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Exercise F-1.2.1-xy

Practice Exercise in Technical Mechanics 2, Mechanics of Materials

Topic: Plane Stress State.

Practice Exercise F-1.2.1-xy

Plane Stress State: Arbitrary Section Angle, Coordinate Transformation x,y-Coordinate System

Problem Statement

In a sheet, the stresses \(\sigma_x = -250\mathrm{~MPa}\), \(\sigma_y = 80\mathrm{~MPa}\) and \(\tau_{xy} = 50\mathrm{~MPa}\) are given.

This illustration depicts a square sheet metal in two positions. In one instance, it lies with two sides in an x, y-coordinate system with the origin at the bottom-left corner of the coordinate system. In the second position, it is tilted at an angle phi to the x-axis. The x-axis is positive to the right, and the y-axis is positive upward. Normal and shear stresses are represented for each side of the sheet, with arrows indicating their respective directions.
Fig. 1: Sheet with given stresses

What normal and shear stresses occur at a section angle of \(\varphi=30°\)?

Short Solution
What normal and shear stresses occur at a section angle of \(\varphi=30°\)?
$$ \begin{aligned} \sigma_\xi &= -124,199\mathrm{~MPa}~(\nearrow)\\ \sigma_\eta &= -45,801\mathrm{~MPa}~(\nwarrow)\\ \tau_{\xi\eta} &= 167.894\mathrm{~MPa}~(\nwarrow) \end{aligned} $$
Comprehensive Solution

For the given \(x\), \(y\)-coordinate system, describing a plane where the x-axis points to the right in the positive direction and the y-axis points upward in the positive direction, and the given stresses \(\sigma_x\), \(\sigma_y\), and \(\tau_{xy}\), we can use the formula 1.7xy to determine the normal and shear stresses at any arbitrary angle \(\varphi\):

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