a. A four-node quadrilateral element is having the following Cartesian coordinates
in cm. Node 1 (1, 1), Node 2 ( 5, 1), Node 3 (6, 6), Node 4 (1, 4). The element
displacement vector in cm 15 given as {ul ‏الا‎ ,, ... 114 V4) '=(00 0.2 0 0.15 0.1 0 0.05).
Determine (i) the x, ۷ co-ordinates of a point 'P' whose location in the parent element
is given by = q = 0.5. (ii). The ‏ہنا‎ ۷ displacement of point 'P’. (iii). Ihe Jacobian matrix
at =0.5. (5)

b. Under what circumstances a 3D problem can be idealised as a plane stress
problem. Give examples. Also write down the respective constitutive relation
for a linearly elastic isotropic material. http:/'www.ktuonline.com

a. In an one dimensional quadratic bar element, the nodal displacements at nodes 1
to 3 are 0.002 cm, 0.004 cm and 0.001 cm respectively, node 3 being the centre of
the element. The length of the element is 10 cm. Determine the strain at a point 'lY
which is at a distance of L/6 from the left end node (node 1). (4)

a. Develop the shape functions for a quadratic triangular element and plot the shape
functions for representative nodes. (5)

a. Evaluate the consistent nodal load vector for a beam element of length 3 m and
subjected to point loads of 5 KN and 10 KN at distance of ന from left end and 1m
from right end respectively. (4)

b. Develop the stiffness maheix for an one-dimensional quadratic bar element of
length, 'L' and cross-sectional area ൧. (5)

PART C

a. Number the nodal points of a rigid-jointed multi-storey frame with 3 bays and
5 storeys so as to attain minimum semi-bandwidth and calculate the same. (2)

0. Explain the penalty approach for imposing the boundary conditions 1௦ 8 problem.

(4)

‎Discuss the different types of hexahedral fi_nite elements that can be used for a‏ ع
‎3D problem and derive the shape functions for the simplest hexahedral element.‏

‎(6)

‎a. Determine the displacements at the points of application of loads for the bar

‎shown in figure by finite element method. E = 200 GPa.
# = 17

‎75 cm 75cm

‎(8)