Mechanics of Materials: Axial Loading and Torsion

Axial loading and torsion sit at the core of mechanics of materials because they describe how common structural members and machine elements carry force and transmit power. A steel tie rod in tension, a column in compression, and a rotating shaft driving a pump are all governed by a small set of ideas: internal force resultants, stress and strain relationships, deformation compatibility, and design checks against strength and serviceability limits.

This article develops those ideas in a practical way, moving from axially loaded members to statically indeterminate systems and then to torsion and shaft design for power transmission.

Axial Loading: Tension and Compression

An axially loaded member carries a force along its centroidal axis, producing normal stress over its cross-section. Ideal axial loading assumes the load is concentric so that bending is negligible. In real components, small eccentricities can introduce bending, but axial theory remains the starting point.

Normal Stress Under Axial Force

For a prismatic member with cross-sectional area $A$ subjected to an internal axial force $N$ (positive in tension), the average normal stress is

$\sigma =\frac{N}{A}$

This expression is widely used in design because many members are sized so that stress stays below allowable levels. When the stress distribution is uniform (as in an ideal bar loaded concentrically), the average stress equals the actual stress.

Axial Strain and Elongation

Within the linear-elastic range, stress and strain relate through Hooke’s law:

$\sigma =Eϵ$

where $E$ is Young’s modulus and $ϵ$ is axial strain. The axial deformation of a prismatic bar of length $L$ is

$\delta =\frac{NL}{AE}$

This is as important as the stress check. In many applications, elongation or shortening controls performance: bolt stretch provides clamping force, tie rods must limit deflection, and machine frames must hold alignment.

Nonprismatic Members and Stepped Bars

If $A$ or $N$ varies along the length, deformation is found by summing (or integrating) contributions:

$\delta ={\int }_0^L\frac{N(x)}{A(x)E(x)}dx$

In practice, stepped bars are common: a rod with two diameters, a member with different materials bonded in series, or a segment that has been machined thinner. Each segment is treated separately, then deformations are added with sign convention.

Compression Members and Buckling Awareness

Compression introduces a critical distinction: a member can fail by material crushing (stress exceeds allowable compressive stress) or by instability (buckling) at stresses far below the material strength. Pure axial stress calculations do not predict buckling. For slender columns, stability checks become essential. Even when buckling is not the focus, designers treat compressive members with caution, controlling slenderness and end conditions.

Statically Indeterminate Axial Systems

Many axial problems are statically indeterminate, meaning equilibrium equations alone cannot determine internal forces. This occurs when there are redundant supports or multiple load paths.

Common examples include:

• A bar fixed between two rigid walls that is heated or loaded.
• A stepped bar with multiple supports.
• Two parallel rods sharing a load through a rigid plate.

Why Indeterminacy Matters

In indeterminate systems, load distribution depends on stiffness. A stiffer member attracts more force. Stiffness in axial loading is

$k=\frac{AE}{L}$

So, for two parallel rods connected by rigid end plates and loaded by a force $P$, equilibrium requires $N_1+N_2=P$, while compatibility requires equal elongations, ${\delta }_1={\delta }_2$. With $\delta =N/k$, the force split follows stiffness:

$N_1=\frac{k_1}{k_1+k_2}P,N_2=\frac{k_2}{k_1+k_2}P$

This principle explains real behavior in built-up members, bolted joints, and composite tie systems.

Compatibility and Support Settlement

Indeterminate axial members are sensitive to imposed deformations, not only applied forces. If a support settles by a known amount, or temperature changes cause free expansion, the restraint generates internal forces. For a bar fixed at both ends, a temperature increase $\Delta T$ produces a free thermal strain $\alpha \Delta T$, but restraint forces it back to zero net elongation:

$$0={\delta }_{\text{mechanical}}+{\delta }_{\text{thermal}}$$

This introduces stress even when no external axial load is applied, a common issue in pipelines, rails, and rigidly clamped assemblies.

Torsion: Twisting of Circular Shafts

Torsion describes how shafts and other members resist twisting under applied torque. For most rotating machinery, torsion is the primary loading mode, and it governs both strength and angle of twist.

Shear Stress Due to Torque

For a circular shaft subjected to torque $T$, the shear stress varies linearly with radius $r$:

$\tau (r)=\frac{Tr}{J}$

where $J$ is the polar moment of inertia of the cross-section. The maximum shear stress occurs at the outer surface, $r=c$:

${\tau }_{\max }=\frac{Tc}{J}$

For solid circular shafts, $J=\frac{\pi d^4}{32}$. For hollow shafts, $J=\frac{\pi (d_o^4-d_i^4)}{32}$. Hollow shafts are often preferred in power transmission because they provide high torsional stiffness and strength relative to weight, and they accommodate passages or splines.

Angle of Twist and Torsional Stiffness

The torsion equivalent of axial deformation is the angle of twist:

$\theta =\frac{TL}{JG}$

where $G$ is the shear modulus. Torsional stiffness is $k_t=\frac{JG}{L}$, and it controls how much a shaft winds up under load. Excessive twist can cause timing errors in drivetrains, misalignment in couplings, and vibration issues.

Validity of the Simple Torsion Formula

The classic torsion relations above assume:

• Circular cross-sections
• Linear-elastic material behavior
• Uniform torque along the member segment
• No warping restraint effects (important mainly for noncircular sections)

For noncircular cross-sections (rectangular bars, thin-walled open sections), shear stress is not linear in radius and warping can be significant. In many machine design contexts, shafts are circular precisely to take advantage of the predictable torsion behavior.

Shaft Design for Power Transmission

A power-transmitting shaft converts rotational speed into torque. The relationship between power $P$, torque $T$, and angular speed $\omega$ is

$P=T\omega$

Since $\omega =2\pi n$ where $n$ is rotational speed in revolutions per second, higher speed for a given power means lower torque. This is why many systems use gearboxes: run motors fast to reduce torque on upstream shafts, then increase torque where needed.

Practical Design Checks

Shaft design typically balances several requirements:

1. Strength (shear stress limit)

Ensure ${\tau }_{\max }$ remains below an allowable value based on material strength and a chosen factor of safety.

1. Stiffness (angle of twist limit)

Cap $\theta$ to meet functional needs, such as limiting backlash or maintaining alignment.

1. Geometry and stress concentrations

Keyways, shoulders, splines, and fillets introduce stress concentrations. Even if nominal torsional stress is acceptable, local peaks can govern. Good design uses generous fillet radii, proper keyway proportions, and surface finishes suitable for the duty.

1. Combined loading reality

Many shafts also experience bending from gears, pulleys, and belt tensions. While this article focuses on torsion, real shaft sizing often considers combined bending and torsion using an appropriate failure theory.

Solid vs Hollow Shafts

For the same material and weight, hollow shafts can provide improved torsional performance because more material is placed away from the center where it contributes most to $J$. Manufacturing complexity, cost, and connection details often determine which option is best.

Putting It Together: A Consistent Analysis Workflow

Whether analyzing a tie rod or a rotating shaft, the workflow is similar:

1. Draw a clear free-body diagram and identify external loads.
2. Cut sections to find internal resultants: axial force $N$ or torque $T$.
3. Compute stresses using $\sigma =N/A$ or $\tau =Tr/J$.
4. Compute deformations using $\delta =NL/(AE)$ or $\theta =TL/(JG)$.
5. For indeterminate systems, add compatibility conditions and stiffness relationships.
6. Check both strength and serviceability, then adjust geometry or material.

Axial loading and torsion are often taught early because they are clean, solvable, and remarkably predictive. Mastering them builds the intuition needed for more complex behavior, including bending, combined stresses, fatigue, and stability, all of which start from the same core idea: internal resistance must satisfy equilibrium, and deformation must satisfy compatibility.