## Finite Element Analysis - May 2016

### Mechanical Engineering (Semester 6)

TOTAL MARKS: 80

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **three** from the remaining questions.

(3) Assume data if required.

(4) Figures to the right indicate full marks.

### Write the short note on (ANY FIVE)

**1(a)** Pre-Processing, Processing and Post-Processing in FEA.(4 marks)
**1(b)** Sub parametric, Iso-parametric and super parametric element in FEA.(4 marks)
**1(c)** Geometric and Forced boundary condition(4 marks)
**1(d)** Advantages and limitations of the FEM.(4 marks)
**1(e)** Write element matrix equation in the following fields. Explain each term properly.

(i) ID steady state, heat transfer by conduction.

(ii) ID, steady state flow of fluid in a pipe(4 marks)
**1(f)** Sources of Error in FEA.(4 marks)
**2(a)** Solve the following differential equation using Gelerkin Method. $$-\dfrac{d}{dx}\left [ (x-1)\dfrac{du}{dx} \right ]=x^2\ ;3\leq x \leq 5$$

Boundary conditions: u(5)=10 and u'(3)=5

Compare the answers with exact solution at x=4 and 5.(10 marks)
**2(b)** Compute the temperature at point P(2.5, 2.5) using natural coordinates system for quadrilateral element shown in the figure. Take: T_{1}=100°C, T_{2}=60°C, T_{3}=50°C, T_{4}=90°C

**3(a)**A copper fin of diameter 20mm, length 60mm bad thermal conductivity is 100 W/m

^{0}C and is exposed to ambient air at 30°C with a heat transfer coefficient 25 W/m

^{2}0 C. If one end of the fin is maintained at temperature 500°C and other end is at 200°C. Solve the following differential equation for obtaining the temperature distribution over the length of a fin $$kA.\dfrac{d^2\theta}{dx^2}-hp\theta=0$$

θ= Temperature difference=Tx-Ta

Use Rayleigh-Ritz mathod, mapped over general element, taking Lagrange's linear shape functions and two linear elements.

Write all the steps clearly. Compare your answer with exact at x=20,40 mm(15 marks)

**3(b)**What do you mean by consistent and lumped mass matrices? Derive the same for linear bar element.(5 marks)

**4(a)**Find the natural frequency of axial vibration of a bar having cross sectional area as 30 × 14

^{-4}m

^{2}, 1 m length with left end fixed. Take E=2 × 10

^{11}N/m

^{2}. Density of the material is 7800 Kg/m

^{3}. Take two linear elements.(10 marks)

**4(b)**Analyze the truss completely for displacement and stress as shown in figure.

Take: E = 2× 10

^{5}Mpa.

ELEMENT | AREA, mm^{2} |
LENGTH, m |

1 | 20 | 6 |

2 | 20 | 3 |

(10 marks)
**5(a)** Using FEM, analyze the taper bar as shown in figure. The cross sectional area to the left and right to is equal to 80 mm^{2} and 20 mm^{2}. Take length of bar is equal to 60 mm, Take E=210 Gpa.

**5(b)**Evaluate the shape function and prove its property, for triangular element as shown in figure. Also sketch the variation of shape function for each node. (10 marks)

**6(a)**Determine the displacement at nodes by using the principal of minimum potential energy and find the support reaction.

Use, k

_{1}= 100 N/mm, k

_{2}= 300 N/mm,

k

_{3}= 150 N/mm, k

_{4}= 300 N/mm. (10 marks)

**6(b)**A CST element has nodal coordinates (10,10), (70,35) and (75,25) for nodes 1,2 and 3 respectively. The element is 2 mm thick and is of material with properties E = 70 Gpa. Poission's ratio is 0.3. After applying the load to the element the nodal deformation were found to be u1 = 0.01 mm, v1 = -0.04 mm, u2 = 0.03 mm, v2 = 0.02 mm, u3 = -0.02 mm, v3 = -0.04 mm. Determine the strains ex, ey, exy and corresponding element stresses.(10 marks)