Transcript Document

MAE 343 - Intermediate Mechanics of
Materials
Thursday, Sep. 23, 2004
Textbook Section 4.5
Multiaxial States of Stress and Strain
Stress Transformations
Principal Stresses and Max. Shear Stresses
Triaxial State of Stress at any Critical
Point in a Loaded Body
• Cartesian stress components are found first
in selected x-y-z coordinate axes (Fig. 4.1)
• Three mutually perpendicular principal
planes are found at unique orientations:
– NO Shear stresses on these planes
– Principal Normal Stresses: 1, 3, 2, one of
which is maximum normal stress at the point
• Three mutually perpendicular Principal
Shearing Planes (Planes of max. shear)
– Principal Shearing Stresses: 1, 2, 3, one of
which is the maximum shear stress at the point
– Normal stresses are NOT zero, NOT principal
stresses and depend on the type of loading
3-D Stress Transformations Equations
• Relate Known Cartesian Stress Components at Any Point with
Unknown Stress Components on Any Other Plane through the
SAME Point
– From equilibrium conditions of infinitesimal pyramid
– Important for the Unique Orientations of the Principal Normal and Shearing
Planes
• Stress Cubic Equation- three real roots are the Principal Normal
Stresses, 1, 2, 3.
   ( x   y   z )   ( x y   y z   z x   xy   yz   zx ) 
3
2
2
2
2
 ( x y z  2 xy yz zx   x yz   y zx   z xy )  0
2
2
2
• Principal Shearing Stresses can be calculated from the Principal
Normal Stresses as follows:
1 
 2 3
2
, 2 
 3  1
2
, 3 
1   2
2
Mohr’s Circle Analogy for Stress – Graphical
Transformation of 2-D Stress State
• Principal stress solution of stress cube eq. for stresses in the x-y
plane (Fig. 4.12)
 
   
2
 3   2 ( x   y )   ( x y   xy 2 )  0,  1 
2 
 x  y
2
x
y
2
 

x
y
2
   xy 2

  x  y 
   xy 2 ,  3  0
 
 2 
2
• Analogy with the equation of a circle plotted in the - plane leads
to Mohr’s circle for
biaxial
stress:
2
2
2
( x  h)  ( y  k )  R , for circle of radius R, centered at (h, k )
where : h 
 x  y
2
  x  y 
   xy 2
, k  0, R  
 2 
2
• Sign convention for plotting the Mohr’s circle (normal stress is
positive for tension, shear is positive for clockwise (CW) couple)
• Two additional Mohr’s circle for triaxial stress states
• Find orientation of principal axes from the Mohr’s circle
2  tan
1
2 xy
 y  x
Strain Cubic Equation and Principal Strains
• STRAIN – a measure of loading severity, defining
the intensity and direction of deformation at a point,
w.r.t. specified planes through that point.
• Strain state at a point is completely defined by:
– Three normal and three shearing strain components in the
selected x-y-z coordinate system, OR
– Three PRINCIPAL strains and their directions from
Strain Cubic Equation (similar to Stress Cubic Equation,
where ’s are replaced by ’s, and shear stresses, ’s, are
replaced by one-half of ’s.)
Summary of Example Problems
• Example 4.8 – Principal Stresses in Beam
– Hollow cylindrical member subjected to transverse forces (fourpoint bending), axial force and torque
– Sketch state of stress at critical point (bottom edge)
– Use “stress cubic equation” to find principal stresses from
calculated Cartesian stresses at critical point
– Find principal shear stresses from principal normal ones
• Example 4.9 –Mohr’s Circle for Stress
– Semi-graphical analysis of biaxial stress state at critical point of
previous example cube is replaced by 2-D sketch in x-y plane
– Principal normal and shear stresses are found graphically from
the basic and the two additional Mohr circles, respectively
Summary of Textbook Problems –
Problem 4.31, Principal Stresses
• Identify critical points for each of the three types of loading
applied on the bar
– Locations where stresses are amplified by superposition of effects from
different loads – top end of vertical diameter and left end of horizontal one
– Sketch infinitesimal cube elements for the states of stress at critical points
• Calculate stresses at critical points, in the given system of
Cartesian coordinates
• Use “stress cubic equation” to find the principal stresses at
each of the two critical points
– Top edge: 1=26,047 psi, 2=0, 3=-7683 psi
– Left edge: 1=31,124 psi, 2=0, 3=-31,124 psi
• Calculate the maximum shearing stress at each point
– Top edge: max= 16,865 psi, while at the left edge: max= 31,124 psi
MAE 343-Intermediate Mechanics of Materials
Homework No. 4 - Thursday, Sep. 23, 2004
1) Textbook problems required on Thursday, Sep. 30, 2004:
Problems 4.22 and 4.30
2)Textbook problems recommended for practice before Sep. 30, 04:
Problems 4.22 – 4.35 (except 4.22 and 4.30)