Question

Mohammed Abdurahman Active Now The defection slong s uniform beam with fecual rigdity B-andapplied lond f(x)ossatisfis the equation (a) Evaluate the deflection y (a). (b) Find the influence function (Green's function) G(z,0, where 0 < ξ < 2 for this problem. Hint: Since 0 < ξ < 2, H(0-E)=0, H(2-E)-1. (e) Henoe write the deflection of this beam as a definite integral. Do not attempt tp evaluate the integral. ((7+2+2)+(6+6+2)-25 marks) Repl Crop Share Scroll Draw capture

Answer 1

Solution: 2. Given that
20
and applied load
f(x) = cos-) COS
satisfies the equations

Ely's) = f(x) = cos

with $y(0)=y'(0)=0;y''(2)=y'''(2)=0$

(a)

Ely's) = f(x) = cos

2 2 y") = cos EI =

Integrating both sides, we have

$\frac{1}{2}\int y^{(4)}=\int \cos\frac{x}{2}$

$\Rightarrow \frac{1}{2}y^{'''}=2\sin\frac{x}{2}+C_{1}$

Now,
y",(2) = 0
gives,

C1-2s2in(1)

So
「y" = 2sin-_ 2 sin(1)

Again, Integrating both sides, we have

$\frac{1}{2}y^{''}=-4\cos\frac{x}{2}-2\sin(1)x+C_{2}$

Now,
y"(2) = 0
, so

4 cos(1)

Therefore,

$\frac{1}{2}y^{''}=-4\cos\frac{x}{2}-2\sin(1)x+4\cos(1)$

By integration again, we have

$\frac{1}{2}y^{'}=-8\sin\frac{x}{2}-\sin(1)x^2+4\cos(1)x+C_{3}$

Now, $y'(0)=0$ gives,

$C_{3}=0$

Therefore,

$\frac{1}{2}y^{'}=-8\sin\frac{x}{2}-\sin(1)x^2+4\cos(1)x$

and by integration again, we have

$\frac{1}{2}y=16\cos\frac{x}{2}+\frac{1}{3}\sin(1)x^3+2\cos(1)x^2+C_{4}$

Now, $y(0)=0$ gives,

$C_{4}=-16$

Therefore,

$\frac{1}{2}y=16\cos\frac{x}{2}+\frac{1}{3}\sin(1)x^3+2\cos(1)x^2-16$

$\Rightarrow y(x)=32\cos\frac{x}{2}+\frac{2}{3}\sin(1)x^3+4\cos(1)x^2-32$

Therefore, deflection

$y(x)=32\cos\frac{x}{2}+\frac{2}{3}\sin(1)x^3+4\cos(1)x^2-32$