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Chapter 3: Load and Stress Analysis
The careful text-books measure (Let all who build beware!)
The load, the shock, the pressure Material can bear.
So when the buckled girder Lets down the grinding span The blame of loss, or murder
is laid upon the man. Not on the stuff - The Man!
Rudyard Kipling, “Hymn of
Breaking Strain”
Introduction • Statics (and Dynamics) • Mechanics of Materials
Understanding the Relationships between external loads and internal reactions on a body.
Deals with the integrity, deformation and stability of a body.
• Pressure, Force & Moment – Stresses
• Displacement, Deformation & Distortions – Strains
• Material Behavior
Displacement Deformation
Distortion
Deformable Body
Stress Strain
Pressure
Force Moment
Cause:
Effect:
Statics
• Equation of Equilibrium Σ Fx = 0
• Balance of Force: Σ F = 0 Σ Fy = 0 Σ Fz = 0 Σ Mx = 0
• Balance of Moment: Σ Mp = 0 Σ My = 0 Σ Mz = 0
2-D
2-D
• External Force • Surface Force & Traction • Body Force (weight) • Reactions
• Free Body Diagram
Ax
Ay
Bx
By
A
B
P
Cut
2
Types of Loads
Axial Load (tensile)
Axial Load (compressive)
Shear Load
Bending (Moment) Load
Torsion (Twisting Moment) Load
Combined loads
Boundary Conditions
θ
θ
Two-force member In equilibrium
Shear Force and Bending Moment in Beam
( ) ( )
VdxdM
dMMVdxdxqdxMM
=
=++−+−=Σ 0k
( )
qdxdV
dVVdxxqVFy
=
=++−−=Σ ;0)(
qdxMd
dxdV
== 2
2
Summary on Beam )(xyy =
θ=dxdy
EIM
dxyd=2
2
EIV
dxyd=3
3
EIq
dxyd
−=4
4
dxd
dxd
dxd
dxd
( )dx∫
( )dx∫
( )dx∫
( )dx∫
3
Singularity Functions defined ME222
( ) 1
1
0
1101
0)(
−
+
−=−
−∫+
=−
≤=
>=−
≤=
>−=−
nn
nn
nn
axnaxdxd
axn
dxax
axifaxifaxNote
axifaxifaxax
1
a x
Stress element showing general state of three-dimensional stress
Stress element showing two-dimensional state of stress. (a) Three-dimensional view; (b) plane view.
Stress Elements
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=
zyzxz
yzyxy
xzxyx
σττ
τστ
ττσ
σTensor:
Stress on an Oblique Plane
x
y x’
y’
( ) ( )( ) ( ) 0coscossincos
sinsincossin''
=−−
−−=∑
θθσθθτ
θθσθθτσ
ΔA ΔAΔAΔAΔAF
xxy
yxyxx
( ) ( )( ) ( ) 0sincoscoscos
cossincossin'''
=+−
−+=∑
θθσθθτ
θθσθθττ
AAAAAF
xyx
yxyyxy
ΔΔ
ΔΔΔ
θτθσσσσ
σ
θτθσσ
τ
θτθσσσσ
σ
2sin2cos22
2cos2sin2
2sin2cos22
'
''
'
xyyxyx
y
xyyx
yx
xyyxyx
x
−−
−+
=
+−
−=
+−
++
=
1xσ
xσ
yσ
xyτyxτ
+90°
( ) ( )22cos1cos;
22cos1sin
;cossin22sin
22 θθ
θθ
θθθ
+=
−=
=Using trigonometry Identities:
Stress Plot
8ksi
40o
Stresses vs. Angle
-10
-5
0
5
10
15
-50 0 50 100 150 200 250 300
Angle
Stre
sses
Series1 Series2 Series3σx’ σy’ τx’y’
θs
θp
5ksi
3ksi 40o
4
Mohr’s Circle-Plane Stress
2
''
2
'
2cos2sin2
2sin2cos22
⎥⎦
⎤⎢⎣
⎡+
−−=
⎥⎦
⎤⎢⎣
⎡+
−=
+−
θτθσσ
τ
θτθσσσσ
σ
xyyx
yx
xyyxyx
x
22
2;
2 xyyxyx
aver RC τσσσσ
σ +⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
+==
22
2''
2
' 22 xyyx
yxyx
x τσσ
τσσ
σ +⎟⎟⎠
⎞⎜⎜⎝
⎛ −=+⎟⎟
⎠
⎞⎜⎜⎝
⎛ +−
Equation for a circle: ((x-C)2+y2=R2) ( ) 22
''2
' RC yxx =+− τσ
θτθσσ
τ
θτθσσσσ
σ
2cos2sin2
2sin2cos22
''
'
xyyx
yx
xyyxyx
x
+−
−=
+−
++
=
• Clockwise shear stress is positive. • Counterclockwise shear stress is negative.
where
+ shear - shear
Mohr’s Circle Convention • Plot two points representing normal
and shear stresses • Draw a straight line • Find the center, C • Draw a circle with the radius of R
τ
σ sy
sx
txy
R
C
(sx, txy)
(s y, txy)
Mohr’s Circle for Plane Stress
Applications
Sy Sy
Axial Load
Torsion Load
P P
Cylindrical Vessel Spherical Vessel
Bending Load
Mohr’s circle for triaxial stress state. (a) Mohr’s circle representation; (b) principal stresses on two planes.
Three Dimensional Mohr’s Circle
5
Mohr’s circle diagrams. (a) Triaxial stress state when σ1=23.43 ksi, σ2 = 4.57 ksi, and σ3 = 0; (b) biaxial stress state when σ1=30.76 ksi, σ2 = -2.76 ksi; (c) triaxial stress state when σ1=30.76 ksi, σ2 = 0, and σ3 = -2.76 ksi.
Examples 4-7 3-D Stress
( )( )( )
( ) ( ) ( )( ) ( )
( ) 02
det
222
22223
222
=−−−+−
−−−+++++−=
=−−−−−−
++−−−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
xyzxzyyzxxzyzxyzyx
xzyzxyzyzxyxzyx
xyzyzxxzy
xzyzxyxzyzxyzyx
zyzxz
yzyxy
xzxyx
τστστστττσσσ
στττσσσσσσσσσσσ
τσστσστσσ
ττττττσσσσσσ
σσττ
τσστ
ττσσ
Eigen Value Problem
General Three-Dimensional Stress
( )
( )( ) 02 222
222
23
=−−−+−
−−−+++
++−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−
−
xyzzxyyzxxzyzxyzyx
xzyzxyzxzyyx
zyx
zyzxz
yzyxy
xzxyx
τστστστττσσσ
στττσσσσσσ
σσσσσ
σσττ
τσστ
ττσσ
Principal Stresses and Directions: Eigen-values and -vectors
Example ( )( )
( )
( )323
322
11
3&2,1
2
251
252
5&5,2
0252340430002
een
een
en
+−±=
+±=
±=
−+=
=−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−
−
σ
σσ
σ
σ
σ
Relationship between Stresses and Strains
( )
( )
( )
GGG
E
E
E
xzxz
yzyz
xyxy
yxzz
zxyy
zyxx
τγ
τγ
τγ
νσνσσε
νσνσσε
νσνσσε
===
−−=
−−=
−−=
,,
1
1
1
( )
( )
( )2133
3122
3211
1
1
1
νσνσσε
νσνσσε
νσνσσε
−−=
−−=
−−=
E
E
E
σx, σy, σz τxy, τyz, τxz
Transformation (Mohr’s Circle)
Transformation (Mohr’s Circle)
εx, εy, εz γxy, γyz, γxz
σ1, σ2, σ3 ε1, ε2, ε3
6
Plane Stress Plane Strain ( )
( )
( )
.
,
,
1
1
1
G
G
G
E
E
E
xzxz
yzyz
xyxy
yxzz
zxyy
zyxx
τγ
τγ
τγ
νσνσσε
νσνσσε
νσνσσε
=
=
=
−−=
−−=
−−=( )
( )
( )
.
,
,
1
1
1
G
G
G
E
E
E
xzxz
yzyz
xyxy
yxzz
zxyy
zyxx
τγ
τγ
τγ
νσνσσε
νσνσσε
νσνσσε
=
=
=
−−=
−−=
−−=
( )ν+= 12GE
3-8 Elastic Strain ( )
( )
( )
,
1
1
1
G
E
E
E
xyxy
yxz
xyy
yxx
τγ
νσνσε
νσσε
νσσε
=
−−=
−=
−=
εσ E=
( )
( )( )
,
1
1
G
E
E
xyxy
yxz
zxyy
zyxx
τγ
σσνσ
νσνσσε
νσνσσε
=
+=
−−=
−−=
( )
( )
( )
.
,
,
1
1
1
G
G
G
E
E
E
xzxz
yzyz
xyxy
yxzz
zxyy
zyxx
τγ
τγ
τγ
νσνσσε
νσνσσε
νσνσσε
=
=
=
−−=
−−=
−−=
( )ν+= 12GE
3-10 Normal Stresses
ρ
x dx
dθ
θθρ
θρ
yddxdyLddxL
ef
ab
−=−−=
==
)(
After Bending,
ρθρθ
εy
dyd
LL
ef
efx −=−=
Δ=
θθρ yddydxLef =−−=Δ )(
maxσρ
εσ ⎟⎠
⎞⎜⎝
⎛−=−==cyyEE xx
θρdLL efab ==Before Bending,
c
dθ ρ
y
e f d a b
Important Formula • No resultant force on the surface: Neutral Plane (N.P.)
• The integral of the elemental moment over the surface must equal to the bending moment
• Moment of Inertia:
• Flexure Formula:
0;0 max =−== ∫∫∑ dAyc
dAFAA
xxσ
σ
dAyEydAMMA A
x∫ ∫∑ −=−== 2;0ρ
σ
∫=A
dAyI 2
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=−=
cyEy
IMy
xmaxσ
ρσ
7
Moments of Areas
( )yx,
xAxdAQ
yAydAQ
Ay
Ax
==
==
∫
∫
( )
( )22
222
'
2
'
hbbbhxAQ
bhhbhyAQ
y
x
===
===
• First Moment of An Area: Centroid of An Area A
dA
x
y
First Moment of Area A w.r.t. x-axis:
First Moment of Area A w.r.t. y-axis:
Centroid of Area A: C
x
y
h
b
Example: Rectangle
x
y
C
x’
y’
( )( )( )( ) 00
00===
===
bhxAQbhyAQ
y
x
First Moment of An Area
11
11
xAxdAQ
yAydAQ
n
ii
Ay
n
ii
Ax
∑∫
∑∫
=
=
==
==
∑
∑
∑
∑
=
=
=
=
=
=
n
ii
i
n
ii
n
ii
i
n
ii
A
xAy
A
yAx
1
1
1
1
1y
Centroid
A1 1y
1x
A2
2y
2x
A3
3y
3x
( )yx,
Second Moment (Moment of Inertia)
yxA
o
Ay
Ax
IIdAJ
dAxI
dAyI
+==
=
=
∫
∫
∫
2
2
2
ρ
Second Moment w.r.t. x-axis:
Second Moment w.r.t. y-axis:
Polar Moment of Inertia:
Parallel-Axis Theorem
( )
( ) 00'2
'2'
'
2'
2''
2222
===
+=++=
++=+== ∫∫∫∫∫
AyAQAdIAddQI
dAddAyddAydAdydAyI
x
xxx
AAAAAx
Mom
ent o
f Ine
rtia
3212
323
'bhhbhbhI x =⎟
⎠
⎞⎜⎝
⎛+=
442sin
2
cos
4
0
33
0
2
0
0
2
0
222
Rdxrdxr
rdrdrdAyII
RR
R
Ayx
ππ
θθ
θθ
π
π
==⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+=
===
∫∫
∫ ∫∫
12123
32/
2/
32/
2/
2/
2/
3
2/
2/
2/
2/
22
bhdxhdxy
dydxydAyI
b
b
b
b
h
h
b
b
h
hAx
==⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
==
∫∫
∫ ∫∫
−− −
− −
8
4rII yxπ
==π34 ry =
b
h
y
x
x’
yx
R x
244
444
0RRRIIJ yxπππ
=+=+=