Inputs
Axial force F, rod length L, cross-sectional area A and the material's elastic modulus E fully describe the rod.
Statics & strength · Linear elasticity
Tension/compression rod: deformation and axial stiffness
Calculate the elongation ΔL, strain ε, normal stress σ, axial stiffness k=EA/L, and the strain energy U stored in a centrically loaded tension/compression rod, from axial force F, rod length L, cross-sectional area A and elastic modulus E.
This covers a slender, prismatic rod under pure axial load in the linear-elastic range. A buckling check for compression members is not included.
ΔLk=EA/L
02
Deformation and stiffness
Set the inputs and calculate.
Inputs and method
Tension/compression rod: deformation and axial stiffness
Calculate elongation, strain, stress, axial stiffness and strain energy for a centrically loaded tension/compression rod.
Calculation
From Hooke's law: σ=F/A, ΔL=F·L/(E·A), ε=ΔL/L=σ/E, axial stiffness k=E·A/L=F/ΔL, and strain energy U=F·ΔL/2=F²·L/(2·E·A).
Example
F=10,000 N, L=2,000 mm, A=100 mm², E=210,000 MPa give σ=100 MPa, ΔL≈0.952 mm, k=10,500 N/mm and U≈4.76 J.
Sources and limits: Classical linear elasticity (Hooke's law); valid only in the linear-elastic range, no buckling check for compressive rods.
Tension/compression rod: deformation and axial stiffness in detail
Calculate elongation, strain, stress, axial stiffness and strain energy for a centrically loaded tension/compression rod.
What is a tension/compression rod?
A tension/compression rod is a slender, prismatic member loaded only by a force acting centrically along its longitudinal axis -- without bending, torsion or transverse shear. In the linear-elastic range, Hooke's law fully describes the relationship between stress and strain.
Formula and variables
ΔL = F·L / (E·A)
ε = ΔL/L = σ/Ek = E·A/L = F/ΔLU = F·ΔL/2 = F²·L/(2·E·A)
| Symbol / input | Meaning |
|---|---|
| F | Axially applied force, in N; positive for tension, negative for compression. |
| L | Unstretched original length of the rod, in mm or m. |
| A | Cross-sectional area perpendicular to the rod axis, assumed constant along the length. |
| E | Elastic (Young's) modulus of the material, in MPa or GPa. |
| ΔL, ε, k, U | Elongation, strain, axial stiffness and strain energy -- the four result quantities of the rod model. |
Choose the inputs correctly
Axial force F, rod length L, cross-sectional area A and the material's elastic modulus E fully describe the rod.
How to use the calculator
Enter the axial force F, the unstretched rod length L, the cross-sectional area A and the material's elastic modulus E. The area can be taken directly from the section-properties calculator if it is not already known.
Worked example
F=10,000 N, L=2,000 mm, A=100 mm², E=210,000 MPa give σ=100 MPa, ΔL≈0.952 mm, k=10,500 N/mm and U≈4.76 J.
How are ΔL, k and U related?
ΔL and k describe the same physical behaviour from two angles: ΔL answers 'how much does the rod stretch under this force', k answers 'what force would be needed for a given elongation' -- k is thus directly the rod's spring constant, as used, for example, in an equivalent spring for an assembly. U is the area under the linear force-elongation curve.
Force in N or kN, length in mm or m, area in mm² or cm², elastic modulus in MPa or GPa; results in mm, dimensionless (strain), N/mm and joules.
Typical applications
Estimating how much a tension or compression member (e.g. a tie rod, a cable or a pendulum strut) stretches or shortens under a known axial force, or comparing the stiffness of different sections or materials.
Assumptions, limits and common mistakes
Classical linear elasticity (Hooke's law); valid only in the linear-elastic range, no buckling check for compressive rods.
Common mistake: Don't confuse the axial stiffness k = E·A/L with the elastic modulus E itself -- E is a pure material constant, k additionally depends on the specific rod's area and length. No buckling check (Euler buckling) is performed here for compressive forces; a slender compression member can buckle before reaching the yield strength.
Frequently asked questions
What is the difference between strain ε and elongation ΔL?
ΔL is the absolute change in length in mm or m; ε=ΔL/L is the dimensionless strain referred to the original length. Two rods of different length with the same ΔL have different ε.
How are axial stiffness k and elastic modulus E related?
k=E·A/L is the stiffness of the specific component (force per unit of elongation); E is a pure material property. Two rods of the same material (same E) can still have very different k due to different area or length.
What is the strain energy U used for?
U describes the elastic energy stored in the rod and is needed, for example, for structural energy methods (such as Castigliano's theorem) or for impact problems where kinetic energy converts into strain energy.
Does this calculation also apply to compression members?
The formulas apply mathematically identically to compression (negative force), but only give the material deformation. Whether a slender compression member buckles first must be checked separately with a buckling analysis (Euler cases).