3. Failure Theories and Design for Static Strength

Introduction and Context

Mechanical components are designed to carry loads without unacceptable deformation, fracture, instability, leakage, or loss of function. A component may be subjected to tension, compression, bending, torsion, internal pressure, contact loading, or several of these simultaneously. The central design question is not merely whether stress exists, but whether the combination of stress, material, geometry, manufacturing process, environment, and uncertainty can produce a critical failure mode.

Static-strength design concerns loads that are constant or vary sufficiently slowly that inertia and fatigue effects are not dominant. It is therefore distinct from fatigue design, although the two are often required for the same component. Static failure may occur through yielding, brittle fracture, crushing, excessive elastic deformation, buckling, contact damage, or plastic collapse. Selection of an appropriate failure theory is essential because materials do not respond identically to multiaxial stress states.

Ductile metals usually tolerate substantial plastic deformation before rupture. Their design is commonly based on yielding criteria such as the maximum-shear-stress theory or distortion-energy theory. Brittle materials, including many cast irons, ceramics, and hardened materials, exhibit limited plasticity and are more appropriately assessed using principal-stress or fracture-based criteria. The designer must also distinguish between a calculated failure criterion and a complete engineering decision involving safety factors, standards, inspection, manufacturability, and consequences of failure.

Learning Objectives

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

• identify the principal static failure modes of mechanical components;
• distinguish ductile yielding from brittle fracture and instability;
• apply Tresca and von Mises criteria to multiaxial stress states;
• use maximum-principal-stress and Coulomb–Mohr concepts for brittle materials;
• explain stress concentration, fracture mechanics, and notch sensitivity;
• design or check thin- and thick-walled pressure vessels;
• evaluate column buckling and contact stresses;
• select safety factors using uncertainty, consequence, and reliability considerations;
• use limit-state and plastic-collapse concepts in design decisions; and
• interpret the role of standards, numerical analysis, testing, and digital tools.

3.1 What Constitutes Static Failure?

Failure is the inability of a component to perform its intended function safely. This definition is broader than rupture. A shaft that twists excessively, a pressure vessel that develops permanent distortion, or a column that buckles before reaching material yield has failed in service even if it remains in one piece.

The principal static failure modes are:

1. Yielding: permanent plastic deformation begins when the stress state reaches a material-dependent yield condition.

2. Fracture: a crack propagates through the component, either suddenly or progressively.

3. Excessive elastic deformation: deflection, rotation, or dimensional change causes loss of fit or function.

4. Buckling: a compressive member loses stability through a lateral or local deformation mode.

5. Crushing or indentation: local compressive stress exceeds the capacity of a surface or material.

6. Plastic collapse: a sufficient region forms plastic hinges or a fully yielded section, producing mechanism-like deformation.

7. Leakage or rupture in pressure equipment: membrane or local stresses exceed allowable limits, or a defect becomes unstable.

A sound design begins with a failure-mode assessment. It is poor practice to calculate only an equivalent stress and assume that the largest numerical result identifies the governing failure. For example, von Mises stress is useful for ductile yielding but does not by itself establish safety against buckling or brittle fracture.

3.2 Stress States and Design Representation

At a point, the stress tensor can be transformed to three mutually perpendicular principal stresses, sigma_1, sigma_2, and sigma_3. Principal stresses act on planes where shear stress is zero. They provide a convenient basis for failure theories.

For plane stress, the principal stresses are obtained from

sigma_1,2 = (sigma_x + sigma_y)/2 ± sqrt[((sigma_x - sigma_y)/2)^2 + tau_xy^2],

with sigma_3 = 0 when the out-of-plane stress is negligible. In three-dimensional analysis, the full stress tensor or finite-element result is required.

Stress components may be nominal or local. Nominal stress is calculated from an idealized geometry, such as sigma = F/A or sigma_b = Mc/I. Local stress includes the effect of geometric discontinuities. A sharp shoulder, keyway, hole, thread, weld toe, or casting defect can intensify stress. The elastic stress concentration factor is

K_t = maximum local elastic stress / nominal stress.

For fatigue, the fatigue notch factor is often lower than K_t because material behavior is influenced by notch sensitivity. For static yielding in a ductile material, local plasticity can redistribute stress, so using K_t without considering elastic–plastic behavior may be conservative or misleading. Brittle materials generally have little capacity for redistribution and are strongly affected by stress concentration.

3.3 Failure Criteria for Ductile Materials

Maximum-Shear-Stress Theory: Tresca Criterion

The Tresca criterion states that yielding begins when the maximum shear stress in the component equals the maximum shear stress at yield in a uniaxial tensile test. The maximum shear stress is

tau_max = (1/2) max(|sigma_1 - sigma_2|, |sigma_2 - sigma_3|, |sigma_3 - sigma_1|).

For a uniaxial tensile test at yield stress S_y, tau_max = S_y/2. Therefore, yielding is predicted when

max(|sigma_1 - sigma_2|, |sigma_2 - sigma_3|, |sigma_3 - sigma_1|) = S_y.

A factor of safety n may be introduced by replacing S_y with S_y/n. Tresca is relatively simple and generally conservative compared with von Mises. It is especially useful for quick screening and for designs in which conservative prediction is desirable.

Distortion-Energy Theory: von Mises Criterion

The von Mises criterion is based on the distortion component of elastic strain energy. Hydrostatic stress changes volume but does not, in the idealized model, cause yielding in a ductile metal. The equivalent stress is

sigma_vm = sqrt{[(sigma_1 - sigma_2)^2 + (sigma_2 - sigma_3)^2 + (sigma_3 - sigma_1)^2]/2}.

Yielding is predicted when sigma_vm reaches S_y. With a safety factor n, the design condition is

sigma_vm <= S_y/n.

For plane stress with sigma_3 = 0,

sigma_vm = sqrt(sigma_x^2 - sigma_x sigma_y + sigma_y^2 + 3 tau_xy^2).

Von Mises generally correlates well with experimental behavior of isotropic ductile metals and is widely implemented in finite-element software. It predicts no yielding under pure hydrostatic compression, which is consistent with the idealized yielding behavior of many ductile metals but not necessarily with porous, pressure-sensitive, or highly anisotropic materials.

Comparison of Tresca and von Mises

For pure shear, Tresca predicts yielding at tau = S_y/2, whereas von Mises predicts tau = S_y/sqrt(3), approximately 0.577 S_y. Thus Tresca predicts yielding at a lower shear stress and is more conservative. The difference is usually modest, but it may affect sizing near an allowable limit. The choice should be consistent with material data, design standards, and the consequences of yielding.

For a shaft under pure torsion, the von Mises design condition becomes sqrt(3) tau <= S_y/n. For combined bending and torsion, a common elastic estimate is

sigma_vm = sqrt(sigma_b^2 + 3 tau_t^2),

where sigma_b is bending normal stress and tau_t is torsional shear stress. This relation is appropriate when the stress state is represented by one normal and one shear component at the critical point.

3.4 Criteria for Brittle Materials

Brittle materials often fail with little warning and exhibit different tensile and compressive strengths. Their failure is governed by the largest principal tensile stress, crack-like flaws, or combinations of normal and shear stress.

Maximum-Principal-Stress Criterion

The Rankine criterion predicts failure when the largest principal tensile stress reaches the tensile strength S_ut or when the most compressive principal stress reaches the compressive strength S_uc. With different tensile and compressive strengths, the checks are

sigma_1 <= S_ut/n,

and

|sigma_3| <= S_uc/n,

where sigma_1 is the greatest tensile principal stress and sigma_3 is the most compressive principal stress. This criterion is easy to apply but does not directly account for interaction between principal stresses.

Coulomb–Mohr and Modified Mohr Criteria

Coulomb–Mohr theory accounts for unequal tensile and compressive strengths and is useful for brittle materials. A simplified form for a stress state with one positive and one negative principal stress is

sigma_1/S_ut - sigma_3/S_uc <= 1/n.

The exact form depends on the signs and ordering of the principal stresses. Modified Mohr theory provides a piecewise envelope based on tensile and compressive test strengths and can give better correlation for some cast and brittle materials. The designer must use the formulation adopted by the relevant textbook, standard, or organization because notation and regions differ.

Neither principal-stress criteria nor ordinary von Mises analysis is sufficient when a known crack or crack-like flaw is present. In that situation, fracture mechanics is the more appropriate framework.

3.5 Fracture Mechanics and Notch Effects

Classical strength calculations assume a continuous material and represent defects through stress concentration factors. Fracture mechanics treats a crack explicitly. In linear-elastic fracture mechanics, the stress intensity factor is commonly written as

K_I = Y sigma sqrt(pi a),

where Y is a geometry factor, sigma is the applied nominal stress, and a is a characteristic crack size. Unstable fracture is expected when K_I reaches the plane-strain fracture toughness K_IC, subject to validity requirements.

A design condition is therefore

K_I <= K_IC/n,

or, in damage-tolerant design, the allowable crack size is determined for a specified stress and inspection interval. Fracture toughness is not the same as tensile strength. A high-strength material may have lower toughness, making it more sensitive to flaws and stress concentrations.

Important fracture-related considerations include crack orientation, residual stress, temperature, loading rate, thickness, corrosion, welding defects, and inspection capability. The presence of a notch does not automatically imply crack behavior, but a sharp notch with a small root radius can approach crack-like conditions. Engineering judgment is required when deciding whether a conventional stress-concentration approach remains valid.

3.6 Safety Factors and Allowable Strength

The factor of safety may be defined as the ratio of failure strength to working demand. In a yielding design using von Mises stress,

n = S_y/sigma_vm.

This numerical ratio is meaningful only when the failure mode, material properties, loads, and analysis method are appropriate. A safety factor is not a substitute for accurate modeling.

Selection should consider:

• uncertainty in loads and dimensions;
• scatter in strength and toughness;
• manufacturing defects and residual stresses;
• environmental degradation and temperature;
• inspection and maintenance capability;
• consequences of failure;
• quality of the analytical model; and
• applicable codes and regulatory requirements.

For high-consequence systems, reliability-based design may use statistical distributions of load and strength rather than a single deterministic factor. Load and resistance factors can be applied separately, allowing greater transparency about uncertainty. The target reliability should be established at the system level and not chosen solely by habit.

3.7 Pressure Vessels: Membrane Strength and Local Effects

For a thin cylindrical vessel with internal pressure p, internal diameter d, and wall thickness t, where d/t is sufficiently large and stress gradients through the wall are small, the principal membrane stresses are approximately

hoop stress: sigma_h = p d/(2t),

longitudinal stress: sigma_l = p d/(4t).

The hoop stress is twice the longitudinal stress and commonly governs the required thickness. A preliminary thickness based on an allowable stress S_allow is

t >= p d/(2 S_allow),

before adding corrosion allowance, manufacturing tolerance, weld efficiency, and code-specific terms. Actual pressure-vessel design must follow the applicable standard, such as a recognized national or international pressure-vessel code.

At closed ends, end-cap loading produces longitudinal stress. Openings, nozzles, supports, welds, discontinuities, and changes in thickness create local bending and peak stresses not represented by the membrane equations. A thin-wall result must therefore be treated as a first design estimate, not a complete certification.

For thick-walled cylinders, radial stress varies substantially through the wall. Lamé’s equations are

sigma_r = A - B/r^2,

sigma_theta = A + B/r^2,

where A and B are determined from the internal and external pressure boundary conditions. The maximum circumferential stress often occurs at the inner wall. End effects and plastic collapse may need separate assessment.

3.8 Buckling and Stability

A component under compression can fail by instability before material yield. For a slender, ideal, pin-ended column, Euler’s critical load is

P_cr = pi^2 E I/(K L)^2,

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

sigma_cr = pi^2 E/(L_e/r)^2,

where L_e = KL and r = sqrt(I/A) is the radius of gyration.

Euler theory is applicable to sufficiently slender columns within the elastic range. Short and intermediate columns may fail through inelastic buckling, for which empirical or tangent-modulus relations are used. Real columns have initial crookedness, eccentric loading, residual stress, and imperfect end conditions. Consequently, the theoretical Euler load is not a permissible working load without an appropriate safety margin.

Local plate or shell buckling can govern pressure vessels, thin webs, panels, and stiffened structures. A finite-element eigenvalue buckling load often overpredicts real capacity because it assumes ideal geometry and loading. Geometric imperfections and nonlinear post-buckling analysis should be considered when stability is critical.

3.9 Contact Stress and Surface Failure

When two curved bodies transmit force over a small area, the local pressure may be much greater than the nominal component stress. Hertzian contact theory provides an elastic estimate of contact pressure and subsurface stresses for spheres, cylinders, and related geometries.

Contact failure may appear as plastic indentation, pitting, spalling, brinelling, fretting, or subsurface cracking. In gears, rolling bearings, cams, and rail contacts, surface durability and subsurface shear stress may govern before gross yielding of the component. Contact analysis should therefore consider material hardness, lubrication, conformity, roughness, contamination, alignment, and repeated loading.

A common mistake is to check only the average bearing pressure F/A. Average pressure can be useful for preliminary sizing but may conceal severe edge loading or a highly nonuniform pressure distribution.

3.10 Limit-State and Plastic-Collapse Design

Elastic allowable-stress design limits stress to a specified fraction of yield or ultimate strength. Limit-state design instead identifies a defined boundary between acceptable and unacceptable performance. A limit state may correspond to first yield, plastic collapse, fracture, buckling, leakage, excessive deformation, or loss of function.

For a beam, first yield occurs when the extreme fiber reaches yield stress. Plastic collapse requires a fully yielded plastic section and may occur at a higher moment. The plastic moment is

M_p = S_p S_y,

where S_p is the plastic section modulus. The ratio of plastic to elastic moment capacity is the shape factor. A ductile structure may redistribute moment after local yielding, but such redistribution must be permitted by geometry, connections, material ductility, and code provisions.

Limit-state methods are particularly valuable for pressure equipment, frames, reinforced structures, and components in which local yielding is acceptable but mechanism formation is not. They should not be used to justify uncontrolled deformation or to ignore fracture, instability, fatigue, or serviceability requirements.

3.11 An Integrated Design Procedure

A practical static-strength assessment can be organized as follows:

1. Define the function, operating envelope, failure consequences, and applicable standards.

2. Establish load cases, including pressure, thermal effects, assembly loads, impact-like transients, and support reactions.

3. Identify plausible failure modes before selecting equations.

4. Determine nominal and local stresses, using hand calculations, analytical models, or finite-element analysis.

5. Obtain principal stresses and select a criterion appropriate to the material and failure mode.

6. Check yielding, fracture, deformation, buckling, contact, and local or code-specific limits as applicable.

7. Include material variability, geometry tolerances, surface condition, residual stress, corrosion, and inspection assumptions.

8. Calculate margins or factors of safety and compare with code requirements.

9. Validate critical assumptions through sensitivity studies, tests, peer review, or independent calculations.

10. Document the design basis, including load combinations, material condition, allowable values, mesh quality, convergence, and acceptance criteria.

Illustrative Example: Shaft Under Combined Bending and Torsion

Consider a solid circular steel shaft subjected to a bending moment M and torque T. At the outer surface, the nominal stresses are

sigma_b = 32M/(pi d^3),

tau_t = 16T/(pi d^3).

Using von Mises theory, the equivalent stress is

sigma_vm = sqrt(sigma_b^2 + 3 tau_t^2).

For a specified yield strength S_y and safety factor n, the required diameter satisfies

sqrt{[32M/(pi d^3)]^2 + 3[16T/(pi d^3)]^2} <= S_y/n.

The calculation should be supplemented by checks on keyways, shoulders, threads, bearing fits, torsional angle, critical speed, and fatigue if the shaft rotates under variable loading. If the shaft contains a sharp keyway, the local stress state may govern even when the smooth-shaft calculation appears satisfactory.

Illustrative Case Study: Pressure Vessel Nozzle Reinforcement

Background. An illustrative manufacturer is developing a small cylindrical compressed-air receiver. The preliminary vessel is made from a ductile steel plate and includes a circular nozzle near a head-to-shell junction.

Situation. A thin-wall membrane calculation gives acceptable hoop and longitudinal stresses. However, a nozzle opening removes load-carrying material and introduces local stress concentration. The vessel also experiences repeated pressure cycles, although the present assessment is focused on static strength.

Analysis. The design team first calculates membrane stresses using the pressure-vessel equations and applies the code-specified allowable stress. It then evaluates reinforcement area around the opening, weld quality, local bending, and the effect of the head geometry. A finite-element model is used to identify peak stresses, but the team distinguishes mesh-sensitive peaks from structural stresses. A linear-elastic peak alone is not accepted as proof of failure because local yielding may redistribute stress; conversely, plastic redistribution is not used to excuse an inadequate nozzle attachment or fatigue concern.

Approach and solution. The opening is relocated away from the most highly constrained region, a reinforcement pad is sized according to the governing pressure-vessel code, and weld inspection requirements are specified. The design includes corrosion allowance and a hydrostatic proof test. A separate fatigue assessment is initiated because pressure cycling can cause crack growth even when static stresses remain below yield.

Outcome. The revised design meets the code-based membrane, local-strength, and reinforcement requirements, while the manufacturing and inspection plan addresses the principal defect risks.

Learning points. Membrane equations are necessary but insufficient near openings. Code compliance, local stress interpretation, weld integrity, proof testing, and fatigue assessment must be integrated. The governing design decision is determined by the complete failure-mode set, not by the largest value from one equation.

Common Errors, Limitations, and Professional Responsibilities

Several errors recur in static-strength design. Designers may apply von Mises theory to brittle materials, use ultimate strength where yield strength is required, neglect the third principal stress, treat a stress singularity as a physical infinite stress, or use Euler buckling for a stocky column. Other errors include ignoring load eccentricity, assuming ideal supports, omitting residual stress, using handbook properties without checking material condition, and selecting a safety factor without considering consequence or uncertainty.

Finite-element results require engineering interpretation. Mesh refinement near a sharp re-entrant corner may cause the peak elastic stress to increase without convergence. This is a signal to assess a finite-radius geometry, structural stress, elastic–plastic analysis, fracture mechanics, or a code-specific classification. Boundary conditions and contact assumptions can be more influential than the element type.

Professional practice also requires traceability. Material certificates, weld procedures, inspection records, design calculations, software versions, assumptions, and review decisions should be controlled. Safety-critical design must not rely on undocumented spreadsheet formulas or unverified automated optimization. Ethical responsibility requires conservative treatment of uncertainty, honest reporting of limitations, and escalation when available evidence is inadequate.

Digital Analysis and Emerging Practice

Modern design workflows combine symbolic calculations, finite-element analysis, nonlinear material models, fracture assessment, optimization, and digital documentation. Parametric scripts can evaluate many load cases and perform sensitivity studies. Probabilistic methods can propagate uncertainty in loads, dimensions, material strength, and defect size. Machine-learning tools may assist surrogate modeling or anomaly detection, but they do not replace validated mechanics, appropriate training data, or engineering accountability.

Emerging practice emphasizes digital twins, structural-health monitoring, probabilistic fracture mechanics, additive-manufactured materials, and topology optimization. These developments create new requirements: anisotropy and defects in additively manufactured parts, model validation for digital twins, uncertainty quantification for automated decisions, and explainability where software influences safety-critical design.

Best-Practice Checklist

A robust static-strength design should:

• begin with failure-mode identification rather than equation selection;
• use a material-appropriate criterion;
• distinguish nominal, local, peak, and structural stresses;
• check both strength and serviceability;
• assess imperfections and realistic boundary conditions;
• use code requirements for pressure vessels and other regulated equipment;
• validate critical analytical and numerical assumptions;
• document safety factors and uncertainty sources; and
• arrange inspection, maintenance, and damage tolerance when failure consequences justify them.

Chapter Summary

Static-strength design requires more than comparing one calculated stress with one material value. The designer must identify yielding, fracture, buckling, contact damage, plastic collapse, deformation, and pressure-related failure as distinct possibilities. Tresca and von Mises criteria are primarily used for ductile yielding, while principal-stress and Coulomb–Mohr approaches are useful for brittle materials with unequal tensile and compressive strengths. Known cracks require fracture-mechanics concepts based on stress intensity and toughness.

Thin-wall pressure-vessel equations provide membrane stresses, whereas thick-wall vessels, openings, welds, and supports require more advanced analysis and code compliance. Euler theory addresses ideal slender-column buckling, but imperfections and inelastic behavior must be included in practical design. Contact stresses and limit-state methods extend the assessment to localized loading and plastic redistribution. Reliable design combines mechanics, standards, inspection, numerical analysis, uncertainty management, and professional judgment.

Key Terms

Allowable stress: Permissible design stress obtained by applying specified limits or safety factors to material strength.

Brittle fracture: Sudden crack propagation with little plastic deformation.

Buckling: Instability involving a change in structural deformation under compression or other destabilizing loads.

Coulomb–Mohr criterion: A brittle-failure criterion accounting for different tensile and compressive strengths.

Distortion energy: The portion of elastic strain energy associated with change in shape rather than volume.

Factor of safety: Ratio of a relevant failure strength to the corresponding working demand.

Fracture toughness: A material’s resistance to crack extension under specified loading conditions.

Hertzian contact stress: Local elastic stress produced when curved bodies transmit force through a small contact region.

Limit state: A defined boundary separating acceptable performance from a specified failure or unacceptable condition.

Maximum-principal-stress theory: A criterion predicting failure when a principal stress reaches a material strength limit.

Membrane stress: Stress distributed approximately uniformly through the thickness of a thin shell.

Plastic collapse: Formation of a mechanism after sufficient sections or regions have yielded.

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

Stress concentration factor: Ratio of maximum local elastic stress to nominal stress.

Stress intensity factor: Parameter characterizing the intensity of the crack-tip stress field.

Tresca criterion: A ductile-yielding criterion based on maximum shear stress.

Von Mises stress: Equivalent stress based on distortion energy and used widely to predict ductile yielding.

Yielding: Onset of permanent plastic deformation.

Review Questions

Short Answer Questions

1. Distinguish between yielding, fracture, buckling, and plastic collapse.

2. Why are principal stresses useful in failure-theory calculations?

3. State the von Mises equivalent-stress equation for three-dimensional stress.

4. What assumptions underlie the thin-wall pressure-vessel equations?

5. Why can an Euler buckling calculation overestimate the capacity of a real column?

Long Answer Questions

1. Explain and compare the Tresca and von Mises criteria, including their predictions in pure shear.

2. Discuss the selection of failure criteria for ductile and brittle materials.

3. Develop a systematic procedure for static-strength assessment of a mechanically loaded component.

4. Explain the role of fracture mechanics in the assessment of notched or cracked components.

5. Discuss membrane stress, local stress, reinforcement, and code considerations in pressure-vessel design.

Analytical / Application Questions

1. A shaft is subjected to simultaneous bending moment and torque. Derive the required diameter using the von Mises criterion and explain which additional geometric checks are necessary.

2. A compressed column has a length-to-radius-of-gyration ratio near the transition between elastic and inelastic buckling. Explain why Euler theory alone is inadequate and propose a design approach.

3. A brittle casting contains a sharp fillet and experiences combined tension and bending. Select an appropriate failure assessment method and explain how tensile–compressive strength asymmetry and flaw sensitivity affect the decision.

22. Multiple Choice Questions with Answers

Question 1

For a ductile material under a general three-dimensional stress state, which quantity is directly compared with yield strength in the von Mises criterion?

A. Maximum principal stress

B. Maximum shear stress

C. Distortion-energy equivalent stress

D. Hydrostatic pressure

Correct Answer: C. Distortion-energy equivalent stress

Question 2

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

A. S_y

B. S_y/2

C. S_y/sqrt(3)

D. 2S_y

Correct Answer: B. S_y/2

Question 3

A brittle material has substantially different tensile and compressive strengths. Which approach explicitly accounts for this asymmetry?

A. Coulomb–Mohr criterion

B. Euler theory

C. von Mises criterion only

D. Hertz theory

Correct Answer: A. Coulomb–Mohr criterion

Question 4

For a thin cylindrical pressure vessel, the hoop membrane stress is approximately:

A. pd/(4t)

B. pd/(2t)

C. 2pt/d

D. pt/(2d)

Correct Answer: B. pd/(2t)

Question 5

A known crack in a component is most appropriately assessed initially using:

A. Average bearing pressure only

B. Fracture mechanics

C. Euler buckling theory

D. Plastic section modulus alone

Correct Answer: B. Fracture mechanics

Question 6

Which change generally increases the Euler critical load of an ideal column, all other variables remaining constant?

A. Increasing unsupported length

B. Reducing Young’s modulus

C. Increasing the least second moment of area

D. Increasing effective-length factor

Correct Answer: C. Increasing the least second moment of area

Question 7

A finite-element model shows an ever-increasing stress peak at a perfectly sharp re-entrant corner as the mesh is refined. The most appropriate interpretation is that:

A. The material necessarily has infinite strength

B. The result may represent a stress singularity requiring a different assessment

C. The mesh is automatically proven accurate

D. Buckling has been eliminated

Correct Answer: B. The result may represent a stress singularity requiring a different assessment

Question 8

For a solid circular shaft under bending stress sigma_b and torsional shear stress tau, the common von Mises expression is:

A. sigma_b + tau

B. sqrt(sigma_b^2 + 3tau^2)

C. sqrt(3sigma_b^2 + tau^2)

D. sigma_b^2 + tau^2

Correct Answer: B. sqrt(sigma_b^2 + 3tau^2)

Question 9

Which statement best describes a limit-state design approach?

A. It checks only nominal tensile stress

B. It identifies a defined boundary such as yielding, collapse, fracture, or buckling

C. It assumes all local yielding is unacceptable

D. It eliminates the need for standards

Correct Answer: B. It identifies a defined boundary such as yielding, collapse, fracture, or buckling

Question 10

Why can the theoretical Euler load be unsafe as a direct working load for a real column?

A. Euler theory includes excessive material hardening

B. Real columns contain imperfections, eccentricity, residual stress, and uncertain end conditions

C. Euler theory applies only to tensile members

D. The critical load is independent of geometry

Correct Answer: B. Real columns contain imperfections, eccentricity, residual stress, and uncertain end conditions