For a given $15m$ beam with a uniform load of $20K\frac{N}{m}$ over a length of $2m$ was placed $2m$

from left support. At the quarter point from the right support, a concentrated gravity load was

placed having a value of $15KN$ and $4m$ to the left of it is another concentrated load with value

of $30KN$ was placed on the beam. If the flexural rigidity for the given beam is constant,

calculate

$($a$)$ slope at the third point from the left support,

$($b$)$ deflection at third point from the right support, and

$($c$)$ Deflection at the second quarter point from right support

Provide the depiction of deflection in equation form, while keeping flexural rigidity constant,

for a simply supported beam supporting a uniformly distributed load had spread across its entire

span.

Establish the representation of the deflection equation, with constant flexural rigidity, for a

cantilever beam supporting a concentrated load applied at $"$a$"$ distance from the support.

A cantilever beam of length $L$ with a stepwise constant cross section carries a vertical load $P$

at its free end. The section of the beam changes midway along its length so that the second

moment of area of its cross section is reduced by half with the small section towards the free end.

$($a$)$ If $E$ is constant, determine the deflection at the free end.

$($b$)$ If $E$ is constant, determine the rotation at the free end

The separating distance between hinges A and $C,$ mounted vertically, is $1\mathrm{.}5m$ being hinge $C$ is

over hinge A$.$ A $4\mathrm{.}5m$ bar stretches horizontally from joint $B$ to hinge while a distinct bar $BC$ is

connected to joint $B$ and hinge $C.$ If the structure supports a $100KN$ load at point $B,E=70$

GPa, and $A=4000m{m}^2,$ compute the horizontal and vertical deflection at $B.$

For a given $20m$ simply supported beam, concentrated load with values of $5KN,10KN,$ and

$15KN$ was placed at every quarter point starting from left to right. Together with it a uniform

load with a value of $20K\frac{N}{m}$ was placed at the first quarter point up to the right support. If the

value of the moment of inertia varies such that it starts at $E$ from left support up to the last

quarter point and begins to double to the remaining length of the beam, calculate the

$($a$)$ deflection at $3m$ from right support,

$($b$)$ deflection at $2m$ to the right of midspan,

$($c$)$ slope at $2m$ from left support, and

$($d$)$ slope at first quarter point front the left support