Formula & Calculator
Axial Stress
Calculates the normal stress in a member subjected to a purely axial (tension or compression) load.
Interpretation
Axial stress (normal stress) is the force per unit area acting perpendicular to a cross‑section. It is given by σ = F / A. It can be tensile (pulling) or compressive (pushing). This is the most basic form of stress in mechanics of materials.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| sigma | Axial stress | MPa |
| F | Applied axial force | N |
| A | Cross-sectional area | mm2 |
What it means
Axial stress, also called normal stress, is the intensity of force acting normal to a surface. It is defined as the force divided by the area over which it acts: σ = F / A. When the force pulls the member apart, it is tensile stress; when it pushes together, it is compressive stress. Axial stress is uniform across the cross‑section if the force is applied at the centroid and the member is prismatic. It is the simplest case in strength of materials and is used for columns, ties, and bolts. The stress must be kept below the material's yield strength to avoid permanent deformation. In design, allowable stress is determined by dividing the yield strength by a factor of safety. Axial stress also appears in pressure vessels, where hoop and longitudinal stresses are derived from axial and tangential components. The concept extends to combined loading, where axial stress contributes to the total stress state.
Worked example
Axial Stress – Two Examples
Real‑World| Parameter | Value |
|---|---|
| F | 25,000 N |
| A | 113 mm² |
| Parameter | Value |
|---|---|
| F | 60,000 N |
| A | 200 mm² |
Common mistakes
- Same as for formula ID 11 – see that entry.
- Additional: Ensure the area is the actual cross‑sectional area perpendicular to the load; for non‑uniform sections, stress varies.
Applications
Axial stress (normal stress) is the force per unit area acting perpendicular to the cross‑section of a member, either tensile or compressive. It is the most basic stress type and appears in countless engineering situations. In structural steel design, axial stress determines the required cross‑sectional area of columns and tension members. In machine design, it is used to size bolts, rods, and cables. In civil engineering, axial stress governs the load‑bearing capacity of piles and piers. The formula also underpins the design of pressure vessels, where internal pressure creates hoop and longitudinal stresses. Moreover, axial stress is central to material testing, where stress‑strain curves are generated to characterise material behaviour. By understanding axial stress, engineers can ensure components are strong enough to carry applied loads without excessive deformation or failure.
- Design of tension members (cables, rods)
- Column and compression member sizing
- Bolted and riveted joint analysis
- Pressure vessel and pipe wall thickness design
- Material testing and quality control
Frequently Asked Questions
Axial stress (also called normal stress) is the internal force per unit area acting perpendicular to the cross‑section of a member when the load is applied along the member’s longitudinal axis (tension or compression). It occurs in truss members, columns, rods, and cables. The formula is σ = F / A, where F is the axial force and A is the cross‑sectional area.
- σ = axial stress (Pa or N/m² in SI; psi or lb/in² in imperial)
- F = axial force (N or lb) – positive for tension, negative for compression
- A = cross‑sectional area (m² or in²)
- Using the wrong area – for hollow sections, you must subtract the inner area (net area). Use A = π/4 (D² – d²).
- Confusing force and stress – stress is not force; it is force per area.
- Ignoring sign convention – tension is positive, compression is negative. This matters for design codes and combining stresses.
- Using the total area of a bolted connection – the area should be the net area (subtracting bolt holes) for tensile members.
In the elastic region, axial stress is linearly related to strain (ε) by Hooke’s Law: σ = E · ε, where E is Young’s modulus (elastic modulus) of the material. Strain is the change in length per unit length (ε = ΔL/L). This relationship is the foundation of linear elasticity and is used to calculate deformations.
- Tensile stress occurs when the member is pulled apart (force tends to elongate it). It is positive and often causes failure by fracturing.
- Compressive stress occurs when the member is pushed together (force tends to shorten it). It is negative and may cause buckling in slender members.
If the cross‑section changes gradually, the stress varies along the length. You must calculate the stress at each section using the local area: σ(x) = F / A(x). The maximum stress occurs where the area is the smallest (assuming constant force). For design, you typically check the minimum area.
Stress concentration occurs at geometric discontinuities (holes, notches, fillets) where the stress is locally higher than the average nominal stress. The stress concentration factor (K_t) multiplies the nominal stress: σ_max = K_t · σ_nominal. This can lead to premature failure, so designers use rounded fillets and avoid sharp corners.
- Structural steel (A36): yield ≈ 250 MPa, ultimate ≈ 400 MPa
- Aluminium 6061‑T6: yield ≈ 276 MPa, ultimate ≈ 310 MPa
- Stainless steel 304: yield ≈ 215 MPa, ultimate ≈ 505 MPa
- Concrete (compression): compressive strength ≈ 20‑40 MPa
Temperature changes cause thermal expansion or contraction. If a member is restrained, thermal strain (α·ΔT) creates thermal stress (σ = E·α·ΔT). This can be significant and must be considered in design (e.g., pipelines, railway tracks).
The factor of safety (FoS) accounts for uncertainties in loads, material properties, and manufacturing. The allowable stress is σ_allow = σ_yield / FoS (or σ_ultimate / FoS for brittle materials). The design criterion is σ ≤ σ_allow. FoS typically ranges from 1.5 to 4 depending on the application.