1

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

=+=+=