Find maximum principal stress70 MPa50 MPa-40 MPа — х0, =-40 MPaa, = 70 MPa--50 MPa

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Find maximum principal stress 70 MPu 50 MPa - 40 MPa – x a, =-40 MPa o, = 70 MPa T --50 MPa

Mechanics of Materials (MindTap Course List)

9th Edition

ISBN:9781337093347

Author:Barry J. Goodno, James M. Gere

Publisher:Barry J. Goodno, James M. Gere

Chapter11: Columns

Section: Chapter Questions

Problem 11.5.7P: Solve the preceding problem for a column with e = 0.20 in,, L = 12 ft, I = 2L7in4, and E = 30 ×...

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Concept explainers

Slope and Deflection

The slope in a deflected beam is described as an angle in radians made by the tangent of the section with the original axis of the beam. The deflection at any point on the axis of the beam is defined as the lateral displacement of the point before and after loading.

Question

100%

Find maximum principal stress
70 MPa
50 MPa
-40 MPа — х
0, =-40 MPa
a, = 70 MPa
--50 MPa

Expert Solution

Step 1

As given in question,

Normal stress in x direction = σ_x = -40 MPa

Normal stress in x direction = σ_y = 70 MPa

Shear stress in x-y plane = τ_xy = -50 MPa

Maximum principal stress is given by,

$\begin{array}{rcl} σ_{1} & = & (\frac{σ_{x} + σ_{y}}{2}) + \sqrt{{(\frac{σ_{x} - σ_{y}}{2})}^{2} + {(τ_{x y})}^{2}} \\ = & (\frac{- 40 + 70}{2}) + \sqrt{{(\frac{- 40 - 70}{2})}^{2} + {(- 50)}^{2}} \\ = & 15 + \sqrt{3025 + 2500} \\ = & 15 + 74.33 \\ = & 89.33 M P a \end{array}$

Hence, maximum principal stress = 89.33 MPa

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