Loads, Stress, Strain and Deflection

Introduction

Machine components rarely experience only one idealized load. Shafts transmit torque while bending under gears, bolts carry tension and shear, frames experience impact and thermal expansion, and bearings impose concentrated reactions. Machine design therefore begins with a disciplined description of loading and a reliable prediction of stress, deformation, and stored energy.

This chapter develops the mechanics needed to analyze axial, bending, torsional, transverse, impact, thermal, and combined loading. The equations are based primarily on linear elastic mechanics. They are appropriate when material behavior is elastic, deformations are small, geometry is adequately represented by idealized models, and loads and supports are sufficiently well defined.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Classify loads and construct free-body diagrams.
2. Calculate normal, shear, bending, torsional, bearing, and thermal stresses.
3. Relate stress to strain using elastic constitutive equations.
4. Determine deflection and angle of twist using standard analytical methods.
5. Explain stress concentration, principal stress, and combined loading.
6. Use strain energy and impact concepts to estimate peak response.
7. Select appropriate models while recognizing their assumptions and limitations.

1. Loads and idealized machine members

A load is an external force, couple, pressure, acceleration effect, or imposed deformation applied to a component. Loads may be:

• Axial: force through the centroidal axis, producing tension or compression.
• Shear or transverse: force normal to the member axis.
• Bending: a couple or transverse loading that produces curvature.
• Torsional: a twisting couple about the member axis.
• Bearing or contact: localized compressive action between mating surfaces.
• Impact or dynamic: rapidly applied or time-varying loading.
• Thermal: stress caused by restrained expansion or contraction.

A free-body diagram separates a component from its surroundings and shows all applied loads and reactions. Equilibrium requires

where ∑F is the vector sum of forces and ∑M is the vector sum of moments. These equations determine external reactions, but internal stresses require a section cut. At a cut, the internal resultants are axial force N, shear force V, bending moment M, and torque T.

A useful distinction is between force-controlled and displacement-controlled loading. A force-controlled load is prescribed independently of deformation; a displacement-controlled load is imposed by geometry or movement. Thermal expansion is commonly displacement-controlled. This distinction strongly affects the resulting stress.

2. Stress and strain

Stress is internal force intensity. For an average normal stress,

where σ is normal stress in pascals (Pa), P is axial force in newtons (N), and A is cross-sectional area in square metres (m²). Tension is conventionally positive and compression negative.

Average shear stress is

where τ_avg is average shear stress and V is transverse shear force. This expression is useful for simple connections, but the actual distribution is often nonuniform.

Normal strain is deformation per original length:

where ε is dimensionless strain, δ is axial deformation, and L is original length. Shear strain γ is the angular distortion, measured in radians.

For a homogeneous, isotropic, linearly elastic material, Hooke’s law gives

where E is Young’s modulus. In pure shear,

where G is the shear modulus. For isotropic materials,

where ν is Poisson’s ratio. Under uniaxial tension, lateral strain is ε_lat = −νε. These relations apply below the proportional limit and do not describe plasticity, creep, or significant material anisotropy.

2.1 Axial deformation

For a prismatic member with constant axial force P, area A, and length L,

For a stepped or continuously varying member, the general elastic expression is

The integral emphasizes that deformation depends on stiffness distribution, not merely total length. Axial rigidity is AE.

Worked example 1: stepped tensile bar. A steel bar consists of two segments in series. Segment 1 has P = 20 kN, L₁ = 0.30 m, A₁ = 400 mm²; segment 2 has L₂ = 0.50 m and A₂ = 250 mm². Take E = 200 GPa.

Using consistent N and mm units, E = 200,000 N/mm²:

δ₁ = (20,000 × 300)/(400 × 200,000) = 0.075 mm.

δ₂ = (20,000 × 500)/(250 × 200,000) = 0.200 mm.

Thus total extension is δ = 0.275 mm. The stresses are 50 MPa and 80 MPa, respectively; the smaller section governs the axial stress.

3. Bending stress and beam deflection

In elementary beam theory, plane sections remain plane and normal to the neutral axis after bending. For an initially straight, slender beam with small deflection,

where σ_b is bending stress, M is internal bending moment, y is distance from the neutral axis, and I is the second moment of area about the bending axis. The maximum magnitude occurs at the extreme fibre, y = c:

where Z = I/c is the section modulus.

For common sections:

The flexure formula assumes elastic behavior, a prismatic or locally representative section, and bending about a principal centroidal axis. It becomes unreliable near abrupt geometry changes, deep beams, or plastic collapse.

For an Euler–Bernoulli beam, curvature is related to bending moment by

with sign depending on the chosen convention. Here v is transverse deflection. Integrating twice and applying support boundary conditions gives the deflection curve.

For a simply supported beam with a central point load F:

For a cantilever with an end load F:

For a cantilever with an end moment M₀:

These formulas apply to slender, prismatic beams with constant E and I and small elastic deflection.

Deflection is often a functional requirement. Excessive displacement may cause misalignment, gear tooth interference, seal failure, or fatigue even when stress is below yield strength. Stiffness can be increased by increasing E, shortening the span, or increasing I; because beam deflection commonly varies with L³, span reduction is particularly effective.

4. Transverse shear

For a beam section, the elastic shear stress distribution is

where V is shear force, Q is the first moment of the area above or below the point of interest about the neutral axis, I is the second moment of area, and b is the local width at that point.

For a rectangular section, the maximum shear stress is

and occurs at the neutral axis. The average value V/A is therefore not the maximum. For a circular section, the maximum is 4V/(3A). Transverse shear is usually less important than bending stress in long slender beams, but it becomes significant in short, deep beams and near supports or load application points.

5. Torsion

For a circular shaft subjected to torque T, the torsion equation is

where r is radial position and J is the polar second moment of area. The maximum stress is

For a solid circular shaft, J = πd⁴/32; for a hollow circular shaft, J = π(D⁴ − d⁴)/32.

The angle of twist is

for constant torque, length, and material. For variable shaft properties,

The torsion formulas assume a circular shaft, Saint-Venant torsion, elastic behavior, and small twist. Noncircular sections develop warping and require specialized torsion constants.

Worked example 2: shaft sizing. A solid steel shaft transmits T = 600 N·m. The allowable torsional shear stress is 50 MPa. Determine the minimum diameter using the strength criterion.

For a solid shaft,

τ_max = 16T/(πd³), so

d = [16T/(πτ_allow)]¹ᐟ³.

With T = 600,000 N·mm and τ_allow = 50 N/mm²,

d = [16(600,000)/(π × 50)]¹ᐟ³ = 39.4 mm.

A practical design would select a larger standard diameter after checking angle of twist, keyways, fatigue, stress concentration, manufacturing tolerance, and combined bending.

6. Strain energy and impact loading

Elastic strain energy is the work stored in a deforming component. For a linearly elastic member under gradually applied load,

For an axially loaded prismatic bar,

For a torsion member,

For a beam under bending,

These expressions assume linear elasticity and quasistatic loading. Castigliano’s theorem states that the displacement in the direction of a force is

and the rotation in the direction of a moment is θ_i = ∂U/∂M_i.

Impact loading

An impact load can produce stresses much greater than the same load applied gradually. Consider a weight W falling through height h onto a linear elastic member whose static deflection under W is δ_s. Conservation of energy gives

where k is stiffness and δ_max is maximum deflection. Since W = kδ_s,

For h = 0, the suddenly applied load produces δ_max = 2δ_s. This idealization neglects damping, wave propagation, local contact deformation, plasticity, and loss of contact. It is useful for preliminary design, not for high-speed impact certification.

7. Thermal stress

A free uniform temperature change produces expansion without stress:

where α is the coefficient of thermal expansion, ΔT is temperature change, and L is original length. If axial expansion is completely restrained, total strain is zero and the thermal stress is

The sign indicates compression for heating under full restraint. Partial restraint requires compatibility with the surrounding structure. If a spring or adjacent member shares the expansion, stiffness compatibility must be used rather than the full-restraint formula.

Temperature gradients can also cause bending. Thermal analysis must consider temperature distribution, support flexibility, assembly sequence, friction, and whether material properties vary with temperature. Thermal stresses are especially important in engines, brakes, pressure vessels, welded structures, and high-temperature rotating machinery.

8. Stress concentration and notch effects

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

for normal stress, with an analogous definition for shear stress. Nominal stress is calculated using a reference area or section without the local discontinuity. Values of K_t are obtained from analytical solutions, charts, standards, or finite-element analysis.

A large theoretical concentration does not always produce the same fatigue severity. Notch sensitivity is represented approximately by a fatigue stress-concentration factor K_f, often less than K_t for ductile materials:

where q is notch sensitivity, 0 ≤ q ≤ 1. This relation is empirical and should not replace validated material data. Good design practice uses generous fillet radii, avoids abrupt stiffness changes, removes surface damage, and locates holes away from peak bending regions.

9. Combined loading and principal stresses

Machine components commonly experience simultaneous normal and shear stresses. For plane stress with σ_x, σ_y, and τ_xy, the principal stresses are

The maximum in-plane shear stress is

The principal direction θ_p satisfies

A common shaft example combines bending stress σ_b and torsional shear τ. If no second normal stress exists, σ_x = σ_b and σ_y = 0, giving

For ductile materials, the distortion-energy or von Mises equivalent stress under plane stress is

Yielding is predicted when σ_vm reaches the material yield strength in a simple static model. For brittle materials, maximum principal stress or a fracture-based criterion may be more appropriate. The selected failure criterion must match material behavior, loading history, temperature, and design standard.

10. A systematic design procedure

A reliable analysis follows these steps:

1. Define function, geometry, material, supports, operating speed, temperature, and load cases.
2. Draw the free-body diagram and determine reactions by equilibrium.
3. Cut the member to obtain N, V, M, and T.
4. Identify critical sections, including discontinuities and contacts.
5. Calculate nominal stresses and deformations using appropriate models.
6. Apply stress-concentration factors where justified.
7. Combine stresses and calculate principal or equivalent stress.
8. Check strength, stiffness, stability, fatigue, wear, and thermal compatibility.
9. Compare results with allowable values using an explicitly selected factor of safety.
10. Validate assumptions with hand calculations, finite-element analysis, testing, or relevant standards.

A factor of safety is not a substitute for sound modeling. Uncertainty may arise from load variation, material scatter, manufacturing defects, corrosion, residual stress, and model error. Safety-critical components require traceable assumptions, independent review, and applicable design codes.

Applied case: illustrative gearbox output shaft

Background

Consider a steel gearbox output shaft carrying a gear at its overhung end. The gear transmits torque T = 600 N·m and applies a transverse force F = 2.0 kN at a bearing distance of 0.20 m. The shaft diameter at the critical section is 45 mm. Assume, initially, that bending and torsional stresses are elastic and that the section is solid and circular.

Problem and analysis

The bending moment at the bearing is

M = FL = 2,000 × 0.20 = 400 N·m.

For a solid circular section,

σ_b = 32M/(πd³), τ = 16T/(πd³).

Using d = 45 mm, M = 400,000 N·mm, and T = 600,000 N·mm:

σ_b = 32(400,000)/[π(45)³] ≈ 44.6 MPa,

τ = 16(600,000)/[π(45)³] ≈ 33.5 MPa.

The von Mises stress is

σ_vm = √(σ_b² + 3τ²)

= √(44.6² + 3(33.5)²)

≈ 69.1 MPa.

If a shoulder produces K_t = 1.8 in bending and K_ts = 1.5 in torsion, the local elastic stresses are approximately 80.3 MPa and 50.3 MPa. A refined fatigue analysis would use K_f and K_fs, surface condition, size effect, mean stress, and endurance data rather than simply applying theoretical factors.

Approach and outcome

The preliminary result suggests that static yielding may not govern if the shaft steel has a substantially higher yield strength. However, the design is incomplete. The engineer must check angle of twist, bearing reactions, keyway effects, fatigue under rotating bending, stress concentration at shoulders, shaft critical speed, corrosion, assembly tolerances, and lubrication. If deflection or fatigue is excessive, increasing diameter or reducing the overhang may be more effective than merely selecting a stronger material.

Learning points

The case demonstrates that torque alone does not define shaft stress. A modest transverse force can create a large bending moment, and geometric discontinuities can dominate local fatigue behavior. Strength and stiffness must be checked together.

Limitations, technology, and best practice

Analytical formulas provide insight and efficient preliminary sizing, but they idealize geometry, supports, contact, and material behavior. Finite-element analysis is valuable for complex geometry, thermal gradients, and local stress fields; nevertheless, it requires suitable mesh refinement, realistic boundary conditions, correct material data, and independent verification. A very high stress at an idealized sharp corner may be a mathematical singularity rather than a physically meaningful finite stress.

Modern design workflows combine hand calculations, CAD-integrated simulation, experimental strain measurement, and digital monitoring. AI-assisted design can help explore geometries or identify patterns in test data, but it cannot replace conservation laws, validated material models, engineering judgment, or responsibility for safety. Results should be reproducible, documented, and reviewed against standards.

Chapter Summary

Loads are represented through axial force, shear force, bending moment, and torque obtained from equilibrium and section cuts. Normal stress is associated with axial force and bending; shear stress is associated with transverse shear and torsion. Hooke’s law connects stress and strain in the linear elastic range, while AE, EI, and JG describe axial, bending, and torsional stiffness.

The principal formulas include σ = P/A, σ = My/I, τ = Tr/J, δ = PL/(AE), φ = TL/(JG), and beam-deflection relations derived from d²v/dx² = M/(EI). Strain energy supports deflection calculations and impact analysis. Thermal expansion is stress-free when unrestrained but produces σ_T = −EαΔT under complete axial restraint. Geometric discontinuities create stress concentrations, and combined loading requires principal-stress or equivalent-stress analysis. Reliable machine design checks strength, stiffness, fatigue, stability, thermal compatibility, manufacturability, and uncertainty rather than relying on a single stress number.

Key Terms

Load: External force, moment, pressure, acceleration effect, or imposed deformation.

Stress: Internal force per unit area.

Strain: Deformation per unit original dimension.

Young’s modulus: Elastic modulus relating normal stress to normal strain.

Shear modulus: Elastic modulus relating shear stress to shear strain.

Axial rigidity: Product AE governing axial deformation.

Bending rigidity: Product EI governing beam curvature and deflection.

Second moment of area: Geometric property governing resistance to bending.

Section modulus: I/c, relating bending moment to extreme-fibre stress.

Polar second moment: Geometric property J governing circular-shaft torsion.

Torsional rigidity: Product GJ governing angle of twist.

Principal stress: Normal stress acting on a plane with zero shear stress.

Von Mises stress: Distortion-energy equivalent stress used commonly for ductile materials.

Stress concentration: Local amplification of nominal stress caused by geometry.

Notch sensitivity: Degree to which a material responds to a geometric notch.

Strain energy: Elastic energy stored during deformation.

Impact loading: Rapid loading involving kinetic-energy transfer.

Thermal stress: Stress caused by restrained thermal deformation.

Free-body diagram: Isolated-body representation showing loads and reactions.

Factor of safety: Ratio relating capacity or allowable strength to design demand.

Review Questions

Short Answer Questions

1. What is the difference between axial rigidity and bending rigidity?
2. State the assumptions behind the flexure formula σ = My/I.
3. Why is the maximum transverse shear stress in a rectangular beam greater than V/A?
4. Under what condition does a uniform temperature change produce no thermal stress?
5. What physical feature causes a stress concentration?

Long Answer Questions

1. Derive the axial deformation of a prismatic bar and explain how the result changes for a stepped bar.
2. Explain the derivation and application of the torsion equation for a solid circular shaft.
3. Compare strength-based and stiffness-based shaft design, including the effects of bending and torsion.
4. Explain principal stresses and the von Mises criterion for a component under combined normal and shear stress.
5. Discuss the assumptions and limitations of analytical beam-deflection formulas in machine design.

Analytical / Application Questions

1. A 1.0 m steel cantilever of rectangular section 20 mm wide and 40 mm deep carries a 500 N end load. Using E = 200 GPa, determine the maximum bending stress and end deflection.
2. A solid shaft of diameter 50 mm transmits 800 N·m while also carrying a bending moment of 500 N·m. Determine the bending stress, torsional shear stress, and von Mises stress at the surface.
3. A fully restrained steel bar experiences a temperature increase of 60 °C. Using E = 200 GPa and α = 12 × 10⁻⁶/°C, determine the thermal stress and state whether it is tensile or compressive.

Multiple Choice Questions with Answers

Question 1

For a prismatic bar under constant axial force, which expression gives elastic axial deformation?

A. δ = PL/(AE)

B. δ = EI/(PL)

C. δ = TL/(JG)

D. δ = My/I

Correct Answer: A. δ = PL/(AE)

Question 2

The maximum bending stress in a beam occurs where?

A. At the neutral axis

B. At the centroid only

C. At the extreme fibre farthest from the neutral axis

D. At the support reaction only

Correct Answer: C. At the extreme fibre farthest from the neutral axis

Question 3

For a solid circular shaft, the polar second moment of area is:

A. πd²/4

B. πd³/32

C. πd⁴/64

D. πd⁴/32

Correct Answer: D. πd⁴/32

Question 4

A rectangular beam’s maximum transverse shear stress is:

A. V/A

B. 3V/(2A)

C. 2V/(3A)

D. 4V/(3A)

Correct Answer: B. 3V/(2A)

Question 5

A freely expanding uniform bar subjected to a temperature increase develops, ideally:

A. Maximum tensile stress

B. Maximum compressive stress

C. Zero thermal stress

D. Torsional stress

Correct Answer: C. Zero thermal stress

Question 6

Which quantity represents elastic bending stiffness?

A. AE

B. JG

C. EI

D. I/E

Correct Answer: C. EI

Question 7

For plane stress, principal stresses are obtained by combining normal stresses with:

A. Mass density

B. Shear stress

C. Thermal conductivity

D. Poisson’s ratio only

Correct Answer: B. Shear stress

Question 8

A fully restrained bar heated uniformly develops thermal stress equal in magnitude to:

A. EαΔT

B. αΔT/E

C. E/(αΔT)

D. αLΔT

Correct Answer: A. EαΔT

Question 9

If a cantilever end load remains constant and its length is doubled, the ideal end deflection changes by a factor of:

A. 2

B. 4

C. 6

D. 8

Correct Answer: D. 8

Question 10

The theoretical stress-concentration factor is defined as:

A. Nominal stress divided by maximum stress

B. Maximum stress divided by nominal stress

C. Yield stress divided by elastic modulus

D. Shear stress divided by normal stress

Correct Answer: B. Maximum stress divided by nominal stress