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$124$ FINITE ELEMENT FORMULATION FOR ONE$-$DIMENSIONAL PROBLEMS

Problem $5\mathrm{.}4$

Given a one$-$dimensional elasticity problem as shown in Figure $5\mathrm{.}20.$ The bar is constrained at both ends

$($A and C$).$ Its cross$-$sectional area is constant $(A=0\mathrm{.}1m^2)$ on segment AB and varies linearly

$A=0\mathrm{.}5(x-1)m^2$ on BC$.$ The Young's modulus is $E=2{10}^{7}Pa.$ A distributed load $b=10N{m}^{-1}$ is

applied along the left portion of the bar AB and a point force $P=150N$ acts at point B $.$ The geometry,

material properties, loads and boundary conditions are given in Figure $5\mathrm{.}20$a$.$Use a three$-$node element on

$AB(n_{en}=3)$ and a two$-$node element on $BC(n_{en}=2)$ as shown in Figure $20$b$.$ The dimensions in Figure $5\mathrm{.}20$

are in meters.

a$.$ Construct the element body force matrices and assemble them to obtain the global force matrix.

b$.$ Construct the element stiffness matrices and assemble them to obtain the global stiffness matrix.

c$.$ Find and sketch the finite element displacements.

d$.$ Find and sketch the finite element stresses.

$(A$ and $C).$ Its cross$-$sectional area $is$ constant $(A0$:$1m2)on$ segment $AB$ and varies linearly

$A0$:$5x1m2$ onBC.TheYoung$s$ modulus $isE2107Pa.$ Adistributed load $b10Nm1is$

applied along the left portion $of$ the bar $AB$ and a point force $P150N$ acts $at$ point $B.$ The geometry,

material properties, loads and boundaryconditions aregivenin Figure $5\mathrm{.}20a.$Use athree$-$node element $on$

$AB(nen3)andatwo-$nodeelementonBC $(nen2)asshowninFigure20b.$ThedimensionsinFigure$5\mathrm{.}20$

are $in$ meters.