Stress and Strain: The Basics for Building Structures

Every structural calculation eventually comes back to two questions: how hard is the material being worked, and how much does it deform as a result? Those two quantities are stress and strain. They are the common language that lets an engineer compare a steel bar, a concrete column and a timber joist on equal terms, regardless of their size.

This article builds the ideas from intuition, then gives the definitions and formulas in plain text, explains the stress-strain curve, works through a simple example with real numbers, and lists the mistakes students most often make. It is written for students and for homeowners who want to understand what engineers mean when they use these words.

The intuition: why force alone is not enough

Hang a 10 kg weight from a thin wire and it may snap. Hang the same weight from a thick rope and nothing happens. The force is identical; what differs is how much material shares it. A thin wire concentrates the force on a small area, so each part of the material works much harder.

Stress captures this idea. It is force per unit area, a measure of intensity rather than total load. Strain captures the other half of the story. A 1 mm stretch matters a great deal in a 10 mm specimen and hardly at all in a 10 m cable. Strain expresses deformation relative to the original size, so behaviour can be compared regardless of length.

Definitions and formulas

Stress

σ = F / A, where σ (sigma) is the normal stress, F is the force acting perpendicular to the cross-section, and A is the cross-sectional area. In structural work stress is usually expressed in N/mm², which is the same as megapascals (MPa).

Tensile stress pulls the material apart, as in a hanger rod or the bottom of a sagging beam.
Compressive stress pushes it together, as in a column or the top of a sagging beam.
Shear stress, written τ (tau), acts parallel to a surface, as in a bolt being sliced by two plates sliding past each other. Our sibling article on shear force develops this further.

Strain

ε = ΔL / L, where ε (epsilon) is the strain, ΔL is the change in length and L is the original length. Because it is length divided by length, strain has no units. It is often very small, so engineers sometimes quote it as a percentage or in microstrain (millionths).

Linking the two: Young’s modulus

For many materials, at moderate loads, stress is proportional to strain. This is Hooke’s law, and the constant of proportionality is Young’s modulus:

E = σ / ε

E measures stiffness. Structural steel has an E of roughly 200,000 to 210,000 N/mm². Normal-weight concrete is typically in the order of 30,000 N/mm², and structural timber lower still, often around 10,000 N/mm² along the grain. A higher E means less strain for the same stress. Our sibling article on Young’s modulus covers this property in depth.

The stress-strain curve

If you pull a steel bar in a testing machine and plot stress against strain, you get a characteristic shape that tells you most of what you need to know about the material.

Elastic region. The first part is a straight line. Stress and strain are proportional, and if the load is removed the bar returns to its original length. The slope of this line is E.
Yield point. At a certain stress the steel begins to deform permanently. This is the yield strength, a central value in design. Common reinforcing steel in Singapore and many other markets has a characteristic yield strength of around 500 N/mm². See the sibling article on yield strength.
Plastic region and strain hardening. Beyond yield, strain increases greatly for modest increases in stress, and the steel stretches visibly. This ductility is valuable: it gives warning before failure.
Ultimate strength and fracture. The curve peaks at the ultimate tensile strength, after which the bar necks down locally and eventually breaks.

Other materials produce very different curves. Concrete is strong in compression but weak and brittle in tension, which is why it is reinforced with steel. Glass is almost perfectly elastic and then fails suddenly with little warning. Timber behaves differently along and across the grain. Understanding these differences is the foundation of material choice; our guide to reinforced concrete shows how two materials with complementary curves are combined.

A worked example

A 20 mm diameter steel bar, 3 m long, carries a tensile force of 50 kN. Take E = 200,000 N/mm².

Area: A = π × 20² / 4 ≈ 314 mm².
Stress: σ = 50,000 N / 314 mm² ≈ 159 N/mm².
Strain: ε = σ / E = 159 / 200,000 ≈ 0.00080, or 0.08 per cent.
Elongation: ΔL = ε × L = 0.00080 × 3,000 mm ≈ 2.4 mm.

The stress of about 159 N/mm² is well below a typical yield strength of 500 N/mm², so the bar is in its elastic range and will return to its original length when the load is removed. The stretch of 2.4 mm over 3 m is small, but in a long hanger or tie it may still need to be considered for alignment and finishes.

Now try the same load on a 10 mm bar. The area falls to about 79 mm², the stress rises to roughly 637 N/mm², and the bar would exceed a 500 N/mm² yield strength. Halving the diameter quarters the area, which is why bar size matters so much.

Where stress and strain matter in buildings

Stress governs strength. Designers check that stresses, after factoring loads and materials, stay within what the material can resist. Strain governs deformation and compatibility. In reinforced concrete, for instance, steel and concrete are bonded, so they experience the same strain at the same location. Because steel is far stiffer, it carries much higher stress for that shared strain, which is exactly why it is effective as reinforcement.

Strain also explains many defects. Concrete shrinks as it dries, and if that shrinkage strain is restrained, tensile stress builds up and the concrete cracks. Temperature changes cause thermal strain in roofs and facades; if movement joints are missing, the resulting stress shows up as cracking or bowing. Many items in a building survey are, at root, strain problems.

Common mistakes

Confusing stress with force. A larger force does not necessarily mean higher stress if the area is larger too.
Mixing units. Working in kN and mm without converting, or in N and m², gives answers off by factors of a thousand or a million. N and mm, giving N/mm², is the safest habit.
Using diameter instead of area. Stress depends on area, which scales with diameter squared.
Assuming Hooke’s law always applies. Beyond yield, or for concrete at high compressive stress, stress and strain are no longer proportional.
Thinking stiff means strong. E describes stiffness, not strength. Two steels with very different yield strengths have nearly the same E.

Frequently asked questions

What is the difference between stress and strain?

Stress is the internal force per unit area in a material, measured in units such as N/mm². Strain is the resulting deformation expressed as change in length divided by original length, which has no units. Stress describes how hard the material is working; strain describes how much it deforms.

What are the units of stress and strain?

Stress is measured in force per area, commonly N/mm² or MPa in structural engineering, which are equivalent. Strain is a ratio of two lengths and is therefore dimensionless, though it is often quoted as a percentage or in microstrain.

What does the stress-strain curve show?

It shows how a material responds as load increases: an initial elastic region where it springs back, a yield point where permanent deformation begins, a plastic region and, finally, ultimate strength and fracture. Its shape reveals stiffness, strength and ductility at a glance.

Why is steel used to reinforce concrete?

Concrete is strong in compression but weak and brittle in tension. Steel is strong and ductile in tension and much stiffer, so where the two are bonded and share the same strain, the steel carries the tensile stresses that would otherwise crack and fail the concrete.

Can a material be strained without being stressed?

Yes. A bar free to expand with temperature or a slab free to shrink experiences strain without significant stress. Stress develops only when that movement is restrained, which is why unrestrained movement is usually harmless and restrained movement causes cracking.

Seeing cracks or movement you cannot explain? Talk to our consultancy or get in touch for an accountable assessment.