Principal Stresses in Plane Stress
Find the two in-plane normal stresses on planes with zero in-plane shear.
Try this formula
Calculate larger in-plane principal stress or choose another supported quantity.
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Try this formula
Calculate larger in-plane principal stress or choose another supported quantity.
Use decimals or scientific notation, such as 1.2e3. Calculations update as you type.
Larger in-plane principal stress
Enter the known values to see the result.
Calculates the larger in-plane principal stress. The smaller value uses the minus sign in the lesson. The out-of-plane principal stress is zero and must be included when comparing absolute three-dimensional extrema.
Difficulty
Intermediate
Read time
5 min
Prerequisites
Basic algebra, mechanics, and consistent SI units
Source
raymond212 — MECH 260 — Mechanics of MaterialsReference equation: Stress Transformation and Failure Criteria — Principal Stresses. FormuLab wrote new explanations, assumptions, applications, and numerical examples. The calculator returns the larger in-plane value; the example also reports the smaller value. The out-of-plane principal stress is zero.
Definition
Rotating the coordinate axes changes the stress components but does not change the physical stress state. The principal planes remove in-plane shear; the two principal values are the center of Mohr’s circle plus or minus its radius.
Variables & Units
| Symbol | Meaning | Units |
|---|---|---|
| Larger in-plane principal stress | ||
| Smaller in-plane principal stress | ||
| Signed normal stress on the x face | ||
| Signed normal stress on the y face | ||
| Signed in-plane shear stress |
Real-World Applications
Combined loading
Interpret simultaneous normal and shear stress.
Yield checks
Supply principal stresses for a failure criterion.
Worked Example
Problem
At a point, σx = 80 MPa, σy = 20 MPa, and τxy = 30 MPa. Find both in-plane principal stresses. The calculator evaluates the larger value.
Given
Solution
Final Answer
Contents
Info
Tools
Source & review
Reference equation: Stress Transformation and Failure Criteria — Principal Stresses. FormuLab wrote new explanations, assumptions, applications, and numerical examples. The calculator returns the larger in-plane value; the example also reports the smaller value. The out-of-plane principal stress is zero.
Assumptions
- • The stress state is plane stress; the third principal stress is zero.
- • Use signed normal stresses with tension positive and a consistent shear convention.
- • The ± signs give the larger and smaller in-plane values; include the zero third stress when finding absolute three-dimensional extrema.
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