the point load and 0.000 mm from the distributed load, totaling 0.800 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.11: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.12: Midspan deflection in a water-tank frame

A simply supported beam, span 2 m, has E=200 GPa, I=50×10⁶ mm⁴, a central point load 12 kN and uniform load 3 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.200 mm from the point load and 0.062 mm from the distributed load, totaling 0.263 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.12: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.13: Midspan deflection in a portal-frame member

A simply supported beam, span 3 m, has E=200 GPa, I=120×10⁶ mm⁴, a central point load 12 kN and uniform load 1 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.281 mm from the point load and 0.044 mm from the distributed load, totaling 0.325 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.13: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.14: Midspan deflection in a cable termination

A simply supported beam, span 4 m, has E=210 GPa, I=120×10⁶ mm⁴, a central point load 3 kN and uniform load 1 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.159 mm from the point load and 0.132 mm from the distributed load, totaling 0.291 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.14: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.15: Midspan deflection in a maintenance walkway

A simply supported beam, span 6 m, has E=210 GPa, I=50×10⁶ mm⁴, a central point load 8 kN and uniform load 2 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 3.429 mm from the point load and 3.214 mm from the distributed load, totaling 6.643 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.15: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.16: Midspan deflection in a girder

A simply supported beam, span 5 m, has E=210 GPa, I=75×10⁶ mm⁴, a central point load 3 kN and uniform load 1 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.496 mm from the point load and 0.517 mm from the distributed load, totaling 1.013 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.16: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.17: Midspan deflection in a roof tie

A simply supported beam, span 4 m, has E=210 GPa, I=75×10⁶ mm⁴, a central point load 10 kN and uniform load 3 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.847 mm from the point load and 0.635 mm from the distributed load, totaling 1.481 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.17: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 6.18: Midspan deflection in a machine anchor

A simply supported beam, span 4 m, has E=210 GPa, I=75×10⁶ mm⁴, a central point load 8 kN and uniform load 0 kN/m. Convert I to m⁴ and use consistent N and m. Euler–Bernoulli midspan deflection is 0.677 mm from the point load and 0.000 mm from the distributed load, totaling 0.677 mm. The superposition applies only for elastic small-deflection response.

Extension task 6.18: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Engineering case study: Deflection-sensitive laboratory equipment platform

A steel supporting beam meets an initial bending stress target, yet an optical instrument suffers vibration and alignment errors. Increasing section modulus can help strength; increasing EI helps static deflection. Natural frequencies and dynamic excitation require an additional dynamics model. Floor finishes, partition details and instrument requirements may impose serviceability limits stricter than generic rule-of-thumb span limits.

Deliverable: Compute static midpoint deflection for point and distributed loads, compare different member depths, and identify a suitable measured acceptance criterion. Explain why elastic static deflection alone cannot establish satisfactory vibration performance.

Case-study discussion

Start by differentiating data from assumptions: measured loads and geometry should be separated from guessed restraints and material properties. Build one simple calculation whose result can be checked independently. List at least three mechanisms by which the idealization could fail, and prioritize them by whether they would change the engineering conclusion. Translate the numerical demand into a practical question about safety, serviceability, testing or maintenance. Keep a clear audit trail so a reviewer can follow the unit conversions, load path and model limitations.

Calculation verification

• Confirm that elastic deflection is zero at x=0 and x=L for ideal simple supports.
• At fixed EI, test point load scaling with L cubed and uniform load scaling with L to the fourth power.
• Check positivity, zero-load behavior, symmetry, and asymptotic scaling whenever applicable.
• Do not use a graph or slider response as the only evidence that an equation is correct.

Structured unit assessment

Answer the following without access to the provided worked solutions. Each question needs a diagram, assumptions and a clear interpretation.

1. Use a load or material value from Case 6.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 6.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 6.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 6.3 and compare with a limiting setting in any associated simulation.
5. A reviewer reports a one-thousandfold discrepancy. Show one plausible N–kN, m–mm or N·m–N·mm conversion error that could create it for a calculation in this unit.
6. For the case study in Unit 6, distinguish two possible failure modes and explain why demonstrating adequate performance against one does not guarantee the other.
7. Create an experimental plan to validate one of the numerical predictions in Unit 6; name the sensor, its calibration requirement and an expected source of uncertainty.
8. State two ways that manufacturing tolerance, support flexibility or temperature could alter a result computed from one equation in Unit 6.

Associated interactive laboratories

Laboratory 05: Elastic Beam Deflection. The plotted response combines a central point load and a full-span uniform load for a simply supported prismatic Euler–Bernoulli beam. Deflections superpose only in the adopted small-deflection elastic model.

Investigation: Double the span with other parameters unchanged. Compare the point-load and uniformly distributed load contributions; explain their different length exponents.

Open the [working HTML model](../labs/lab-05-deflection.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Laboratory 06: Section Geometry Studio. The section modulus relates bending stress to moment. Removing a centered fraction of the interior removes less stiffness than the same material taken from the outer fibers. The model uses a centered geometrically similar rectangular opening.

Investigation: Vary the opening ratio from zero to 0.5. Record relative area and second moment. Explain why a light section can remain efficient in bending despite reduced area.

Open the [working HTML model](../labs/lab-06-section.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Laboratory 13: Rectangular Beam Shear Distribution. For an ideal elastic prismatic rectangular beam, the transverse shear traction follows a parabola through depth; surface traction vanishes and the peak is 1.5 times the average V/A.

Investigation: Change depth while keeping section width and transverse force fixed. Compare changes in average and maximum shear, then discuss support-region limitations.

Open the [working HTML model](../labs/lab-13-beamshear.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Unit research project

Prepare a technical notebook on Flexure, shear stress and beam deflection for a civil engineering structure of your choice. Include a drawing with geometric dimensions, a carefully numbered input table and a free-body diagram. Develop two load or restraint scenarios, compute the most relevant stress or deformation variables, cross-check them with an independent limiting case, and identify all omissions that would prevent formal design approval. Where an associated laboratory exists, export a CSV series and include an annotated plot. Write a short conclusion that distinguishes what your simplified model genuinely predicts from what would need a professional code-based check.

Project marking rubric