Allowable Stress & Safety Factor Calculator (Multiaxial)

This industrial-grade calculator evaluates structural integrity using advanced failure theories. Unlike simple uniaxial tools, this calculator handles Multiaxial Stress States ($\sigma_x, \sigma_y, \tau_{xy}$) to calculate Principal Stresses and **Equivalent Stresses** (Von Mises, Tresca). It determines Safety Factors against Yield (Ductile) and Ultimate (Brittle) limits.

1. Material & Settings

Material
Configuration

2. Applied Stress State (2D Plane Stress)

Normal Stresses
Shear Stress

Engineering Insights: Failure Theories Decoded

1. Understanding Stress State

Real-world components are rarely subject to simple pulling (tension). A drive shaft experiences torsion ($\tau$) and bending ($\sigma$). A pressure vessel wall experiences hoop stress ($\sigma_1$) and longitudinal stress ($\sigma_2$).

To analyze safety, we must combine these multidirectional stresses into a single Equivalent Stress that can be compared to the material's yield strength (which is measured in simple tension). This is the job of Failure Theories.

2. Principal Stresses ($\sigma_1, \sigma_2$)

No matter how complex the loading (shear, tension combined), there is always an orientation where the shear stress vanishes, and only normal stresses remain. These are the Principal Stresses. They represent the maximum and minimum normal stresses experienced by the material.

$$ \sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{ \left(\frac{\sigma_x - \sigma_y}{2}\right)^2 + \tau_{xy}^2 } $$

These values are the inputs for all modern failure theories.

3. Distortion Energy Theory (Von Mises)

Best for: Ductile materials (Steel, Aluminum).
Concept: It assumes yielding begins when the distortion energy (energy changing shape, not volume) reaches a critical value.

$$ \sigma' = \sqrt{\sigma_1^2 - \sigma_1 \sigma_2 + \sigma_2^2} $$

The Von Mises yield surface is an ellipse. It is generally the most accurate theory for ductile metals and allows slightly higher loads than Tresca.

4. Maximum Shear Stress Theory (Tresca)

Best for: Ductile materials (Conservative Design).
Concept: Yielding occurs when the maximum shear stress in the material reaches the shear yield strength ($\tau_{yield} = S_y / 2$).

$$ \sigma_{Tresca} = \max(|\sigma_1 - \sigma_2|, |\sigma_1|, |\sigma_2|) $$

The Tresca yield surface is a hexagon inside the Von Mises ellipse. It is more conservative (predicts failure earlier) and is often used in codes like ASME B&PV because it is simpler and safer.

5. Maximum Normal Stress Theory (Rankine)

Best for: Brittle materials (Cast Iron, Glass, Concrete).
Concept: Failure occurs when the maximum principal stress exceeds the Ultimate Strength ($S_{ut}$). Brittle materials fail by cracking, usually in tension, without yielding.

$$ \sigma_{Rankine} = \max(|\sigma_1|, |\sigma_2|) $$

Never use Rankine for steel or ductile metals, as it ignores shear failure modes.