3 Point Bending Stress Calculator

Analyze center loaded beams with confidence. See stress, deflection, inertia, and section modulus instantly today. Export results, compare examples, and inspect response curves visually.

Calculator Inputs

Section Shape

Applied Load

Load Unit

Support Span

Length Unit

Width

Depth

Diameter

Section Modulus

Section Modulus Unit

Second Moment of Area

Second Moment Unit

Young’s Modulus

Young’s Modulus Unit

Yield Strength

Yield Strength Unit

Example Data Table

Case	Shape	Load	Span	Section Input	Young’s Modulus	Stress	Deflection
Example 1	Rectangular	500 N	300 mm	20 mm × 10 mm	69 GPa	56.25 MPa	2.4457 mm
Example 2	Circular	1000 N	400 mm	Diameter 20 mm	200 GPa	127.324 MPa	0.8488 mm
Example 3	Custom	1500 N	600 mm	Z = 5000 mm³, I = 60000 mm⁴	210 GPa	45 MPa	0.5357 mm

Formula Used

For a simply supported beam with a single load at midspan, the maximum bending moment is:

M = PL / 4

The bending stress at the outer surface is:

σ = M / Z

The midspan deflection is:

δ = PL³ / (48EI)

For a rectangular section:

I = bd³ / 12 and Z = bd² / 6

σ = 3PL / (2bd²)

For a circular section:

I = πd⁴ / 64 and Z = πd³ / 32

σ = 8PL / (πd³)

Each support reaction equals P / 2. Outer fiber strain is σ / E. Factor of safety is Yield Strength / Bending Stress.

How to Use This Calculator

Select the beam section type.
Enter the applied center load and choose its unit.
Enter the support span and choose the matching length unit.
For rectangular beams, enter width and depth. For circular beams, enter diameter. For custom beams, enter section modulus and second moment directly.
Enter Young’s modulus and yield strength.
Press Calculate to show stress, moment, deflection, strain, safety factor, and the response graph above the form.
Use the export buttons to download the calculated result set as CSV or PDF.

Frequently Asked Questions

1. What does three point bending stress represent?

It represents the highest bending stress at the outer surface of a beam loaded at midspan between two supports. It helps estimate whether the beam remains below allowable stress.

2. When should I use the rectangular option?

Use it when the beam cross section is a rectangle and the entered width and depth describe the loaded orientation correctly. Depth strongly affects the result because it is squared or cubed in beam formulas.

3. Why does span length change the result so much?

Longer spans increase bending moment directly and deflection even faster. In a center loaded beam, stress grows with span, while deflection grows with the cube of span.

4. Does this page calculate deflection too?

Yes. It calculates midspan deflection using the elastic beam equation with the entered load, span, Young’s modulus, and second moment of area.

5. Can I use this for plastics, wood, or composites?

Yes, if the material behaves close to linear elastic in the tested range and you know its modulus and strength. For strongly nonlinear materials, results become approximate.

6. What does factor of safety mean here?

It is the ratio of yield strength to calculated bending stress. Values above one indicate the elastic stress is below yield, while larger values indicate more margin.

7. Why is there a custom section option?

Custom input lets you use nonstandard beam shapes when you already know section modulus and second moment of area from design tables, CAD, or hand calculations.

8. Are results valid after yielding starts?

No. These equations assume elastic beam theory. Once yielding, cracking, or large deformation begins, actual stress distribution and deflection can differ from the calculator result.