Embed Size (px)
Citation preview
Lecture 34 – Principal stressesMaximum shear stress - Mohr’s circle
Instructor: Prof. Marcial Gonzalez
Spring, 2021ME 323 – Mechanics of Materials
Reading assignment: 8.1-8.4
Last modified: 1/14/21 7:22:36 PM
2
Principal stresses - Maximum shear stress
Three-dimensional state of stress (review, Lecture 5)
requirements forequilibrium?
… recall simple shear
Note: these are Cartesian coordinates
Integrate over thecross-section
Transformation of stress
Problem 67 (from last lecture): Given the components of a state of plane stress, find the maximum normal and shear stresses, and the planes on which they act.
3
’
’
’
’’ ’
’ ’
’ ’
Principal stressesFind the orientation such that the normal stress is maximum:
4
Principal stresses:
Principal stresses - Maximum shear stress
with
Principal directions:
no shear stress on the principal planes!!
maximumin plane
normal stress
minimumin plane
normal stress
shear stress onprincipal planes
Maximum in-plane shear stressesFind the orientation such that the normal stress is maximum:
5
the planes of maximum shear stress are not free of normal stress!!
Principal stresses - Maximum shear stress
Maximum in-plane shear stress Planes of maximum shear:
with
normal stress onplanes of max. shear
Principal stresses & maximum in-plane shear stresses
6
Principal stresses - Maximum shear stress
Mohr’s circle for plane stress
7
Goal: a graphical representation of the plane stress transformation equations
is constant for any angle .
… after basic manipulation:
with
Q: Where are stresses on x- and y-planes?
Q: Where are the principal stresses?
Q: Where is the maximum shear stress?
The radius of Mohr’s circle is
Mohr’s circle
Positive angle
(from slide 3)
(from slide 4)
Mohr’s circle for plane stress
8
Goal: a graphical representation of the plane stress transformation equations
Mohr’s circle
Always form a right-handed
coordinate systems
Mohr’s circle for plane stress
9
Goal: a graphical representation of the plane stress transformation equations
Mohr’s circle
Always form a right-handed
coordinate systems
negative
Plane stress: draw the Mohr’s circle
10
Mohr’s circle
Problem 69)
Problem 70)
Problem 71)
Problem 72)
Any questions?
12
Principal stresses - Maximum shear stress
