University of Mumbai 2007-5th Sem M.E Mechanical Engineering Finite Elent Analysis - - Question Paper
(?.) Answer any Jour questions Trom remaining ix i|uvuunt>.
(Ct) Assunption made should be clearly stated and justified.
m. a
1. ifte governing differential equation, representing steady laminar (low of a visoous tiuio tmougn a wng 20 circular cylindrical tube, is given as ; f
t o.
(")
d
37
Where, w is the axial component of velocity, yfcis the constant viscosity,
f0 Is the constant pressure gradient (which includes the combined effect of static pressure
and gravitational force),
Tha E-oundary conditions are-
dw _
(i) at r
(i) at r 0, r
R0w
dr
Using symmetry, and two linear elements and using RaylBigh Pltz Method over general element, determine;
(I) Element Matrix Equation. (Ill) Velocity Distribution.
(II) Global Matrix Equation.
Compare the velocltydistribution with exact solution, at least al the nodes.
Write all the steps clearly reflecting preprocessing, processing and post processing therein.
(a) State True* or False* and justify your answer In brief. Rectify the statement If It is wrong.
(1} Degrees of freedom te the primary variables for the element.
(Ii) Weight function value is zero at a node whore NBC is defined for all nonweak form type melhods.
(III) Node numbering has got no effect on the element or global matrix equations.
Using ay two methods out 0*:
(I) Collocation method. (ill) Galerkin method.
(ii) Petrov - Galerkin method.
Solvo.the deferential equation.
2.
(b)
14
d*i
OiXi t
J 0 COS RX
dx
Boundary conditions are (i) u(0)0
ro 3?L.
Compare the answer with the exact one oIXb 0-2$, 0-5, 0 75 ana 1.
What are the advantages of weak fo/m methods ? Compare these methods with lha nonweak form methods. Solve th* following equation by Unite difference method (4 subintervals) or Rltzmethod mapped over entire domain (2 parameters).
A
dx5
Boundary conditions :
(a)
<b>
5
10
3.
-16u + 3x* -0 0 x 1
du
d7
1
U{0) * 0
X-1
Compare tha answers with exact at salient points, (c) Explain the sources of error, in brief.
What are th* methods to reduce the error ?
Analyse completely any two :
(h) One dimensional fluid flow In porous medium.
4.
20
7
Take 4 elements.
d
EME
It
I
permeability, cm/s 1 25 cm/s.
Fluid held at the top is 25 cm and that atthe bottom 1$ 2*5 cm. Area ol cross-section Is 5.0m2.
Determine : (i) Fluid head distribution. rriioM aupb
(ii) Velocity at the upper part. Liutiw uvcn
(iii) Volumetric flow rate in upper pari.
Writ* global Matrix aquation. Assume suitable values oi primary variables at entry and exit and solve for remaining.
(c) Tapered bar:
D, m 12 cms
|
lir<ar vaiitljr |
 |
0j * 06 cms P, 20 KN P2 - 50 kN
E, 200-Gpa
E, .100 GPa
Le : (i) Ispfacements at node points, {ii) Reaction.
Stresses and strains Jn the elements.
Determ
(a) Oeflne . Transformation Matrix, Plane Truss, Semibandwidth, Aspect ratio. 6
(b) Analyse the plane truss for nodal displacements, reactions, elemental stresses and strains. Verify 14 tor force eoulllbrlum conditions.
L, * 0*6, l2 b 0*4 m
P, 5kN, P2 2 kN
E a 160 QPa, A * 6cm2 for ail elements.
Data:
(a) Using directly the Element Matrix Equation for beam element analyse the beam completely. /V 1/ 10 lc Aiff?)
r
.0232222022
E 100 GPa |.4v 10*4 m4
S' rv>-.J,* q m -]
(b) Find out (Mi) ] using Guass Quadrature Method.
I end I 1, 2 .. and are Lagrange's linear shape (unction,
|
r |
r 1 1 1 * |
w |
|
2 |
-Xl3 |
to |
|
3 |
00 1 JoJ |
8/9 5/0 |
I
Mij
(a) Write a note on, any one ; 10
(i) Plane stress problem using CST element, (ii) Transient Analysis used in FEM
(b) Evaluate stiffness matrix for the element shown, assumfrrg plane stress case. Find element stresses. 101
E * 210 GPa u poisson'6 ratio * 0*3 t * thickness a 1 cm Co-ordinate in cms.
Nodal displacement:
|
Attachment: |
| Earning: Approval pending. |
