Stress Transformation and Combined Loads

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

Stress is a tensor, which means the traction on a plane depends on its orientation. Mohr circle is a visual tool for this transformation, not a separate physical phenomenon. A principal plane has zero in-plane shear, while maximum in-plane shear occurs on another plane. The circle radius captures the combined effect of normal-stress difference and shear. In a uniaxial tensile coupon, a plane inclined at 45 degrees carries shear even though the original axial cross-section does not. That does not imply the coupon is externally twisted.

Combined loading illustrates how several simple models interact. A column subjected to eccentric compression experiences both axial force and bending, so one edge may become more compressed than the average and the other may be less compressed or even tensile. This may trigger cracking, instability or contact separation before a gross-section yield check is exceeded. Superposition assumes linear elastic response and no important change in geometry or boundary conditions. Once the section cracks or contact opens, redistribution is nonlinear.

Failure theories compress multiaxial demand into a scalar for a specified material failure mechanism. Von Mises equivalent stress is relevant to yielding of many isotropic ductile metals, but brittle fracture, reinforced concrete, fatigue and local instability have their own criteria. Even for ductile steel, design verification also involves buckling, weld detailing and adequate deformation capacity. Interpretation matters more than collecting formulas.

Lesson 7.1 — Stress at an inclined plane

Physical mechanism and governing idea

Normal and shear components depend on plane orientation. A plane-stress state defined by sigma_x, sigma_y and tau_xy can be rotated to find stresses on another surface. The transformation has twice-angle trigonometric terms because an oriented plane returns to the same physical plane after a half turn.

Governing relationship

sigma_theta=(sigma_x+sigma_y)/2+(sigma_x-sigma_y)/2 cos(2theta)+tau_xy sin(2theta)

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

Equal biaxial normal stresses with zero shear produce the same normal stress on every rotated plane.

What the simplified formula misses

Rotating the body sketch and rotating the coordinate axes use opposite angular conventions. State explicitly which one your equations implement.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.1 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 7.2 — Mohr circle and principal stresses

Physical mechanism and governing idea

Mohr circle provides a geometric map of normal and shear components on planes through a material point. The center is mean normal stress; the radius combines half-difference of normal stresses and in-plane shear. Principal planes have zero in-plane shear, and maximum in-plane shear equals the circle radius.

Governing relationship

C=(sigma_x+sigma_y)/2; R=sqrt(((sigma_x-sigma_y)/2)^2+tau_xy²)

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 sigma_x=80 MPa, sigma_y=20 MPa and tau_xy=40 MPa, principal stresses are approximately 100 and 0 MPa.

What the simplified formula misses

The in-plane maximum shear is not necessarily the absolute three-dimensional maximum unless the third principal stress is considered.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.2 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 7.3 — Combined axial load and bending

Physical mechanism and governing idea

When linear elasticity and small deformations hold, a concentric axial force and bending moment produce stresses that may be superposed. At a section, sigma=N/A ± M c/I. One face becomes more tensile while the opposite becomes more compressive. The location of maximum tension changes with the sign of the moment.

Governing relationship

sigma(y)=N/A - M y/I

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

An axial stress of 20 MPa plus or minus 35 MPa bending gives extreme fiber stresses of 55 MPa tension and 15 MPa compression under one sign convention.

What the simplified formula misses

The method assumes the load acts through a known section, plane sections remain plane and stress concentration is not controlling.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.3 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 7.4 — Equivalent stress and failure theories

Physical mechanism and governing idea

A ductile-metal yield screen often uses the von Mises equivalent stress computed from stress components. For plane stress, sigma_VM=sqrt(sigma_x²-sigma_x sigma_y+sigma_y²+3tau_xy²). The expression is not a universal fracture or concrete design criterion, and its interpretation depends on the material and loading state.

Governing relationship

sigma_VM = sqrt(sigma_x²-sigma_x sigma_y+sigma_y²+3 tau_xy²)

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

Pure shear of 50 MPa produces a von Mises equivalent stress near 86.6 MPa.

What the simplified formula misses

Brittle fracture, fatigue, buckling, weld details and nonmetallic failure mechanisms require other checks even if equivalent stress is below yield.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.4 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 7.01: Plane stress transformation at a maintenance bracket

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-60 MPa and τxy=45 MPa. Mohr-circle center is C=20.0 MPa and radius R=91.79 MPa. The two in-plane principal values are 111.79 and -71.79 MPa. Von Mises plane-stress equivalent is 160.23 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.01: 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 7.02: Plane stress transformation at a temporary support

At a selected material point, the estimated plane-stress components are σx=20 MPa, σy=20 MPa and τxy=20 MPa. Mohr-circle center is C=20.0 MPa and radius R=20.00 MPa. The two in-plane principal values are 40.00 and 0.00 MPa. Von Mises plane-stress equivalent is 40.00 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.02: 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 7.03: Plane stress transformation at a pipe rack

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=0 MPa. Mohr-circle center is C=20.0 MPa and radius R=30.00 MPa. The two in-plane principal values are 50.00 and -10.00 MPa. Von Mises plane-stress equivalent is 55.68 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.03: 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 7.04: Plane stress transformation at a solar panel rail

At a selected material point, the estimated plane-stress components are σx=0 MPa, σy=-10 MPa and τxy=45 MPa. Mohr-circle center is C=-5.0 MPa and radius R=45.28 MPa. The two in-plane principal values are 40.28 and -50.28 MPa. Von Mises plane-stress equivalent is 78.58 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.04: 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 7.05: Plane stress transformation at a transit shelter

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=0 MPa. Mohr-circle center is C=45.0 MPa and radius R=55.00 MPa. The two in-plane principal values are 100.00 and -10.00 MPa. Von Mises plane-stress equivalent is 105.36 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.05: 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 7.06: Plane stress transformation at a lightweight crane hook

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=45 MPa. Mohr-circle center is C=20.0 MPa and radius R=54.08 MPa. The two in-plane principal values are 74.08 and -34.08 MPa. Von Mises plane-stress equivalent is 95.79 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.06: 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 7.07: Plane stress transformation at a water-tank frame

At a selected material point, the estimated plane-stress components are σx=80 MPa, σy=0 MPa and τxy=20 MPa. Mohr-circle center is C=40.0 MPa and radius R=44.72 MPa. The two in-plane principal values are 84.72 and -4.72 MPa. Von Mises plane-stress equivalent is 87.18 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.07: 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 7.08: Plane stress transformation at a portal-frame member

At a selected material point, the estimated plane-stress components are σx=-40 MPa, σy=-60 MPa and τxy=20 MPa. Mohr-circle center is C=-50.0 MPa and radius R=22.36 MPa. The two in-plane principal values are -27.64 and -72.36 MPa. Von Mises plane-stress equivalent is 63.25 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.08: 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 7.09: Plane stress transformation at a cable termination

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=10 MPa. Mohr-circle center is C=45.0 MPa and radius R=55.90 MPa. The two in-plane principal values are 100.90 and -10.90 MPa. Von Mises plane-stress equivalent is 106.77 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.09: 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 7.10: Plane stress transformation at a maintenance walkway

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=20 MPa. Mohr-circle center is C=45.0 MPa and radius R=58.52 MPa. The two in-plane principal values are 103.52 and -13.52 MPa. Von Mises plane-stress equivalent is 110.91 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.10: 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 7.11: Plane stress transformation at a girder

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=35 MPa. Mohr-circle center is C=20.0 MPa and radius R=46.10 MPa. The two in-plane principal values are 66.10 and -26.10 MPa. Von Mises plane-stress equivalent is 82.31 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.11: 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 7.12: Plane stress transformation at a roof tie

At a selected material point, the estimated plane-stress components are σx=80 MPa, σy=20 MPa and τxy=10 MPa. Mohr-circle center is C=50.0 MPa and radius R=31.62 MPa. The two in-plane principal values are 81.62 and 18.38 MPa. Von Mises plane-stress equivalent is 74.16 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.12: 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 7.13: Plane stress transformation at a machine anchor

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=20 MPa and τxy=20 MPa. Mohr-circle center is C=35.0 MPa and radius R=25.00 MPa. The two in-plane principal values are 60.00 and 10.00 MPa. Von Mises plane-stress equivalent is 55.68 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.13: 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 7.14: Plane stress transformation at a laboratory frame

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=20 MPa and τxy=10 MPa. Mohr-circle center is C=35.0 MPa and radius R=18.03 MPa. The two in-plane principal values are 53.03 and 16.97 MPa. Von Mises plane-stress equivalent is 46.90 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.14: 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 7.15: Plane stress transformation at a small footbridge

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=35 MPa. Mohr-circle center is C=45.0 MPa and radius R=65.19 MPa. The two in-plane principal values are 110.19 and -20.19 MPa. Von Mises plane-stress equivalent is 121.55 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.15: 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 7.16: Plane stress transformation at a service gantry

At a selected material point, the estimated plane-stress components are σx=-40 MPa, σy=0 MPa and τxy=45 MPa. Mohr-circle center is C=-20.0 MPa and radius R=49.24 MPa. The two in-plane principal values are 29.24 and -69.24 MPa. Von Mises plane-stress equivalent is 87.61 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.16: 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 7.17: Plane stress transformation at a construction prop

At a selected material point, the estimated plane-stress components are σx=20 MPa, σy=-10 MPa and τxy=10 MPa. Mohr-circle center is C=5.0 MPa and radius R=18.03 MPa. The two in-plane principal values are 23.03 and -13.03 MPa. Von Mises plane-stress equivalent is 31.62 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.17: 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 7.18: Plane stress transformation at a testing rig

At a selected material point, the estimated plane-stress components are σx=0 MPa, σy=20 MPa and τxy=0 MPa. Mohr-circle center is C=10.0 MPa and radius R=10.00 MPa. The two in-plane principal values are 20.00 and 0.00 MPa. Von Mises plane-stress equivalent is 20.00 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.18: 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: Offset load on a column base plate

A nominal axial force arrives at a base plate with a measurable eccentricity. The force can be represented as an equivalent centric force and an applied moment about the section centroid. A linear stress distribution may show tension at one edge, but a concrete-bearing base with no tensile contact responds nonlinearly; the initial superposition no longer predicts the actual pressure distribution.

Deliverable: Calculate the elastic linear stress estimate, then state how a compression-only support changes equilibrium and contact area. Record the distinctions among material stress, plate bending and anchor forces.

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

• Check that the Mohr-circle radius squared equals ((σx−σy)/2)²+τxy².
• At a principal orientation, the transformed in-plane shear must vanish numerically.
• 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 7.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 7.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 7.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 7.3 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 7, 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 7; 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 7.

Associated interactive laboratories

Laboratory 07: Mohr’s Circle and Stress Rotation. Normal and shear tractions transform with twice the physical angle. The circle visual maps these components, with center mean stress and radius determined by normal difference and shear. The angle control represents an oriented outward plane normal.

Investigation: Set normal stresses equal and shear to zero. Observe the degenerate circle. Increase shear; predict the changes in principal stresses and principal direction.

Open the [working HTML model](../labs/lab-07-mohr.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 08: Combined Axial + Bending Stress. Axial stress N/A adds to an ideal linear bending distribution M y/I. The result can change sign across a member; the neutral axis moves when axial load is applied. The outer fibers carry the largest absolute bending contribution.

Investigation: At zero axial load, compare the top and bottom extreme stresses. Apply tensile axial load until one edge becomes entirely tensile, then explain the shift.

Open the [working HTML model](../labs/lab-08-combined.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 Stress transformation and combined loads 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

Unit 07 — Stress transformation and combined loads

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

Stress is a tensor, which means the traction on a plane depends on its orientation. Mohr circle is a visual tool for this transformation, not a separate physical phenomenon. A principal plane has zero in-plane shear, while maximum in-plane shear occurs on another plane. The circle radius captures the combined effect of normal-stress difference and shear. In a uniaxial tensile coupon, a plane inclined at 45 degrees carries shear even though the original axial cross-section does not. That does not imply the coupon is externally twisted.

Combined loading illustrates how several simple models interact. A column subjected to eccentric compression experiences both axial force and bending, so one edge may become more compressed than the average and the other may be less compressed or even tensile. This may trigger cracking, instability or contact separation before a gross-section yield check is exceeded. Superposition assumes linear elastic response and no important change in geometry or boundary conditions. Once the section cracks or contact opens, redistribution is nonlinear.

Failure theories compress multiaxial demand into a scalar for a specified material failure mechanism. Von Mises equivalent stress is relevant to yielding of many isotropic ductile metals, but brittle fracture, reinforced concrete, fatigue and local instability have their own criteria. Even for ductile steel, design verification also involves buckling, weld detailing and adequate deformation capacity. Interpretation matters more than collecting formulas.

Lesson 7.1 — Stress at an inclined plane

Physical mechanism and governing idea

Normal and shear components depend on plane orientation. A plane-stress state defined by sigma_x, sigma_y and tau_xy can be rotated to find stresses on another surface. The transformation has twice-angle trigonometric terms because an oriented plane returns to the same physical plane after a half turn.

Governing relationship

sigma_theta=(sigma_x+sigma_y)/2+(sigma_x-sigma_y)/2 cos(2theta)+tau_xy sin(2theta)

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

Equal biaxial normal stresses with zero shear produce the same normal stress on every rotated plane.

What the simplified formula misses

Rotating the body sketch and rotating the coordinate axes use opposite angular conventions. State explicitly which one your equations implement.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.1 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 7.2 — Mohr circle and principal stresses

Physical mechanism and governing idea

Mohr circle provides a geometric map of normal and shear components on planes through a material point. The center is mean normal stress; the radius combines half-difference of normal stresses and in-plane shear. Principal planes have zero in-plane shear, and maximum in-plane shear equals the circle radius.

Governing relationship

C=(sigma_x+sigma_y)/2; R=sqrt(((sigma_x-sigma_y)/2)^2+tau_xy²)

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 sigma_x=80 MPa, sigma_y=20 MPa and tau_xy=40 MPa, principal stresses are approximately 100 and 0 MPa.

What the simplified formula misses

The in-plane maximum shear is not necessarily the absolute three-dimensional maximum unless the third principal stress is considered.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.2 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 7.3 — Combined axial load and bending

Physical mechanism and governing idea

When linear elasticity and small deformations hold, a concentric axial force and bending moment produce stresses that may be superposed. At a section, sigma=N/A ± M c/I. One face becomes more tensile while the opposite becomes more compressive. The location of maximum tension changes with the sign of the moment.

Governing relationship

sigma(y)=N/A - M y/I

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

An axial stress of 20 MPa plus or minus 35 MPa bending gives extreme fiber stresses of 55 MPa tension and 15 MPa compression under one sign convention.

What the simplified formula misses

The method assumes the load acts through a known section, plane sections remain plane and stress concentration is not controlling.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.3 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 7.4 — Equivalent stress and failure theories

Physical mechanism and governing idea

A ductile-metal yield screen often uses the von Mises equivalent stress computed from stress components. For plane stress, sigma_VM=sqrt(sigma_x²-sigma_x sigma_y+sigma_y²+3tau_xy²). The expression is not a universal fracture or concrete design criterion, and its interpretation depends on the material and loading state.

Governing relationship

sigma_VM = sqrt(sigma_x²-sigma_x sigma_y+sigma_y²+3 tau_xy²)

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

Pure shear of 50 MPa produces a von Mises equivalent stress near 86.6 MPa.

What the simplified formula misses

Brittle fracture, fatigue, buckling, weld details and nonmetallic failure mechanisms require other checks even if equivalent stress is below yield.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 7.4 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 7.01: Plane stress transformation at a maintenance bracket

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-60 MPa and τxy=45 MPa. Mohr-circle center is C=20.0 MPa and radius R=91.79 MPa. The two in-plane principal values are 111.79 and -71.79 MPa. Von Mises plane-stress equivalent is 160.23 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.01: 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 7.02: Plane stress transformation at a temporary support

At a selected material point, the estimated plane-stress components are σx=20 MPa, σy=20 MPa and τxy=20 MPa. Mohr-circle center is C=20.0 MPa and radius R=20.00 MPa. The two in-plane principal values are 40.00 and 0.00 MPa. Von Mises plane-stress equivalent is 40.00 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.02: 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 7.03: Plane stress transformation at a pipe rack

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=0 MPa. Mohr-circle center is C=20.0 MPa and radius R=30.00 MPa. The two in-plane principal values are 50.00 and -10.00 MPa. Von Mises plane-stress equivalent is 55.68 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.03: 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 7.04: Plane stress transformation at a solar panel rail

At a selected material point, the estimated plane-stress components are σx=0 MPa, σy=-10 MPa and τxy=45 MPa. Mohr-circle center is C=-5.0 MPa and radius R=45.28 MPa. The two in-plane principal values are 40.28 and -50.28 MPa. Von Mises plane-stress equivalent is 78.58 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.04: 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 7.05: Plane stress transformation at a transit shelter

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=0 MPa. Mohr-circle center is C=45.0 MPa and radius R=55.00 MPa. The two in-plane principal values are 100.00 and -10.00 MPa. Von Mises plane-stress equivalent is 105.36 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.05: 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 7.06: Plane stress transformation at a lightweight crane hook

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=45 MPa. Mohr-circle center is C=20.0 MPa and radius R=54.08 MPa. The two in-plane principal values are 74.08 and -34.08 MPa. Von Mises plane-stress equivalent is 95.79 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.06: 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 7.07: Plane stress transformation at a water-tank frame

At a selected material point, the estimated plane-stress components are σx=80 MPa, σy=0 MPa and τxy=20 MPa. Mohr-circle center is C=40.0 MPa and radius R=44.72 MPa. The two in-plane principal values are 84.72 and -4.72 MPa. Von Mises plane-stress equivalent is 87.18 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.07: 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 7.08: Plane stress transformation at a portal-frame member

At a selected material point, the estimated plane-stress components are σx=-40 MPa, σy=-60 MPa and τxy=20 MPa. Mohr-circle center is C=-50.0 MPa and radius R=22.36 MPa. The two in-plane principal values are -27.64 and -72.36 MPa. Von Mises plane-stress equivalent is 63.25 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.08: 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 7.09: Plane stress transformation at a cable termination

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=10 MPa. Mohr-circle center is C=45.0 MPa and radius R=55.90 MPa. The two in-plane principal values are 100.90 and -10.90 MPa. Von Mises plane-stress equivalent is 106.77 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.09: 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 7.10: Plane stress transformation at a maintenance walkway

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=20 MPa. Mohr-circle center is C=45.0 MPa and radius R=58.52 MPa. The two in-plane principal values are 103.52 and -13.52 MPa. Von Mises plane-stress equivalent is 110.91 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.10: 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 7.11: Plane stress transformation at a girder

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=-10 MPa and τxy=35 MPa. Mohr-circle center is C=20.0 MPa and radius R=46.10 MPa. The two in-plane principal values are 66.10 and -26.10 MPa. Von Mises plane-stress equivalent is 82.31 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.11: 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 7.12: Plane stress transformation at a roof tie

At a selected material point, the estimated plane-stress components are σx=80 MPa, σy=20 MPa and τxy=10 MPa. Mohr-circle center is C=50.0 MPa and radius R=31.62 MPa. The two in-plane principal values are 81.62 and 18.38 MPa. Von Mises plane-stress equivalent is 74.16 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.12: 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 7.13: Plane stress transformation at a machine anchor

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=20 MPa and τxy=20 MPa. Mohr-circle center is C=35.0 MPa and radius R=25.00 MPa. The two in-plane principal values are 60.00 and 10.00 MPa. Von Mises plane-stress equivalent is 55.68 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.13: 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 7.14: Plane stress transformation at a laboratory frame

At a selected material point, the estimated plane-stress components are σx=50 MPa, σy=20 MPa and τxy=10 MPa. Mohr-circle center is C=35.0 MPa and radius R=18.03 MPa. The two in-plane principal values are 53.03 and 16.97 MPa. Von Mises plane-stress equivalent is 46.90 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.14: 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 7.15: Plane stress transformation at a small footbridge

At a selected material point, the estimated plane-stress components are σx=100 MPa, σy=-10 MPa and τxy=35 MPa. Mohr-circle center is C=45.0 MPa and radius R=65.19 MPa. The two in-plane principal values are 110.19 and -20.19 MPa. Von Mises plane-stress equivalent is 121.55 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.15: 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 7.16: Plane stress transformation at a service gantry

At a selected material point, the estimated plane-stress components are σx=-40 MPa, σy=0 MPa and τxy=45 MPa. Mohr-circle center is C=-20.0 MPa and radius R=49.24 MPa. The two in-plane principal values are 29.24 and -69.24 MPa. Von Mises plane-stress equivalent is 87.61 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.16: 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 7.17: Plane stress transformation at a construction prop

At a selected material point, the estimated plane-stress components are σx=20 MPa, σy=-10 MPa and τxy=10 MPa. Mohr-circle center is C=5.0 MPa and radius R=18.03 MPa. The two in-plane principal values are 23.03 and -13.03 MPa. Von Mises plane-stress equivalent is 31.62 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.17: 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 7.18: Plane stress transformation at a testing rig

At a selected material point, the estimated plane-stress components are σx=0 MPa, σy=20 MPa and τxy=0 MPa. Mohr-circle center is C=10.0 MPa and radius R=10.00 MPa. The two in-plane principal values are 20.00 and 0.00 MPa. Von Mises plane-stress equivalent is 20.00 MPa. Principal stresses and equivalent yielding checks answer different questions; a brittle-material fracture model may require other criteria.

Extension task 7.18: 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: Offset load on a column base plate

A nominal axial force arrives at a base plate with a measurable eccentricity. The force can be represented as an equivalent centric force and an applied moment about the section centroid. A linear stress distribution may show tension at one edge, but a concrete-bearing base with no tensile contact responds nonlinearly; the initial superposition no longer predicts the actual pressure distribution.

Deliverable: Calculate the elastic linear stress estimate, then state how a compression-only support changes equilibrium and contact area. Record the distinctions among material stress, plate bending and anchor forces.

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

• Check that the Mohr-circle radius squared equals ((σx−σy)/2)²+τxy².
• At a principal orientation, the transformed in-plane shear must vanish numerically.
• 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 7.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 7.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 7.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 7.3 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 7, 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 7; 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 7.

Associated interactive laboratories

Laboratory 07: Mohr’s Circle and Stress Rotation. Normal and shear tractions transform with twice the physical angle. The circle visual maps these components, with center mean stress and radius determined by normal difference and shear. The angle control represents an oriented outward plane normal.

Investigation: Set normal stresses equal and shear to zero. Observe the degenerate circle. Increase shear; predict the changes in principal stresses and principal direction.

Open the [working HTML model](../labs/lab-07-mohr.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 08: Combined Axial + Bending Stress. Axial stress N/A adds to an ideal linear bending distribution M y/I. The result can change sign across a member; the neutral axis moves when axial load is applied. The outer fibers carry the largest absolute bending contribution.

Investigation: At zero axial load, compare the top and bottom extreme stresses. Apply tensile axial load until one edge becomes entirely tensile, then explain the shift.

Open the [working HTML model](../labs/lab-08-combined.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 Stress transformation and combined loads 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

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