Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314...

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Principle and Maximum Shearing Stresses 1 Principle and Maximum Shearing Stresses (7.1- 7.3) MAE 314 – Solid Mechanics Yun Jing

Transcript of Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314...

Page 1: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

Principle and Maximum Shearing Stresses 1

Principle and Maximum Shearing Stresses (7.1-7.3)

MAE 314 – Solid Mechanics

Yun Jing

Page 2: Principle and Maximum Shearing Stresses1 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.

Page 3: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 4: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 5: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 6: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 7: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 8: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 9: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 10: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 11: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

Mohr's Circle 11

Mohr’s Circle (7.4-7.6)

MAE 314 – Solid Mechanics

Yun Jing

Page 12: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) 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

Page 13: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

Mohr's Circle 13

Sign Convention

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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

Page 15: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 16: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 17: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 18: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 19: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 20: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 21: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

Mohr's Circle 21

Example ProblemFor the state of stress shown, determine two values of σy for which themaximum shearing stress is 75 MPa.

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Thin-Walled Pressure Vessels 22

Thin-Walled Pressure Vessels (7.9)

MAE 314 – Solid Mechanics

Yun Jing

Page 23: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) 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

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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

Page 25: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 26: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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

Page 27: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 28: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.

Page 29: Principle and Maximum Shearing Stresses1 Principle and Maximum Shearing Stresses (7.1-7.3) MAE 314 – Solid Mechanics Yun Jing.

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.