Enter the bending moment (at failure) and the section modulus into the calculator to determine the modulus of rupture. The modulus of rupture is a measure of flexural strength (the nominal extreme-fiber stress at failure in bending) and can be calculated when the moment and section modulus are known.

Modulus Of Rupture Calculator

Pick your test type, enter the dimensions and load at failure.

3-Point Bend
4-Point Bend
M & S
Assumes third-point loading (load points at L/3 from each support).
Modulus of Rupture
— MPa
— psi
— kPa
How this is calculated ▾

Related Calculators

Modulus Of Rupture Formula

The modulus of rupture (MOR) is the nominal extreme-fiber stress at failure in bending. It is commonly referred to as flexural strength and is used to estimate how much bending stress a material can sustain before it breaks.

MOR = \frac{M}{S}
  • MOR = modulus of rupture
  • M = bending moment at failure
  • S = section modulus of the cross section about the bending axis

Because the equation divides bending moment by section modulus, MOR increases when the failure moment increases and decreases when the section modulus increases. For the same moment, a section with a larger section modulus develops less bending stress.

What the Calculator Does

This calculator finds the modulus of rupture from the moment at failure and the section modulus. It is useful when analyzing beams, slabs, wood members, concrete specimens, plastics, ceramics, and other components loaded in flexure.

  1. Enter the failure moment.
  2. Enter the section modulus.
  3. Calculate the result to obtain the modulus of rupture.

If you already know the maximum moment at failure from a test or structural analysis, this is the fastest way to convert that bending action into a nominal failure stress.

Unit Consistency Matters

The formula only works correctly when the moment and section modulus use a consistent length basis. Stress units come from force per unit area, so the length unit inside the moment must match the length unit used in the section modulus.

For example, if the section modulus is in in3, then the moment should be in lb·in to obtain MOR in psi.

M_{lb \cdot in} = 12 \times M_{lb \cdot ft}
  • psi comes from lb·in divided by in3, which simplifies to lb/in2.
  • MPa is commonly obtained from N·mm divided by mm3, which simplifies to N/mm2.
  • If your calculator handles unit conversion internally, the underlying principle is still the same: use compatible dimensions.

Finding the Section Modulus

If you do not already have the section modulus, it can be found from the cross-sectional geometry. Section modulus is a geometric property that tells you how efficiently a section resists bending.

S = \frac{I}{c}
  • I = area moment of inertia about the bending axis
  • c = distance from the neutral axis to the outermost fiber

For common shapes, the section modulus can often be determined directly from standard formulas or steel/wood/aluminum shape tables.

Cross Section Section Modulus Formula Notes
Rectangle 
S_{rect} = \frac{b d^2}{6}
 b = width, d = depth; bending about the strong axis
Solid Round 
S_{solid\ round} = \frac{\pi d^3}{32}
 d = diameter
Hollow Round 
S_{tube} = \frac{\pi \left(D^4 - d^4\right)}{32D}
 D = outside diameter, d = inside diameter

For a rectangular section, combining the rectangle formula with the main MOR equation gives:

MOR = \frac{6M}{b d^2}

Example

Assume a specimen fails at a bending moment of 120 lb·ft and has a section modulus of 8 in3. First convert the moment to lb·in, then divide by the section modulus.

M = 120 \times 12 = 1440 \, lb \cdot in
MOR = \frac{1440}{8} = 180 \, psi

The modulus of rupture is 180 psi.

How to Interpret the Result

MOR is a failure-level bending stress. A higher value generally means the material or member can sustain a larger bending stress before fracture under the specific test or loading condition used to obtain the failure moment.

  • Higher MOR = greater nominal resistance to bending failure.
  • Lower MOR = failure occurs at a lower extreme-fiber bending stress.
  • MOR is most meaningful when comparing specimens tested under similar geometry, span, loading configuration, and environmental conditions.

Modulus Of Rupture vs. Related Terms

Term What It Represents Why It Matters
Modulus of Rupture Stress at failure in bending Used for ultimate flexural strength comparisons
Modulus of Elasticity Material stiffness in the elastic range Used for deflection and deformation calculations
Section Modulus Geometric resistance of a cross section to bending Connects bending moment to outer-fiber stress
Bending Moment Internal moment caused by applied loads Drives flexural stress in the member

Common Mistakes

  • Using lb·ft with in3 without converting the moment to lb·in.
  • Using the wrong section modulus for the actual bending axis.
  • Confusing MOR with stiffness; MOR is a failure property, not an elastic property.
  • Comparing results from different specimen sizes, moisture conditions, or test setups as if they were directly equivalent.
  • Treating MOR as an allowable design stress without applying the proper design methodology or safety factors.

When This Calculation Is Useful

  • Checking the nominal flexural strength from a beam test
  • Comparing material performance in bending
  • Back-calculating stress at failure from a known collapse moment
  • Evaluating how cross-section geometry affects bending strength
  • Estimating failure stress for laboratory specimens and structural samples

Practical Notes

The modulus of rupture is especially helpful for brittle or quasi-brittle materials because bending tests often provide a clearer failure measure than direct tension tests. However, it should still be interpreted carefully: the reported value depends on specimen geometry, support conditions, load placement, and the direction of bending.

Use the calculator whenever you know the moment at failure and the section modulus, and want a quick, reliable estimate of the corresponding flexural failure stress.