Learning topic
Tension and Compression
Comprehensive guide to axial loading: normal stress, axial strain, Hooke's law, and diagrams.
Tension and compression are forms of axial deformation of a bar in which the external forces act along its longitudinal axis. In the simplest model of centric tension or compression, the cross-section carries one internal force resultant: the axial force $N$.
Internal force and stress
The axial force is determined by the method of sections from the equilibrium equations of a cut portion of the bar. Tension is commonly taken as positive and compression as negative. For a centrally loaded prismatic bar, sufficiently far from local disturbances, the normal stress is the axial force divided by the cross-sectional area.
For centric tension or compression of a straight member with a uniform cross-section, the normal stress is:
$$\sigma=\frac{N}{A}$$
- $\sigma$ — normal stress, MPa;
- $N$ — axial force, N;
- $A$ — cross-sectional area, mm².
Since $1\ \text{N/mm}^2=1\ \text{MPa}$, using $N$ in newtons and $A$ in mm² gives the result directly in megapascals.
Deformation
Under axial force, the length of the bar changes. It elongates in tension and shortens in compression. Axial strain characterizes the change in length relative to the initial length.
For a straight prismatic bar under centric tension or compression, when $N$, $E$, and $A$ are constant over the segment and the material behaves linearly elastically:
$$\Delta l=\frac{NL}{EA}.$$
- $\Delta l$ — change in bar length;
- $N$ — axial force;
- $L$ — initial segment length;
- $E$ — Young's modulus;
- $A$ — cross-sectional area.
The axial strain is $\varepsilon=\Delta l/L$. For a stepped bar, the total change in length is the sum of segment deformations: $\Delta l=\sum_i N_iL_i/(E_iA_i)$.
The relation between stress and strain in the linear-elastic range is described by Hooke's law.
Within the linear-elastic range, normal stress is proportional to axial strain:
$$\sigma=E\varepsilon.$$
- $\sigma$ — normal stress, Pa or MPa;
- $E$ — Young's modulus, Pa or MPa;
- $\varepsilon$ — axial strain, dimensionless.
The law applies to the linear portion of the stress–strain curve while $\sigma$ and $\varepsilon$ remain proportional.
Strength and stiffness
Axial design is not limited to calculating stress. A member must satisfy strength requirements and, where relevant, stiffness requirements. The strength condition limits dangerous stresses, while the stiffness condition limits excessive deformation.
For centric tension or compression, the strength condition is:
$$|\sigma_{\max}|\le[\sigma].$$
For a uniform cross-section, $\sigma=N/A$, so the required area for a known axial force can be estimated as:
$$A_{\mathrm{req}}\ge\frac{|N|}{[\sigma]}.$$
The stiffness check limits deformation or displacement, for example:
$$|\Delta l|\le[\Delta l].$$
- $\sigma_{\max}$ — maximum absolute normal stress;
- $[\sigma]$ — allowable stress;
- $A_{\mathrm{req}}$ — required cross-sectional area;
- $\Delta l$ — calculated change in length;
- $[\Delta l]$ — allowable change in length.
Topics covered in this section
The child topics address internal forces and normal stresses, mechanical properties of materials, axial deformations, strength and stiffness checks, thermal deformation, and statically indeterminate axial systems. This sequence moves from equilibrium and stress toward deformation and compatibility.
Basic calculation procedure
- Determine external loads and reactions.
- Use the method of sections to find the axial force $N$ in each segment.
- Calculate normal stresses.
- Determine strains and displacements when required.
- Check strength and stiffness conditions.
For stepped bars or systems made of different materials, perform the calculation segment by segment using the appropriate values of $N$, $A$, $E$, and $L$.
About this topic
Tension and compression are basic forms of deformation where the only internal force factor in a structural member's cross-section is the axial force N. This section describes procedures for constructing normal force diagrams, calculating normal stresses, and linear strains. You will study Hooke's law under axial load, Young's modulus, and Poisson's ratio. Special attention is given to strength conditions and cross-sectional dimension design.