Failure Criteria and Design Factors

Introduction

A component rarely fails because a single stress value exceeds a single material property. Real design involves multiaxial loading, imperfect knowledge of loads and dimensions, manufacturing variation, residual stress, corrosion, fatigue, temperature, and uncertainty in material data. A failure criterion provides a rational rule for converting a complex stress state into a prediction of yielding or another specified limit state. A factor of safety provides additional design margin between the predicted service condition and the allowable condition.

This chapter develops four classical criteria used primarily for ductile materials: maximum principal stress, maximum principal strain, Tresca maximum shear stress, and von Mises distortion-energy theory. It then explains the selection of factors of safety using uncertainty, reliability, material variability, consequences of failure, and design codes. The emphasis is on choosing a criterion appropriate to the failure mechanism rather than applying a formula mechanically.

Learning objectives

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

• describe the purpose and assumptions of common failure criteria;
• calculate principal stresses, maximum shear stress, and equivalent stresses;
• apply maximum-stress, maximum-strain, Tresca, and von Mises criteria;
• distinguish yielding, fracture, buckling, fatigue, creep, and wear as different limit states;
• select and justify a factor of safety using uncertainty, reliability, variability, consequence, and codes;
• communicate a defensible design decision in a professional engineering context.

1. Failure is a limit-state problem

A failure criterion specifies a boundary between acceptable and unacceptable states. For a ductile metal under static loading, the relevant limit state is commonly initial yielding. For a brittle material, fracture may occur with little plastic deformation, so a tensile or compressive stress criterion is often more appropriate. Under repeated loading, fatigue life—not static yield—controls. Slender compression members may buckle while their stresses are below the yield strength. Thus, “failure” must first be defined.

Typical limit states include:

The stress state at a point is represented by three principal stresses, σ₁, σ₂, and σ₃, ordered so that σ₁ ≥ σ₂ ≥ σ₃. Principal stresses are normal stresses on planes where shear stress is zero. They may be tensile or compressive. The maximum shear stress is

where all stresses use the same units. This expression is valid for a three-dimensional stress state after the principal stresses have been identified. In plane stress, the third principal stress is normally σ₃ = 0, but it must still be included when ordering the values.

For a plane-stress element with normal stresses σ_x, σ_y and shear stress τ_xy, the principal stresses are

The equation assumes small deformation and a correctly defined Cartesian stress state. It is especially useful for thin plates and other situations in which the out-of-plane stress is negligible.

2. Maximum principal-stress theory

The maximum principal-stress theory, often called Rankine theory, predicts failure when the largest tensile or compressive principal stress reaches the corresponding uniaxial strength. For a material with tensile yield strength S_yt and compressive yield strength S_yc, the safe conditions are

where n is the factor of safety. Equivalently, the design stress ratio is

for positive tensile and compressive demand values. If one of the relevant stresses is not tensile or compressive, that branch is not controlling.

Rankine theory is simple and often conservative for brittle materials, in which normal stress is closely related to crack opening. It is generally unsuitable as the primary yield criterion for ductile metals under combined loading because ductile yielding is strongly influenced by shear and deviatoric stress. A hydrostatic stress can be large without causing yielding in an ideal ductile metal, whereas Rankine theory may predict failure from the principal stress magnitude alone.

3. Maximum principal-strain theory

The maximum principal-strain theory predicts yielding when the greatest principal strain reaches the uniaxial yield strain. In linear isotropic elasticity,

and similarly by cyclic permutation for ε₂ and ε₃, where E is Young’s modulus and ν is Poisson’s ratio. The uniaxial yield strain is

for an elastic, isotropic material with tensile yield strength S_y. The maximum-strain criterion is therefore

or, in stress form for equal tensile and compressive yield strength,

where i, j, and k are distinct indices.

This criterion accounts for Poisson contraction and can be useful when deformation itself is the design concern. However, it is not a generally accepted modern yield law for ductile metals. It depends on elastic constitutive behavior and may give misleading predictions when plastic flow begins. It should not be confused with a serviceability requirement that limits total deflection or strain to protect function.

4. Tresca maximum-shear-stress theory

Tresca theory states that yielding begins when the maximum shear stress reaches the shear stress at yield in a uniaxial tensile test. In uniaxial tension, the principal stresses are S_y, 0, and 0; hence the maximum shear stress at yield is S_y/2. The criterion is

or

for equal tensile and compressive yield strengths.

Tresca is a maximum-deviatoric-stress criterion. Hydrostatic pressure, which adds equally to all principal stresses, does not change σ₁ − σ₃ and therefore does not cause yielding in the idealized model. Tresca is computationally simple and usually conservative relative to von Mises for ductile metals.

For pure shear, σ₁ = τ, σ₃ = −τ, so Tresca predicts yielding when τ = S_y/2. This explains the familiar relationship between tensile yield strength and shear yield strength under Tresca theory.

5. von Mises distortion-energy theory

The von Mises criterion predicts yielding when the distortional, or shape-changing, energy reaches the value at uniaxial yield. The equivalent stress is

The safe condition is

and the corresponding factor of safety is

For plane stress with σ₃ = 0, an equivalent Cartesian form is

The formula assumes isotropic, homogeneous material, small elastic strains, and a ductile yielding mechanism. Von Mises theory is widely used in machine design and finite-element post-processing because it agrees well with experimental yielding of many ductile metals. It is less appropriate for brittle fracture, strongly anisotropic materials, polymers outside their applicable range, porous materials, and situations dominated by pressure-sensitive plasticity.

For pure shear, the von Mises criterion gives

and yielding occurs at τ = S_y/√3, which is less conservative than Tresca’s S_y/2 prediction because S_y/√3 ≈ 0.577S_y whereas S_y/2 = 0.5S_y.

Comparison of ductile criteria

A useful design practice is to use von Mises as the principal criterion for ductile metals and check Tresca when a conservative comparison is warranted. The choice must be documented rather than hidden in software settings.

6. Worked example: a combined-stress shaft

A solid steel shaft is subjected at a critical surface point to a bending normal stress of σ_x = 120 MPa and a torsional shear stress of τ_xy = 80 MPa. Assume σ_y = 0, σ₃ = 0, and yield strength S_y = 360 MPa. Determine the Tresca and von Mises factors of safety.

First calculate the principal stresses:

Thus, σ₁ = 160 MPa, σ₂ = −40 MPa, and σ₃ = 0 MPa. In ordered form they are 160, 0, and −40 MPa.

For Tresca,

For von Mises,

Therefore,

The Tresca result is more conservative. If the design requirement is n ≥ 2.0, the von Mises result just satisfies it, while the Tresca result does not. This difference is not an arithmetic disagreement; it reflects two different idealizations of yielding.

7. Factors of safety and reliability

A factor of safety is not a universal correction for poor engineering. It is a ratio between a resistance measure and a demand measure, provided both refer to the same failure mode and compatible conditions:

For von Mises yielding, this becomes n = S_y/σ_vm. An allowable-stress design may instead define

and require σ_vm ≤ σ_allow. The factor must be applied consistently; multiplying the load and separately reducing the strength can unintentionally double-count or omit safety margins.

Important sources of uncertainty include:

• variation in applied loads, impact, misalignment, and operating speed;
• uncertainty in dimensions, surface finish, residual stress, and stress concentration;
• scatter in yield strength, toughness, fatigue properties, and weld quality;
• inaccuracies in analytical models, numerical meshes, and boundary conditions;
• deterioration caused by corrosion, wear, temperature, radiation, or aging;
• consequences of injury, environmental release, loss of service, or financial damage.

Reliability is the probability that the component performs its required function for a specified time under stated conditions. A high factor of safety does not automatically establish high reliability, especially when the governing failure mode is fatigue or brittle fracture. Reliability-based design treats resistance and demand as random variables. If R is resistance and S is demand, a basic reliability statement is

where P_f is probability of failure. The reliability is Rel = 1 − P_f. If both variables are normally distributed and statistically independent, the mean safety margin is μ_Z = μ_R − μ_S and its standard deviation is σ_Z = √(σ_R² + σ_S²). The reliability index is

and, under these normality assumptions, P_f = Φ(−β), where Φ is the standard normal cumulative distribution function. Real engineering variables are often non-normal or correlated, so more advanced methods such as FORM, Monte Carlo simulation, or code-specific reliability formats may be required.

Selecting a factor of safety

A defensible selection considers:

1. Failure mode: yielding, fracture, fatigue, buckling, creep, and wear require different models.
2. Load knowledge: steady, measured loads justify less margin than uncertain impact loads.
3. Material and manufacturing control: certified material, process capability, inspection, and traceability reduce uncertainty.
4. Geometry and analysis quality: validated models and verified finite-element results reduce model uncertainty.
5. Ductility and warning: ductile yielding may provide warning; brittle fracture may be sudden.
6. Consequences: injury, environmental harm, inaccessible repair, and mission loss require greater reliability.
7. Redundancy and detectability: fail-safe design, monitoring, proof testing, and planned inspection can change the required margin.
8. Regulatory requirements: mandatory codes override informal preference.

The numerical value should therefore be traceable to a standard, company design basis, reliability target, or documented engineering judgment. “Use a large factor because the component is important” is not a sufficient justification.

8. Design codes and professional governance

Design codes translate research, field experience, safety policy, and accepted practice into enforceable or widely recognized requirements. Examples include ASME pressure-vessel rules, API equipment standards, AISC structural provisions, Eurocodes, and ISO machinery standards. The applicable code depends on jurisdiction, equipment class, material, loading, fabrication method, inspection regime, and service environment.

Codes may use allowable-stress design, limit-state design, load factors, resistance factors, fatigue curves, fracture control, or combinations of these approaches. A designer must verify:

• the current edition and jurisdictional adoption;
• material specifications and permitted design properties;
• load combinations and environmental conditions;
• weld categories, joint efficiency, and inspection requirements;
• corrosion allowance, temperature derating, and fatigue provisions;
• documentation, certification, testing, and professional approval.

A code minimum is not permission to ignore an unusual hazard. Conversely, adding an arbitrary extra factor to a code calculation may produce an uneconomical design or double-count uncertainty. Engineering ethics requires competent analysis, truthful reporting of assumptions, protection of public safety, and escalation of unresolved risks.

9. Illustrative case: pump-shaft redesign

Background. A chemical-processing pump has experienced two shaft failures near a keyway. The shaft is a ductile alloy steel, and the operating torque varies with process demand. The replacement design must avoid recurrence while remaining compatible with the existing coupling.

Problem. The original calculation considered torsional shear stress only and used the nominal shaft diameter. It omitted bending from coupling misalignment, the keyway stress concentration, corrosion thinning, and torque fluctuations.

Analysis. The redesign team first defined the limit states: static yielding, fatigue initiation, and fracture after a possible crack. A three-dimensional finite-element model was checked against hand calculations for a smooth shaft. The local alternating and mean stresses were obtained at the keyway root. Von Mises stress was selected for static ductile yielding, while an appropriate fatigue method was used for cyclic loading. The shaft material certificate, hardness data, surface condition, and corrosion environment were reviewed.

A notch factor and surface-condition correction were included in the fatigue assessment, but they were not incorrectly inserted into the nominal static yield equation. The team also checked the key, coupling, bearing reactions, critical speed, and shaft deflection. Inspection access and a replacement interval were included in the design basis.

Approach and solution. The keyway geometry was modified to increase fillet radius, the shaft diameter was increased within coupling limits, and alignment controls were improved. A corrosion allowance and protective coating were specified. The design factor was selected from the company standard and applicable machinery requirements, with additional reliability justification because failure could release hazardous chemicals. A prototype underwent overspeed, vibration, and endurance testing.

Outcome. The redesigned shaft met static and fatigue requirements, and the endurance test showed no crack initiation. The principal improvement was not simply a larger factor of safety; it was a more complete definition of loads, local stress concentration, environment, and inspection.

Learning points. Failure criteria are only as useful as the stress state and limit state supplied to them. A nominal stress calculation can be misleading near discontinuities. Reliability comes from the combined effect of sound modeling, material control, geometry, inspection, maintenance, and appropriate safety margins.

10. Practical workflow and contemporary tools

A robust design workflow is:

1. Define function, life, environment, and consequences of failure.
2. Identify every credible limit state.
3. Establish loads and load combinations, including transients.
4. Calculate nominal stresses and principal stresses.
5. Account for concentrations, residual stresses, contact, and boundary effects.
6. Select the criterion consistent with the material and failure mechanism.
7. Apply the required factor or reliability method.
8. Check deformation, fatigue, buckling, fracture, temperature, and code rules.
9. Verify results through mesh convergence, hand calculations, testing, or inspection planning.
10. Record assumptions, units, material source, code edition, and decision rationale.

Finite-element software can calculate principal and equivalent stresses, but a contour plot is not a failure diagnosis. Singularities at sharp corners, unrealistic constraints, poor mesh quality, and incorrect material models can produce meaningless peaks. Results should be interpreted away from mathematical singularities or assessed using a physically realistic radius and a suitable local model.

Digital monitoring, sensors, automated inspection, and digital twins can support condition-based maintenance. Machine-learning systems may identify patterns in vibration or strain, but they do not replace a validated failure model, code compliance, or professional accountability. Training data may omit rare catastrophic events, and automated predictions require uncertainty bounds and human review.

Challenges and limitations

Classical criteria are valuable abstractions, not universal laws. They generally assume homogeneous isotropic materials and well-defined stress states. They do not by themselves predict fatigue life, crack growth, creep, fracture toughness, local plastic collapse, residual stress effects, or progressive damage. Material anisotropy, welds, composites, additive-manufactured defects, and temperature-dependent behavior may require specialized models.

The largest practical error often lies in the input rather than the criterion. An accurate von Mises calculation based on an underestimated load remains unsafe. Similarly, a high factor of safety cannot compensate for an unrecognized resonance, corrosion mechanism, or brittle transition temperature. Sound design combines theory, experiment, standards, inspection, and engineering judgment.

Chapter Summary

Failure criteria convert a multiaxial stress state into a design decision. Maximum principal-stress theory is most naturally associated with brittle fracture; maximum principal-strain theory compares elastic principal strain with the uniaxial yield strain but has limited modern use as a ductile yield law. Tresca uses maximum shear stress and is conservative for ductile yielding. Von Mises uses distortion energy and is widely accepted for isotropic ductile metals.

Principal stresses must be calculated and ordered correctly, especially in plane stress. The factor of safety is a resistance-to-demand ratio tied to a specified limit state. Its selection should reflect load uncertainty, material variability, model uncertainty, reliability, deterioration, consequences, redundancy, inspection, and applicable codes. Code compliance, verification, and ethical documentation are essential. Most successful redesigns improve the entire system—loads, geometry, material, environment, manufacturing, monitoring, and maintenance—not merely the numerical factor of safety.

Key Terms

Failure criterion: Rule relating a stress or strain state to a specified failure or yielding condition.

Limit state: Condition beyond which a required performance criterion is no longer satisfied.

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

Maximum shear stress: Greatest shear stress at a point, equal to (σ₁ − σ₃)/2.

Factor of safety: Ratio of resistance to demand for a stated failure mode.

Allowable stress: Permitted design stress after applying a prescribed safety factor.

Rankine theory: Maximum principal-stress theory.

Maximum principal-strain theory: Criterion based on the largest elastic principal strain.

Tresca theory: Maximum-shear-stress theory for ductile yielding.

Von Mises stress: Equivalent stress based on distortional energy.

Hydrostatic stress: Equal normal stress acting in all three principal directions.

Deviatoric stress: Stress component responsible for shape change after hydrostatic stress is removed.

Ductile material: Material capable of substantial plastic deformation before fracture.

Brittle material: Material that fractures with little plastic deformation.

Reliability: Probability of performing the required function under stated conditions and time.

Probability of failure: Probability that resistance is less than demand.

Reliability index: Standardized separation between resistance and demand distributions under stated assumptions.

Stress concentration: Local stress amplification caused by a geometric discontinuity.

Design code: Published technical requirements governing design, materials, fabrication, testing, or operation.

Limit-state design: Design method that checks specified ultimate and serviceability conditions, often with separate load and resistance factors.

Review Questions

Short Answer Questions

1. What physical quantity does Tresca theory compare with the shear stress at uniaxial yield?
2. Why is von Mises theory generally more suitable than maximum principal stress for ductile metals?
3. State the plane-stress von Mises equation.
4. Name four factors that influence selection of a factor of safety.
5. Why must the applicable design-code edition and jurisdiction be verified?

Long Answer Questions

1. Explain the assumptions, strengths, and limitations of Rankine, maximum principal-strain, Tresca, and von Mises criteria.
2. Derive the Tresca and von Mises factors of safety for a general three-dimensional principal-stress state.
3. Discuss why a large factor of safety cannot compensate for an incomplete load model.
4. Explain how reliability-based design differs from selecting an arbitrary deterministic factor of safety.
5. Describe a professional workflow for designing and verifying a shaft subject to combined bending and torsion.

Analytical / Application Questions

1. A component has principal stresses 150 MPa, 20 MPa, and −80 MPa and yield strength 360 MPa. Calculate its Tresca and von Mises factors of safety and identify the more conservative criterion.
2. A thin plate has σ_x = 90 MPa, σ_y = 30 MPa, and τ_xy = 40 MPa. Calculate the principal stresses and von Mises equivalent stress.
3. Propose a design and inspection strategy for a brittle, safety-critical pressure component in a corrosive environment. Justify the failure criterion, reliability approach, and role of the governing code.

Multiple Choice Questions with Answers

Question 1

For a three-dimensional principal-stress state, the maximum shear stress is:

A. (σ₁ + σ₃)/2

B. (σ₁ − σ₃)/2

C. σ₂/2

D. √(σ₁² + σ₃²)

Correct Answer: B. (σ₁ − σ₃)/2

Question 2

Which criterion is most widely used for predicting initial yielding of isotropic ductile metals?

A. Maximum principal stress

B. Maximum principal strain

C. Tresca only

D. von Mises

Correct Answer: D. von Mises

Question 3

For pure shear stress τ, the von Mises equivalent stress is:

A. τ/2

B. τ

C. √3τ

D. 2τ

Correct Answer: C. √3τ

Question 4

A ductile material has yield strength 300 MPa and a calculated von Mises stress of 150 MPa. The factor of safety against yielding is:

A. 0.5

B. 1.0

C. 2.0

D. 4.0

Correct Answer: C. 2.0

Question 5

Which failure mode is most directly associated with repeated fluctuating loading?

A. Fatigue

B. Hydrostatic compression

C. Elastic recovery

D. Static equilibrium

Correct Answer: A. Fatigue

Question 6

In ideal ductile yielding theories such as Tresca and von Mises, a purely hydrostatic stress primarily:

A. Controls distortional yielding directly

B. Does not change the deviatoric stress

C. Always causes shear yielding

D. Makes Young’s modulus zero

Correct Answer: B. Does not change the deviatoric stress

Question 7

For plane stress, the third principal stress is commonly taken as:

A. σ₃ = 0

B. σ₃ = σ_x + σ_y

C. σ₃ = τ_xy

D. σ₃ = S_y

Correct Answer: A. σ₃ = 0

Question 8

If resistance and demand are random variables, probability of failure is represented by:

A. P(R > S)

B. P(R = S)

C. P(R ≤ S)

D. P(R + S)

Correct Answer: C. P(R ≤ S)

Question 9

Which statement about design codes is correct?

A. Any older edition is automatically acceptable

B. Codes eliminate the need to define the failure mode

C. Codes should be checked for edition, jurisdiction, materials, loads, and inspection requirements

D. Codes apply only to numerical simulation

Correct Answer: C. Codes should be checked for edition, jurisdiction, materials, loads, and inspection requirements

Question 10

For equal tensile and compressive yield strengths, Tresca predicts yielding when the principal-stress range equals:

A. S_y/2

B. S_y

C. 2S_y

D. S_y/√3

Correct Answer: B. S_y