How to Calculate Maximum Stress

Understanding maximum stress is critical in infrastructure design.
••• the first two steel beams for the freedom tower ro image by Gary from Fotolia.com

"Stress," in everyday language, can mean any number of things, but in general implies urgency of some sort, something that tests the resilience of some quantifiable or perhaps unquantifiable support system. In engineering and physics, stress has a particular meaning, and relates to the amount of force a material experiences per unit area of that material.

Calculating the maximum amount of stress a given structure or single beam can tolerate, and matching this to the expected load of the structure. is a classic and everyday problem facing engineers every day. Without the math involved, it would be impossible to construct the wealth of enormous dams, bridges and skyscrapers seen the world over.

Forces on a Beam

The sum of the forces ​Fnet​ experienced by objects on Earth include a "normal" component pointing straight down and attributable to the gravitational field of earth, which produces an acceleration ​g​ of 9.8 m/s2, combined with the mass m of the object experiencing this acceleration. (From Newton's second law, ​Fnet​ = m​a.​ Acceleration is the rate of change of velocity, which is in turn the rate of change of displacement.)

A horizontally oriented solid object such as a beam that has both vertically and horizontally oriented elements of mass experiences some degree of horizontal deformation even when subjected to a vertical load, manifested as a change in length ΔL. That is, the beam ends.

Young's Modulus Y

Materials have a property called ​Young's modulus​ or the ​elastic modulus Y​, which is particular to each material. Higher values signify a higher resistance to deformation. Its units are the same as those of pressure, newtons per square meter (N/m2), which also is force per unit area.

Experiments show the change in length ΔL of a beam with an initial length of L0 subjected to a force F over a cross-sectional area A is given by the equation

\Delta L=\bigg(\frac{1}{Y}\bigg)\bigg(\frac{F}{A}\bigg)L_0

Stress and Strain

Stress​ in this context is the ratio of force to area F/A, which appears on the right-side of the length change equation above. It is sometimes denoted by σ (the Greek letter sigma).

Strain​, on the other hand, is the ratio of the change in length ΔL to its original length L, or ΔL/L. It is sometimes represented by ε (the Greek letter epsilon). Strain is a dimensionless quantity, that is, it has no units.

This means that stress and strain are related by

\frac{Delta L}{L_0}=\epsilon =\bigg(\frac{1}{Y}\bigg)\bigg(\frac{F}{A}\bigg)=\frac{\sigma}{Y}

or stress = Y × strain.

Sample Calculation Including Stress

A force of 1,400 N acts on an 8-meter by 0.25-meter beam with a Young's modulus of 70 × 109 N/m2. What are the stress and the strain?

First, calculate the area A experiencing the force F of 1,400 N. This is given by multiplying the length L0 of the beam by its width: (8 m)(0.25 m) = 2 m2.

Next, plug your known values into the equations above:

Strain:

\epsilon = (1/(70\times 10^9))(1400)=1\times 10^{-8}

Stress:

\sigma = \frac{F}{A}=Y\epsilon = (70\times 10^9)(1\times 10^{-8})=700\text{ N/m}^2

I-Beam Load Capacity Calculator

You can find a steel beam calculator free online, like the one provided in the Resources. This one is actually an indeterminate beam calculator and can be applied to any linear support structure. It allows you to, in a sense, play architect (or engineer) and experiment with different force inputs and other variables, even hinges. Best of all, you can't cause any construction workers any "stress" in the real world in so doing!

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