Problem 1 ( 3 0 points ) Stress field within a loaded continuum Given the stress tensor, at a given point M , ( M ) ( a , b , 0 , b , c , 0 , 0 , 0 , d ) ( v e c ( e ) 1 , v e c ( e ) 2 , v e c ( e ) 3 ) MPa with a 2 0 , b 3 4 , c 8 8 , and d 1 1 Describe the nature of the load on surfaces of normal vec ( e ) 1 and vec ( e ) 2 , What is the characteristic of the surface of normal vec ( e ) 3 Considering a surface, containing M , of normal vec ( n ) 4 vec ( e ) 1 4 vec ( e ) 2 4 vec ( e ) 3 A Determine the stress vector acting on this surface B Determine the normal stress acting on this surface C Determine the magnitude of the shear stress acting on this surface We are now studying the ultimate state of the continua at M A Determine the eigenvalues and eigenvectors of the stress tensor at M B Determine the orientation of the surface experiencing the highest normal stress

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Question: Problem 1 ( 3 0 points ) : Stress field within a loaded continuum Given the stress tensor, at a given point M , ?

Problem

1 (30

points

)

: Stress field within a loaded

continuum

Given the stress tensor, at a given point

M,_{?_{?}} (M) = ([a, b, 0], [b, c, 0], [0, 0, d])_{(v e c (e)_{1}, v e c (e)_{2}, v e c (e)_{3})}

MPa

with

a = 20, b = 34, c = 88,

and

d = 11

Describe the nature of the load

on surfaces of normal vec

(e)_{1}

and vec

(e)_{2},

What is the characteristic of the surface of normal vec

(e)_{3}

Considering a surface, containing

M,

of normal vec

(n) = 4

vec

(e)_{1} + 4

vec

(e)_{2} + 4

vec

(e)_{3}

.

Determine the stress vector acting on this surface

.

Determine the normal stress acting on this surface

.

Determine the magnitude of the shear stress acting on this surface

We are now studying the ultimate state of the continua at

M

.

Determine the eigenvalues and eigenvectors of the stress tensor at

M

.

Determine the orientation of the surface experiencing the highest normal stress

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