Principal Stresses - Plane Stress

PRINCIPAL STRESSES - PLANE STRESS

The digest of plane attemptes is not through upon the isolation of an piece only but it extends upon the analysis of all told stresses existent in an isolated plane. A further analysis of the element is done to see all existent critical state and all probable stress combination.

Consider the element as shown,

[pic]
but [pic]

[pic]

from the retell angle formula
[pic]

then
[pic]
rearranging terms

[pic] eqtn 1

[pic]

using the double angle formula and rearranging

[pic] eqtn 2

NOTE: (x, (y, (xy and ( are positive in the directions shown in the Plane Stress figure.
To determine where the supreme cling to of (n occurs: [pic]
[pic]
[pic]
settlement for the angle (, [pic]eqtn 3
where (p locates the plane where (n is extremum.

Equating eqtn 2 to zero,
[pic]
and solving for the angle (,[pic]eqtn 4

NOTE:
1.



On planes of maximum or token(prenominal) normal stresses, shear stresses are zero.
2. Planes having zero shear stress are ace planes.
3. Normal stresses on principal planes are principal stresses.
4. Two values of principal stresses are 90( apart.

From trigonometry, convert ( = tan (( + 180)
tan 2(p1 = tan (2(p1 + 180) = tan (2(p2)

Consider the triangle:[pic]

[pic]

substituting equations 5 & 6 to equation 1

[pic]
[pic]

where (p1 and (p2 are called principal stresses.

Actual analysis would involve 3 principal stresses. A simplification reduces the third plane stress to zero, ((p3 = (z = 0).
To determine where the maximum in-plane shearing stress occurs: [pic]
[pic]
[pic]
where (( locates the plane where (nt is maximum.

[pic](negative reciprocals)

NOTE:
2(( & 2(p are 90(...If you requisite to get a full essay, order it on our website: Orderessay



If you want to get a full essay, wisit our page: write my essay .