Exercise F-1.2.2-xy TechMech 2 Mechanics of Materials

Practice Exercise F-1.2.2-xy

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

Problem Statement

In a sheet, the stresses $σ_{x} = 20 MPa$ , $σ_{y} = 30 MPa$ and $τ_{x y} = 10 MPa$ are given.

This illustration depicts a square sheet of metal. It lies with two sides in an x, y-coordinate system with the origin at the bottom-left corner of the sheet. 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

Determine the magnitude and direction of the principal stresses.

Short Solution

Determine the Magnitude and Direction of the Principal Stresses.

\begin{aligned} σ_{1} & = 36, 180 MPa (↗) \\ φ_{1}^{*} & = 58, 283 ° \\ σ_{2} & = 13, 820 MPa (↘) \\ φ_{2}^{*} & = - 31, 717 ° \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 $σ_{x}$ , $σ_{y}$ , and $τ_{x y}$ , we can use formula 1.10_xy to determine the principal stresses:

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