1. USN 06ME63
Sixth Semester B.E. f)egree Examination, December 2Ol2
Modeling and Finite Element Analysis
Tirne: 3 hrs. Max. Marks:100
Note: Answer FIVE full questions, selecting
at leust TWO questions from eoch part.
!
.! PART _ A
a
u
e la. Using Rayleigh-Ritz method, derive an expression for maximum det.lection of the simply
= b. supported beam with point load P at centre. Use trigonometric function. (08 Marks)
a Solve the following system of simultaneous equations by Gauss elimination method.
O X-l Y 'l Z:9
aX, x-2y+32:8
=D- 2x+ Y - z:3 (08 Marks)
c. Explain the principle of minimum potential energy and principle of virtual work. (04 Marks)
3
otll
=co
.= a.l
2a. Explain the basic steps involved is FEM. (10 Marks)
b. Explain the concepts of iso, sub and super parametric elements. (05 Marhs)
Etf c. Define a shape function. What are the properlies that the shape functions should satisly?
-O (05 Marks)
=ts
a2
3a. What are the convergence requirements? Discuss three conditions of
convergence i
6= requirements. (05 Marks)
1
OO b. What are the considerations for choosing the order of the polynomial functions? (05 Marks) I
Derive the shape functions for CST element.
rl
-1 c. (10 Marks) {
boi
:'
2G 4a. Derive the Hermite shape function tbr a 2-noded beam element. (10 Marks)
b. Derive the shape functions fbr a four noded quadrilateral element in natural coordinates.
3u (10 Marks)
AE
6X I
:
o --: PART _ B
,i .9.
6E
oLE 5a. Derive an expression for stifthess matrix for a2-D truss element. (10 Marks)
a,-
b Derive the strain displacement matrix tbr 1-D linear element and show that o: E[B]{u}
>(k (10 Marks)
cno bIr
=
0=
so
F>
6a. Discuss the various steps involved in the finite element analysis of a one dirnensional heat
o transfbr problem with refbrence to a straight unifbrm fin. (10 Marks)
(-) <
b. Derive the element matrices, using Galerkin for heat conduction in one dimensional element
with heat generation Q. (l0 Marks)
-N
o
Z 7 a. A bar is having uniform cross sectional area of 300 mm2 and is subjected to a load
6 P : 600kN as shown in Fig.Q7(a). Determine the displacement field, stress and support
o reaction in the bar. Consider two element and rise elimination method to handle boundary
conditions. Take E :200 GPa. (10 Marks)
I of 2

2. a
I
,
,
,
/.
t
Fig.Q7(a)
b. For the two bar truss shown in Fig.Q7(b), determine the nodal displacements and stress in
each number. Also find the support reaction. Take E :200 GPa. (10 Marks)
, SotlN
Fig.Q7(b)
8a. For the beam shown in Fig.Q8(a), determine the end reaction and deflection at mid span.
Take E :200 GPa,I:4x106 mma. (10 Marks)
TYc
24hNlm
h,,|tl,lfi
Fig.Q8(a) Fie.Q8(b)
Determine the temperature distribution through the composite wall subjected to convection
heat loss on the right side surface with convection heat transfer coefficient shown in
Fig.Q8(b). The ambient temperature is *5oC. (10 Marks)
**+*8
2 of2

3. 06ME63
sixth semester B.E. Degree Examinatlon, December 2011
Modelling and Fisrite Elememt Analysis
Time:3 hrs" Max. Marks:100
Note: Answer uny FIYE full questions, selecting
at least TWO questions from each part'
PART-A
equatio, for ffrtate of stress and state the terms involved"
(04 Marks)
oi
I a. write the equilibrium
o
o b. solve the following system of equations by Gaussian elimination rnethod :
(d
a x1*x2*x:=6
(6
Xr-Xz*2x3=5
Marksi (08
rd x1* 2x2-x3=2.
{)
c. Determine the displacements of holes of the spring system shown in the figure using
iE
e) (08 Marks)
principle of minimum potential en?rg{;
_o? o t{" trln"*
Srtcll.t"r
Sorf
(!u
!.,
Fig.Q.1(c). 6-s rr lx't
ll
ao
traP
.=N
d+
i. 60 number and
otr 2a. Explain the discreti zationprocess of a given domain based on element shapes
-o (06 Marks)
slze.
a structural
o= b. Explain basic steps involved in FEM with the heip of an example involving
Es
member subjected to axial loads.
(08 Marks)
od
vd
0. Why FEA is widely accepted in engineering? List various appiications of FEA in
(06 Marks)
6o engineering
o'o
boc
3a. Derive interpolation model for 2-D simplex element in global co - ordinate system'
"o! (10 Marks)
26
!s=
'd(g b. What is an interpolation function? Write the interpolation functions for:
.a
EO
o€
(,
i) 1 -Dlinearelement ; ii) 1 -Dquadraticelement'
2O
iiU 2-D linearelement ; iv) 2-Dquadraticelement'
tro. v) 3-Dlinearelement. (06 Marks)
(04 Marks)
oj c. Explain "complete" and "conforming" elements'
AE
Derive shape function for 1 - D quadratic bar element in neutral co-ordinate tVttelm
5L)
olE 4a. Marks)
LO
o.E
>.9 b. Derive shape functions for CST element in NCS. (08 Marks)
on-
troo c. What ur. rhup. functions and write their properties. (any two). (04 Marks)
qo
:a)
EE PART -B
-h
U< 5a. Derive the body force load vector for I - D linear bar etrement.
(04 Marks)
(06 Marks)
--.; c'i b. Derive the Jacobian matrix for CST element starting from shape function'
(10 Marks)
o c. Derive stiffness matrix for a beam element starting from shape function'
o
z
d 6a, Explain the various boundary conditions in steady state heat transfer problems with simple
o
o, sketches. (06 Marks)
b. Derive stiffness matrix for 1 - D heat conduction problem using either functional approach
or Galerkin's approach
l
(08 Marks) .j
l'
I of Z ij
ii

4. 06M863
r
I c. For the composite wall shown in the figure, derive the global stifftress matrix. (06Marts)
Take
Ar: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.
i) Steel Ar = 1000 mm2
Er :200 GPa
ii) Bronze Az:2000 mm2
Ez: 83 GPa' (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
Fie.Q.7(b)
8 a. Determine the temperature distribution in 1 - D rectangular cross - section fin as shown in
figure. Assume that convection heat loss occurs from the end of the fin. Take
'3w
K=-.
CmoC'
- = 0.1w , T*:20oC. Consider two elements
h "" (10 Marks)
=
Cm'oC
fo v 5 E*fr.,r
Fr z reg,il
.f.y-tol nnll
Fie.Q.8(a)
b. For the cantilever beam subjected to UDL as shown in Fig.Q.8(b), determine the deflections
of the free end. Consider one element. (10 Marks)
fo;5s1.
Fs z rD qlt
.t-tot$fltt
Fie.Q.8(b)
,*****
2 of?

5. r-
I
I
06ME63
USN
SixthSemesterB.E.DegreeExamination,December2010
Modeling and Finite Element Analysis
Max. Marks:100
Time:3 hrs. selecting
Note: Answer ony FIVE futl questions'
at least TWO questions from each part'
PART _ A
,9
o
ffi'"-" tt for two dimensions' (06 Marks)
H
I a. Explain, with a sketch, plain stress
p*"'ii"f energy' Explain the potential energy' with usual
"in
a
(g b. State the principles of minimu* (06 Marks)
notations.
c. t^hTT: the steps invotved in Ravleigh-Ritz methog?
DeTnTl:,'X ul?l]":::'*'
d
shown in Fig.l(c). use second degree
::#::
()
d
0) ffii il'r'ffi:i:.1il;J*_d#;,I;J;;g-^
approximation, for the displacement'
(08 Marks)
39 iolynomial
d9
-o ,,
ao"
Fm
.=+
'E-f
b?p
Fig.l (c).
Pfr
method with finite element methocl'
o> (04 Marks)
!1 a 2a. Bring out the four differences in continuum with example'
b. What do you underrtuod FEM?
eri"ny e*piain the steps involved in FEM'
acd
the generai node numbering *d t"l-ff:;Tl
5(J
do
Write properries of stiffness matrix K. Show - (06 Marks)
the half bandwidth-
6d
{tz Marks)
}E 3a. What is an interpolation function? ,..r - of convergence ,
-r ^^--.^- requirements'
tr5 b. what are convergence requirements? Discuss three
conditions
(08 Marks)
Write a shot notes on :
!O
oe c. -
o- gt
Eo. i1 C.o*etrical isotropy for 2D Passal triangle (CST) elernent' with a sketch' (lG Marks)
ii) Shapg function for constant strain triangrilar
si ^9
bar eiernent, in natural co-ordinates"
'@q 4a. Derive the shape functions for the one-dimensional (08 Marlcs)
quadrilateral eler'rent, in natural co-'rdinates'
Derive the shape functions for a four-node
L0
b. (08 Marks)
>.k
mo (04 Marksi
c. Write four properties of shape functions'
g0
=(6
tr> PART - B
59
o-
U<
5a. Derive the following :
1) Element stiffness matrix (K")'
il Element load vector (f)
c.i
-i
() (12 Manlis)
o Uy aire"t method for one-dimensional
bar etrement'
Z (l-1) for constant strain triangle (csr)'
b. K:ff:"Iffi::f the Jocabian transformation matrix (08 Marks)
(08
d
o
a (06 Marks)
6a.Explainwithasketch,one-dimensionalheatconduction. for heat conduction in one
b.Derivetheelementmatrices,usingGalerkinapproach, (10 Marks)
dimensional element' (04 Marks)
dimension'
c. Explain heat flux boundary condition in one

6. I
06M863
7 a. Solve for nodal displacements and elemental stresses for the following. Fig.Q,7(a), shows a
thin plate of uniform lmm thickness, Young's modulus E : 200 Gfa, weighr aensity of the
plate : 76.6 x 10-6 N/mm2. In addition to its weight, it is subjected to a point load of 1 kN at
its mid point and model the plate with 2 bar elements. (10 Marks)
r
Fig.Q.7(a). I
I
+
t
b. For the pin-jointed configuration shown in Fig.e.7(b), formulate the stiffness matrix. Also
determine the nodal displacements. (10 Marks)
IKN
fiIom'rJ.
{.
Es = E2=26r6$Pr.
Fig-Q.7(b).
8a. Solve for vertical deflection and slopes, at points 2 and,3, using beam elements, for the
structure shown in Fig.Q.8(a). Also determine the deflection at the centre of the
the beam carrying UDL.
ir"rtffi
E:z.o06P(
Fig.Q.8(a).
J = 4x lob**ti
b. Determine the temperature distribution through the composite wall, subjected to convection
heat transfer on the right side surface, with convective heat transfer co-efficient shown in
Fig.Q.8(b). The ambient temperature is -5oC. (10 Marks)
a-
t*
[2 looo $ly*tt
Fig.Q.8(b).
Kz=***
****,r
2 of2

7. 06M863
USN
May/June 20L0
Sixth semester B.E. Degree Examination,
Modeling and Finite Element Analysis
Max' Marks:100
Time: 3 hrs.
d questions, seleeting atleast TIYO from each part'
o
o
Note: Answer any FIVE futl
a PART - A
of a simply supported beam with
(g
i
! a. Using Rayleigh Ritz method, find tt. ,il*l*--a"flection (10 Marks)
'o
() point load at center' . . -- ,--- ^^-.^+r^-- L., I],,,oci;rn eli method'
by Gaussian elimination
b. Solve the following system of simultaneous equations
(B
o
B9
qp- 4xr f 2W+ 3x3:4
2xr * 3x2* 5x3:2 (10 Marks)
'=h
Zxr * 7xz: 4
aoll
t-6
.= e'l 2a.Explainthedescretizationprocess'sketchthedifferenttypesofelementslD'2D'3D
(06 M'::Y)
cdS
c^ bI)
!i {) elements used in the finite element
analysis' -..,-- L-- A:-^
!'a
otr
rA b. considerinil;gton"rrt,.ou.oio it. "tl*.rrt
stiffness matrix by direct stiffiress
fir;111*;
eE
Comment on its characteristics' tudti9,}*rr.ut
o7 the properties that the shape flrnction should
8z a. De{ine
"
J# #ffi;. irh;;*"
oid
?d
6o criteria with suitable examples and compatibil*
*o*T#H:i;
do 3 a. Explain the convergence
boc
.dd
b.
FEM.
Explain simplex, complex and multiplex
elements using element shapes' (06 Marks)
rk coordinates for one dimensionai
}E
!o= c. Explain linear interpolatiorr, potyrro*ials
in terms
"igilu"r - (06 Marks)
!rg
-2" ts
simPlex element'
irO
oe
E3 ."t,,rte'Pr vr revr*r*.
4a,Explaintheconceptofisoparametric,subparametricandsuperparametricelementsanrj
(06Marks)
o9'
tro theirurrr'to a ,1 r:--ri^^^*a
o-i
b.DerivetheshapefunctionsforaCsTelementandalsothedisplacementmatrix.(08Marks)
for abeam element'
(06 Marks)
9E
A,E
c. Derive the Hermite shape nn.ti*
=9
LO PART. B
shown in fig' Qs(a)'
s a. Find the shape tunctions forpgintp ruiffi"lement
o.<
>.(I at
g";o ^t11r11r*;
(10 Marks)
6E area and Jacobian the eiement'
matrix
AE 61 8)
tr>
=6J
Ek
'P*
_h C6rsJ
o< I
..I e.i Fig.Qs(a) I
(trt{
C' C$,
o
z (10 l![arks)
(l
b.Derivethestiffrressmatrixfota2_dimensionaltrusselement.
o
+ analysis of a one dimensional heat
6 a. Discuss the various steps involvedain the finite element
unjform fin' (10 Marks)
transfer problem with reference to straight for linear interpolation of
b. Explain the finite element *oa"rirrg La rrrrp" functions (10 Marks)
tieat trunsfeielement)'
temperature field (one - dimensional
1of?

8. 7 a. Determine the nodal displacement and stresses in the erement shown in
fig. e7(a).1r0 Marks)
Ar = 500 mm2
Fie.Q7(a)
lokN Az = 2000mm2
:
Er 100 GPa
E2:200 GPa
300mm 300mm
3::::_1"":::rl-::tg::r^ *"gx oferementr (1);;
b.
truss etements shown in fig.
of 200mmz and .the e7(b). Au the
elements have an area
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 outer temperature
To = 200c' convective heat transfe, tuk., place
on the inner surface of the wall with Too =
8000c and h :25 wrmz o.[""ir.
it . ,.rp.rature distribution on the wall.
'C. (20 Marks)
f*: I, otc
^*Jli kr:20 WmoC
kz:30 WmoC
k3 = 50 WimoC
Fig.Q8 ' h-25WlmzoC
T*:8000C
*****
2 of2

9. 06M863
USN
sixth semester B.E. Degree Examination, June-July 2009
Modeling and Finite Element Analysis
Time: 3 hrs. Max. Marks:100
Note: Answer any FIVE full questions, selecting
at least TWO questions from each part.
PART _ A
L a. Explain the principle of minimum potential energy and principle of virtual work. (06 Marks)
+l
b. Evaluate the integral 1= J{fE' +2z ++2F by using 2 point and 3 point Gauss
-t
(06 Marks)
quadrature.
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)
2a. Explain briefly about node location system' (06 Marks)
Explain preprocessing and preprocessing in FEM. (06 Marks)
b.
Explain the basic steps involved in FEM. (08 Marks)
c.
3a. What are the considerations for choosing the order of polynomial functions? (06 Marks)
b. Explain convergence requirements of a polynomial displacement model. (06 Marks)
c. Derive the linear interpolation polynomial in terms of natural co-ordinate for 2-D triangular
elements. (08 Marks)
4a. What are Hermite shape functions of beam element? (06 Marks)
b. Derive the shape function for a quadratic bar element using Lagrangian method. (06 Marks)
c. Derive the shape function for a nine noded quadrilateral element. (08 Marks)
PART _ B
5a. Derive the element stiffness matrix for truss element. (10 Marks)
b. Derive the Jacobian matrix for 2D triangular element. (I0 Marks)
6a. Explain the types of boundary conditions in heat transfer problems. (r0 Marks)
b. Discuss the Galerkin approach for l-D heat conduction problem. (10 Marks)
la. Using the direct stiffness method, determine the nodal displacements of stepped bar shown
in figure Q7 (a). (lo Marks)
Er :200 GPa
Ez:70 GPa
Ar : 150 mm2
Az: i00 mm2
Fr:l0kW
Fz:5 kW
Fie. Q7 (a)
I ofZ

10. 06M863
7 b. For the truss shown in figure Q7 (b), find the assembled stiffness matrix. (10 Marks)
lkN
T
5oo ' E1 : E2:200 GPa
I
L
Fig. Q7 (b)
8 a. Determine the temperature distribution through the composite wall subjected to convection
heat loss on the right side surface with convective heat transfer coefficient shown if figure
Q8 (a). The ambient temperature is -5"c. (r0 Marks)
t:rdc
k= 6 =j_
Kr: )*cr.l/*.r-lt
, v+K
prq6.b- -ri._o,94
F
Fig. 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 I:4x10-a ma
----* l
Fig. Q8 (b)
**{.**
2 of2

11. u
ME6Fl
-
USN
OLD SCI{EME -;?
l--/--
sixth semester B.E. Degree Examination, July 20A6
Mechanical Engineering
Finite Element Methods
Time:3 hrs.l [Max. Marks:100
Note: Answer any FIVE full questions'
(03 Marks)
Define functional.
(10 Marks)
Derive Euler's Langranges's equation'
(07 Marks)
Expiain principle of minimum potential energy'
Briefly explain the steps involved in FEM'
(10 Marks)
(10 Marks)
Derive shape functions for CST triangular element in local co-ordinater.
Explain Banded matrix. Write an algor'ithm for Guass elimination technique'
(10 Marks)
Explain Raieigh's Ritz method in detail' (10 Marks)
4 What do you understand by weak form of differential equation. (05 Marks)
a*,..'u_-,
. 3j!.j
ft, ="lS$*
tY,,Y.'d.,c
:1
u
'.j
'-"F,
'-lt
ffi':gnr*:.et*r-** bar whose cross
i) For the above problem compute [B] and [c] matrix. It is^tapered
- section area decreases linearly from t-000 m*2 to 500 Take E:2x 10s N/mm2'
mm2.
ii) Use two elements and findthe nodal displacements. (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
(10 Marks)
elements.
b. Explain "penalty approach" for handling the boundary conditions' (10 Marks)
a. Discuss the requirements to be fulfilled for the convergence of FEM solution' Marks)
(10
b. Derive FEM equation by variational principle' (10 Marks)
Write short notes on anY four :
a. Pascal's triangle d. Truss element
b. Local - co - ordinate sYstem e. Shell element
c. Patch test. f. EliminationaPProach'
:t:t:k* *

12. Poge No,,. I ME6FI
Reg. No.
Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006
Mechonicol Engineering
(Old Scheme)
Finite Elemenl Methods
Time: 3 hrs.) (Mox.Morks: 100
NOtg: Answer any FVE lull questions.
l. (o) Find the inverse of
[lt] (5 Morks)
ror a:
[3 ]]
,:l; {l
Find : i) AB ii1 gT 4T (5 Morks)
Solve by Gouss eliminotion
2q * * rJ: -7
3x'2
5r1 * n2 * a3: Q (10 Morks)
321 *2x214x3:11
2. @t Whot is finite element method? Whot ore the odvontoges of FEM over finite
difference method? (4 Morks)
(b) Exploin boundory volue ond initiol volue problems using suitoble exomples.
' (8 Morks)
(c) Exploin the steps involved in the finite element onolysis of solids ond structures.
(8 Morks)
.
3. (o) whot is meont by 'Bclnd width' of o motrix? Give on exomple. Exploin why it
should be minimized. (6 Morks)
(b) Stote the principle of minimum potentiol energy, ond derive on expression for totol
potentiol energy of o solid bor under compression. (6 Morks)
(c) Exploin the Royleigh-Rit method with on exomple, (8 Morks)
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)ond
P;(80.5,10.2) ofter looding. Determine normol ond sheor stroins. (10 Morks)
Confd.,.. 2

13. Poge No,,, 2 ME6FI
5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the
following
D Element ond ossembled stiffness motrix
iD Displocement of eoch node
iii) Reoction force ot fixed end (20 Morks)
2-
A,=t0O mm.
*r=1-Oo mm' Lku
E = 200G Pa
I t'r= ro Q Po
6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.
(10 Morks)
,, l]
(b) Derive shope function CST triongulor element. (10 Morks)
7. @) Derive shope functions for o l-D quodrotic element with 3 nodes. (10 Mofts)
. (b) Exploin convergence criterio ond potch test in brief, (10 Morks)
8. Write short note on ony FOUR:
o) Voriotionol opprooch
'b) Hermition shope functions
c) Penolty opprooch for hondling boundory conditions
d) Logronge ond serendipity fomily of elements
e) ISO porometric: elements (5x4 Morks)

14. Page N0... 1
ME6F1
USN
Sixth Semester B.E. Degree Examination, July/August 2005
Mechanical En gineering
Finite Element Methods
Time: 3 hrs.I [Max.Marks : 100
Note: 1. Answer any FIVE full questions.
2. Missing data may be suitable assumed.
1. (a) Define positive definite matrix. (2 Marks)
(b) Solve the system of simultaneous equations given below by Gaussian elimination method.
2c1 * 2n2 * ns :9
n1*n2+fry:6 (10 Marks) '
2a1 * a2: 4
(c) Determine the inverse and eigen values of the given matrix A
. I 4 -2.286 (8 Marks)
^: -z.zJG
L
8
2. (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)
F tS , a.z ir.
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
at the free end by Rayleigh-Ritz method. (10 Marks)
Contd.... 2

15. Page N0... 2 ME6Fi
5. (a) For the one dimensional truss element, develop the element stiffness matrix in the global
coordinate system. (10 Marks)
(b) Determine the nodal displacement and stress by using truss element. (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
an example. (5 Marks)
(b) Evaluate the following by Gaussian quadrature
i) /: /]i (s"* + *, + #)da by one point and two point formula. (3 Marks)
ii) I : I: * OV 3-point formula. (8 Marks)
8. 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. (5x4=20 Marks)
*****

16. Page N0,. 1 ME6F1
USN
Sixth Semester B.E, Degree Examination, January/February 2005
Mechanical Engineering
Finite Element Methods
Time: 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:4
3*t-4xz-2r3-2
5r1l3r2*5r3- -7
either by Gaussian elimination method or malrix inversion method. (6 Marks)
(d) Find the eigen values of the matrix A
lz B
A- lr 4 -21
-2lr (6 Marks)
Lz 10 ,r j
2. (a) What is the basis of the Finite Element Method? Explain the basic steps involved in the
finite element method. (10 Marks)
(b) Determine the true displacement field for a two noded one dimensional tapered elemenl
shown in Fig.1. Also compute the stiffness matrix for this elemerit.
o c.n^--*J
I"t-eJ
At= loo
n;r'o[, "
q2&) , Ftq' t'
At :700rnz
-t12 :900mm2
A
. '2 (10 Marks)
An : ('* #)
Contd.... 2

17. Pase N0... 2 MEOF1
3. (a) What are the principles of continuum method? Compare this method with finite element
method 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 ils
span '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) Derive the strain displacement relations. (2 Marks)
(b) b<plain the concepts of plane stress and plane strain with suitable examples, Also derive
the corresponding equations. (8 Marks)
(c) A uniform rod of lengh I fixed al both ends is subjected to a constant axial load of w
kN/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)
5. (a) What are interpolation rnodels? Give reasons for choosing polynomial funclions for such
npdels. (5 Marks)
(b) 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
A l,^v, i*1,^r"
3oo t'tt'T 4 OO x^1^4
Area of (1):2400mm2
Area of (2) :6A0mm2
(10 Marks)
EAL:o'7 xTosNfrnrnz
Esteel:2x705Nlmrnz
6. (a) Explain the concept of isoparametric formulation. (5 Marks)
(b) Derive an elemenl stiffness matrix of a constant strain triangular element using the above
concept. (15 Marks)
Contd.... 3

18. 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 ** * **

19. Page No., 1
ME6Fl
USN
Sixth Semester B,E. Degree Examinatlon, July/August 2004
Mechanical Engineerlng
Finite Element Methods
Time: 3 hrs.I lMax.Marks : lOO
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 for
a general matrix, (10 Marks)
2, (a) With suitable examples explain.
i) Essential (geometric) boundary condition
ii) Natural (force) boundary condition. (5 Marks)
(b) Outline the steps in finite element analysis. (5 Marks)
(c) State the. principle. of minimum potential energy. Obtain the equilibrium equation of the
system 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 fixed
at one end and is subjected to axial load at the other end. Obtain the Euler equation
governing 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
with in the element. (10 Marks)
(b) Explain the criteria for monotonic convergence. (10 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, (12 Marks)
(b) What are characteristics of stiffness matrix ? (8 Marks)
6. (a) For a pin jointed configuration shown in Fig 6.a determine the stilfness matrix. Also
determine g, interms of g,. (10 Marks)
(b) Derive the Hermite shape functions of a beam. (r0 Marks)
Contd.... 2

20. :
Fage No... 2 ME6FT
7. (a) Evaluate
I
I [r,,* ;r*ffif*
-1
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 as
matrix.
to minimize the half band width of the resulting stilfness (5 Marks)
(c) For the beam shown in fig Q.8c. obtain the global stiffness matrix. (10 Marks)
F
fi?. qL. c
vf, ts*' +'ol
)I o'. g-- 7oxto3 ^'/t",'ol
-/-
I fZ A= l3oo ss +ozn'
I J CLo,P) L= S m
*
fi3. Q6.o
''l s0
ooo
c+1) mm
. / A.: 5oo ms ,
gnz Qoo
c z 3.5) ri , too 6Pa
Ct.gr.l L'; zoo a'oo-
63' QB'o- FS' E(")
+R
=l qe. b
ng.
7* l'rD ---+L t -o ----l
|[---6--G-re-Z-
/,r-'----=---'---i{--a----v
-7.,
I ,=2-ooePd
Fs, Ee .c ?=- +^iie *'"-4
; - r -,r^O
*****

21. Page No... 1 ME6Fl
Heg. No.
Sixth Semester B.E. Degree Examination, January/February 2009
Mechanical Engineering
Finite Elembnt Methods
Tirne: 3 hrs.I lMax.Marks : IOO
Note: Answer any FIVE questions. l
1. (a) Solve the following system of simultaneous equations by Gaussian Elimination Method.
t1 -2n2 f 613 - 0 l
l
2a1*2c2*3n3-3
-rr*3r2-2 (10 Marks)
(b) Find the inverse of the following matrices
l0 1 21
f1 2
', Ll?il ilL;:, ll -21
(5+5 Marks)
2. (a) Explain the theorem,of minimum potential energy. Distinguish between minimum potential
-
theorem 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
Tq-Ke E = L1q6J- (Yo*,,.3'S ^".U.*[-y
A' luniF (A tea
fl(r.,1-r a-h.orr
(10 Marks)
(b) Explain plane stress and plane strain methods with rerevant equations. (10 Marks)
4. (a) Explain the penalty approach for handling the specified displacement boundary conditions.
(10 Marks)
Contd.... 2

22. Page N0... 2 ME6F1
(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)
(a) Explain convergence criteria in detail, (10 Marks)
to) Derive shape functions for 'CST' element from
generalized co-ordinates. (10 Marks)
(a) Derive the stiffness matrix for a two noded beam element (12 Marks)
i
(b) Distinguish between isoparametric, sub-parametric and super-parametric elements.la uarxsl"
l
[^ 7. Consider the 4 -bur truss shown below, Determine.
i) Element stiffness matrix for each element
I
I
ii) Using eliminations approach to solve for the nodal displacements.
(iiD Calculate stresses in each element. (20 Marks)
+v 2-gooor..t (n.+i5 Ja-svr*,; ll
Qg Ar
t
3otv t"t
@
I 20,0001.; >1
rQrC)
4-4O
I
* *r
I
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 ** * ** (5x&20 Marks)

23. a
Poge No.,. I ME6FI
Reg. No.
Sixth Semesler B.E. Degree Exominolion, Jonuory/Februory 2006
Mechonicol Engineeilng
(Old Scheme)
Finiie Elemenl Methods
'1.
Time: 3 hrs.) ':.
(Mox.Morks: 100
NOle: Answer ony FIVE tuil queslions.
I. (o) Find the inverse of
[r ol
lo rl (5 Morks)
,o, a:
[3 1] ,:l; {l
Find : i) AB ii1 BT ar (5 Morks)
(c) Solve by Gouss eliminotion
2*t+3a2*nJ:-1
541*e2*rs:0 (10 Morks)
3rr + 2a2l4a3 -']".1
2. @, Whot is finite element method? Whot ore the odvontoges of FEM over finite
difference method? (4 Morks)
(b) Exploin boundory volue ond initiol volue problems using suitoble exomples.
' (8 Morks)
(c) Exploin the steps involved in the finite element onolysis of solids ond structures.
(S Morks)
: .
3. tol whot is meont by 'Bcind width' of o motrix? Give on exomple, Exploin why it
should be minimized, (6 Morks)
(b) Stote the principle of minimum potentiol energy, ond derive on expression for totol
potentiol energy of o solid bor under compression. (6 Morks)
(c) Exploin the Royleigh-Ritz method with on exompte. (8 Morks)
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)ond
Pj(80.5,10.2) ofter looding. Determine normol ond sheor stroins, (t0 Mql1s)
Confd.... 2

24. Poge No,,, 2 ME6FI
5. A solid stepped bor os shown in fig.l is subjected to on oxiol force. Determine the
following
i) Element ond ossembled stiffness motrix
iD Displocement of eoch''node I
iii) Reoction force of fixed end (20 Morks)
2-
A,= tOo hm ,
*r=LOo mhn- h-k u
g = 2,00G Pa
rt"= lo q Pq
6. (o) Whot is Jocobion Motrix? Derive o Jocobion motrix for Two-Dimensionol element.
(10 Morks)
(b) Derive shope function CST triongulor element, (10 Morks)
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)
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 (5x4 Mqrks)

25. Page No.., 1
ME6Fl
USN
$ixth sernester B"E. Degree Examination, July/August 2004
Mechanical Engineering
Finite Element Methods
3 hrs.l [Max.Marks : 10O
Note: 1. Answer any F|VE full questions.
2. Assume suitable data if necessary.
1. (a) Explain with example,
i) Syrnmetric matarix
ii) Determinant of a matrix
iii) Pcsitive definite matrix
iv) Half band width
v) Partitioning of matrices. (10 Marks)
(b) Give the algorithm for forurard elimination and back substitution of Gauss elimination for
a general matrix. (io Marks)
2. (a) With suitable examples explain.
i) Essential (geometric) boundary mndition
ii) Ndtural (force) boundary condition. (5 Marks)
(b) outline the steps in finite element analysis. (5 Marks)
(c) State the principle of minimum potential energy. Obtain the equilibrium equation ol the
system shown in fig 2.c using the principle of-minimum potentidl energy. (10 Marks)
3. (a) Derive the equilibriqm equation ol 3D.elastic body oc.cypyt"ng a volume V and having a
surface s, subjected to body force and a concentrated lddd. (r0 Marks)
(b) ry elastic bar of length.L, modulusof elasticity E, area of cross section A, which is fixed
at one end and is subjected to axial load at-the other end. Obtain the'Euler equation
governing the bar, and natural boundary conditions. t10 Marks)
4. (a) Fo1 a two noded one dimensional element, show that the strain and stress are constant
with in the element" (ro Marks)
(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. ('t2 Marks)
(b) What are characteristics of stiffness matrix ? (8 Marks)
(a) For a.pin jointed configuration shown in Fig 6.a detennine the stiffness matrix. Also
determine qt interms of g,. (10 Marks)
(b) Derive the Hermite shape functions of a beam. (10 Marks)
Contd.... 2

26. Page Nor, 2 illE6F1
7. (a) Evaluate
1
-1
Using two point Gauss quadrature. (5 Marks)
{b) Derive the expression for shape lunctions of eight noded isoparametric element. (15 Marks)
8. (a) Determine the Jacobian for the triangular element shown in fig eg.a, (5 Marks)
(b) Give thp element number and mode numbers for the structure shown in Fig Q 8.b, so as
to 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)
i
I
fi?. qL. e-
r'
t
vf, ct'o'+c>
)t o', -/" Et 7oxto3^l/t''ol
I {/Clo,rs) A= l3oo ss m"n'
I J.
* V-- S n
t
fr3. Q6.a
le
ooo
, )',+;') mm
/ A.; 5oo mw ,
gn: QOO
C z s's) c : 0o GPa'
L1.51) L'; r-oo aOo.
63' Qe'o- F3 5(a1
s
+R
ft. q8. b
,@
h-
/l
I'ro nD I .tlo -,
^^
,-2oold.
.qc.c i= "^lo6+nYo*
t ,
*****

27. a
Page No... 1 ME6F1
USN
Sixth Semester B.E. Degree Examination, January/February 2004
Mechanical Engineering
Finite Element Methods
Time: 3 hrs.l [Max.Marks : IO0
Note: 1. Answer any FIVE full questions.
2. Missing data may be suitably assumed,
1. (a) Find the eigen values of
A- 4 -{51 (5 Marks)
-,/3 a l
(b) Solve the following system of simultaneous equations by Gaussian elimination method.
2e1*12!3rs:t$
4r1*r21.a3:$
3n1*2r2 * rs:3 (10 Marks)
(c) Define the following with example
i) Skew matrix
ii) Symmetric banded matrix. (5 Marks)
(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 a
vertical load 'P' at its free end by Rayleigh Ritzmethod (i0 Marks)
(c) List the demerits of cantinuum methods. (2 Marks)
4' (a) Derive strain displacement matrix, stiffness matrix for one dimentional bar element.
(8 Marks)
(b) Solve for stresses and strains for the following problem by using bar element.
(12 Marks)
? = loco l.J
/t<_
E:2.7xlA5Nfrrurnz
At :5Omm2
Az :25mm2
P : 100011
Contd.... 2

28. Page N0... 2
ME6F1
5. (a) Derive stiffness matrix for a truss element. (8 Marks)
(b) For a pin jointed configuration shown in figure, determine nodal
displacements and stress
by using truss elemenls.
f : looo;?
T
5oo r
Ar : LAAmmz
t :lSovnr'
Az:125Amm,2
E:200GPa (12 Marks)
6. (a) Compute.the deflection of simply supported beam carrying concentrated load at its centre,
Use two beam elments.
(16 Marks)
(b) ls FEM analysis applicable for highly elastic materials? Explain. (4 Marks)
7. Find the displacement of node 1 in the triangurar element shown using one triangular
element. Also find stress and strain in the elefient.
. 1+----- 3o n
(-3o,o ) loo l,/
l r.-__ 5o
I
2o
I.(,2,o )
I
E:70GPa L
7:0.3 c 3o,
Le : lAmm (20 Marks)
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 (4x5=2Q fYl2Y[s)
*****

29. a
-----
'
-'-t/'
Page N0,,. I ME6F1
USN
Sixth Semester B.E. Degree Examination, July/August 2000
Mechanical Engineering
Finite Element Methods
Time: 3 hrs.I [Max.Marks : 10O
Note: Answer any FIVE futt questions.
1. (a) Given o:l; i], ort.,*in.
i) Inverse of matrix ii) Eigen values. (10 Marks)
(b) lf ,7"r: [€, 1-(2], evaluate /, wT Nag (5 Marks)
(c) Explain symmetric banded matrix. (5 Marks)
2. (a) With an example explain Rayleigh -Ritz method. (10 Marks)
(b) State the principle of minimum potential energy. (4 Marks)
(c) Sketch the quadratic and Hermite shape functions. (6 Marks)
3. (a) Derive the following characteristics of three noded l-D element.
i) Strain displacement matrix [B]
ii) Stiffness matrix [frr] (10 Marks)
(b) Solve for nodal displacements and stresses for the structure shown in fig 1. Use penality
approach to apply boundary csnditions. (10 Marks)
h t"laao n{' 2"17o frrn*
.,€ r 2lo$ pa
*1,€=zo$fo"
?JaoN
4. (a) Derive an expression for
i) Jacobian matrix
ii) Stiffness matrix for axisymmetric element. (10 Marks)
Contd.... 2

30. _ _
, ___:_
Page N0... 2 ME6F1
(b) 0onsider a rectangular element as shown in Fig.2. Evaluate J and B matrices at
(=0, =0, (10 Markr)
+
+
C1i,o,{)
cv>-
t A,>
L -t a)
(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
I
I
"l/
b MvY'
vjup ln7
>}lac?", ,
E-
(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. ** * **
(4x5=20 Marks)

31. I
Page No... l ME6Fl
Reg. No.
sixth serrester B.E. Degree Examflnatlon, Februar5r zooz
Mechanical Englneering
Ftntte 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
lz 0 1l
(c) Evaluate .4.-1 when -d. : lo 4 ol
fr o 2l
(d) Drptain Gauss-elimination method to solve a set of simultaneous equations.
(4X6=20 Marks)
2. (a) What is finite element method? Drplain the basic steps in the formulation of
finite element analysis. (12 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 an
expression for potential energr and hence determine the extreme value of the
potential 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 arrd
thickness of the bar t : Llmm. Also state its equilibrium stability. . ,
l_
{
T
-+
'L
Fta, I
iff
(b) use Rayleigh-Ritz method to find the disptacement and the stress .,tilIill
point of the rod as shown in fig.2. The area of cross section of the bar is 4OO
mmz and. the modulus of elasticity of the material is 7O GPa. Assume the
displacement to be second degree polynomial. (to Marks)
4. (a)
-Explain the elimination approach for handling the specified displacement
boundary conditions (5 Marks)
Contd.... 2