Machine Design: Stress Analysis and Failure Theories

Designing machine components is largely the discipline of predicting what will break, where it will break, and how far a design is from that breaking point. Under static loading, the goal is not to estimate life in cycles, but to ensure that stresses produced by a one-time or slowly applied load remain below a material-dependent failure threshold, with appropriate margin. That margin is expressed through the factor of safety, adjusted for uncertainty, variability, and the consequences of failure.

This article focuses on stress analysis under static loading and three widely used failure criteria: maximum normal stress, distortion energy (von Mises), and Coulomb-Mohr. Along the way, it connects the theory to practical design issues such as stress concentration and conservative selection of allowable stress.

Why stress analysis matters in machine design

A “safe” part is not one that never yields or fractures in any possible circumstance. It is one that meets the intended function with an acceptable probability of survival over its design conditions. Static failures in machine elements typically fall into two broad categories:

• Yielding (mostly in ductile materials): permanent deformation begins when the stress state exceeds the yield strength in the appropriate sense.
• Fracture (often in brittle materials): cracking and catastrophic separation can occur with limited or no plastic deformation.

A sound design process starts by identifying the loads (forces, moments, pressure), translating them into internal stresses (normal and shear), and then comparing the resulting stress state against a failure theory appropriate to the material and the uncertainty level.

Stress state essentials: normal, shear, and principal stresses

Real components often carry combined loading: bending plus torsion, direct tension plus bending, or pressure plus thermal effects. At a point in a part, the stress state can be represented by normal stresses and shear stresses. For many failure criteria, it is convenient to work with the principal stresses ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$, which are the normal stresses on planes where shear stress is zero.

In many machine design problems (shafts, plates, thin-walled parts), a plane stress assumption applies, so one principal stress is approximately zero. Even in that simpler case, combined loading can produce a stress state that looks moderate in terms of any single component but severe when considered as a whole. That is why failure theories exist: they provide a consistent way to convert multiaxial stress into a single “equivalent” measure.

Factor of safety: what it is and how it is used

The factor of safety $n$ expresses how far the design is from the chosen failure limit. In its simplest form:

• For yielding-based design: $n=\frac{S_y}{{\sigma }_{\text{equiv}}}$
• For brittle fracture design: $n=\frac{S_{ut}}{{\sigma }_{\text{equiv}}}$ (or based on compressive strength where relevant)

Here, $S_y$ is yield strength and $S_{ut}$ is ultimate tensile strength. The equivalent stress ${\sigma }_{\text{equiv}}$ depends on the failure theory.

Selecting $n$ is not a purely mathematical decision. It reflects uncertainties in loads, material properties, manufacturing variability, stress concentrations, residual stresses, and inspection quality. Higher factors of safety are typical when:

• loads are poorly known or variable,
• failure consequences are severe,
• the material is brittle or exhibits wide property scatter,
• the environment can degrade strength (corrosion, temperature).

Stress concentration: why geometry often dominates

In machine elements, failure frequently begins at a notch, shoulder, keyway, groove, hole, or thread root. These features raise the local stress above the nominal value predicted by simple formulas. The geometric amplification is quantified by the stress concentration factor $K_t$:

$${\sigma }_{\max }=K_t{\sigma }_{\text{nom}}$$

A fillet radius that is slightly too small, or a groove placed in a high-moment region, can multiply the peak stress enough to trigger yielding or fracture even if the nominal stress appears acceptable. Under static loading, designers often use $K_t$ directly for brittle materials and for ductile materials when localized yielding is unacceptable (for example, in precision fits). In more ductile parts, some local yielding can redistribute stress, but it still may be unacceptable due to distortion, loss of alignment, or crack initiation.

Failure theories for static loading

Different materials fail differently under the same stress state. The three criteria below are common in machine design because they map reasonably well to observed behavior.

Maximum normal stress theory (Rankine)

Concept: Failure occurs when the maximum principal stress reaches a limiting value.

For tensile failure: $${\sigma }_1\ge S_{\text{allow}}$$

For compressive failure (important for brittle materials): $$|{\sigma }_3|\ge S_{\text{allow,comp}}$$

This criterion is simple and tends to be conservative for brittle materials in tension, because brittle fracture is strongly influenced by tensile normal stress that opens cracks. It is generally not appropriate for ductile yielding under multiaxial loading, because ductile yielding correlates better with shear and distortional effects than with the maximum principal stress alone.

Where it’s useful:

• brittle materials (cast iron, ceramics, some hardened steels under certain conditions),
• parts where tensile cracking is the main concern,
• preliminary checks when principal stresses are readily available.

Distortion energy theory (von Mises)

Concept: Ductile materials yield when the distortion energy reaches the same level as at yield in a uniaxial tension test.

The von Mises equivalent stress is commonly written in terms of principal stresses:

$${\sigma }_{vm}=\sqrt{\frac{({\sigma }_1-{\sigma }_2{)}^2+({\sigma }_2-{\sigma }_3{)}^2+({\sigma }_3-{\sigma }_1{)}^2}{2}}$$

Yielding is predicted when: $${\sigma }_{vm}\ge S_y$$

Designers often rearrange this to compute a factor of safety: $$n=\frac{S_y}{{\sigma }_{vm}}$$

Von Mises is widely used in machine design for ductile metals because it matches experimental yielding behavior well for many steels and aluminum alloys under combined loading. It also aligns naturally with finite element analysis outputs, where ${\sigma }_{vm}$ is a standard reported field.

Practical example: A solid circular shaft transmitting torque and carrying bending. Bending creates normal stress; torsion creates shear. Von Mises combines these into a single equivalent stress that can be compared to $S_y$. This avoids underestimating risk when neither bending stress nor torsional stress alone exceeds yield.

Coulomb-Mohr criterion (brittle materials with different tensile and compressive strengths)

Concept: Many brittle materials have significantly different strengths in tension and compression. Coulomb-Mohr accounts for that asymmetry by relating failure to combinations of principal stresses using different allowable limits depending on sign.

In practice, the Coulomb-Mohr approach checks the principal stress state against a piecewise linear failure envelope based on:

• ultimate tensile strength $S_{ut}$,
• ultimate compressive strength $S_{uc}$.

Because the criterion is sign-sensitive, it is particularly relevant when one principal stress is tensile and another is compressive, as occurs in bending with superimposed compressive loads or contact stresses. It generally gives a more realistic assessment than using a single symmetric criterion for materials like cast iron.

Where it’s useful:

• cast irons and other brittle materials,
• situations with mixed tension and compression,
• designs where compressive capacity is substantially higher than tensile capacity.

Choosing the right criterion in real design work

A practical selection rule is:

• Ductile metals under static loading: use von Mises for yielding checks.
• Brittle materials: use maximum normal stress for a conservative tensile-fracture check, and prefer Coulomb-Mohr when tensile and compressive strengths differ materially and the stress state includes both signs.

This choice should be paired with stress concentration evaluation, because brittle materials are notch-sensitive and local tensile peaks dominate failure.

Putting it together: a disciplined design workflow

A robust static-strength workflow usually looks like this:

1. Define loads and constraints

Use realistic worst-case static loads, including assembly preload, press fits, and thermal loads if relevant.

1. Compute nominal stresses

Use classical formulas (bending, torsion, direct stress) or finite element analysis for complex geometry.

1. Apply stress concentration factors

Identify geometric discontinuities and estimate $K_t$ to obtain local peak stresses.

1. Convert to principal stresses or equivalent stress

Determine ${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$, then compute ${\sigma }_{vm}$ for ductile design or apply brittle criteria.

1. Apply the failure theory and factor of safety

Compare against $S_y$, $S_{ut}$, or $S_{uc}$ as appropriate and confirm $n$ meets requirements.

1. Check deformation and function

Even if the part does not fail, excessive deflection can cause misalignment, vibration, leakage, or loss of performance.

Common pitfalls to avoid

• Using nominal stress without stress concentration at shoulders, holes, threads, and keyways.
• Applying a ductile yielding criterion to a brittle material (or vice versa) without considering failure mode.
• Treating factor of safety as a universal constant rather than a decision tied to uncertainty and