'

UNI'I : I

;

ENERCY PRINCIPLES

-

Strain ener\$, arld strain energy density strain energy in traction, shear in flexure and to.sion - castigliano's theorems - principle of virtual work - application of energy theorems for computing defle€tions in beams aIld trusses - Maxwell's reciprocal theorems 1. Define:

Strai[ Energy

When an elastic body is under the il.l,o,r of external fo.ces the body deforms and worl( is rlone by these forces. If a stl.a:ned, perfccth, elastic body is allowed to recover slowly to its unst.ained state. [t is capable ofgiving bar:k.rll thc work done by these external forcos_'This work dorre in straininil such .r body may be rcgar,ied ds encrgy stored in a body and is called sarain e ne rgy U= cr2V

2E

,

2.

Define Resilience- (May-2012) The strain etrergy stored by a body withix tlle elasti. limit, when loaded externally is cailed

Resiiience. 3. Define: ProoI Resilicnce. 'l he ntaximu tn energy, which a bcxly sLor e:i upto clastic lirnit is calied Proof Resilience. U= oCIL '2E

4. Defire Modulus of resilielce. [Nov-2012) 'fhe pt-ool resilience of a body per I nit voluntc. (ie) The maximum energy s[ored in the body within the elastic lirnit pcr unit volunre. u= o-t! 2E 5.

Write the formula to calcllate the stt ain erergy due to axial loads (tension).

U=J Wl

a.r

limir0roL

2AE Where,

w

= Applied reDsile load. L = Length of the menber A = Arca olthc rnerllber

Ii = Young's moclulus. 6. Write the

formula to calculate the strain energy due to bending.

U=

i

Nl,

2tt

!!..

lirnir 0 to L

Where, M = Berrdirrg nronrerrt due to applied loads. E - Yoillrg's rnl)r:lulris i = N{onr,..nt oi i..r-tia

t '

7.

Write the formula to calculate the strain energy due to torsion U=

Where. TC

T1_dr

j

limit

0

to L

2C lp

Applied'lorsion

= Shear modulus or Modulus Poiar moment ofinertia

ofrigidity

Ip =

B.Write the lormula to calculate the strain energy due to pure shear

4* 2AC

u = J_Ei

!Vler.e,

limir

0

to L

F=Shearloacl C = Shear modulns or Modulus ofrjgidiry A - Area ofcross section. K = Constant depends upon shape ofcross section.

'

9. Write down the formula to calculate the strain energy.due to pure shear, if shear stress is given. (Nov-2013)

u=ra 2C.

Wncr

u, r - Shpdr Sl t.ess modultrs or Modulus of rigidity V = Volume .ri the tnaferiai.

C = Shear

10.

W.ite dowlr the tbrmlrla to calctrlate the straio energy, ifthe moment value is given U

-

M'zl. ?

t:.1

Wherc, M=Bendingmoment l, = Length ofrhe beam g = Young's modulus I = Moment of inertia

11. Write down the formula to calculate the strain energy, given.

if the torsion moment value

is

U=1']1 ZClp

Where,'l'=Appliedlbrtion L = l,ength of the beam C

- she.l !nodulus or Modulus of rigidiry

lr = Polar tBon!erlt of inertia

lyrite down tlre fortnula to calculate the strain energ),., if the applied tensio[ Ioad is given. (Nov'2010) Li = WrL 12.

2AE

\ryhere,

W

- Applied

i- = LL,ngth rif rhr' rrerrrber A = Area o[the tncmbel. E

- Young's nrodrrlus.

z

Wlite the Castigliaro's tirst theorem. (Nov-2012J {May-z012) Iftlisthetotal straincncrgyolanJ.Tslructureduetoapplicataonofexternal loadsW1,W2, W3,- -.-Wn at points A1, 42, An respectively in the direction AX1,AX2,--'AXn and due to couplesM1,M2,----MllatpointsBl,ll2.- Bn respectively then the deflection at the points A1, A2,-and the angular position ofthe couples -An in the direction AX1, Axz,..-..Axn are 13.

' :

are At the respective points ofapplication.

y=oL ow

rvvhere, Y=

U=

,

Deflection Slrain I.i'rergy sto.L,d

W = Lo:rd 14. What are uses ofCastigliano's first theolenr? 1. To determine the deflection ofcolrtllicated structure. 2. To deier&ire the deflection of.Urvr'(l beams sp.ings15. State Maxwell's Recipiocal Theorem. (May-2011) (May-z010) (Nov-2013) (May-z013) The work done by the llrst systern of loads due tc displacernent caused by the sccond systern ol loads equals lhe work di)ne by lhc sl).on.l system of loads due to displacement caused by the tlrsl system oads.

o

l

F:-f ti

'\ -

f--_ |

6,

-,_/,-T I

dc

16. Define: Unit load method or Principle ofvirtual work. (May-2011) [May-2013) Principal of virtual lvorl< stJtcs lhat if a system of lorces acting on a body or a system of bodies be in equilibriunl and if lhe sysrem is imagined to undergo a srnall displacement consistent with the geometric.l conditions, then the algebraic sum of the virtual work done by the forces ofthe system is zero. Dellection = XI(FL

i7. Give the pro.edire for unit load lncthod. 1. find thc fr)rces P1. P2, ....... in all tirc mcmbers due lo external loads 2. lle rcre lhc exler-nal loads erd apply the unil verlicai point load at the jolnt ii the vc! tical del'lcction is .equiied and find thc lorce. 3. n pll_v the equation tor verticai a d horizontal detleclicrl.

18. Compare thc unit load method ard Castigliano's first theorem In thc unjt loril rirethod, ollc has to.Ildlyze thc franlc h^'ice to iind the load and deflection 'y'./hile in lhc Ltt.r nrelhod, only ont analrvsis is needcd.

7

iiJf{+:i"':Tfrffiff,"',:,',:,." U= t

2

pcr unit

vorunre' the shear stress ror a material is siven as s0

volume

;

=502/{2x80000) = 0.015625

N/mm 2- per unit volume

vorume' the tensire stress ror a materiar is siven as per unit volume

i)J,lll i':i:Tll:ffiT)Jili"'t U=

o2

rso

2E

= (150)2 / (2\{zx7oz ) = 0.05625 N/nnr 2. per unitvolume. 21. DeIine: Trussed Beam_ A beam strer:gthcneci by providing ties and struts is known as .frussed Beams.

.

22. Haw will you find dctlcci-ror at a point in a beam using castigliano's Theorem? (NovzooT) To find the deflectio! at a poitlt of the structure where there is no load or couple actin& then it maydbe assumed thar a d'rrrr.y t""d w;;;,;;;;;"d:i:ffi; _ *at point and sive zero at 6 =J

[i)M

dx

EJ i)W 23. Caiculate tlre strain energy stored in the beam shown in lig. El = consrant.

"l Mxz rt\ U= 1.. "U 2EI

-

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,,..1_ fiWzxz ax

,,=Ii ["'j ,-y' zt:r Itl

t,'1

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--

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+)

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r

ANNA UNIVERSITY OUESTIONS

UNIT

-I

PART - A

1. Define Moduius ofresilience. pg:No:1 2. Write the Castigliano's first theorem. pg:No:3 3. Define Resilienco. Pg:No:1 4. State the l,r'inuilre of virtual work. pg:No:3 5. State Max\,!cll's Reciprocal I'heorem. pg:No:3 6. Calculate the strain energy stored in the beam shown in figure. El is constant

Pg:No:4

W

L__)

)

k-- L -----y

7.

Write down tjre expression for strair energy storecl in a bar of cross sectional A Iength ! and subjected to a tensile load

B. Howwil!yorr

w' pg:No:2

tirrd deflection at a point in a lleam using castigliano.s theorem?

Pg:No:4

PART - B

1.

Find the strain energy stored in a steel barof 50cm longand ofcross section 5cm x

1cm when

il

is subjected to axial pull of 60kN and to a compressive stress of

l20N/rnrrr .lnd on its narrow

edges. Take E = 2,10s N/mnr2 and 1/m = 0.29. What

will then be the modulus of

resilience

of the rnaterial of the bar. I

t'g:No:7

2. Asimpll srropoitedbeamofspan3n':iscarryingapointloil.!

of 20kNatlmfromthe

left suppor r irt rilditioir to a UDL of 10kN/m spread over the right of spar Using castigli.rn(,'\ theorem, Determine thc deflection undcr the point load. Take EI is col]stant thIi]u!ihout. Pg:No:22

3. Determine

the vertical deflection at the fi.ee end of the caltilever truss shown in Iigure. Take cross section area of compression members as B50mm2 and tension

members as 1000mm2. E= 210Gpa. pg:No:32

F-Zrra ----+-=--

5m -------l

-\

4.

A beam 4m in iength is simply supported at the ends and carries UDL of 6kN/m Iength. Det€rmine the strain energy stored ill the beam. Take E= 200cpa and I = 1440cnra. Pg:No:18

5.

A beam si[)r)iv supported over a span of 3m carries uDL 20kN/m over the entire span. The llcxural rigidity El 2.Z5MNm2. using = Casrigliano,s theorem, determine the deflection ar the ce[tre ofthe bearn. pg:No;23 Fot the trLrss sholvn in f,gure. Find the total str"in encrgy stored. E= 2 x 10sN/mm2, Area AB = 100mr 2, BC = 100mm2, AC B0mrnr. = I

t I

I I

y__ 4tt1_)(L 7.

4r.

-__,a{

For the bean shown in figure, find the cleflectjon at C and slope at D. I = 40 x 107 mma, E=200Cpa. I,g:No:20

5ov^J

40 Xr'l

'-5L ----- J ^llB TtCDH f - ?m )f-- 2n - l

^t.

'- --+

B.

Derive ahe expression fbr strai[ energy in tol sio[ of a circular shaft of length I and radius R subiected to a torque T, producing a twist 0 in the length of the shaft for the

following cases. Pg:No:10

i. ii.

Solid circular shaft

Hollow circular shaft

9. A cantilever of rectangular spread from fiee end

section breadth b, depth d, and length

I carries

to the mid section of the cantilever. Using

UDL

castigliano's

theorems find the slope and deflection due to bcnding at the liee end. Pg:No:26 10.

A simply supportcd beam of span Bm carries two concentrated loads of 32kN and 4BkN at 3m and 6m trom left support. Calculate the deflection at the centre by strain

energy prin, inlP. Pg:No: l6 11. Find the

strain elrergy stored ina steel bar 50cnr long and 3m x 1cm in cross section,

when it is subicctcd simultaneously to an axial pull of 50kN and compressive stress of 100N/mm'] on its nar.ow edge. Pg:No:39

12-A piece of rnatcrial is subjected to t\a/o perpendicula. stresses fr (tensile) and

f2

(compressivel. Find an expression for a strrrn encrgy stored per unit volume. lf the stress of 12tlN/mmz acting along gives thc sanrc value of strain energy as per the

exprcssion already found. Find the value ot f: when fr = 112N/mm2. Take 1/m = 0.30. Pg:No:40 13. Using Castigliano's

first theorem calcul.rte the central deflection and the slope at the

end of a ss heanr carrying a UDL of intensity per unit length over the whole span-Pg.No-

2Z

## som unit 1_NoRestriction.pdf

to.sion - castigliano's theorems - principle of virtual work - application of energy theorems. for computing defleâ¬tions in beams aIld trusses - Maxwell's reciprocal ...

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