Exercise F-1.2.2-xz TechMech 2 Mechanics of Materials

Practice Exercise F-1.2.2-xz

Plane Stress State: Determine the Magnitude and Direction of the Principal Stresses

Problem Statement

In a sheet, the stresses $\sigma_x = 20\mathrm{~MPa}$, $\sigma_z = 30\mathrm{~MPa}$ and $\tau_{xz} = 10\mathrm{~MPa}$ are given.

This illustration depicts a square sheet of metal. It lies with two sides in an x, z-coordinate system with the origin at the bottom-left corner of the sheet. The x-axis is positive to the right, and the z-axis is positive downward. Normal and shear stresses are represented for each side of the sheet, with arrows indicating their respective directions. — Fig. 1: Sheet with given stresses

Determine the magnitude and direction of the principal stresses.

Short Solution

Determine the Magnitude and Direction of the Principal Stresses.

$$ \begin{aligned} \sigma_{1} &= 36,180\mathrm{~MPa}~(\searrow)\\ \varphi_1^* &= -58,283°\\ \sigma_{2} &= 13,820\mathrm{~MPa}~(\nearrow)\\ \varphi_2^* &= 31,717° \end{aligned} $$

Comprehensive Solution

For the given $x$, $z$ coordinate system, describing a plane where the x-axis points to the right in the positive direction and the z-axis points downward in the positive direction, and the given stresses $\sigma_x$, $\sigma_z$, and $\tau_{xz}$, we can use formula 1.10_xz to determine the principal stresses:

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