Transcript 7 - unisa.it

Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Application of Mohr’s Circle to the ThreeDimensional Analysis of Stress
• If A and B are on the same side of the
origin (i.e., have the same sign), then
a) the circle defining smax, smin, and
tmax for the element is not the circle
corresponding to transformations within
the plane of stress
b) maximum shearing stress for the
element is equal to half of the
maximum stress
c) planes of maximum shearing stress are
at 45 degrees to the plane of stress
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-1
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Yield Criteria for Ductile Materials Under Plane Stress
• Failure of a machine component
subjected to uniaxial stress is directly
predicted from an equivalent tensile test
• Failure of a machine component
subjected to plane stress cannot be
directly predicted from the uniaxial state
of stress in a tensile test specimen
• It is convenient to determine the
principal stresses and to base the failure
criteria on the corresponding biaxial
stress state
• Failure criteria are based on the
mechanism of failure. Allows
comparison of the failure conditions for
a uniaxial stress test and biaxial
component loading
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-2
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Yield Criteria for Ductile Materials Under Plane Stress
Maximum shearing stress criteria:
Structural component is safe as long as the
maximum shearing stress is less than the
maximum shearing stress in a tensile test
specimen at yield, i.e.,
s
t max  t Y  Y
2
For sa and sb with the same sign,
t max 
sa
2
or
sb
s
 Y
2
2
For sa and sb with opposite signs,
t max 
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s a  sb
2
s
 Y
2
7-3
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Yield Criteria for Ductile Materials Under Plane Stress
Maximum distortion energy criteria:
Structural component is safe as long as the
distortion energy per unit volume is less
than that occurring in a tensile test specimen
at yield.
ud  uY
1 2
1 2
s a  s as b  s b2 
s Y  s Y  0  02
6G
6G




s a2  s as b  s b2  s Y2
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-4
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Fracture Criteria for Brittle Materials Under Plane Stress
Brittle materials fail suddenly through rupture
or fracture in a tensile test. The failure
condition is characterized by the ultimate
strength sU.
Maximum normal stress criteria:
Structural component is safe as long as the
maximum normal stress is less than the
ultimate strength of a tensile test specimen.
s a  sU
s b  sU
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-5
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses in Thin-Walled Pressure Vessels
• Cylindrical vessel with principal stresses
s1 = hoop stress
s2 = longitudinal stress
• Hoop stress:
 Fz  0  s12t x   p2r x 
s1 
pr
t
• Longitudinal stress:
 
2
 Fx  0  s 2 2 rt   p  r
pr
s2 
2t
s1  2s 2
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-6
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses in Thin-Walled Pressure Vessels
• Points A and B correspond to hoop stress, s1,
and longitudinal stress, s2
• Maximum in-plane shearing stress:
1
2
t max( in plane)  s 2 
pr
4t
• Maximum out-of-plane shearing stress
corresponds to a 45o rotation of the plane
stress element around a longitudinal axis
t max  s 2 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
pr
2t
7-7
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Stresses in Thin-Walled Pressure Vessels
• Spherical pressure vessel:
s1  s 2 
pr
2t
• Mohr’s circle for in-plane
transformations reduces to a point
s  s1  s 2  constant
t max(in -plane)  0
• Maximum out-of-plane shearing
stress
t max  12 s1 
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
pr
4t
7-8
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Transformation of Plane Strain
• Plane strain - deformations of the material
take place in parallel planes and are the
same in each of those planes.
• Plane strain occurs in a plate subjected
along its edges to a uniformly distributed
load and restrained from expanding or
contracting laterally by smooth, rigid and
fixed supports
components of strain :
 x  y  xy
 z   zx   zy  0
• Example: Consider a long bar subjected
to uniformly distributed transverse loads.
State of plane stress exists in any
transverse section not located too close to
the ends of the bar.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7-9
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Transformation of Plane Strain
• State of strain at the point Q results in
different strain components with respect
to the xy and x’y’ reference frames.
     x cos 2    y sin 2    xy sin  cos
 OB   45  12  x   y   xy 
 xy  2 OB   x   y 
• Applying the trigonometric relations
used for the transformation of stress,
 x 
 y 
 xy
2
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x   y
2
x   y
2



x   y
2
x   y
x   y
2
2
cos 2 
cos 2 
sin 2 
 xy
2
 xy
2
 xy
2
sin 2
sin 2
cos 2
7 - 10
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Mohr’s Circle for Plane Strain
• The equations for the transformation of
plane strain are of the same form as the
equations for the transformation of plane
stress - Mohr’s circle techniques apply.
• Abscissa for the center C and radius R ,
 ave 
2
x   y
  x   y    xy 
  

R  
 2   2 
2
2
• Principal axes of strain and principal strains,
 xy
tan 2 p 
x   y
 max   ave  R
 min   ave  R
• Maximum in-plane shearing strain,
 max  2R 
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 x   y 2   xy2
7 - 11
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Three-Dimensional Analysis of Strain
• Previously demonstrated that three principal
axes exist such that the perpendicular
element faces are free of shearing stresses.
• By Hooke’s Law, it follows that the
shearing strains are zero as well and that
the principal planes of stress are also the
principal planes of strain.
• Rotation about the principal axes may be
represented by Mohr’s circles.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7 - 12
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Three-Dimensional Analysis of Strain
• For the case of plane strain where the x and y
axes are in the plane of strain,
- the z axis is also a principal axis
- the corresponding principal normal strain
is represented by the point Z = 0 or the
origin.
• If the points A and B lie on opposite sides
of the origin, the maximum shearing strain
is the maximum in-plane shearing strain, D
and E.
• If the points A and B lie on the same side of
the origin, the maximum shearing strain is
out of the plane of strain and is represented
by the points D’ and E’.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7 - 13
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Three-Dimensional Analysis of Strain
• Consider the case of plane stress,
s x  s a s y  sb s z  0
• Corresponding normal strains,
a 
s a s b
E
b  

E
s a
s
 b
E
E


 a   b 
 c   s a  s b   
E
1 
• Strain perpendicular to the plane of stress
is not zero.
• If B is located between A and C on the
Mohr-circle diagram, the maximum
shearing strain is equal to the diameter CA.
© 2006 The McGraw-Hill Companies, Inc. All rights reserved.
7 - 14
Fourth
Edition
MECHANICS OF MATERIALS
Beer • Johnston • DeWolf
Measurements of Strain: Strain Rosette
• Strain gages indicate normal strain through
changes in resistance.
• With a 45o rosette, x and y are measured
directly. xy is obtained indirectly with,
 xy  2OB   x   y 
• Normal and shearing strains may be
obtained from normal strains in any three
directions,
1   x cos2 1   y sin 2 1   xy sin 1 cos1
 2   x cos2  2   y sin 2  2   xy sin  2 cos 2
 3   x cos2 3   y sin 2 3   xy sin 3 cos3
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7 - 15