Static Strength and Failure Theories

Introduction

Machine components are commonly subjected to loads that are constant, slowly applied, or sufficiently slow that inertial effects are negligible. Such loading is treated as static loading. Static-strength design ensures that a component does not yield excessively, fracture, buckle, crush, or suffer unacceptable local contact damage during its intended service.

Static design is not simply a comparison between one calculated stress and one material strength. Real components may experience combined tension, compression, bending, torsion, bearing pressure, stress concentrations, residual stress, surface defects, and uncertainty in loading or material properties. The designer must therefore select a suitable failure criterion, determine a defensible factor of safety, and verify all relevant failure modes.

The distinction between ductile and brittle materials is central. Ductile materials usually undergo substantial plastic deformation before fracture; yielding is therefore the usual design limit. Brittle materials may fracture with little warning, so fracture strength and the state of principal stresses are more important.

Learning Objectives

After studying this chapter, the student should be able to:

1. Distinguish normal, shear, principal, bearing, crushing, and contact stresses.
2. Explain the assumptions and applicability of the maximum-stress, maximum-shear-stress, and distortion-energy theories.
3. Calculate principal stresses and use failure criteria for ductile and brittle components.
4. Design against yielding, fracture, buckling, crushing, bearing, and contact failure.
5. Select a defensible factor of safety and recognize the limitations of idealized theories.
6. Apply a systematic static-strength design process to a machine component.

1. Stress, Strength, and Failure

Stress is the intensity of internal force. For a uniform axial load, the average normal stress is

where σ is normal stress, P is axial force, and A is the loaded cross-sectional area. Tensile stress is conventionally positive and compressive stress negative. The equation assumes a prismatic member, an axial load through the centroid, and a sufficiently uniform stress distribution away from load introduction regions.

Average direct shear stress is

where τ is average shear stress and V is transverse shear force. Actual shear stress may be nonuniform. For example, in a rectangular beam the maximum transverse shear stress is 1.5 times the average value.

For a circular shaft under torque, the torsion relation is

where T is torque, r is radial distance, and J is the polar second moment of area. For a solid circular shaft of diameter d, J = πd⁴/32, and the maximum torsional stress at the surface is τ_max = 16T/(πd³). These relations assume linear-elastic, Saint-Venant torsion of a prismatic circular shaft away from discontinuities.

For a beam subject to bending moment M,

where y is distance from the neutral axis and I is the second moment of area. The maximum bending stress occurs at the largest absolute value of y.

Stress is measured in pascals, usually megapascals (MPa), where 1 MPa = 1 N/mm². Strength must be expressed in the same units as stress.

Stress concentration

A geometric discontinuity such as a keyway, shoulder, hole, thread, or sharp corner changes the local stress distribution. The theoretical stress-concentration factor is

for normal stress, with an analogous definition for shear. Here σ_nom is the nominal stress calculated using the unmodified geometry. In elastic analysis, K_t describes the local peak stress. For fatigue design, a fatigue notch factor is usually required; a static ductile component may redistribute stress through local yielding, so blindly applying a fatigue factor can be overly conservative or inappropriate.

Factor of safety

A factor of safety compares an available strength with the design demand. For a simple yield check,

where n is the factor of safety, S_y is yield strength, and σ_eq is an equivalent stress from a selected failure theory. Design is acceptable when n is at least the specified required value. The chosen value must reflect uncertainty in loading, material data, manufacturing quality, consequences of failure, inspection, and applicable standards. A factor of safety is not a substitute for accurate modelling.

2. Combined Stress and Principal Stresses

A general three-dimensional stress state contains three normal stresses and three shear stresses. At any point, there exist three mutually perpendicular planes on which shear stress is zero. The normal stresses on these planes are the principal stresses, ordered as σ₁ ≥ σ₂ ≥ σ₃.

For plane stress with normal stresses σ_x and σ_y and shear stress τ_xy,

The maximum in-plane shear stress is

For plane stress, the third principal stress is σ₃ = 0, but it must still be included when determining the absolute maximum shear stress. The principal-stress equations assume small deformation and a stress state represented at one material point.

A useful invariant is the hydrostatic stress, the mean normal stress:

Hydrostatic stress changes volume but, in an ideal isotropic ductile metal, does not directly cause yielding. Deviatoric stress changes shape and is primarily responsible for yielding.

3. Failure Theories for Ductile Materials

3.1 Maximum principal-stress theory

The maximum principal-stress theory, often associated with Rankine, predicts yielding when the largest absolute principal stress reaches the uniaxial yield strength:

For ductile metals this theory is simple but generally conservative or inaccurate under combined loading because it ignores the interaction among principal stresses. It is more appropriate as a screening check or for materials whose behaviour is strongly governed by normal stress.

3.2 Maximum-shear-stress theory

The maximum-shear-stress theory, or Tresca theory, predicts yielding when the maximum shear stress equals the shear stress at yielding in a uniaxial tensile test. Since the latter is approximately S_y/2,

and the equivalent stress for design is

Yielding is predicted when σ_eq,Tresca ≥ S_y. Tresca is independent of hydrostatic stress and is generally conservative for ductile metals. It is particularly useful when a conservative yield boundary is desired.

3.3 Distortion-energy theory

The distortion-energy theory, or von Mises theory, predicts yielding when the distortion energy per unit volume equals that at uniaxial yield. Its equivalent stress is

Yielding is predicted when σ_eq,VM ≥ S_y. For plane stress with σ₃ = 0,

For uniaxial stress, both Tresca and von Mises reduce to the applied normal stress. Under pure shear, Tresca predicts yielding at τ = S_y/2, while von Mises predicts τ = S_y/√3. Thus von Mises generally predicts a slightly higher allowable pure shear stress and is often a good representation of isotropic ductile metals under static loading.

Comparison

4. Failure of Brittle Materials

Brittle materials such as cast iron, ceramics, glass, and some hardened materials have limited plastic deformation. Their tensile and compressive strengths may differ substantially. A single ductile yield criterion is therefore inappropriate.

A basic maximum-principal-stress approach compares tensile principal stresses with tensile strength and compressive principal stresses with compressive strength. If S_ut is ultimate tensile strength and S_uc is ultimate compressive strength, a simple safety check is

for a positive tensile principal stress, and

for a compressive principal stress, provided the stress signs are interpreted consistently and the relevant principal stress is the governing one. This simple approach is most defensible when one principal stress dominates.

For combined tensile and compressive principal stresses, the modified Mohr approach is often used because it accounts for unequal tensile and compressive strengths. Its failure envelope is constructed in principal-stress space from experimentally determined strengths. Because brittle fracture is sensitive to flaws, surface finish, size, loading rate, residual stress, and environmental conditions, material-specific test data and design standards should take precedence over a generic envelope.

Fracture mechanics may be required when a crack or crack-like defect exists. The mode-I stress-intensity factor is commonly written as

where Y is a dimensionless geometry factor, σ is applied tensile stress, and a is crack size. Crack growth or fracture is assessed by comparing K_I with fracture toughness K_IC, under the assumptions of linear-elastic fracture mechanics and an appropriate crack geometry. A nominal-strength calculation cannot replace fracture assessment when defects are significant.

5. Local and Instability Failure Modes

Bearing and crushing

Bearing stress is a local compressive contact stress, such as the pressure between a pin and a hole. A commonly used average estimate is

where P is transmitted force and A_b is projected bearing area. For a pin of diameter d in a plate of thickness t, A_b = dt, so

This is an average pressure, not the exact Hertzian pressure distribution. The design value must be compared with an allowable bearing or crushing stress obtained from material data, standards, or testing. Net-section tension, shear-out at an edge, pin bending, and hole elongation must also be checked.

Contact stress

When curved bodies contact over a small area, the local stress can be much greater than the nominal force divided by the apparent area. Hertz contact theory is used for elastic, smooth, nonconforming contacts. For two spheres, the effective radius is

with signs selected according to the contact geometry. The effective elastic modulus is

where E₁, E₂ are Young’s moduli and ν₁, ν₂ are Poisson ratios. Detailed Hertz formulas depend on whether contact is spherical, cylindrical, or elliptical. In gear, cam, bearing, and wheel applications, contact stress is compared with an allowable value that includes material hardness, lubrication, surface finish, and life requirements.

Column buckling

A slender component in compression may fail by lateral instability before compressive yielding. For an ideal pin-ended, slender, straight column, Euler’s critical load is

where E is Young’s modulus, I is the least second moment of area, L is unsupported length, and K is the effective-length factor. The slenderness ratio is

where r_g = √(I/A) is the radius of gyration. Euler theory assumes elastic behaviour, an initially straight member, ideal end conditions, concentric loading, and negligible residual stress. It is not valid for short or intermediate columns that yield or experience inelastic instability. Real design uses code-based column curves or the Johnson parabolic relation where applicable.

A compression member should be checked for both crushing and buckling. The lower allowable load governs. Eccentricity produces bending, and imperfections can substantially reduce capacity.

6. Worked Example: Ductile Shaft Under Bending and Torsion

A solid steel shaft of diameter 30 mm carries a bending moment of 180 N·m and torque of 240 N·m at a critical section. The yield strength is 360 MPa. Determine the von Mises factor of safety, neglecting stress concentration.

Convert moments to N·mm: M = 180,000 N·mm and T = 240,000 N·mm. For a solid circular shaft, the outer-surface bending stress is

and the torsional shear stress is

Since d³ = 30³ = 27,000 mm³,

σ_b = 32(180,000)/(π×27,000) ≈ 67.9 MPa

and

τ_t = 16(240,000)/(π×27,000) ≈ 45.3 MPa.

At the surface, take σ_x = 67.9 MPa, σ_y = 0, and τ_xy = 45.3 MPa. The plane-stress von Mises equation gives

Therefore,

σ_eq = √(67.9² + 3(45.3)²) ≈ 112.9 MPa.

The factor of safety is

n = 360/112.9 ≈ 3.19.

Thus, under the stated idealized assumptions, the shaft has a von Mises static yield factor of safety of approximately 3.2. A final design must include keyway or shoulder stress concentration, surface condition, overloads, fatigue, deflection, and any applicable standard.

7. Worked Example: Pin Bearing and Net-Section Checks

A 12 mm diameter steel pin transmits 18 kN through a 10 mm thick plate. The plate is 40 mm wide and contains a central hole. Its tensile yield strength is 250 MPa. Assume an allowable average bearing stress of 150 MPa and an allowable net-section tensile stress of 125 MPa. Determine whether the plate passes these two checks.

The projected bearing area is

A_b = dt = 12×10 = 120 mm².

Thus,

p_b = 18,000/120 = 150 MPa.

The bearing check is exactly at the stated allowable and therefore passes only if the allowable is not required to be exceeded and all assumptions are satisfied.

The net tensile area through the hole is

A_net = (w − d)t = (40 − 12)×10 = 280 mm².

The average net-section tensile stress is

σ_net = 18,000/280 ≈ 64.3 MPa.

Since 64.3 MPa is less than 125 MPa, the net-section check passes. The design still requires checks for tear-out, pin shear, pin bending, hole deformation, edge distance, and local stress concentration.

8. Applied Case: Support Bracket for a Machine Actuator

Background

The following is an illustrative engineering case. A fabricated steel bracket supports an actuator that applies a 20 kN force through a clevis pin. The bracket has two 8 mm thick lugs, a 16 mm pin, and an edge distance of 24 mm from the hole centre to the free edge. The material has yield strength 280 MPa and ultimate tensile strength 430 MPa. The actuator force is inclined, producing both direct shear and a bending moment in the lug.

Problem definition

A preliminary design based only on average pin bearing stress appears adequate. However, the bracket may fail by bearing, net-section tension, shear-out, pin bending, or yielding at the lug root. The design objective is not merely to show that one stress is below yield; it is to identify the governing mode with a suitable margin.

Analysis and approach

The engineer first constructs a free-body diagram and resolves the actuator force into components. Each lug is assigned its share of the load only if symmetry and stiffness justify that assumption. The projected bearing area per lug is A_b = dt, with t = 8 mm and d = 16 mm. If the load is shared equally, each lug carries 10 kN and the average bearing stress is

p_b = 10,000/(16×8) = 78.1 MPa.

This value is compared with the allowable bearing stress, not automatically with the tensile yield strength. The net-section area per lug is A_net = (w−d)t, where w is the lug width. If the lug width is 40 mm, A_net = (40−16)×8 = 192 mm², giving an average net tension of approximately 52.1 MPa per lug.

The edge tear-out area is estimated from the two shear planes extending from the hole to the edge. Its adequacy depends on the material shear strength, edge distance, and load direction. A finite-element model may then be used to locate stress concentrations at the hole and lug root, but the model must be verified using mesh refinement, realistic constraints, and hand calculations. A highly peaked elastic stress at a sharp, idealized corner is not automatically a physical failure prediction.

Because the bracket is ductile, von Mises stress is appropriate for general yielding, while Tresca may be used as a conservative comparison. The pin is checked separately for double shear, bending, bearing, and combined stress. Welds, if present, require weld-group analysis and inspection requirements.

Outcome and learning points

The illustrative design is acceptable only when every relevant limit state meets the required safety margin. The governing mode may be local yielding at the lug root rather than average pin bearing. The case demonstrates four professional practices: define load paths, check multiple failure modes, distinguish average from peak stress, and validate computational results with mechanics-based calculations.

9. Selecting a Design Method and Safety Margin

A defensible design workflow is:

1. Define loads, supports, load paths, operating environment, and required life.
2. Identify critical sections and all plausible failure modes.
3. Calculate nominal stresses and apply appropriate elastic concentration factors.
4. Determine principal stresses or an equivalent stress at critical points.
5. Select the material model: ductile yielding, brittle fracture, contact, crushing, or buckling.
6. Apply a suitable failure theory and compare with allowable strength.
7. Check deflection, permanent set, clearances, and function as well as strength.
8. Validate assumptions by testing, standards, peer review, or numerical analysis.

Use von Mises for many isotropic ductile metals under static multiaxial loading; use Tresca when a conservative yield estimate is preferred. For brittle materials, compare tensile and compressive principal stresses with appropriate strengths and consider fracture mechanics for known flaws. For columns, use an elastic or inelastic buckling model based on slenderness. For contacts, use bearing or Hertz theory only within their assumptions.

Safety factors should be selected in accordance with applicable codes and organizational practice. High consequence, poorly known, variable, or difficult-to-inspect systems require greater reliability assurance, but increasing n does not correct a wrong load path or an invalid failure model.

10. Limitations, Standards, and Contemporary Practice

Ideal failure theories assume homogeneous, isotropic materials and accurately known stresses. Actual components may be anisotropic, residual-stressed, temperature-dependent, corroded, damaged, or assembled with interference. Static strength may also interact with fatigue, creep, wear, vibration, and thermal stress.

Finite-element analysis is valuable for complex geometry, but it should not replace free-body diagrams, convergence studies, material-model selection, or independent checks. Digital image correlation, strain gauges, load testing, and non-destructive inspection can provide evidence about real behaviour. AI-assisted design and topology optimization can generate efficient geometries, but the engineer remains responsible for boundary conditions, manufacturability, explainability, material traceability, and compliance with standards. Safety-critical designs require documented review and configuration control.

Chapter Summary

Static-strength design protects machine components against yielding, fracture, buckling, crushing, bearing damage, and contact failure. Normal and shear stresses are first determined from equilibrium and compatibility, then combined through principal stresses or an equivalent-stress measure. Tresca uses maximum shear stress and is conservative for ductile metals; von Mises uses distortion energy and is widely applicable to isotropic ductile materials. Maximum principal stress is simple but less reliable for ductile multiaxial states and is useful for screening or brittle-oriented assessment.

Brittle components require separate consideration of tensile and compressive strengths, modified-Mohr-type envelopes, and fracture mechanics when cracks are present. Local bearing and crushing stresses, Hertzian contact stresses, and column buckling are distinct limit states that cannot be replaced by a nominal tensile-stress check. A professional design identifies every credible failure mode, uses consistent units and defensible safety margins, and validates analytical or computational results.

Key Terms

Stress: Internal force per unit area.

Normal stress: Stress acting perpendicular to a plane.

Shear stress: Stress acting tangentially to a plane.

Principal stress: Normal stress on a plane where shear stress is zero.

Hydrostatic stress: Mean normal stress that primarily changes volume.

Deviatoric stress: Stress component associated mainly with shape change.

Yield strength: Stress at which specified permanent deformation begins.

Ultimate strength: Maximum nominal stress reached in a tensile or compressive test.

Factor of safety: Ratio of available strength to calculated design demand.

Stress concentration: Local stress amplification caused by geometry.

Tresca theory: Ductile-yield criterion based on maximum shear stress.

von Mises theory: Ductile-yield criterion based on distortion energy.

Bearing stress: Average local compressive pressure between contacting parts.

Crushing: Local compressive failure caused by excessive contact pressure.

Hertz contact stress: Elastic local stress in curved-body contact.

Buckling: Instability involving lateral deformation of a compression member.

Slenderness ratio: Effective length divided by radius of gyration.

Fracture toughness: Resistance to crack extension under specified conditions.

Net-section stress: Average stress over the remaining area beside a hole or notch.

Limit state: Specific condition representing unacceptable performance or failure.

Review Questions

Short Answer Questions

1. Why are principal stresses useful in static-strength design?
2. State the main physical distinction between ductile and brittle failure.
3. What does the factor of safety represent?
4. Why must a column be checked for both crushing and buckling?
5. Distinguish bearing stress from Hertz contact stress.

Long Answer Questions

1. Explain the assumptions, equations, and relative conservatism of Tresca and von Mises theories.
2. Derive the plane-stress von Mises equation from the three-dimensional principal-stress form.
3. Describe a complete design procedure for a pin-connected bracket.
4. Explain why maximum principal stress is not generally the preferred theory for ductile metals under combined loading.
5. Discuss the limitations of Euler buckling theory and the factors that affect real column capacity.

Analytical / Application Questions

1. A solid 25 mm diameter shaft carries M = 120 N·m and T = 200 N·m. For a material with S_y = 300 MPa, calculate the surface bending stress, torsional shear stress, von Mises stress, and yield factor of safety.
2. A 14 mm pin transmits 24 kN through a 12 mm plate. Determine the average bearing stress and state what additional failure modes must be checked before approving the connection.
3. A compression member has E = 200 GPa, I = 1.5×10⁶ mm⁴, effective-length factor K = 1, and length 2 m. Calculate the Euler critical load, identify the assumptions involved, and explain why the result may not govern a stocky member.

Multiple Choice Questions with Answers

Question 1

Which failure theory is based on the maximum difference between principal stresses?

A. Maximum principal-stress theory

B. Maximum-shear-stress theory

C. Distortion-energy theory

D. Hertz contact theory

Correct Answer: B. Maximum-shear-stress theory

Question 2

For an isotropic ductile metal under general static multiaxial loading, which theory is commonly preferred?

A. von Mises distortion-energy theory

B. Euler buckling theory

C. Modified Mohr theory only

D. Maximum principal-strain theory

Correct Answer: A. von Mises distortion-energy theory

Question 3

A 20 mm diameter solid shaft carries a torque of 100 N·m. What is the approximate maximum torsional shear stress?

A. 15.9 MPa

B. 31.8 MPa

C. 63.7 MPa

D. 127.3 MPa

Correct Answer: C. 63.7 MPa

Question 4

The projected bearing area of a pin in a plate is calculated as:

A. πd²/4

B. dt

C. 2πdt

D. d/t

Correct Answer: B. dt

Question 5

Euler’s critical-load equation is most applicable to a:

A. Short, heavily crushed column

B. Slender, elastic column

C. Brittle plate under tension

D. Surface contact pair

Correct Answer: B. Slender, elastic column

Question 6

Under pure shear, the Tresca criterion predicts yielding when the shear stress reaches approximately:

A. S_y

B. S_y/4

C. S_y/2

D. S_y/√3 only for all theories

Correct Answer: C. S_y/2

Question 7

Which stress component does not directly cause yielding in the ideal von Mises model for an isotropic ductile metal?

A. Hydrostatic stress

B. Deviatoric stress

C. Bending stress

D. Torsional stress

Correct Answer: A. Hydrostatic stress

Question 8

A 10 kN load acts on a 5 mm thick plate through a 20 mm diameter pin. What is the average bearing stress?

A. 50 MPa

B. 100 MPa

C. 200 MPa

D. 400 MPa

Correct Answer: B. 100 MPa

Question 9

For a brittle material with substantially different tensile and compressive strengths, which approach is more appropriate than assuming equal yield strengths?

A. Modified Mohr or strength-specific principal-stress assessment

B. Euler buckling alone

C. Average shear stress alone

D. The circular-shaft torsion equation alone

Correct Answer: A. Modified Mohr or strength-specific principal-stress assessment

Question 10

The radius of gyration used in column slenderness is defined by:

A. r_g = A/I

B. r_g = √(I/A)

C. r_g = I/A²

D. r_g = √(A/I)

Correct Answer: B. r_g = √(I/A)