Home | | **Strength of Materials I** | | **Strength of Materials for Mechanical Engineers** | Triaxial Stress, Biaxial Stress, and Uniaxial Stress

Triaxial stress refers to a condition where only normal stresses act on an element and all shear stresses (txy, txz, and tyz) are zero. An example of a triaxial stress state is hydrostatic pressure acting on a small element submerged in a liquid.

**Triaxial Stress, Biaxial Stress, and Uniaxial Stress**

*Triaxial stress *refers to a condition where only normal stresses act on an element and all* *shear stresses (txy, txz, and tyz) are zero. An example of a triaxial stress state is hydrostatic pressure acting on a small element submerged in a liquid.

A two-dimensional state of stress in which only two normal stresses are present is called *biaxial stress*. Likewise, a one-dimensional state of stress in which normal stresses act along one direction only is called a *uniaxial stress* state.

**Pure Shear**

Pure shear refers to a stress state in which an element is subjected to plane shearing stresses only, as shown in **Figure 3**. Pure shear occurs in elements of a circular shaft under a torsion load.

**Figure 3. Element in pure shear**

**Thin cylindrical and spherical shells**

**Thin-walled assumption**

For the thin-walled assumption to be valid the vessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius. This allows for treating the wall as a surface, and subsequently using the Young–Laplace equation for estimating the hoop stress created by an internal pressure on a thin wall cylindrical pressure vessel:

where

· *P *is the internal pressure

· *t *is the wall thickness

· *r *is the inside radius of the cylinder.

· Ro Deta is the hoop stress.

The hoop stress equation for thin shells is also approximately valid for spherical vessels, including plant cells and bacteria in which the internal turgor pressure__ __may reach several atmospheres.

Inch-pound-second system (IPS) units for *P* are pounds-force per square inch (psi). Units for *t*, and *d* are inches (in). SI units for *P* are pascals (Pa), while *t* and *d*=2*r* are in meters (m).

When the vessel has closed ends the internal pressure acts on them to develop a force along the axis of the cylinder. This is known as the axial stress and is usually less than the hoop stress.

Also in this situation a radial stress is developed and may be estimated in thin walled cylinders as:

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