Reg No

Question Paper Code: 41413

B.E, B.Tech DEGREE EXAMINATION. APRIL/MAY - 2018

Sixth/Seventh Semester

Mechanical Engineering

ME 6603-FINITE ELEMENT ANALYSIS

(Common to Mechanical Engineering (Sandwich/Automobile

Engineering Manufacturing Engineering Mechanical and Automation Engineering)

(Regulations 2013)

Time: Three hours maximum: 100 marks

Answer ALL questions

PART – A(10x 2 =20 marks)

1. Compare the Ritz technique with the nodal approximation technique.

2. Differentiate between primary and secondary variables with suitable examples.

3. What are the properties of stiffness matrix?

4. Write the conduction, convection and thermal load matrices for 1D heat transfer through a fin.

5. Write down the shape functions for a 4 noded quadrilateral element.

6. Distinguish between scalar and vector variable problems in 2D.

7. Write the Strain Displacement matrix for a 3 noded triangular element.

8. Distinguish between plate and shell elements.

9. What are the advantages of natural coordinates?

10. Derive the Jacobian of transformation for a 1D quadratic element.

PART- B (5 x 13 = 65 marks)

11. (a) A tapered bar made of steel is suspended vertically with the larger end rigidly clamped and the smaller
end acted on by a pull of 105N. The areas at the larger and smaller ends are 80cm2 and 20cm2 respectively. The
length of the bar is 3m. The bar weighs 0.075 N/cc. Young’s modulus of the bar material is E=2x107N/cm2.
Obtain an approximate expression for the deformation of the rod using Ritz technique. Determine the
maximum displacement at the tip of the bar.

(Or)
b) The Governing Equation for one dimensional heat transfer through a fin of length l attached to a hot source
as shown in fig.11 b is given by
𝑑 𝑑𝑇
[−𝑘𝐴 𝑑𝑥 ] + ℎ𝑝(𝑇 − 𝑇∞) = 0
𝑑𝑥
 k=3W/cmoC h= 0.1W/cm2oC T∞=20oC

To 1cm

l=8cm 4cm

Fig. 11b

If the free end of the fin is insulated, give the boundary conditions and determine using the Collocation
technique the temperature distribution in the fin. Report the temperature at the free end.

12 (a)Determine the deflection in the beam, loaded as shown in fig.12a, at the mid-span and at a length of 0.5m
from left support. Determine also the reactions at the fixed ends. E=200GPa, I1=20x10-6m4, I2=10x10-6m4.

(Or)

(b) Determine the first two natural frequencies of longitudinal vibration of the stepped steel bar shown in
fig.12b and plot the mode shapes. All dimensions are in mm. E=200GPa and ρ=0.78kg/cc, A=4cm2, length
l=500mm.

13. (a) i) Determine three points on the 50oC contour line for the rectangular element shown in the Fig.13a. The
nodal values are T1=42oC, T2=54oC, T3=56oC and T4=46oC.
 (8)
ii) Derive the conductance matrix for a 3 noded triangular element whose nodal coordinates are known.
The element is to be used for two dimensional heat transfer in a plate fin. (5)
(Or)
(b)A square shaft of cross section 1cm x 1cm as shown in Fig.13b is to be analysed for determining the stress
distribution. Considering geometric and boundary condition symmetry 1/8th of the cross section was modeled
using four equisized triangular elements as shown. The element stiffness matrix and force vector for a triangle
whose nodal coordinates are (0,0), (0.25,0) and (0.25,0.25) are given below. Carry out the assembly and
determine the assembled stiffness matrix. Impose the boundary conditions and explain how the unknown stress
function values at the nodes can be used to determine the shear stress.

1 −1 0 29.1
1
Stiffness matrix(K) = 2 [−1 2 −1] Load vector(f) = {29.1} (13)
0 −1 1 29.1

14 (a) i) A thin plate of thickness 5mm is subjected to an axial loading as shown in the Fig.14.a. It is divided
into two triangular elements by dividing it diagonally. Determine the Strain displacement matrix [B], load
vector and the constitutive matrix. How will you derive the stiffness matrix? (Need not be determined). What
will be the size of the assembled stiffness matrix? What are the boundary conditions? E=2x107 N/cm2, µ=0.3.
 (8)
ii) Differentiate between plane stress and plane strain analysis. (5)
(Or)

(b) i) With at least two examples explain what is meant by axisymmetric analysis. For the 3 noded triangular
axisymmetric element shown in fig. 14b. Derive the strain displacement matrix [B] and also the constitutive
matrix [D]. (9)

ii) Derive the stiffness matrix for a ID linear element. (4)

15 (a) i) Using Gauss Quadrature evaluate the following integral. (7)

+1
I=∫−1 (4𝜉 3 − 2𝜉 2 + 3𝜉 + 6)𝑑𝜉
ii) Evaluate the shape functions for one corner node and one mid side node of a nine noded quadrilateral
element. (6)
(Or)

(b) i) Differentiate between subparametric, isoparametric and superparametric elements. (5)

ii) For the four noded elements shown in Fig.15.b determine the Jacobian and evaluate its value at the
point (0,0) (8)
 PART-C (1X15=15 Marks)
16. a) A Gantry crane as shown in Fig.16a, of overall length 10m is designed to carry a maximum load of
50kN. The beam is of I section whose Ixx is 40 x 10-6 m4. E=200 GPa. If the maximum deflection of the beam
and the support reactions and moments are to be determined using Finite Element Analysis, discuss how the
member of length L is to be modeled. What element will be used and what are the boundary conditions? What
is the maximum deflection of the beam? (4+4+7)

12 6𝑙 −12 6𝑙 1
c 𝐸𝐼 6𝑙 4𝑙 2 −6𝑙 2𝑙 2 𝑞𝑙 𝑙/6
Stiffness Matrix [K] = 𝑙3 [ ] {f}c = 2 { }
−12 −6𝑙 12 −6𝑙 1
6𝑙 2𝑙 2 −6𝑙 4𝑙 2 −𝑙/6
3𝑥 2 2𝑥 3
N1 = 1-(
𝑙2
) + ( 𝑙3 )
2𝑥 2 𝑥3
N2 = x-(
𝑙
) + ( 𝑙2 )
3𝑥 2 2𝑥 3
N3 = (
𝑙2
) − ( 𝑙3 )
𝑥2 𝑥3
N4 = -( ) + ( 2 )
𝑙 𝑙

156 22𝑙 54 −13𝑙

𝜌𝐴𝐼 22𝑙 4𝑙 2 13𝑙 −3𝑙 2
[M]c = 420 [ ]
54 13𝑙 156 −22𝑙
−13𝑙 −3𝑙 2 −22𝑙 4𝑙 2
No. of Points Location Weight Wi
1 ξ1= 0.00000 2.00000
2 ξ1, ξ2± 0.57735 1.00000
ξ1, ξ3 = ±0.77459 0.55555
3
ξ2 = 0.00000 0.88888
(or)
b) Consider a cylindrical pin fin as shown in fig.16 (b) for which hwater = 567 w/m2k, kfin = 207
w/m.k, hair = 284 w/m2k.
The right face of the fin is in contact with water at 4.5oC. The left face of the fin is subjected to
a constant temperature of 82.2oC, while the exterior surface of the pin is in contact with moving
air at 22.2oC, Using four equal length two node elements to obtain a finite element solution for
the temperature distribution across the length of the fin.