Calculate the force to start bending a solid round rod from its length and diameter with typical steel or aluminum yield strengths, or find stress or size.

Finds the load that brings the rod to the chosen yield strength or stress limit.

Cantilever: fixed end to load. Simply supported: full support span, with load at its center.

Use an applicable elastic stress limit. A yield value estimates first outer-fiber yield, not full plastic bending.

Optional settings

Rod Bending Force Formula

For a straight solid circular rod under the selected ideal point-load model, the nominal elastic stress relation gives the force at a user-entered target stress. It does not determine the machine force for a permanent bend.

F = σ × π × D³ / (32 × g × L)
  • Where F is the transverse point-load magnitude (N)
  • σ is the entered applicable elastic stress limit (Pa = N/m²); using yield stress estimates first outer-fiber yield
  • D is the solid rod diameter (m); I = πD⁴/64 and the elastic section modulus is Z = πD³/32
  • L is the fixed-end-to-load distance for a cantilever, or the full support span for a simply supported midpoint load (m)
  • g = 1 for a cantilever end load, or 1/4 for a simply supported midpoint load, so maximum moment M = gFL

Calculate Z = πD³/32, then F = σZ/(gL). Reverse the same relationship for stress, σ = gFL/Z, or diameter, D = [32gFL/(πσ)](1/3). Use N, mm and MPa together, or lbf, inches and psi; the calculator converts ksi to psi internally through a coherent unit conversion. See the elastic bending and support-load derivations in MIT Structural Mechanics Lecture 4 and Lecture 5.

What is a Rod Bending Force?

Definition:

A rod bending force is a force that causes a rod (or beam) to flex or bend, creating an internal bending moment and internal stresses.

The force associated with a target elastic bending stress depends on the loading/support geometry, section modulus and chosen stress limit. Any transverse load can cause elastic deflection; reaching first yield only begins yielding at the outer fibers. A permanent bend requires plastic deformation, with tooling, springback, material hardening and changing contact geometry outside this model. No safe working load or design-code capacity is established.

How to Calculate Rod Bending Force?

Example Problem:

The following example outlines the steps and information needed to calculate the Rod Bending Force.

First, choose the cantilever end-load model and a target stress of 250 MPa. This is a numerical stress-limit example, not a recommended material allowable.

Use a solid rod diameter D = 10 mm. Its I = πD⁴/64 ≈ 490.8739 mm⁴ and Z = πD³/32 ≈ 98.1748 mm³; these properties come from the same circular section.

Use the fixed-end-to-load distance L = 200 mm (0.2 m). For this cantilever, g = 1.

The outer-fiber distance is D/2 = 5 mm. All example calculations use coherent N, mm and MPa units.

Finally, calculate the rod bending force using the formula above:

F = σZ / (gL)

F = (250 N/mm²) × (π × 10³ / 32 mm³) / (1 × 200 mm)

F ≈ 122.7185 N. With the same full length as a simply supported midpoint span, g = 1/4 and the corresponding force is approximately 490.8739 N.

FAQ

What materials can be used for rod bending?
The elastic calculation assumes a straight, homogeneous solid round section. Use an applicable elastic stress limit for the actual material and conditions. Yield, fracture, ductility, hardening and springback all matter for permanent forming; this calculator does not select a material or a forming process.

How does the thickness of a rod affect its bending force?
For this solid-circle model at fixed target stress, load length and support type, force scales with diameter cubed because Z = πD³/32. Doubling diameter multiplies the nominal target-stress force by eight. Other failure modes and forming limits still require separate checks.

Can the rod bending force formula be used for any shape of the rod?
The general elastic relation σ = Mc/I can apply to other suitable beam sections with their correct bending properties. This calculator specifically uses a solid circular section; it does not cover tubes, arbitrary shapes or additional axial/torsional loads. Ideal supports, small deflection and elastic material behavior must also be applicable.