adv mech of structures - module1

7/23/2019 Adv Mech of Structures - Module1

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Arun MenonDepartment of Civil Engineering, IIT Madras

E-mail: [email protected]; !one: "#$$% &&'( $&))

CE*(+#: Advanced tructural Mec!anics

Module : T!eor of Elasticit and Inelasticit


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ANALYSIS OF STRESS & STRAIN

T!eor of Elasticit and Inelasticit


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Analysis of Stress and Strain

Definition of tress at a oint

Plane cut in general loaded body

&

/

$

'

0&

0

0/3


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Definition of tress at a oint

Force transmitted through incremental area andcomponents: normaland shearforce

Average stresses:

Stress vector defined at a point:

DA

12

1

1

A

A

F

A

F

A

F SN

DD ,,

A

F

A DD

0

lim

A

F

A

NN

DD

0

lim

A

F

A

ss

DD

0

lim

Normal stressvector:

Shear stressvector:

S

N

4


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tress Components at a oint

Infinitesimal volume around Point P

4

44

45

4

45

5

54

55

4

44ositive faces

2egative faces

tress Tensor

zzzyzx

yzyyyx

xzxyxx

T

44Direction of stress

Direction of normal of t!e planeon 6!ic! t!e stress acts

2egative faces

6


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4

44

45

4

45

5

54

55

4

44ositive faces

2egative faces

mmetricaltress Tensor

zzyzzx

yzyyxy

xzxyxx

zzzyzx

yzyyyx

xzxyxx

T

27TE: 7nl * components are re8uired to descri9e t!e state of stress, if t!eonl forces t!at act on t!e free 9od are surface forces and 9od forces , asin t!e volume element 6it! uniform stress components (in case of bodycouples or surface couples, all 9 components are required for theunsymmetrical state of stress). 7


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4

44

45

4

45

5

54

55

4

44ositive faces

2egative faces

mmetricaltress Tensor

zzyzzx

yzyyxy

xzxyxx

zzzyzx

yzyyyx

xzxyxx

T

E8ualit of cross s!ears can 9e esta9lis!ed 9 taing moment e8uili9riuma9out t!e "4, , 5% a4es:

Refer to solution to tutorial no.

zxxzzyyzyxxy

!


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tress 2otations

Frequently used symmetric stress notations

Indicial or inde4notations

44 55 44 55 4554

4 5 t4t4 t5t5 t45t54


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tress Acting on an Ar9itrar lane

Stress vectors x yand !on planes perpendicular to xy ! axes:

i, ? and are unit vectors relativeto "4, , 5% a4es.

4

5

44i

4?

45

i

#

$

#

kji

kjikji

zzzyzxz

yzyyyxy

xzxyxxx

%


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Stress vector Pacting on an arbitrary oblique plane

through point "# nit normal vector to t!e plane is:kjiN nml 4

5

%

2

-

-5

-4

!ere "l, m, n% are direction cosines of 2.Bectorial summation of forces action ontetra!edral element ields:

Tetra!edral element 0ABC

zyxP nml 4, , 5are t!e pro?ections of t!e stressvector along t!e "4, , 5% a4es:

kji PzPyPxP T!erefore,

zzyzxzPz

zyyyxyPy

zxyxxxPx

nml

nml

nml

tress components on an o9li8ue plane defined 9 unitnormal 2: "l, m, n% can 9e calculated if * stresscomponents at 0are no6n.

&

'

B

Cauchys stress

formula:


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Normal and Shear Stress on an $blique Plane:

2ormal stress 2on t!e plane is t!epro?ection of vector in t!e direction of 2:

xzyz

xyzzyyxxPN

zxxzzyyz

yxxyzzyyxxPN

PPN

nlmn

lmnml

nlmn

lmnml

22

2222

222

N4

5

%

2

-

-5

-4

Tetra!edral element 0ABC

T!e magnitude of t!e s!ear stress on t!eplane :

&

'

B

222222

PNPyPyPxPNPPS --

(


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Transformation of tress0otation of A4es

Stress components relative to ne% reference axes:

T6o rectangular coordinate sstems 6it!common origin, #: "4, , 5% and "%.Cosines of t!e angles 9et6een t!ecoordinate a4es "4, , 5% and "%:

T!e normal stress component on a planeperpendicular to < a4is, l! m! n!

tress components on plane perpendicularto transformed a4es

=

t


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imilarl, =, >

xz

yzxyzyxZ

xz

yzxyzyxY

ln

nmmlnml

ln

nmmlnml

t

tt

t

tt

33

3333

2

3

2

3

2

3

22

2222

2

2

2

2

2

2

2

22

2

22

!earing stress component t


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xzyz

xyzyx

YXXY

nllnmnnm

lmmlnnmmll

tt

t

t

21212121

2121212121

12

NN

!earing stress component tand tareo9tained 6it! scalar products 6it!respective unit normal:

5

!earing stress componentt


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To define using matrix notation:

6

321

321

321

333

222

111

nnn

mmm

lll

nml

nml

nml

zyzzx

yzyxy

xzxyx

ZYZZX

YZYXY

XZXYX

tt

tt

tt

tt

tt

tt


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rincipal tresses and Directions

&ritical questions:

're there any +lanes +assin, throu,h the +oint on -hich theresultant stresses are -holly normal i.e. Shear stresses vanish

/hat is the +lane on -hich the normal stresses reach a

ma)imum i.e. 0rinci+al stresses and 0rinci+al +lanes

/hat is the +lane on -hich the shear stresses reach a ma)imum

7


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Assume a plane %ith unit normal 2%ith direction

cosines 'l m n( on %hich the stress Pis %holly normal:

)he pro*ections of Palong 'x y !( axes are:

Normal and shear stresses on any plane %ith unit normal2can be defined %ith Cauchys Stress Formula

!

kjiN nml

N P

nml PzPyPx

zyzxzPz

zyyxyPy

zxyxxPx

nml

nml

nml

tt

tt

tt


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Subtracting

)hese are linear homogeneous equations in 'l m n(

)he trivial solution l+ m+ n+ " is not possible:

)herefore the determinant of the coefficients of l m nshould vanish:

"

0

0

0

-

--

tt

tt

tt

zyzxz

zyyxy

zxyxx

nml

nml

nml

1

222

nml

-

-

-

0

0

0

n

m

l

zyzzx

yzyxy

xzxyx

tt

tt

tt

0

-

-

-

tt

tt

tt

zyzzx

yzyxy

xzxyx


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Expanding

)he , roots of the cubic equation at point " are: - . , )he direction cosines of the , principal axes are obtained

from the follo%ing relation by setting in turn equal to- . , and considering

(%

02 222

22223

--------

xyzzxyyzxzxyzxyzyx

zxyzxyxzzyyxzyx

tttttt

ttt

0

0

0

-

-

-

zzyzxz

zyyyxy

zxyxxx

nml

nml

nml

1222 nml


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tress Invariants

/xamining the determinant:

Introducing terms I- I.and I,: 1nvariants of stress

(

02 222

22223

--------

xyzzxyyzxzxyzxyzyx

zxyzxyxzzyyxzyx

tttttt

ttt

032

2

1

3 -- III

222

3

222

2

1

2 xyzzxyyzxzxyzxyzyx

zyzzx

yzyxy

xzxyx

zxyzxyxzzyyx

zxz

xzz

zyz

yzy

xxy

xyx

zyx

I

I

I

tttttt

tt

tt

tt

ttt

t

t

t

t

t

t

---

---


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tress Invariants

)he magnitudes and directions of - . , for a given

member depend only on the loads applied and not onthe choice of coordinate axes used to specify the stateof stress at point "#

)herefore I- I.and I,are 1nvariants of stress and musthave the same magnitude for all coordinate axes#

An invariant is one %hose value does not change %hen

the frame of reference is changed# 'l m n( is 2i,en vector of the stress matrix

Principle stresses - . , are the 2i,envalues#

((

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