Bending Stress Calculator
Free Bending stress Calculator for structural engineering projects. Enter dimensions to get material lists and cost estimates.
Formula
sigma = M * c / I = M / S
Bending stress sigma equals the bending moment M multiplied by the distance c from the neutral axis to the extreme fiber, divided by the moment of inertia I. Equivalently, sigma equals M divided by the section modulus S, where S = I/c. The moment must be in consistent units (kip-inches when I is in in^4) to get stress in ksi.
Worked Examples
Example 1: Rectangular Timber Beam
Problem: Find the bending stress in an 8x12 inch timber beam carrying a moment of 100 kip-ft.
Solution: I = 8 * 12\u00B3 / 12 = 1,152 in\u2074\nc = 12 / 2 = 6 in\nS = 1,152 / 6 = 192 in\u00B3\nsigma = (100 * 12) / 192 = 6.25 ksi = 6,250 psi
Result: Bending stress = 6.25 ksi (6,250 psi)
Example 2: Steel Beam Check
Problem: A steel beam with S = 50 in\u00B3 carries a moment of 80 kip-ft. Check against Fb = 24 ksi.
Solution: sigma = M / S = (80 * 12) / 50 = 19.2 ksi\nUtilization = 19.2 / 24 = 80%\n19.2 < 24, so the beam is adequate.
Result: Bending stress = 19.2 ksi, 80% utilized
Frequently Asked Questions
What is bending stress and how is it calculated?
Bending stress is the normal stress induced at a point in a beam due to an applied bending moment. It is calculated using the flexure formula: sigma = M * c / I, where M is the bending moment, c is the perpendicular distance from the neutral axis to the point of interest (usually the extreme fiber), and I is the moment of inertia about the neutral axis. The stress varies linearly from zero at the neutral axis to a maximum at the outermost fibers. The ratio I/c is called the section modulus S, so sigma = M/S is an equivalent form.
What is an allowable bending stress for steel and wood?
For structural steel with Fy = 50 ksi, AISC allows a bending stress of 0.66 * Fy = 33 ksi for compact sections in ASD, or uses a resistance factor of 0.90 in LRFD giving phi-Mn = 0.90 * Fy * Z. For dimensional lumber, the National Design Specification (NDS) provides reference bending values Fb ranging from about 750 to 2,400 psi depending on the species and grade, which are then adjusted by multiple factors for size, duration, moisture, and other conditions.
What happens if bending stress exceeds the yield strength?
If bending stress reaches the yield strength, the outermost fibers begin to yield plastically while the interior fibers remain elastic. As the moment increases further, yielding progresses inward toward the neutral axis, forming a plastic hinge. The fully plastic moment Mp equals Fy times the plastic section modulus Z. For steel beams, the shape factor Z/S typically ranges from 1.1 to 1.5. Beyond Mp, the beam cannot carry additional moment and undergoes large rotations, potentially leading to collapse if not properly braced.
How does the cross-section shape affect bending stress?
The cross-section shape determines the moment of inertia I and the extreme fiber distance c, which together control the bending stress. I-beams and wide-flange shapes are efficient because they concentrate material at the flanges far from the neutral axis, maximizing I while minimizing weight. A rectangular section has I = bh-cubed/12 and S = bh-squared/6. Doubling the depth quadruples I and doubles S, making deeper beams much more efficient in bending.
How accurate are the results from Bending Stress Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.