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Bài giảng Strength of materials - Chương 4: Stress transformation - Nguyễn Sỹ Lâm

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CHAPTER 4: STRESS TRANSFORMATION  
4.1 Introduction  
4.2 Plane Stress State  
4.3 Transformation of plane stress  
4.4 Morh’s circle for plane stress  
4.5 Hooke’s Laws  
4.6 Transformation of plane strain  
4.7 Morh’s circle for plane strain  
4.1 INTRODUCTION  
Stress state at a point is the set of all stresses acting on all faces passing through  
this point.  
• The most general state of stress at a point may be  
represented by 6 components,  
,,  
z
normalstresses  
x
y
, ,   
shearing stresses  
xy yz zx  
(Note : , , )  
xy  
yx yz  
zy zx  
xz  
• Same state of stress is represented by a different set  
of components if axes are rotated.  
• The first part of the chapter is concerned with how  
the components of stress are transformed under a  
rotation of the coordinate axes. The second part of  
the chapter is devoted to a similar analysis of the  
transformation of the components of strain.  
4.1 INTRODUCTION  
SIGN CONVENTION:  
Normal stress:  
Tension is positive  
Compression is Negative  
Shear stress: two subscripts  
+ First subscript denotes the face on which  
the stress acts  
+ Second gives the direction on the stress  
vector  
Positive face (+): normal axis follows the  
positive direction of the original axis  
Negative face (-): normal axis follows the  
negative direction of the original axis  
4.1 INTRODUCTION  
SIGN CONVENTION:  
Positive direction (+): stress vector follows  
positive direction of the axis  
Negative direction (-): stress vector follows  
negative direction of the axis  
positive direction - positive face = positive stress  
negative direction-negative face = positive stress  
positive direction-negative face = negative stress  
negative direction-negative face = negative stress  
4.2 PLANE STRESS STATE  
• Plane Stress - state of stress in which two faces of  
the cubic element are free of stress. For the  
illustrated example, the state of stress is defined by  
, , and 0.  
x y xy  
z zx zy  
4.2 PLANE STRESS STATE  
• State of plane stress also occurs on the free surface  
of a structural element or machine component, i.e.,  
at any point of the surface not subjected to an  
external force.  
4.2 PLANE STRESS STATE  
• State of plane stress occurs in a thin plate subjected to forces acting in the  
midplane of the plate.  
4.3 TRANSFORMATION OF PLANE STRESS  
• Consider the conditions for  
equilibrium of a prismatic element  
with faces perpendicular to the x,  
y, and x’ axes.  
F 0   A  
Acos  
sin   
cos   
xy  
Acossin  
Asin  
x
x
x
  
Asin  
Acos  
Acoscos  
cos  
y
xy  
F 0 A  
   
x y  
Asin  
sin   
xy  
cos   
Asinsin  
y
x
  
y
xy  
4.3 TRANSFORMATION OF PLANE STRESS  
θ is positive if the rotation is counter  
clockwise from x to x’  
• The equations may be rewritten to yield  
x y x  y  
   
cos2 xy sin 2  
cos2 xy sin 2  
x
2
2
x y x  y  
   
y
2
x y  
2
    
   
x y  
sin 2 xy cos2  
2
4.3 TRANSFORMATION OF PLANE STRESS  
Principal Stresses  
• The previous equations are combined to  
yield parametric equations for a circle,  
2
2
2
  
  
R  
   
x y  
x
ave  
where  
2
  
  
x
y
x
y
2
R   
  
xy  
ave  
2
2
• Principal stresses occur on the principal  
planes of stress with zero shearing stresses.  
2
  
  
x
y
x
y
2
  
xy  
max,min  
2
2
2  
xy  
tan2  
p
  
x
y
o
Note : defines two angles separatedby 90  
4.3 TRANSFORMATION OF PLANE STRESS  
Maximum Shearing Stresses  
2
  
x
y
2
R   
  
xy  
max  
2
  
x
y
tan2   
s
2  
xy  
o
Note :defines two angles separatedby 90 and  
o
offset fromby 45  
p
  
x
y
  
ave  
2
4.3 TRANSFORMATION OF PLANE STRESS  
4.3 TRANSFORMATION OF PLANE STRESS  
EXAMPLE 4.01  
SOLUTION:  
• Find the element orientation for the principal  
stresses from  
2  
xy  
tan2  
p
  
x
y
• Determine the principal stresses from  
2
  
  
x
y
x
y
2
  
xy  
max,min  
2
2
• Calculate the maximum shearing stress with  
For the state of plane stress shown,  
determine (a) the principal panes,  
(b) the principal stresses, (c) the  
maximum shearing stress and the  
corresponding normal stress.  
2
  
x
y
2
  
xy  
max  
2
  
x
y
  
2
4.3 TRANSFORMATION OF PLANE STRESS  
EXAMPLE 4.01  
SOLUTION:  
• Find the element orientation for the principal  
stresses from  
2  
2
40  
xy  
tan 2  
1.333  
p
  
50   
10  
x
y
253.1, 233.1  
p
26.6, 116.6  
p
 50 MPa  
 40 MPa  
xy  
x
• Determine the principal stresses from  
 10 MPa  
x
2
  
  
x
y
x
y
2
  
xy  
max,min  
2
2
2
2
20   
30  
40  
70 MPa  
max  
min  
 30 MPa  
4.3 TRANSFORMATION OF PLANE STRESS  
EXAMPLE 4.01  
• Calculate the maximum shearing stress with  
2
  
x
y
2
  
xy  
max  
2
2
2
30  
40  
50MPa  
max  
 50 MPa  
 40 MPa  
xy  
45  
x
s
p
 10 MPa  
x
 18.4, 71.6  
s
• The corresponding normal stress is  
  
50 10  
x
y
  
ave  
2
2
20MPa  
4.4 MORH’S CIRCLE FOR PLANE STRESS  
• With the physical significance of Mohr’s circle  
for plane stress established, it may be applied  
with simple geometric considerations. Critical  
values are estimated graphically or calculated.  
,,  
• For a known state of plane stress  
x
y xy  
plot the points X and Y and construct the  
circle centered at C.  
2
  
  
x
y
x
y
2
R   
  
xy  
ave  
2
2
• The principal stresses are obtained at A and B.  
R  
max,min  
ave  
2  
xy  
tan2  
p
  
x
y
The direction of rotation of Ox to Oa is  
the same as CX to CA.  
4.4 MORH’S CIRCLE FOR PLANE STRESS  
• With Mohr’s circle uniquely defined, the state  
of stress at other axes orientations may be  
depicted.  
• For the state of stress at an angle with  
respect to the xy axes, construct a new  
diameter X’Y’ at an angle 2with respect to  
XY.  
• Normal and shear stresses are obtained  
from the coordinates X’Y’.  
4.4 MORH’S CIRCLE FOR PLANE STRESS  
• Mohr’s circle for centric axial loading:  
P
P
, 0  
  
x
y
xy  
x
y
xy  
A
2A  
• Mohr’s circle for torsional loading:  
Tc  
J
Tc  
  
0  
xy  
0   
x
y
x
y
xy  
J
4.4 MORH’S CIRCLE FOR PLANE STRESS  
EXAMPLE 4.02  
For the state of plane stress shown,  
(a) construct Mohr’s circle, determine  
(b) the principal planes, (c) the  
principal stresses, (d) the maximum  
shearing stress and the corresponding  
normal stress.  
SOLUTION:  
• Construction of Mohr’s circle  
  
50  
2
10  
x
y
  
20 MPa  
ave  
2
CF 50 20 30 MPa FX 40 MPa  
2
2
R CX   
30  
40  
50 MPa  
4.4 MORH’S CIRCLE FOR PLANE STRESS  
EXAMPLE 4.02  
• Principal planes and stresses  
OA OC CA 20 50  
70MPa  
max  
max  
max  
max  
OB OC BC 20 50  
 30 MPa  
FX 40  
tan 2  
p
CP 30  
253.1  
p
26.6  
p
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