Problem $4(30$ points$)-$ Internal force analysis

A simply supported beam subjected to the distributed load and point loads is illustrated below. We will analyze

internal force distributions along the length of the beam, draw diagrams $($AFD: Axial force diagram; SFD: Shear force

diagram; BMD: Bending moment diagram$)$ and assess a deformed shape.

Figure $4.$ Simply supported beam $($left$)$ and its reactions $($right$)$

a$.$ Calculate the reactions exerted by supports using the given initial guess of reactions $(R_{AY},R_{Bx},R_{BY})$ in Figure

$4($left$).$ Although the solutions to this problem are already given in Figure $4($right$),$ to receive full credit,

provide explanations and steps that validate the provided answer, which should include horizontal force,

vertical force, and moment equilibrium conditions.

b$.$ Draw axial force, shear force, and bending moment diagrams in Figure $5.$ For full credit, show all steps with

beam segments as discussed in class and discussion. Please note that internal forces in the structural

member change dramatically where external forces are applied. Therefore, you can start analyzing internal

forces with a beam sectioned at an arbitrary distance $x$ from support A$.$ Considering the applied forces,

determine the internal forces and draw diagrams with section ranges as follows:

I. $0ftx<mn>8</mn><mi>f</mi><mi>t</mi>$

II$.8ftx<mn>1</mn><mn>2</mn><mi>f</mi><mi>t</mi>$

c$.$ Sketch the deformed shape of the beam. We can predict the deformed shape of the beam based on the

bending moment diagram and boundary conditions. It is essential to note that the pin support does not

allow horizontal and vertical displacement, and the given roller support does not allow the vertical

movement of the beam at the support. However, both the pin and roller supports permit rotations.