Post on 19-Dec-2015
Principle and Maximum Shearing Stresses 1
Principle and Maximum Shearing Stresses (7.1-7.3)
MAE 314 – Solid Mechanics
Yun Jing
Principle and Maximum Shearing Stresses 2
Transformation of Stress Recall the general state of stress at a point can be written in terms of 6 components: σx, σy, σz, τxy= τyx, τxz= τzx, τyz= τzy This general “stress state” is independent of the coordinate system used. The components of the stress state in the different directions do depend on the coordinate system.
Principle and Maximum Shearing Stresses 3
Transformation of Stress Consider a state of plane stress: σz=τxz=τyz=0 Where does this occur?
Outer surfaceThin plate
Principle and Maximum Shearing Stresses 4
Transformation of Stress What do we want to calculate?
Principle stresses (σ maximum and σ minimum) Principle planes of stresses (orientation at which they occur)
Slice cube at an angle θ to the x axis (new coordinates x’, y’). Define forces in terms of angle and stresses.
Principle and Maximum Shearing Stresses 5
Transformation of Stress Sum forces in x’ direction.
Sum forces in y’ direction.
sincoscossin
sinsincoscos'
AA
AAA
xyxy
yxx
cossin2sincos 22' xyyxx
coscossinsincossinsincos'' AAAAA xyxyyxyx
22'' sincossincos xyxyyx
Principle and Maximum Shearing Stresses 6
Transformation of Stress cossin2sincos 22
' xyyxx
22'' sincossincos xyxyyx
2
2cos1sin2cossincos
2
2cos1cos2sincossin2
222
2
Trig identities
2sin2cos22' xy
yxyxx
2cos2sin2'' xy
yxyx
To get σy’, evaluate σx’ at θ + 90o.
2sin2cos
22' xyyxyx
y
Principle and Maximum Shearing Stresses 7
Transformation of Stress
2sin2cos
22' xyyxyx
x
2cos2sin2'' xy
yxyx
2sin2cos
22' xyyxyx
y
Now, let’s perform some algebra:
2sin2cos2
22cos2sin2
2sin2cos2
22sin2cos22
222
2
222
2
2''
2
'
yxxy
yx
yxxy
yxyx
yxx
2
2
2''
2
' 22 xyyx
yxyx
x
Constants (we can find these stresses).
Variables
Principle and Maximum Shearing Stresses 8
Principle and Max Shearing Stress Define Plug into previous equation
Which is the equation of a circle with center at (σave,0) and radius R.
yxave 2
1 2
2
2
2 xyyxR
2
2
2''
2
' 22 xyyx
yxyx
x
22''
2' Ryxavex
Principle and Maximum Shearing Stresses 9
Principle and Max Shearing Stress We learned before that the principle stresses (maximum and minimum σ) occur when τx’y’ = 0.
' ' sin 2 cos 2 02
cos 2 sin 22
2tan 2
x yx y P xy P
x yxy P P
xyP
x y
2
2
minmax,
2
2
minmax,
2
22
xyyx
xyyxyx
R
R
R
R
ave
ave
min
max
min
max
tan 22x y
sxy
Mohr's Circle 10
Example ProblemFor the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress.
Mohr's Circle 11
Mohr’s Circle (7.4-7.6)
MAE 314 – Solid Mechanics
Yun Jing
Mohr's Circle 12
Constructing Mohr’s Circle Given: σx, σy, τxy at a particular orientation Draw axes Plot Point O, center, at (σave,0) Plot Point X (σx,τxy),this corresponds to θ = 0° Plot Point Y (σy,-τxy), this corresponds to θ = 90° Draw a line through XY (passes through O): This is the diameter of the circle Draw circle
yxave 2
1
2
2
2
2 xyyxR
X
Y
O
2P
Mohr's Circle 13
Sign Convention
Mohr's Circle 14
Find stresses on any inclined plane. Rotate by 2θ1 to point X’. This points gives the stress state (σx1, τxy1). The point Y’ gives (σy1, -τxy1).
Find principle stresses. Q1 is the point with the highestnormal stress. Q2 is the point with the smallestnormal stress. S1 is the highest shearstress = R. S2 is the lowest shearstress = -R.
Uses of Mohr’s Circle
X
O
Y’
X’
2 Q1
S1
Q2
S2
2Q1
2Q2
18022 12 QQ 9022 12 QS
(y1, -xy1)
(min, )
(ave, min)
(ave, max)
(x1, xy1)
(max, )2P
Mohr's Circle 15
Example ProblemFor the state of plane stress shown below, determine (a) the principal planes, (b) the principal stresses, (c ) the maximum shearing stress and the corresponding normal stress.
Mohr's Circle 16
Special Cases of Mohr’s Circle Uniaxial tension
σmax = σx σmin = 0 τmax = σx / 2
Hydrostatic pressure σmax = σx σmin = σx τmax = 0
Pure torsion σmax = σx σmin = -σx τmax = σx
2xR
0R
2xR
Mohr's Circle 17
Example ProblemFor the given state of stress, determine the normal and shearingstresses after the element shown has been rotated (a) 25o clockwiseand (b) 10o counterclockwise.
Mohr's Circle 18
Example ProblemFor the element shown, determine the range of values of xy for which the maximum tensile stress is equal to or less than 60 Mpa.
Mohr's Circle 19
3D Stress States Mohr’s circle can be drawn for rotation about any of the three principle axes. These axes are in the direction of the three principle stresses. The maximum shear stress is found from the largest diameter circle.
minmaxmax 2
1
x-y plane
y-c planex-c plane
Mohr's Circle 20
Applied to Plane Stress Recall σz = 0 (z-axis is one of the principle axes) The third σ value is always at the origin. Where does τmax occur?
2 possibilitiesmax is in the x-y plane max is out of the x-y plane
Mohr's Circle 21
Example ProblemFor the state of stress shown, determine two values of σy for which themaximum shearing stress is 75 MPa.
Thin-Walled Pressure Vessels 22
Thin-Walled Pressure Vessels (7.9)
MAE 314 – Solid Mechanics
Yun Jing
Thin-Walled Pressure Vessels 23
Thin-Walled Pressure Vessels
Assumptions Constant gage pressure, p = internal pressure – external pressure Thickness much less than radius (t << r, t / r < 0.1) Internal radius = r Point of calculation far away from ends (St. Venant’s principle)
Cylindrical vessel with capped ends Spherical vessel
Thin-Walled Pressure Vessels 24
Cylindrical Pressure VesselCircumferential (Hoop) Stress: σ1Sum forces in the vertical direction.
0)2(2 1 xrpxt
t
pr1Longitudinal stress: σ2Sum forces in the horizontal direction:
0)(2 22 rprt
t
pr
22
Thin-Walled Pressure Vessels 25
Cylindrical Pressure Vessel There is also a radial component of stress because the gage pressure must be balanced by a stress perpendicular to the surface.
σr = p However σr << σ1 and σ2 , so we assume that σr = 0 and consider this a case of plane stress.
Mohr’s circle for a cylindrical pressure vessel: Maximum shear stress (in-plane)
Maximum shear stress (out-of-plane)t
pr
422
max
t
pr
22max
Thin-Walled Pressure Vessels 26
Spherical Pressure Vessel
Sum forces in the horizontal direction: 0)(2 22 rprt
t
pr
221
In-plane Mohr’s circle is just a pointt
pr
421
max
Thin-Walled Pressure Vessels 27
Example Problem A basketball has a 9.5-in. outer diameter and a 0.125-in. wall thickness. Determine the normal stress in the wall when the basketball is inflated to a 9-psi gage pressure.
Thin-Walled Pressure Vessels 28
Example Problem A cylindrical storage tank contains liquefied propane under a pressure of 1.5MPa at a temperature of 38C. Knowing that the tank has an outer diameter of 320mm and a wall thickness of 3mm, determine the maximum normal stress and the maximum shearing stress in the tank.
Thin-Walled Pressure Vessels 29
Example ProblemThe cylindrical portion of the compressed air tank shown is fabricated of 8-mm-thick plate welded along a helix forming an angle β = 30o with the horizontal. Knowing that the allowable stress normal to the weld is 75 MPa, determine the largest gage pressure that can be used in the tank.