Stress & Strain

1 · Start with the idea

Force is the total push or pull. Stress describes how that load is distributed internally; strain describes deformation relative to original size. Think of people sharing a table: the same total load spread over more supporting area is less demanding on each portion. That analogy helps with average stress, but local stresses near holes and corners need more care.

2 · Why it matters in real machines

A part may remain below yield and still stretch or bend too much to function. Strength and stiffness answer different questions. A tie rod needs enough resistance to failure, but its extension may also affect alignment, travel, sealing or load sharing.

3 · Core concepts

Core definitions

Average normal stress: σ = F/A
Engineering axial strain: ε = ΔL/L₀
Average direct shear stress: τ_avg = V/A
Linear axial elasticity: σ = Eε
Uniform axial extension: ΔL = FL/(AE)

Normal stress acts perpendicular to a section; shear acts along it. Strain is dimensionless but can be reported as mm/mm or as a percentage. V/A is an average for direct shear, not the full transverse-shear distribution in a beam or the torsional-stress distribution in a shaft. FL/(AE) assumes a straight, uniform member, centered axial force and constant linear-elastic E. Threads, bending and temperature changes need additional treatment.

Read an engineering stress–strain curve

The initial straight-line slope is Young’s modulus E. Yield describes the onset of significant permanent deformation under the material’s specified definition; some materials have no sharp yield point, so a specified proof/offset method is used. Ultimate tensile strength (UTS) is the maximum engineering stress based on original area. A ductile tensile specimen may neck after reaching UTS, so engineering stress can fall before fracture even while local conditions become more severe.

Ductility describes the ability to deform plastically before fracture, often reported using elongation or reduction of area under a defined test method. A material with high strength is not automatically stiff or ductile. The qualitative curve in this lesson is not a test result, material certificate or universal curve for every metal.

Poisson’s ratio and thermal strain

ν = −ε_transverse/ε_axial
Free thermal strain: ε_thermal = αΔT
Free thermal extension: ΔL_thermal = αLΔT

In ordinary uniaxial tension, many common metals become slightly narrower as they become longer; the minus sign makes their Poisson’s ratio positive. Use the appropriate material value. Under isotropic linear elasticity, the shear modulus is G = E/[2(1 + ν)].

Thermal expansion of an unrestrained member creates strain without requiring stress. Restraint can create stress, but its value depends on compatibility and support stiffness. A temperature difference of 1 K equals a difference of 1 °C [1.8 °F]; absolute Celsius and kelvin temperatures are not interchangeable. Do not use α in 1/K with a Fahrenheit difference without conversion.

4 · Worked example

Uniform tie-rod example: d = 12.0 mm [0.4724 in], L = 1000 mm [39.37 in], F = 12.0 kN [2697.7 lbf] in centered static tension. Use E = 200 GPa = 200,000 N/mm² [29.0 million psi] as an illustrative steel stiffness. No grade or yield strength is specified, so no strength approval is implied.

A = πd²/4 = π(12.0 mm)²/4 = 113.097 mm² [0.1753 in²]
σ = 12,000 N / 113.097 mm²
σ ≈ 106.10 MPa [15.39 ksi]
ε = σ/E = 106.103/200,000 = 0.0005305 = 0.05305%
ΔL = εL = 0.0005305 × 1000 mm
ΔL ≈ 0.531 mm [0.0209 in]

Independent unit-system check: use L = 1.000 m, A = 1.13097 × 10⁻⁴ m² and E = 2.00 × 10¹¹ N/m². Then FL/(AE) = 0.0005305 m = 0.5305 mm. This matches the millimetre calculation. Doubling the length doubles extension but leaves stress unchanged for the same force and area.

Thermal comparison: for this separate teaching case, choose α = 12 × 10⁻⁶ K⁻¹ and a uniform temperature rise ΔT = 40 K [72 °F difference]. An unrestrained 1.000 m rod expands by 0.480 mm [0.0189 in]. That is similar in scale to its calculated mechanical extension. This α is an example input, not certified material data.

Before applying this model to a real threaded tie rod, establish the relevant tensile stress area, actual material properties, connection details, load eccentricity, fatigue, temperature and allowable movement [VERIFY]. Do not use the nominal shank area as a substitute for the threaded tensile stress area.

5 · Common mistakes

• Using kN with N/mm² without converting force to newtons.
• Forgetting that area depends on diameter squared.
• Confusing 0.0005305 strain with 0.0005305 percent; multiply by 100 to obtain percent.
• Assuming a high modulus means a high yield strength.
• Using UTS as an automatically acceptable working stress.
• Predicting thermal stress without describing whether and how expansion is restrained.

6 · In the field

When a connection moves, distinguish elastic stretch from clearance, slip, bending and measurement error. Record temperature and load state before comparing measurements. A hand calculation is a scale check; the instrument and setup must be capable of resolving the movement being investigated.

7 · Quick checks

8 · Free resources & references

1. OpenStax: stress, strain and elastic modulus ↗

   Free conceptual treatment and equations.

2. OpenStax College Physics: elasticity ↗

   Free complementary discussion of extension and stiffness.

3. OpenStax: thermal expansion ↗

   Free explanation of linear thermal expansion.

9 · Mini-project

Use paper and a spreadsheet to compare 12 mm and 24 mm [0.472 and 0.945 in] diameter rods of the same 1.000 m length under the example force. Predict the ratios before calculating, then make a plot of extension against length. Optionally stretch a soft elastic band very gently between your fingers, below eye level, without launching or attaching weights; observe that deformation can be reversible without claiming it behaves like linear-elastic steel. The numerical exercise needs no physical loading.

10 · Next topics