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Consider a taper steel plate of uniform thickness t := 25 mm as shown in the Fig. In addition to its self weight, the plate is subjected to a point load P := 100 N at its mid point. Find the global force vector [F] , global stiffness matrix [K] , displacement in each element (1 and 2) , stresses in each element (1 and 2) and reaction force at the support.Take E := 2*10^5 N/mm²; rho := 8.2*10^(-5) kg/m³;

restart

t__1 := 150:

t__3 := 75:

w := 25:

l := 600:

t__2 := (t__1-t__3)/l*((1/2)*l)+t__3 = 225/2

A__1 := t__1*w = 3750

A__2 := t__2*w = 5625/2

A__3 := t__3*w = 1875

Revised areas:

A__1e := (A__1+A__2)*(1/2) = 13125/4

A__2e := (A__2+A__3)*(1/2) = 9375/4

E := 2*10^11: m²; F__1 := R__1: is support reaction N; F__2 := 100: N;

rho__1 := 82*10^(-6) = 41/500000 N/mm²

rho__2 := 82*10^(-6) = 41/500000 N/mm²

l := 600:

Number of elements,

n__e := 2:

l__e := 300 = 300

q__0 := 100: N/m ; l := 1: m; n__e := 4: elementsl l__e := l/n__e: m;

We shall consider a two element system as shown in the Fig.
For element 1 Stiffness matrix K is

Vector[row](2, {(1) = 1, (2) = 2})
$K__1 := A__1e*E/l__e.(Matrix(2, 2, {(1, 1) = 1, (1, 2) = -1, (2, 1) = -1, (2, 2) = 1}))$ = Matrix([[2625000000000000, -2625000000000000], [-2625000000000000, 2625000000000000]]) Vector(2, {(1) = 1, (2) = 2})

For element 2 Stiffness matrix K is

Vector[row](2, {(1) = 2, (2) = 3})
$K__2 := A__2e*E/l__e.(Matrix(2, 2, {(1, 1) = 1, (1, 2) = -1, (2, 1) = -1, (2, 2) = 1}))$ = Matrix([[1875000000000000, -1875000000000000], [-1875000000000000, 1875000000000000]]) Vector(2, {(1) = 2, (2) = 3})

Global stiffness matrix obtained by adding all the elemental stiffness matrices and given b

Vector[row](3, {(1) = 0, (2) = 0, (3) = 0})

$K__g := Matrix(3, 3, {(1, 1) = K__1[1, 1], (1, 2) = K__1[1, 2], (1, 3) = 0, (2, 1) = K__1[2, 1], (2, 2) = K__1[1, 2]+K__2[1, 1], (2, 3) = K__2[1, 2], (3, 1) = 0, (3, 2) = K__2[2, 1], (3, 3) = K__2[2, 2]})$ = Matrix([[K__1[1, 1], K__1[1, 2], 0], [K__1[2, 1], K__1[1, 2]+K__2[1, 1], K__2[1, 2]], [0, K__2[2, 1], K__2[2, 2]]]) Vector(3, {(1) = 0, (2) = 0, (3) = 0})