Module 20-26 Stress Transformations and Mohrs Circle

\Thin-Walled Pressure Vessel Assumption

Vessel is considered thin-walled when the inner radius divided by wall thickness is roughly 10 or greater. Stresses are assumed uniform through the thickness.

Spherical Pressure Vessel Stress

In a spherical vessel, the normal stress in the wall is the same in all directions (equibiaxial).

Cylindrical Pressure Vessel Stresses

Hoop stress (circumferential) is twice the longitudinal stress (axial). The hoop stress acts around the circumference, longitudinal stress acts along the length.

Hoop Stress Direction

Acts tangentially around the cylinder wall, trying to split the cylinder along its length.

Longitudinal Stress Direction

Acts along the axis of the cylinder, trying to pull the end caps off.

Principle of Superposition for Combined Loadings

The total stress at a point is the algebraic sum of stresses from individual internal forces and moments, provided the material is linear elastic, deformations are small, and loads do not interact.

Internal Loads at a Cross-Section

All internal forces (normal, shear) and moments (bending, torsion) act through the centroid of the cross-section. Always draw a separate sketch of the section with all internal loads and their correct directions.

Combined Loading Steps

1. Determine internal loads at the section. 2. For each load, identify the type of stress it produces at the point of interest (axial, bending, shear, torsion). 3. Superpose stresses of the same type and direction.

State of Stress

A complete description of normal and shear stresses on three mutually perpendicular planes at a point. For plane stress, only stresses on two perpendicular faces are non-zero.

Complementary Property of Shear Stress

Shear stress on perpendicular planes are equal in magnitude and both point toward or away from the common edge.

Stress Invariants

For a given state of stress, the sum of normal stresses on perpendicular faces is constant regardless of orientation. Also, the determinant of the stress matrix is invariant.

Stress Transformation Purpose

To find stresses on any rotated plane. Used because failure occurs on the plane where stresses are most severe (maximum normal or shear stress).

Sign Convention for Rotation Angle $\theta$

Positive $\theta$ rotates the element counterclockwise. The transformed plane's orientation is measured from the original x-axis.

Principal Stresses

The maximum and minimum normal stresses at a point. On the principal planes, the shear stress is zero.

Principal Planes

The orientations at which the normal stress is either maximum or minimum. These planes are 90° apart.

Maximum In-Plane Shear Stress ($\tau_{maxip}$)

The maximum shear stress that can be obtained by rotating the element within the plane (2D). It acts on planes 45° from the principal planes.

Average Normal Stress on Plane of Max Shear

On the plane where maximum in-plane shear stress occurs, the normal stress equals the average of the two principal stresses.

Relationship Between Principal Angles and Max Shear Angles

The angle between a principal plane and the plane of maximum in-plane shear is 45° (the double-angle values differ by 90°).

Mohr's Circle

A graphical representation of the stress transformation equations. Each point on the circle corresponds to a specific orientation. The circle's center is at average normal stress, and its radius is the maximum in-plane shear stress.

Mohr's Circle Center Location

The center is at the average normal stress, plotted on the horizontal (normal stress) axis.

Mohr's Circle Radius

The radius equals the magnitude of the maximum in-plane shear stress.

Plotting Shear Stress on Mohr's Circle

Clockwise shear stress is plotted as positive on the vertical axis; counterclockwise as negative. This is a special convention for Mohr's circle only.

Reading Principal Stresses from Mohr's Circle

The points where the circle crosses the horizontal axis (zero shear) give the principal stresses: the rightmost point is the larger principal stress, the leftmost is the smaller.

Double Angle Rule in Mohr's Circle

A rotation of the stress element by $\theta$ corresponds to a rotation of $2\theta$ on Mohr's circle, in the same direction.

Absolute Maximum Shear Stress ($\tau_{absmax}$)

The largest shear stress at a point considering all planes (including out-of-plane). For plane stress, if the two principal stresses have opposite signs, $\tau_{absmax}$ equals the in-plane maximum shear stress. If both have the same sign, $\tau_{absmax}$ occurs out-of-plane and equals half the larger principal stress.

Theories of Failure Purpose

To predict failure under multiaxial stress using simple uniaxial test data (tension or torsion). Failure mode: yielding for ductile materials, fracture for brittle materials.

Maximum Shear Stress Theory (MSST / Tresca)

For ductile materials: yielding occurs when the absolute maximum shear stress reaches the shear stress at yield in a tension test (which is half the yield strength).

Maximum Distortion Energy Theory (MDET / von Mises)

For ductile materials: yielding occurs when the distortion energy per unit volume equals that at yield in tension. More accurate for pure shear than MSST.

Maximum Normal Stress Theory (Rankine)

For brittle materials with equal tensile and compressive strengths: failure occurs when the maximum principal stress reaches the ultimate tensile stress.

Mohr's Failure Criterion

For brittle materials with different tensile and compressive strengths. Failure occurs when the Mohr's circle at the point is tangent to or exceeds the failure envelope constructed from tension, compression, and torsion tests.

Tresca Hexagon

The yield envelope for MSST in principal stress space. Points inside the hexagon are safe; points on or outside indicate yielding.

von Mises Ellipse

The yield envelope for MDET in principal stress space. An ellipse oriented at 45° that better matches experimental data for ductile metals.

Common Misconception: Hoop stress is smaller than longitudinal stress in a cylinder

Reality: Hoop stress is twice longitudinal stress for a cylinder, so it governs design (often the critical stress).

Common Misconception: In combined loading, you can add axial and bending stresses without considering sign

Reality: Both are normal stresses, but axial stress is uniform while bending stress varies linearly. Superposition requires careful sign assignment (tension positive, compression negative).

Common Misconception: Principal planes always occur at 45° to the direction of maximum shear

Reality: Principal planes are 45° from the planes of maximum in-plane shear, but the orientation relative to the original axes depends on the stress state.

Common Misconception: Mohr's circle shear stress sign convention matches standard sign convention

Reality: Mohr's circle uses a different sign convention: clockwise shear is positive, counterclockwise negative. This is necessary to make angles consistent.

Common Misconception: For plane stress, the absolute maximum shear stress always equals the in-plane maximum shear stress

Reality: If both principal stresses have the same sign (both tensile or both compressive), the absolute maximum shear stress occurs out-of-plane and is larger than the in-plane value.

Common Misconception: Maximum normal stress theory applies to all brittle materials

Reality: It only works when tensile and compressive strengths are equal. If they differ significantly, Mohr's failure criterion must be used.

Common Misconception: MSST and MDET always give the same factor of safety

Reality: They differ, especially for pure shear. MDET predicts a higher allowable load (less conservative) and is generally more accurate for ductile metals.

Exam-Ready Concept: Pressure vessel joint design

The number of bolts required to seal a spherical tank is determined by the force from internal pressure acting on the projected area, divided by the allowable bolt force.

Exam-Ready Concept: Combined loading on a pipe

A pipe may have axial force, bending moment, torque, and internal pressure simultaneously. Axial plus bending give total normal stress; torque gives shear stress; pressure gives hoop and longitudinal stresses. Superpose carefully.

Exam-Ready Concept: Identifying which loads cause stress at a point

For a point on the outer surface of a circular cross-section: axial force causes uniform normal stress; bending moment causes normal stress proportional to distance from neutral axis; torque causes shear stress maximum at outer radius; shear force causes transverse shear (often zero at extreme fibers for circles).

Exam-Ready Concept: Using stress invariants to check calculations

After transformation, the sum of normal stresses on perpendicular faces should remain constant. Use this to verify your answers.

Exam-Ready Concept: Principal stress order

By convention, the principal stresses are ordered from largest to smallest. For plane stress on a free surface, one principal stress is usually zero.

Exam-Ready Concept: Determining whether to use in-plane or absolute max shear

For ductile failure theories (MSST, MDET), MSST specifically uses the absolute maximum shear stress, not just the in-plane value.

Exam-Ready Concept: Factor of safety for MSST (conceptual)

Compare the absolute maximum shear stress to the yield shear stress (half the tensile yield strength). The factor of safety is the ratio of yield shear stress to actual absolute max shear stress.

Exam-Ready Concept: Factor of safety for MDET (conceptual)

Compute the von Mises stress (a single equivalent stress from the principal stresses). The factor of safety is the ratio of tensile yield strength to the von Mises stress.

Exam-Ready Concept: Mohr's circle construction from known data

Plot the points representing stresses on the x-face and y-face using the special shear sign convention. The line between these points gives the center and radius.

Exam-Ready Concept: Reading transformed stresses from Mohr's circle

To find stresses on a plane rotated by a given angle, rotate from the reference point on the circle by twice that angle. The coordinates of that new point are the normal and shear stresses on the rotated plane.