Tirne: 3 hrs.

USN

c.

2a.b.

c.

3a.

b.

c.

4a.b.

06ME63

(08 Marks)(04 Marks)

(10 Marks)(05 Marhs)

Sixth Semester B.E. f)egree Examination, December 2Ol2Modeling and Finite Element Analysis

!.!aue=

aO

aX,=D-

3otll=co.= a.l

Etf-O=ts

a2

6=

OO

-1

boi

2G

3uAE6X

o --:,i .9.6EoLE

a,->(kcno= bIr

0=soF>o

(-) <-No

Z6

o

la.b.

Max. Marks:100Note: Answer FIVE full questions, selecting

at leust TWO questions from eoch part.

PART _ A

Using Rayleigh-Ritz method, derive an expression for maximum det.lection of the simplysupported beam with point load P at centre. Use trigonometric function. (08 Marks)Solve the following system of simultaneous equations by Gauss elimination method.

X-l Y 'l Z:9x-2y+32:82x+ Y - z:3

Explain the principle of minimum potential energy and principle of virtual work.

Explain the basic steps involved is FEM.Explain the concepts of iso, sub and super parametric elements.Define a shape function. What are the properlies that the shape functions should satisly?

(05 Marks)

What are the convergence requirements? Discuss three conditions of convergencerequirements. (05 Marks)What are the considerations for choosing the order of the polynomial functions? (05 Marks)Derive the shape functions for CST element. (10 Marks)

Derive the Hermite shape function tbr a 2-noded beam element. (10 Marks)Derive the shape functions fbr a four noded quadrilateral element in natural coordinates.

(10 Marks)

i

1

Irl{

:'

I:

5a.b

PART _ B

Derive an expression for stifthess matrix for a2-D truss element. (10 Marks)

6a.

Derive the strain displacement matrix tbr 1-D linear element and show that o: E[B]{u}(10 Marks)

Discuss the various steps involved in the finite element analysis of a one dirnensional heattransfbr problem with refbrence to a straight unifbrm fin. (10 Marks)

b. Derive the element matrices, using Galerkin for heat conduction in one dimensional elementwith heat generation Q. (l0 Marks)

7 a. A bar is having uniform cross sectional area of 300 mm2 and is subjected to a loadP : 600kN as shown in Fig.Q7(a). Determine the displacement field, stress and supportreaction in the bar. Consider two element and rise elimination method to handle boundaryconditions. Take E :200 GPa. (10 Marks)

I of 2

aI

b. For the two bar truss shown in Fig.Q7(b), determine the nodal displacements and stress ineach number. Also find the support reaction. Take E :200 GPa.

, SotlN

For the beam shown in Fig.Q8(a),

Take E :200 GPa,I:4x106 mma.

(10 Marks)

Fig.Q7(b)

determine the end reaction and deflection at mid span.

(10 Marks)8a.

24hNlm TYc

h,,|tl,lfi

Determine the temperature distribution through the composite wall subjected to convectionheat loss on the right side surface with convection heat transfer coefficient shown inFig.Q8(b). The ambient temperature is *5oC.

**+*82 of2

,,,/.t

Fig.Q7(a)

Fig.Q8(a) Fie.Q8(b)

(10 Marks)

sixth semester B.E. Degree Examinatlon, December 2011

Modelling and Fisrite Elememt Analysis

06ME63

Max. Marks:100

(08 Marksi

in the figure using(08 Marks)

Note: Answer uny FIYE full questions, selecting

at least TWO questions from each part'

PART-AI a. write the equilibrium equatio, for ffrtate of stress and state the terms involved" (04 Marks)

b. solve the following system of equations by Gaussian elimination rnethod :

Time:3 hrs"

x1*x2*x:=6Xr-Xz*2x3=5x1* 2x2-x3=2.

c. Determine the displacements of holes

2a.

b.

0.

3a.b.

oioo(d

a(6

rd

{)iEe)

_o? o

(!u!.,

ao ll

traP.=Nd+i. 60

otr-oo=Esodvd6o

o'oboc"o!26!s='d(g.a (,EOo€2O

tro.oj

AE5L)olE

LOo.E>.9on-troo

qo

:a)EE-hU<--.; c'i

oozd

oo,

b.

c.

of the spring system shown

principle of minimum potential en?rg{;t{" trln"*

SorfFig.Q.1(c). 6-s rr lx't

Explain the discreti zationprocess of a given domain based on element shapes number and(06 Marks)

slze.

Explain basic steps involved in FEM with the heip of an example involving a structural

member subjected to axial loads. (08 Marks)

Why FEA is widely accepted in engineering? List various appiications of FEA in

engineering (06 Marks)

Derive interpolation model for 2-D simplex element in global co - ordinate system'(10 Marks)

What is an interpolation function? Write the interpolation functions for:i) 1 -Dlinearelement ; ii) 1 -Dquadraticelement'iiU 2-D linearelement ; iv) 2-Dquadraticelement'v) 3-Dlinearelement.Explain "complete" and "conforming" elements'

Derive shape function for 1 - D quadratic bar element in neutral co-ordinate tVttelm Marks)

Derive shape functions for CST element in NCS. (08 Marks)

What ur. rhup. functions and write their properties. (any two). (04 Marks)

(06 Marks)(04 Marks)

(04 Marks)(06 Marks)(10 Marks)

c.

4a.

5a.b.c.

6a,

b.

PART - BDerive the body force load vector for I - D linear bar etrement.

Derive the Jacobian matrix for CST element starting from shape function'

Derive stiffness matrix for a beam element starting from shape function'

Explain the various boundary conditions in steady state heat transfer problems with simple

sketches. (06 Marks)

Derive stiffness matrix for 1 - D heat conduction problem using either functional approach(08 Marks)or Galerkin's approach

I of Z

l.jl'ijii

Srtcll.t"r

rI

06M863

(06Marts)

TakeAr:Az=A3:A

Fie.Q.6(c)

7 a. The structured member shown in figure consists of two bars. An axial load of P:200 kN is

loaded as shown. Determine the following :

i) Element stiffness matricies.ii) Global stiffness matrix.iii) Global load vector.iv) Nodaldisplacements.

c. For the composite wall shown in the figure, derive the global stifftress matrix.

fo;5s1.Fs z rD qlt.t-tot$fltt

Fie.Q.8(b)

,*****2 of?

i) Steel Ar = 1000 mm2Er :200 GPa

ii) Bronze Az:2000 mm2Ez: 83 GPa'

8 a. Determine the temperature distribution in 1 - D rectangular cross - section

figure. Assume that convection heat loss occurs from the end of the fin. Take

- 0.1wh = "" = , T*:20oC. Consider two elements

Cm'oCfo v 5 E*fr.,r

Fr z reg,il

.f.y-tol nnllFie.Q.8(a)

b. For the cantilever beam subjected to UDL as shown in Fig.Q.8(b), determine the deflections

(08 Marls)

b. For the truss system shown, determine the nodal displacements. Assume E: 210 GPa and A= 500 mm2 for both elements. (I2 Marks)

;f =loovl.rt

fin as shown in

'3wK=-.CmoC'

(10 Marks)

Fie.Q.7(b)

of the free end. Consider one element. (10 Marks)

r-II

USN

Time:3 hrs.

SixthSemesterB.E.DegreeExamination,December2010Modeling and Finite Element Analysis

06ME63

Max. Marks:100

Note: Answer ony FIVE futl questions' selecting

at least TWO questions from each part'

PART _ A

I a. Explain, with a sketch, plain stress ffi'"-" tt "in

for two dimensions' (06 Marks)

b. State the principles of minimu* p*"'ii"f energy' Explain the potential energy' with usual(06 Marks)

o,9H

a(g

d()d0)

39

d9

-o ,,

ao"Fm.=+'E-fb?p

Pfr

o>!1 a

acd

5(Jdo

6d

}Etr5

!Ooe

o- gtEo.

si ^9

'@q

L0

>.kmo

=(6g0tr>59o-U<-i c.i

()oZd

oa

c. t^hTT: the steps invotved in Ravleigh-Ritz methog? DeTnTl:,'X ul?l]":::'*' ::#::ffii il'r'ffi:i:.1il;J*_d#;,I;J;;g-^ shown in Fig.l(c). use second degree

(08 Marks)(08 Marks)iolynomial approximation, for the displacement'

notations.

2a.b.

Fig.l (c).

Bring out the four differences in continuum method with finite element methocl' (04 Marks)

What do you underrtuod FEM? eri"ny e*piain the steps involved in FEM' with example'

Write properries of stiffness matrix K. Show the generai node numbering *d t"l-ff:;Tlthe half bandwidth-

- (06 Marks)

What is an interpolation function? ,..r - -r ^^--.^- , {tz Marks)

what are convergence requirements? Discuss three conditions of convergence requirements'(08 Marks)

Write a shot notes on : -

i1 C.o*etrical isotropy for 2D Passal triangle

ii) Shapg function for constant strain triangrilar (CST) elernent' with a sketch' (lG Marks)

Derive the shape functions for the one-dimensional bar eiernent, in natural co-ordinates"(08 Marlcs)

Derive the shape functions for a four-node quadrilateral eler'rent, in natural co-'rdinates'(08 Marks)

Write four properties of shape functions' (04 Marksi

PART - BDerive the following :

1) Element stiffness matrix (K")'

il Element load vector (f)

6a.Explainwithasketch,one-dimensionalheatconduction.b.Derivetheelementmatrices,usingGalerkinapproach,

dimensional element'

c. Explain heat flux boundary condition in one dimension'

3a.b.

c.

4a.b.

5a.

c.

Uy aire"t method for one-dimensional bar etrement'(12 Manlis)

b. K:ff:"Iffi::f the Jocabian transformation matrix (l-1) for constant strain triangle (csr)'(08 Marks)(08 Marks)

(06 Marks)

for heat conduction in one(10 Marks)(04 Marks)

I06M863

7 a. Solve for nodal displacements and elemental stresses for the following. Fig.Q,7(a), shows athin plate of uniform lmm thickness, Young's modulus E : 200 Gfa, weighr aensity of theplate : 76.6 x 10-6 N/mm2. In addition to its weight, it is subjected to a point load of 1 kN atits mid point and model the plate with 2 bar elements. (10 Marks)

Fig.Q.7(a).

b. For the pin-jointed configuration shown in Fig.e.7(b), formulate thedetermine the nodal displacements.

fiIom'rJ.IKN

{.

Es = E2=26r6$Pr.

Fig-Q.7(b).

8a. Solve for vertical deflection and slopes, at points 2 and,3,structure shown in Fig.Q.8(a). Also determine the deflectionthe beam carrying UDL.

stiffness matrix. Also(10 Marks)

using beam elements, for theat the centre of the

ir"rtffi

Fig.Q.8(a).E:z.o06P(

J = 4x lob**ti

b. Determine the temperature distribution through the composite wall, subjected to convectionheat transfer on the right side surface, with convective heat transfer co-efficient shown inFig.Q.8(b). The ambient temperature is -5oC. (10 Marks)

a-t*

[2 looo $ly*tt

Kz=***

****,r2 of2

r

I

I

+t

Fig.Q.8(b).

USN

06M863

(10 Marks)

(10 l![arks)

doo

a(gi

'o()(Bo

B9qp-

'=h

aollt-6

.= e'lcdSc^!i bI)!'a {)otrrAeE

o78zoid?d6odoboc.ddrk}E!o=!rg-2" tsirOoeE3o9'troo-i

9EA,E=9LO

o.<>.(Ig";o6EAEtr>=6JEk_ho<..I e.i

C'oz(l

o+

Sixth semester B.E. Degree Examination, May/June 20L0

Modeling and Finite Element Analysis

Time: 3 hrs. Max' Marks:100

Note: Answer any FIVE futl questions, seleeting atleast TIYO from each part'

PART - A

! a. Using Rayleigh Ritz method, find tt. ,il*l*--a"flection of a simply supported beam with

point load at center' . . -- ,--- ^^-.^+r^-- L., I],,,oci;rn eli (10 Marks)

b. Solve the following system of simultaneous equations by Gaussian elimination method'

4xr f 2W+ 3x3:42xr * 3x2* 5x3:2Zxr * 7xz: 4

2a.Explainthedescretizationprocess'sketchthedifferenttypesofelementslD'2D'3Delements used in the finite element analysis' -..,-- L-- A:-^

(06 M'::Y)

b. considerinil;gton"rrt,.ou.oio it. "tl*.rrt stiffness matrix by direct stiffiress fir;111*;

Comment on its characteristics'

a. De{ine " J# #ffi;. irh;;*" the properties that the shape flrnction should tudti9,}*rr.ut

3 a. Explain the convergence criteria with suitable examples and compatibil* *o*T#H:i;FEM.

b. Explain simplex, complex and multiplex elements using element shapes' (06 Marks)

c. Explain linear interpolatiorr, potyrro*ials in terms "igilu"r coordinates for one dimensionai

simPlex element' - (06 Marks)

4a,Explaintheconceptofisoparametric,subparametricandsuperparametricelementsanrjtheirurrr'to

."t,,rte'Pr vr revr*r*.

a ,1 r:--ri^^^*a (06Marks)

b.DerivetheshapefunctionsforaCsTelementandalsothedisplacementmatrix.(08Marks)c. Derive the Hermite shape nn.ti* for abeam element' (06 Marks)

PART. B

s a. Find the shape tunctions at pgintp ruiffi"lement shown in fig' Qs(a)' ^t11r11r*;

area and Jacobian matrix for the eiement'(10 Marks)

61 8)

Fig.Qs(a) IC$,

I(trt{

b.Derivethestiffrressmatrixfota2_dimensionaltrusselement.

6 a. Discuss the various steps involved in the finite element analysis of a one dimensional heat

transfer problem with reference to a straight unjform fin' (10 Marks)

b. Explain the finite element *oa"rirrg La rrrrp" functions for linear interpolation of

temperature field (one - dimensional tieat trunsfeielement)' (10 Marks)

1of?

'P*\C6rsJ

7 a. Determine the nodal displacement and stresses in the erement shown in fig. e7(a).1r0 Marks)

Fie.Q7(a)Ar = 500 mm2

lokN Az = 2000mm2Er : 100 GPaE2:200 GPa

300mm 300mm

b.3::::_1"":::rl-::tg::r^ *"gx of

.the truss etements shown in fig. e7(b). Au theelements have an area of 200mmz and erementr (1);; irt;; sil,- f;*.-;:';fi'bil;:

(10 Marks)go l.^1,

Fie.Q7(b) f

6o*t

A composite wall consists of three materials as shown in fig. eg. The outerTo = 200c' convective heat transfe, tuk., place on the inner surface of the wall8000c and h :25 wrmz 'C. o.[""ir. it . ,.rp.rature distribution on the wall.

temperaturewith Too =(20 Marks)

^*Jlif*: I, otc

kr :20 WmoCkz:30 WmoCk3 = 50 WimoC

' h-25WlmzoCT*:8000C

Fig.Q8

*****

2 of2

USN06M863

(06 Marks)(06 Marks)(08 Marks)

(10 Marks)(I0 Marks)

(r0 Marks)(10 Marks)

sixth semester B.E. Degree Examination, June-July 2009

Modeling and Finite Element AnalysisTime: 3 hrs. Max. Marks:100

Note: Answer any FIVE full questions, selectingat least TWO questions from each part.

PART _ AL a. Explain the principle of minimum potential energy and principle of virtual work. (06 Marks)

+l

b. Evaluate the integral 1= J{fE' +2\z +\+2F\ by using 2 point and 3 point Gauss

-tquadrature. (06 Marks)

c. Sotve the following system of simultaneous equations by Gauss Elimination method:

x, -2x, * 6x, = Q

Zxr+Zxr*3x, =l- Xr * 3x, = 0 (08 Marks)

Explain briefly about node location system'

Explain preprocessing and preprocessing in FEM.Explain the basic steps involved in FEM.

What are the considerations for choosing the order of polynomial functions? (06 Marks)

Explain convergence requirements of a polynomial displacement model. (06 Marks)

Derive the linear interpolation polynomial in terms of natural co-ordinate for 2-D triangular

elements. (08 Marks)

What are Hermite shape functions of beam element? (06 Marks)

Derive the shape function for a quadratic bar element using Lagrangian method. (06 Marks)

Derive the shape function for a nine noded quadrilateral element. (08 Marks)

PART _ B

Derive the element stiffness matrix for truss element.

Derive the Jacobian matrix for 2D triangular element.

Explain the types of boundary conditions in heat transfer problems.

Discuss the Galerkin approach for l-D heat conduction problem.

Using the direct stiffness method, determine the nodal displacements of stepped bar shown

in figure Q7 (a). (lo Marks)

Er :200 GPaEz:70 GPa

Ar : 150 mm2

Az: i00 mm2

Fr:l0kWFz:5 kW

4a.b.c.

2a.b.

c.

3a.b.c.

5a.b.

6a.b.

la.

Fie. Q7 (a)

I ofZ

7 b. For the truss shown in figure Q7 (b), find the assembled stiffness matrix.

06M863

(10 Marks)

' E1 : E2:200 GPa

Fig. Q7 (b)

8 a. Determine the temperature distribution through the composite wall subjected to convectionheat loss on the right side surface with convective heat transfer coefficient shown if figureQ8 (a). The ambient temperature is -5"c. (r0 Marks)

k=,

Kr: )*cr.l/*.r-lt

F prq6.b- -ri._o,94Fig. Q8 (a)b' Determine the maximum deflection in the uniform cross section of Cantilever beam shown

in figure Q8 (b) by assuming the beam as a single element. (10 Marks)loe i< Fj

E:7x10e N/m2

I:4x10-a ma

lkN

T5oo

I

L

t:rdc

6 =j_v+K

Fig. Q8**{.**

i

----*l

(b)

2 of2

u-\

USN

OLD SCI{EME -;?

l--/--

sixth semester B.E. Degree Examination, July 20A6

Mechanical EngineeringFinite Element Methods

[Max. Marks:100

Note: Answer any FIVE full questions'

Define functional.Derive Euler's Langranges's equation'

Expiain principle of minimum potential energy'

Briefly explain the steps involved in FEM' (10 Marks)

Derive shape functions for CST triangular element in local co-ordinater. (10 Marks)

Explain Banded matrix. Write an algor'ithm for Guass elimination technique'(10 Marks)

Explain Raieigh's Ritz method in detail' (10 Marks)

What do you understand by weak form of differential equation. (05 Marks)

ft, ="lS$*

'-lt

ffi':gnr*:.et*r-**i) For the above problem compute [B] and [c] matrix. It is^tapered bar whose cross

- section area decreases linearly from t-000 m*2 to 500 mm2.

ii) Use two elements and findthe nodal displacements. Take E:2x 10s N/mm2'(15 Marks)

a. Derive shape functions and stiffness matrix for beam element' (15 Marks)

b. Explain the need of Jacobian transformation matrix. (05 Marks)

a. Explain in detail ISO - parametric, sub - parametric and Super - parametric

elements. (10 Marks)

b. Explain "penalty approach" for handling the boundary conditions' (10 Marks)

a. Discuss the requirements to be fulfilled for the convergence of FEM solution'(10 Marks)

b. Derive FEM equation by variational principle' (10 Marks)

Write short notes on anY four :

a. Pascal's triangleb. Local - co - ordinate sYstem

c. Patch test.

ME6Fl

(03 Marks)(10 Marks)(07 Marks)

Time:3 hrs.l

4

a*,..'u_-,. 3j!. jtY,,Y.'d.,c

:1u'.j

'-"F,

d. Truss elemente. Shell elementf. EliminationaPProach'

:t:t:k* *

Poge No,,. I ME6FI

Reg. No.

2. @t

(b)

(c)

3. (o)

Find : i) AB ii1 gT 4T (5 Morks)

Solve by Gouss eliminotion

2q * 3x'2 * rJ: -75r1 * n2 * a3: Q (10 Morks)

321 *2x214x3:11

Whot is finite element method? Whot ore the odvontoges of FEM over finitedifference method? (4 Morks)

Exploin boundory volue ond initiol volue problems using suitoble exomples.' (8 Morks)

Exploin the steps involved in the finite element onolysis of solids ond structures.

. (8 Morks)

whot is meont by 'Bclnd width' of o motrix? Give on exomple. Exploin why itshould be minimized. (6 Morks)

(Mox.Morks: 100

(5 Morks)

(6 Morks)

(8 Morks)

(b) Stote the principle of minimum potentiol energy, ond derive on expression for totolpotentiol energy of o solid bor under compression.

(c) Exploin the Royleigh-Rit method with on exomple,

Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006

Mechonicol Engineering

(Old Scheme)

Finite Elemenl Methods

Time: 3 hrs.)

NOtg: Answer any F\VE lull questions.

l. (o) Find the inverse of

[lt]ror a:

[3 ]] ,:l; {l

4. (o) Exploin the Golerkin's opprooch for obtoining stiffness motrix of o bor element,(10 Morks)

(b) TwopointsPl(10,5)ondP2(80,10)onosolidbodydisplocesto PlOO.z,b.4)ondP;(80.5,10.2) ofter looding. Determine normol ond sheor stroins. (10 Morks)

Confd.,.. 2

Poge No,,, 2

5. A solid stepped bor os shown in fig.l is subjected to on oxiol force.

following

D Element ond ossembled stiffness motrix

iD Displocement of eoch node

iii) Reoction force ot fixed end

ME6FI

Determine the

(20 Morks)

2-

A,=t0O mm.

*r=1-Oo mm'

E = 200G Pa

t'r= ro Q Po

Lku

I

6. (o)

,, l]

(b)

7. @)

. (b)

8.

o)

'b)c)

d)

e)

Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.(10 Morks)

(10 Morks)

(10 Mofts)

(10 Morks)

(5x4 Morks)

Derive shope function CST triongulor element.

Derive shope functions for o l-D quodrotic element with 3 nodes.

Exploin convergence criterio ond potch test in brief,

Write short note on ony FOUR:

Voriotionol opprooch

Hermition shope functions

Penolty opprooch for hondling boundory conditions

Logronge ond serendipity fomily of elements

ISO porometric: elements

Page N0... 1 ME6F1

USN

Mechanical En gineering

Finite Element Methods

Time: 3 hrs.I

Note: 1. Answer any FIVE full questions.2. Missing data may be suitable assumed.

Sixth Semester B.E. Degree Examination, July/August 2005

1. (a) Define positive definite matrix. (2 Marks)

(b) Solve the system of simultaneous equations given below by Gaussian elimination method.

2c1 * 2n2 * ns :9n1*n2+fry:6

2a1 * a2: 4

(c) Determine the inverse and eigen values of the given matrix A

[Max.Marks : 100

(10 Marks) '

(8 Marks)

(a) Explain basic steps in FEM. (10 Marks)

(b) Explain potential energy of an elastic body. (5 Marks)

(c) Explain isoparametric, subparametric and superparametric element concept. (5 Marks)

(a) Derive the shape functions for a three node 1-D element in natural coordinate. (EMarks)

(b) Determine the displacemenl field, stress and support reactions in the body shown in

fis.Q3(b).

, P :60k/f E : 2O0kN lrrlnlZ : Et : Ez At :2000mm2 Az : l00Omrn2(12 Marks)

4. (a) Explain steps involved in Galerkin method. (10 Marks)

(b) Determine the defleetioh of canlilever beam of length I and loaded with a vertical load P(10 Marks)

Contd.... 2

. I 4 -2.286^: L -z.zJG 8

2.

F tS , a.z ir.

at the free end by Rayleigh-Ritz method.

Page N0...

5. (a)

(b)

2

For the one dimensional truss element, develop the element stiffness matrix in

coordinate system.

Determine the nodal displacement and stress by using truss element.

an example.

(b) Evaluate the following by Gaussian quadrature

ME6Fi

the global

(10 Marks)

(10 Marks)

(a) Derive the stiffness matrix for a two node beam element. (10 Marks)

(b) For the beam shown in fig,Q.No.6(b). Determine the maximum deflectlon and the reaction

at the support. El is constant throughout the beam. (10 Marks)

7. (a) What is the significance of the band width? lllustrate best method of node numbering with

8.

i) /: /]i (s"* + *, + #)da by one

ii) I : I: * OV 3-point formula.

Write short nole on the following :

(a) Coordinate systems

(b) Convergence criteria

(c) Variational method

(d) Plane stress and plane strain conditions

(e) Penalty approach for handling boundary conditions.

(5 Marks)

point and two point formula. (3 Marks)

(8 Marks)

*****(5x4=20 Marks)

Page N0,. 1 ME6F1

USN

Sixth Semester B.E, Degree Examination, January/February 2005

Mechanical Engineering

Finite Element MethodsTime: 3 hrs.l [Max.Marks : lO0

Note: Answer any FIVE full questions.

1. (a) Distinguish between :

Symmetric and skew symmetric matrix, transpose and inverse of a matrix. (4 Marks)

(b) What is a banded matrix? What are its merits? (4 Marks)

(c) Solve the following system of simultanegus equations :

11l2t2lrt:43*t-4xz-2r3-25r1l3r2*5r3- -7

either by Gaussian elimination method or malrix inversion method.

(d) Find the eigen values of the matrix A

lz B -21A- lr 4 -2lrLz 10 ,r j

(a) What is the basis of the Finite Element Method?finite element method.

2.

(6 Marks)

Explain the basic steps involved in the(10 Marks)

(b) Determine the true displacement field for a two noded one dimensional tapered elemenlshown in Fig.1. Also compute the stiffness matrix for this elemerit.

o c.n^--*J

At= loo n;r'o[, "

(6 Marks)

(10 Marks)

AtA

-t12

An

q2&) , Ftq' t'

:700rnz:900mm2

. '2: ('* #)

I"t-eJ

Contd.... 2

Pase N0... 2 MEOF1

3. (a) What are the principles of continuum method? Compare this method with finite elementmethod clearly bringing out their relative merits. (6 Marks)

(b) Stale the variational principle of minimum potential energy. (4 Marks)

A uniform cantilever is subjected to a uniformly distributed load of W kN/m over ilsspan 'l'. The displacement function is given as

y: 4(*, +612a,2 - lrs)where A is fhe displacement at the free end, Compute

the v"a'lue of the deflection A by the principle of minimum polential energy. Compare

this with the exact value. (r0 Marks)

4. (a)

(b)

(c)

Derive the strain displacement relations. (2 Marks)

b<plain the concepts of plane stress and plane strain with suitable examples, Also derive

the corresponding equations. (8 Marks)

A uniform rod of lengh I fixed al both ends is subjected to a constant axial load of wkN/m. Establish the displacement field and compute the stresses at the fixed ends and

rnidspan. What are lhe nalure and magnitude of the reaclions at the lwo ends? Use

Rayleigh-Bitz method. (10 Marks)

What are interpolation rnodels? Give reasons for choosing polynomial funclions for such

(5 Marks)npdels.

Explain briefly the penalty approach for handling displacement boundary conditions. '

(5 Marks)

Using the penalty approach, determine the nodal displacements and lhe stresses in

each material in the axially loaded bar shown in Fig.2

5. (a)

(b)

6. (a)

(b)

Area of (1):2400mm2Area of (2) :6A0mm2

EAL:o'7 xTosNfrnrnz

Esteel:2x705Nlmrnz

Explain the concept of isoparametric formulation.

Derive an elemenl stiffness matrix of a constant strain triangular

concept.

(10 Marks)

(5 Marks)

element using the above

(15 Marks)

Contd.... 3

A l,^v, i*1,^r"

3oo t'tt'T 4 OO x^1^4

Pase N0... 3 MEOF1

7, (a) what is a higher order element? what is its importance? (4 Marks)

(b) Derive the stiffness matrix for an element in the form

K: IW)r t"l tBl d,a

Show that the above matrix is symmetric. (10 Marks)

(c) A beam element carries a concentrated load P af { from one end. Obtain nodal loads

using the formulae of fixed beam. (6 Marks)

8. Write brief explanatory notes on any FOUR: (5x4=!Q [irs*s;

D Advantages and disadvantages of finite element methods

ii) Types of Finite Elements

iii) Boundarycondifions

iv) Principle of virtual work

v) Cohvergence criteria ** * **

\

Page No., 1

USN

ME6Fl

Mechanical Engineerlng

Finite Element Methods

Time: 3 hrs.I

Note: 1, Answer any FIVE futt questions.2. Assume suitable dak if necessiry.

1. (a) Explain with example.

i) Symmetric matarix

ii) Determinant of a matrix

iii) Positive definite matrix

iv) Half band width

v) Partitioning of matrices. (10 Marks)

(b) Give the aigorithm for fonruard elimination and back substitution of Gauss elimination fora general matrix,

2, (a) With suitable examples explain.

i) Essential (geometric) boundary condition

ii) Natural (force) boundary condition.

(b) Outline the steps in finite element analysis. (5 Marks)

(c) State the. principle. of minimum potential energy. Obtain the equilibrium equation of thesystem shown in fig 2.c using the principle of minimum potential energy. (10 Marks)

3. (a) Derive the equilibrium equation of 3D elastic body occupying a volume V and having a: surface S, subjected to body force and a concentiated lodd. (10 Marks)

(b) An elastic bar of length L, modulus of elasticity E, area of cross section A, which is fixedat one end and is subjected to axial load at the other end. Obtain the Euler equationgoverning the bar, and natural boundary conditions. (10 Marks)

4. (a) For a two noded one dimensional element, show that the strain and stress are constant

Sixth Semester B,E. Degree Examinatlon, July/August 2004

lMax.Marks : lOO

(10 Marks)

(5 Marks)

with in the element.

(b) Explain the criteria for monotonic convergence.

(10 Marks)

(10 Marks)

(12 Marks)

(8 Marks)

5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Detennine the lollowoing.ii Element stiffness matrices

iD B - matrices

iii) Dispiacemerrts and strains

iv) Stresses and reactions.

Obtain the stiffness matrix and load vector assuming two elements,

(b) What are characteristics of stiffness matrix ?

6. (a) For a pin jointed configuration shown in Fig 6.a determine the stilfness matrix. Also(10 Marks)determine g, interms of g,.

(b) Derive the Hermite shape functions of a beam. (r0 Marks)

Contd.... 2

:

Fage No... 2

7. (a) Evaluate

ME6FT

I

I-1

[r,,* ;r*ffif*

Using two point Gauss quadrature. (5 Marks)

(b) Derive the expression for shape functions of eight noded isoparametric element. (15Marks)

8, (a) Determine the Jacobian for the triangular element shown in lig Q8.a. (5 Marks)

(b) Give the element number and mode numbers for the structure shown in Fig Q 8.b, so asto minimize the half band width of the resulting stilfness matrix. (5 Marks)

(c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)

fi?. qL. c

vf, ts*' +'ol

) I o'. -/- g-- 7oxto3 ^'/t",'olI fZ A= l3oo ss +ozn'

I J CLo,P) L= S m*

fi3. Q6.o

\c+1). / \

c z 3.5)

Ct.gr.l63' QB'o-

+R=l\

ng. qe. b

F

''l s0\ooo

A.: 5oo mm ,gnz Qoo msri , too 6Pa

L'; zoo a'oo-

FS' E(")

7* l'rD ---+L t -o ----l/,r-'----=---'---i{--a----v|[---6--G-re-Z-I -7.,

,=2-ooePd; - r -,r^O

Fs, Ee . c ?=- +^iie *'"-4

\

*****

Page No... 1

Heg. No.

ME6Fl

Sixth Semester B.E. Degree Examination, January/February 2009

(5+5 Marks)

l

l

l

2. (a) Explain the theorem,of minimum potential energy. Distinguish between minimum potentialtheorem and principle of virtual displacement" - (10 Marks)

(b) Explain the basic steps in the formulation of finite element analysis. (10 Marks)

3. (a) ,Flnd_thg.s!re!9 al w.:0 and displacement at the mid - point of the rod shown below.Use Raleigh Ritz method

A)

I

(b)

4. (a)

Mechanical Engineering

Finite Elembnt Methods

Tirne: 3 hrs.I lMax.Marks : IOO

Note: Answer any FIVE questions.

1. (a) Solve the following system of simultaneous equations by Gaussian Elimination Method.

t1 -2n2 f 613 - 0

2a1*2c2*3n3-3-rr*3r2-2

(b) Find the inverse of the following matrices

l0 1 21 f1 2 -21

', Ll?il ilL;:, ll

(10 Marks)

Tq-Ke E = L1q6J- (Yo*,,.3'S ^".U.*[-y

A' luniF (A tea fl(r.,1-r a-h.orr

Explain plane stress and plane strain methods with rerevant equations.

Explain the penalty approach for handling the specified displacement boundary conditions.(10 Marks)

(10 Marks)

(10 Marks)

Contd.... 2

Page N0... 2

(a)

to)

(a)

(b)

(b) For the {ollowing figure (bar), find the nodal displacements. The cross sectional area

decreases linearly from 1000rnm2 lo 500mm2. Use two elements.

Take E :2x1O5MPa,7:0.3,t 5ooss

lbbo -, looo A1

k- J$o''twr -

4(10 Marks)

Explain convergence criteria in detail, (10 Marks)

Derive shape functions for 'CST' element from generalized co-ordinates. (10 Marks)

Derive the stiffness matrix for a two noded beam element (12 Marks)

Distinguish between isoparametric, sub-parametric and super-parametric elements.la uarxsl"

Consider the 4 -bur truss shown below, Determine.

i) Element stiffness matrix for each element

ii) Using eliminations approach to solve for the nodal displacements.

(iiD Calculate stresses in each element.

rQrC)I4-4O * *rI

Write shorl notes on any FOUR of the following.

a) Eliminationapproach

b) Patch test

c) Galerkin's approach

d) Geometric isotropy

e) Post Processing

f) LST triangular element ** * **

ME6F1

(20 Marks)

2-gooor..t (n.+i5 Ja-svr*,; ll

Ar

20,0001.; >1

i

l

[^

I

I

7.

+vQg

@

t3otv\ t"t

I

(5x&20 Marks)

a

Poge No.,. I ME6FI

Reg. No.

2*t+3a2*nJ:-1541*e2*rs:0

3rr + 2a2l4a3 -']".1

2. @,

(b)

(c)

3. tol

potentiol energy of o solid bor under compression.

(c) Exploin the Royleigh-Ritz method with on exompte.

Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006

Mechonicol Engineeilng

(Old Scheme)

Finiie Elemenl Methods'1.

Time: 3 hrs.) ':.

NOle: Answer ony FIVE tuil queslions.

I. (o) Find the inverse of

[r ollo rl

,o, a: [3 1] ,:l; {l

Find : i) AB ii1 BT ar(c) Solve by Gouss eliminotion

4. (o) Exploin the Golerkin's opprooch for obtoining stiffness motrix of o bor element,(10 Morks)

(b) TwopointsPl(10,8)ondP2(80,10)onosotidbodydisptocesto pl(Lo.z,b.4)ondPj(80.5,10.2) ofter looding. Determine normol ond sheor stroins, (t0 Mql1s)

Confd.... 2

Whot is finite element method? Whot ore the odvontoges of FEM over finitedifference method? (4 Morks)

Exploin boundory volue ond initiol volue problems using suitoble exomples.' (8 Morks)

Exploin the steps involved in the finite element onolysis of solids ond structures.

: . (S Morks)

whot is meont by 'Bcind width' of o motrix? Give on exomple, Exploin why itshould be minimized, (6 Morks)

(b) Stote the principle of minimum potentiol energy, ond derive on expression for totol

(Mox.Morks: 100

(5 Morks)

(5 Morks)

(10 Morks)

(6 Morks)

(8 Morks)

Poge No,,, 2

5. A solid stepped bor os shown in fig.l is subjected to on oxiol force.following

i) Element ond ossembled stiffness motrix

iD Displocement of eoch''node I

iii) Reoction force of fixed end

ME6FI

Determine the

(20 Morks)

2-

A,= tOo hm ,

*r=LOo mhn-

g = 2,00G Pa

rt"= lo q Pq

h-k u

6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.

7. @| Derive shope functions for o l-D quodrotic element with 3 nodes. (t0 Morks)

(b) Exploin convergence criterio ond potch test in brief. (10 Morks)

(b) Derive shope function CST triongulor element,

8. Write short note on ony FOUR:

o) Voriotionol opprooch

6) 'Hermition shope functions

c) Penolty opprooch for hondling boundory conditions

d) Logronge ond serendipity fomily of elements

e) ISO porometric elements

(10 Morks)

(10 Morks)

(5x4 Mqrks)

Page No.., 1

USN

ME6Fl

1.

[Max.Marks : 10O

(10 Marks)

(5 Marks)

3.

4.

(b) Explain the criteria for monotonic convergence. (,l0 Marks)

5. (a) A component shown in fig. 5(a) is subjected to a load 5 kN. Determine the foilowoing.i) Element stiffness matrices

ii) B - matrices

iii) Displaeements and strains

iv) Stresses and reactions.

Obtain the stiffness matrix and load vector assurning two eiements.

(b) What are characteristics of stiffness matrix ?

$ixth sernester B"E. Degree Examination, July/August 2004

Mechanical Engineering

Finite Element Methods

3 hrs.l

Note: 1. Answer any F|VE full questions.2. Assume suitable data if necessary.

(a) Explain with example,

i) Syrnmetric matarix

ii) Determinant of a matrix

iii) Pcsitive definite matrix

iv) Half band width

v) Partitioning of matrices.

(b) Give the algorithm for forurard elimination and back substitution of Gauss elimination fora general matrix. (io Marks)

2. (a) With suitable examples explain.

i) Essential (geometric) boundary mnditionii) Ndtural (force) boundary condition.

(b) outline the steps in finite element analysis. (5 Marks)

(c) State the principle of minimum potential energy. Obtain the equilibrium equation ol thesystem shown in fig 2.c using the principle of-minimum potentidl energy. (10 Marks)

(a) Derive the equilibriqm equation ol 3D.elastic body oc.cypyt"ng a volume V and having asurface s, subjected to body force and a concentrated lddd. (r0 Marks)

(b) ry elastic bar of length.L, modulus of elasticity E, area of cross section A, which is fixedat one end and is subjected to axial load at-the other end. Obtain the'Euler equationgoverning the bar, and natural boundary conditions. t10 Marks)

(a) Fo1 a two noded one dimensional element, show that the strain and stress are constantwith in the element" (ro Marks)

('t2 Marks)

(8 Marks)

(a) For a.pin jointed configuration shown in Fig 6.a detennine the stiffness matrix. Alsodetermine qt interms of g,. (10 Marks)

(b) Derive the Hermite shape functions of a beam. (10 Marks)

Contd.... 2

Page Nor, 2

7. (a) Evaluate

illE6F1

(5 Marks)

{b) Derive the expression for shape lunctions of eight noded isoparametric element. (15 Marks)

1

-1

Using two point Gauss quadrature.

8. (a)

(b)

Determine the Jacobian for the triangular element shown in fig eg.a, (5 Marks)

Give thp element number and mode numbers for the structure shown in Fig Q 8.b, so asto minimize the half band width of the resulting stiffness matrix. (5 Marks)

(c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)

fi?. qL. e-

vf, ct'o'+c>

) t o', -/" Et 7oxto3^l/t''ol

I {/ A= l3oo ss m"n'I J. Clo,rs) V-- S n*

i

I

r't

t

fr3. Q6.a

)',\+;'), / \C z s's)

L1.51)

63' Qe'o-

+Rsft. q8. b

A.; 5oo mm ,gn: QOO mwc : \0o GPa'

L'; r-oo aOo.

F3 5(a1

h- I'ro nD I .tlo -,,@

/l ^^,-2oo\ld..qc.c i=

"^lo6+nYo*\t*****

le

\ooo

,

a

Page No... 1

USN

ME6F1

Time: 3 hrs.l

Note:

Sixth Semester B.E. Degree Examination, January/February 2004

Mechanical Engineering

Finite Element Methods

1. Answer any FIVE full questions.2. Missing data may be suitably assumed,

1. (a) Find the eigen values of

A- 4 -{51-,/3 a l(5 Marks)

(b) Solve the following system of simultaneous equations by Gaussian elimination method.

2e1*12!3rs:t$4r1*r21.a3:$3n1*2r2 * rs:3

(c) Define the following with example

i) Skew matrix

ii) Symmetric banded matrix.

(a) Explain difference between continuum method and finite element method, (5 Marks)

(b) Explain basic steps involved in FEM. (10 Marks)

(c) Explain principle of minimum potential energy and virlual work. (5 Marks)

(a) Expain steps involved in Rayleigh - Ritz method. (B Marks)

(b) Determine the deflection at the free end of a cantilever beam of length '1, carrying avertical load 'P' at its free end by Rayleigh Ritz method (i0 Marks)

List the demerits of cantinuum methods. (2 Marks)

Derive strain displacement matrix, stiffness matrix for one dimentional bar element.(8 Marks)

Solve for stresses and strains for the following problem by using bar element.(12 Marks)

? = loco l.J

/t<_

(c)

4' (a)

(b)

[Max.Marks : IO0

(10 Marks)

(5 Marks)

E:2.7xlA5NfrrurnzAt :5Omm2Az :25mm2P : 100011

Contd.... 2

Page N0...

5. (a)

(b)

2

Derive stiffness matrix for a truss element.

Ar : LAA\mmzAz:125Amm,2E:200GPa

ME6F1

(8 Marks)

(12 Marks)

(16 Marks)

(4 Marks)

using one triangular

(20 Marks)

For a pin jointed configuration shown in figure, determine nodal displacements and stressby using truss elemenls.

f : looo;?

T5oo r

t6. (a) Compute.the deflection of simply supported beam carrying concentrated load at its centre,

Use two beam elments.

:lSovnr'

(b) ls FEM analysis applicable for highly elastic materials? Explain.

Find the displacement of node 1 in the triangurar element shownelement. Also find stress and strain in the elefient.

7.

loo l,/

I.(,2,o )5o

I

E:70GPa L7:0.3Le : lAmm

3o,c

Write short notes on any FOUR of the following :

a) Static condensation

b) lsoparametric, super parametric and subparametrlc element

c) Static and kinematic boundary condition

d) Lagrangian and Hermite shape functions

e) Convergencecriterion*****

. 1+----- 3o n(-3o,o )\l r.-__

I2o

(4x5=2Q fYl2Y[s)

a -----

-'-t/' '

Page N0,,. I

USN

ME6F1

[Max.Marks : 10O

(10 Marks)

(5 Marks)

(5 Marks)

(10 Marks)

(4 Marks)

(6 Marks)

(10 Marks)

Use penality(10 Marks)

.,€ r 2lo$ pa

?JaoN

(10 Marks)

Sixth Semester B.E. Degree Examination, July/August 2000

Mechanical Engineering

Finite Element MethodsTime: 3 hrs.I

Note: Answer any FIVE futt questions.

1. (a) Given o:l; i], ort.,*in.

i) Inverse of matrix ii) Eigen values.

(b) lf ,7"r: [€, 1-(2], evaluate /, wT Nag

(c) Explain symmetric banded matrix.

2. (a) With an example explain Rayleigh -Ritz method.

(b) State the principle of minimum potential energy.

(c) Sketch the quadratic and Hermite shape functions.

3. (a) Derive the following characteristics of three noded l-D element.i) Strain displacement matrix [B] ii) Stiffness matrix [frr]

4. (a) Derive an expression for

i) Jacobian matrix

ii) Stiffness matrix for axisymmetric element.

(b) Solve for nodal displacements and stresses for the structure shown in fig 1.

approach to apply boundary csnditions.

h t"laao n{' 2"17o frrn**1,€=zo$fo"

Contd.... 2

_ _, ___:_

Page N0... 2

(b) 0onsider a rectangular element as shown in Fig.2. Evaluate

(=0, \=0,

ME6F1

J and B matrices at(10 Markr)

+

t A,>-t a)L

(0, ,)

5. (a) Explain with neat sketches the library of elements used in FEM. (10 Marks)

(b) Using Gaussian quadrature, evaluate the following integral by two point formula

d, /], (€2 + zrt€ + rf) dt drt (10 Marks)

6, (a) For the pin jointed_ configuration shown in Fig.3 determine the stiflness values of' ' kn, l*e and,-k2, of global stiffness matrix. (10 Marks)

O hra'tgroivl"nL'L

"l/b MvY'

E- >}lac\?", ,

(b) Derive an expression lor stiffness matrix ol a two noded beam element. (10 Marks)

7. (a) Explain in detail the leatures of any one commercial FEA software package. (l0Marks)

(b) Bring out the differences between continuum methods and FEM. (10 Marks)

Write short notes on any FOUR :

a) State functions

b) Galerkin methods

c) Elimination method of handling boundary conditions.

d) Temperature effects

e) Convergence criteria. ** * **

/LI

I

vjup l\n7

+C1i,o,{)

cv>-

(4x5=20 Marks)

I

Page No... l ME6Fl

Reg. No.

sixth serrester B.E. Degree Examflnatlon, Februar5r zoozMechanical EnglneeringFtntte Element Methods

Time: 3 hrs.l [Max.Marks : I0O

Note: Answer any FIVE full questions,

1. (a) What is a banded matrix and state its advantage?(b) Calculate the eigen values of the matrix A.

o:lt ?,1

(c) Evaluate .4.-1 when -d. :lz 0 1llo 4 olfr o 2l

(d) Drptain Gauss-elimination method to solve a set of simultaneous equations.(4X6=20 Marks)

(b) Differentiate between continuum method and finite element mettrod. (8 Marks)

3. (a) A rectangular bar in subjected to an axial load P as shown in fig.l. Derive anexpression for potential energr and hence determine the extreme value of thepotential 9le-1ry forthe-following data. Modutus of elasticity E :200Gpa,load P - SkNr length of the bar I : L00mm, width of the ba; b :20mm arrdthickness of the bar t : Llmm. Also state its equilibrium stability. . ,

2. (a) What is finite element method?finite element analysis.

Drplain the basic steps in the formulation of(12 Marks)

iff

l_T

{

-+'L

Fta, I

(b) use Rayleigh-Ritz method to find the disptacement and the stress .,tilIillpoint of the rod as shown in fig.2. The area of cross section of the bar is 4OOmmz and. the modulus of elasticity of the material is 7O GPa. Assume thedisplacement to be second degree polynomial. (to Marks)

-Explain the elimination approach for handling the specified displacementboundary conditions (5 Marks)

4. (a)

Contd.... 2

ME6F.1Page No... 2

5. (a)

(b)

(b) Determine the nodal displacements, element stresses, and suPport reactions of'-' thtrnuliy loaded bar ai shown in fig3"^Usi elimination method for handling

the bound.ry;;;;itio.o. rrr." E :"200Gpa aad load P : 300&N. .

Fir"3

state tJre assumptions made in the analysis of trusses.

(15 Marks)

(5 Marks)

For the three-bar t1ass shown in fig.4, determine the nodal {i9pl1rments andror the three-bar tmss shown in fig.4, determine the nodal {i9pl15ntfre stre"s in each member. Take Inodulus of elasticity as 2OO GPa'

lTo ?< rl

1tOoo mm

(15 Marks)

(2O Marks)

Ff a.4I"6. A beam of length 1O m, fixed at one end and suppott""{,!V l l:]b:-:t l*-"^t*I

end carries a 2o kN;;;;A&"lia toao at the ceirier of thti spgn' B;r taking,the

modulus of elasticity "T

*rtoirt as 2OO GPa and moment of inertia of section

7.

as 24 x 10-6m4, determine1. Deflection under the load, and2, Shear force and bending moment on each element'

(a) Derive strain- displacement relation of a cST element. (1O Marks)

(b) For a linear quadrilateral element, derive an expression for Jacobian matrix'

Write short notes on arly four of the following'

i) VariationalPrinciPles.ii) Co-ordinatesYstem.iii) Convergenceiv) Penalty approach for handling the boundary conditions.

v) Quadratic shaPe t r",tot* * **

lro m\ lrlm,/,

8.

(4X 5=2O Marks)

.;Pag* fi,o... ,

Reg, No.

Mechanical Enginering

Finite Element llethods

Time: 3 hrs.I

Note: Answer any F|VE full guestfons,

. ( c'!-t*q,f I r

,.1 SF --^{// fi,lE6F1

Sixth $emester B.E" Degree Examlnation, Juffiugu$t 2002

1" ia) Solve the following system of siriultaneous equaticns by Gaussian Elimination Method.

2*t+*2*a3=l&i -2rz * 343 - ll2*t + 4nz * 3r3 : 19 (10 Marks)

(b) Write a briel note on frontal solution technique for handling large systems of algebraic(6 ffarks)equations,

{c) Define the {ollowing with examples,

i) Symmetric matrix

ii) Banded Euare matrix.

IMax.Marks: lOO

(4 Ma*s)

2" (a) Discuss the advantages and limitations of FEI';I over other numerical method'FDM.i5 l,lark*)

(b) List the various applications of finite element method. (5 Marksi

(c) Explain the steps invoived in finiie element method lvith suitaCIie exampies. (10 Marksi

3" (a) Derive strain-displacement relationship for a two dimensional soiid mechanics problem.

{10 Marks}

(b) Explain plane siress and plane strain problems as applied to solid mechanics problem

with suitable examples. , , i10 Marks)

4. iai ExBlain the theorcm of mjnimum potential enerEy ancj cietive an expression {oi totai

potentiat energy ior a sne-oimensionai bar sub,;ecteci to an axtat iorce. (i 0 f'{arhsi

(b) Using direct stif{ness method, determine the nodal displacements of the bar, as shown in

Fig.1. i10 Ma*s)

$" (a) Define shape function. What are the properties that a shape funciion shoulcl satisfy?

{10 Marks}

ib) Expiain wiih suiiable examples ihe iagrange and serenciipity famiiy o{ eiements. (10 Marks}

6. (a) Distinguish between consisient and lumped load vectors through examples. (5 Marks)

(b) A stepped bar is shown in Fig.2 Calculate Jacobian J for eaeh element. Cbiain the

etemrjrit stiffness matrices and solve {or the nodal displacements by using elirninationapproach for handling the boundary conditions. {15 Mtrks}

[Use one Gauss point for Numerical integration]

Contd.... 2