USN 10ME64

Sixth Semester B.E. Degree Examination, Dec.2013 /Jan.20l4Finite Element Methods

Time: 3 hrs. Max. Marks:100Note: Answer FIVEfull questions, selecting

., ut least TWO questions from each part.

PART _ A1 a,, r Differentiate between plane stress and plain strain problems with examples. Write the stress-

strain relations for both. (08 Marks)b. Explain the node numbering scheme and its effect on the half band-width. (06 Marks)c. List down the basic steps involved in FEM for stress analysis of elastic solid bodies.

(06 Marks)

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2 a. State the piineiple of minimum potential energy. Determine tl.le displacements at nodes forthe spring system qhown in the Fig.a.,2J3). (08 Marks)

b. Determine the deflection of a cantilevel beam of length 'L' subjected 1o uniformly

distributed load (UDL) of Po/unit iength, using the trail function y : a sin [ '* I . Comnare\2L )

Fig.Q.2(a)

the results with analyical solution and comment on accuracy.

Derive an expression for Jacobian matrix for a four-noded quadrilateral element. (10 Marks)For the triangular element shown in the Fig.Q.3(b). Obtain the strain-displacement matrix'B' and determine the strains €r. €y and y*y.

Nodal displacements {q} : {2 | 1 -4 -3 7} x 10-2mm.

(12 Marks)

(10 Marks)

Note; All dimensions in mm.

E r:70 x 10e N/m2,82: 200 x 10eN/m2

Ar :900mm2, Az: l200mm2a1 :23 x 10-6/oC, az: 71.7 x 10'61'C.

Fig.Q.3(b)

4 a. an' axial load P : 300 x 103 N is applied at 20'C to the rod as shown in the Fig.Q.41a.y. The,, , temperature is then raised to 60oC.

D Assemble the global stiffness matrix (K) and global load vector (F).ii) Determine the nodal displacenrents and element stresses.

Fig.Q.a(a)

lsoN/nn

I of2

(12 Marks)

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5a.

b

c.

6a.

10M864b. Solve the following system of equations by Gaussian-Elimination method:

xr -2xz * 6x: : 0

2x1-t 2x2 + 3x3 : 3

-xy * 3x2:2. (08 Marks)

PART _ BUsing Lagrangian method, derive the shape function of a three-noded one-dimension (1D)element [quadratic element]. (06 Marks)-ll rlEvaluate I: ll 3e^+xt+ l.d*Lvuruulv '--lL" rA *G*2)-l''"

For the two-bar truss shown in the Fig.Q.6(a). Determine the nodal displacements andelement stross.es- A force of P: 1000 kN is applied at node 1, Take p,: ZtO GPa andA : 600mm2'for each element. ,,

t '' , ' (12 Marks)

using one-point and two-point Gauss quadrature.Wtite short notes on higher order elements used in FEM.

Fie.Q.6(a)

loads. , ,,"

Fig.Q.7(b)

b. Derive an expression for stiffness matrix for a2-D truss element.

a. Derive the Hermite shape functions of a beam eleiirent.A simply supported beam'of span 6m and uniform flekural rigidity EI : 40000 kN-m2 issubjected to clockwise couple of 300 kN-m at a distanCe of 4m from the left end as shown inthe Fig.Q.7(b). Find the deflection at the point of application of the couple and internal

?wW'n't

F8 a. Find the temperature distribution and heat transfer through an iron fin of thickness 5mm-

'-rr.j---- :-;nrn and uidth 1000mm. The heat transfer coefficient around the fin is 10 Wim2.K and ambient temperature is 28oC. The base of fin is at 108'C. Take K : 50 Wm.K. Usetwo elements.

Fig.Q.8(a)

b.

I

- I flil jr,.nt, {r^rn * rEry

Derive element matrices for heat conductionapproach.

>k*{<**

in one-dimensional element using Galerkin's(10 Marks)

(06 Marks)(08 Marks)

(08 Marks)

(08 Marks)

(12 Marks)

(10 Marks)

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