Flexure Calculator

When a beam is bent there are stresses in the beam corresponding to this. The inner concave side is compressed, the outer convex side is stretched and the length of the thin layer in the middle - the neutral axis - remains unchanged. The bending or flexural stress indicates how much the outside fibres are stretched or squashed.

This calculator allows you to determine these stresses as well as working back from the stresses to loads and dimensions and can also be used to find the modulus of rupture (or breaking factor) in three or four point bend tests.

Formula for bending:

The bending stress is expressed by the following relationship: The stress at a given point is equal to the product of the bending moment and the distance of that point from the neutral axis divided by the second moment of area.

where σ is the bending stress, M is the internal moment of bending, y is the distance from the point being considered to the neutral axis and I is the moment of inertia about the axis of bending. The maximum stress occurs at the outermost fibers, where y equals c, which is the distance to the extreme surface. Engineers usually combine I and c into a section modulus Z. Since Z equals I divided by c, the maximum stress equals M divided by Z.

Simplified formulas for rectangular cross-sections and round bars:

You do not have to calculate the value of I manually each time. For a massive rectangle with width b and height h, c is calculated as the value of h divided by 2, and I as the value of b * h³ divided by 12. The value sigma will then be substituted into the formula Mc/I. The terms regarding the height cancel out, resulting in a simple and neat formula.

For a massive round bar with diameter D the values for c and I can be calculated as follows: c is the value of D divided by 2, and I is the value of pi * D⁴ divided by 64. This gives you a formula for the circular cross-section.

Unlike rectangular shapes where the height is squared and circular shapes where the diameter is cubed, depth provides the most effective leverage. Doubling the height of a rectangular beam for example reduces maximum stress by about one quarter for the same bending moment. This is the geometric basis for tall narrow beams and deep I-beams.

Flexural strength and flexural tests

Flexural strength, also known as the modulus of rupture, bend strength, or transverse rupture strength, is the stress in the outer fibres of a test specimen when it is subjected to bending until failure. It is an important parameter for brittle materials such as ceramics, concrete, wood and many composites because these materials fail in tension before their actual yield point is reached. A standard method of measuring flexural strength is the three-point bend test per ASTM D790 or ISO 178, where a load is applied to the center of a simply supported beam until it fails.

If the cause of the bending moment in a bend test is considered, then the calculation tool calculates the bending moment from load and span length. The maximum bending moment by a central load is the value of F*L divided by four. In a four-point test where the load is applied at the third points, the bending moment f*L divided by 6 and remains constant between the two load points. By entering the breaking load and dimensions of the test object, the breaking factor can be read directly.

Here's how to use this calculator.

Select a cross section. For the general options, you will need to input the moment of inertia for your cross section and the distance y which is applicable for any shape where the value of T, U or I are already known. For the rectangular and circular options, simply enter the dimensions that can be measured with a caliper or ruler. Then select the cause of the bending moment. Either directly input the bending moment or calculate it from the test load and span length.

In each numerical field a back calculation is also possible. If you enter a known value and leave an input field empty the tool will calculate the corresponding value whether it be stress, bending moment, width or height, diameter, breaking load or span length. The units are converted on site so that you can mix Newton-millimeters with millimeters or pound-force with inches without any problems.

Example:

The calculation for the stress turns out as follows:

What does that mean? How do we know what is the magnitude of 75 MPa?

Calculation example:

A small rectangular alumina rod with a width of 12 mm and a depth of 5 mm is tested on a span of 80 mm. A break was found at a load of 320 N in a bending test. The maximum bending moment is 320 × 0.080 ÷ 4, which is approximately 6.4 Newton meters. The breaking coefficient is 3 × 320 × 80 ÷ (2 × 12 × 25), so it's 128 Megapascals. This matches the published values for technical alumina and confirms that the sintering step went well.

Let's look at the general formula as well. If a bending moment of 500 Newton meters is applied to a cross section whose moment of inertia I equals four million millimeters to the fourth power, consider the outermost fiber above the neutral axis in a distance of 50 mm. The stress is 500 × 0.05 ÷ (4 × ten to the minus six), which is 6.25 megapascal, which is well below the yield strength of normal steel at 250 megapascal. This beam has sufficient reserves.

The ratio of span and depth is important.

In a flexural test, pure bending action only occurs if the specimens are long enough between support points. If the span to depth ratio is less than about 16 then shear deformation will occur and simple formulae start to overestimate strength. Both ASTM D790 and ISO 178 require a minimum span to depth ratio of at least 16. This tool displays this value during testing. Use the support length, i.e. the distance between the supports, not the overall length of the specimen.

Typical values for flexural strength

Since the flexural strength of typical materials varies over three orders of magnitude, your results will be plotted on a graph.

Deformation and stiffness

Strength is only half the problem. Even if a beam is safe, it may be unusable due to excessive deflection. When deflection estimation is enabled and the modulus of elasticity added, this tool calculates the midspan deflection in a three-point bending scenario. This deflection (delta) is based on the value F*L³ / (48*E*I), which is then compared to a service limit determined by dividing the typical span by 1000. Even if a joist passes a strength test it may be too slender and tend to springy behavior, so both tests are essential.

Applications of bending

Bends are found everywhere if you look closely. Floor joists, light aircraft wing spars, pickup truck bed sides, I-beams supporting an office floor, tree branches, turbine blades and roof tiles - all of these components experience bending forces. A diving board bends most at the point where the diver stands on it, breaking there because that is where the maximum bending moment occurs. A bookcase bends in the middle. Even when fishing, struggling with a fish, the same formula applies but to a tapering cross-section.

Important notes:

Use the maximum bending moment along the beam and do not use values at support points. For a simply supported span, the bending moment is zero at the support point. Make sure that "I" refers to the value with respect to the axis of bending and not the area moment of inertia. The area moment of inertia is another value. In ductile metals, the bending formula is a useful indicator but since failure occurs by yielding rather than through a distinct crack, compare it against the yield strength and use a safety factor. Remember that this formula is based on linear elasticity. Once the proportional limit is exceeded, the formula no longer applies.