Bending Stress Calculation: A Step-by-Step Guide
Mastering the flexure formula σ = My/I for safe member selection.
Deflection is about how much a beam sags, but Bending Stress is about whether the beam will snap or permanently deform. For structural engineers, calculating the internal stress is the most critical check. If the stress exceeds the material's yield strength, the structure fails. In this guide, we break down the famous flexure formula and show you how to apply it to real-world design.
Step 1: Determine the Maximum Moment (M)
The bending moment is the internal 'twisting' force inside the beam. For a simply supported beam with a center point load (P) and length (L), the maximum moment is M = PL / 4. For a UDL (w), it is M = wL² / 8. You must find the peak moment, which usually occurs where the shear force crosses zero. Our calculator handles these equations for common scenarios.
Step 2: Find the Neutral Axis and 'y'
The neutral axis is the horizontal line through the beam cross-section where there is zero stress. 'y' is the distance from this neutral axis to the furthest point on the beam (usually the top or bottom edge). For a beam 10 inches deep, the neutral axis is in the center at 5 inches, so y = 5. Bending stress is always highest at the extreme fibers.
Step 3: Apply the Moment of Inertia (I)
As discussed in our previous articles, 'I' represents the resistance to bending. By dividing the moment times the distance (M * y) by 'I', you get the stress (σ). The formula σ = My/I tells us that as 'I' increases, the stress decreases—this is why deep beams are safer and more efficient.
Step 4: Selection using Section Modulus (S)
Engineers often use a shortcut: Section Modulus (S = I / y). The flexure formula becomes σ = M / S. Steel catalogs list the 'S' value for every I-beam size. You simply take your calculated moment, divide it by the allowable stress for your steel, and find an I-beam in the catalog with an 'S' value higher than your result.
FAQ
What is the difference between tension and compression?
In a simply supported beam, the top fibers are pushed together (Compression) and the bottom fibers are pulled apart (Tension). Most materials like steel are equally strong in both, but concrete is weak in tension and requires steel reinforcement.
Does shear stress matter?
Yes. While bending stress usually controls the design of long beams, shear stress (τ = VQ/It) can control the design of short, heavily loaded beams. Always check both.
Can I use the flexure formula for plastic bending?
No. The σ = My/I formula is only valid in the 'Elastic' range, where stress is proportional to strain. Once the material starts to yield, you must use more complex plastic analysis methods.