Question

Determine the axial forces, shears, and bending moments at points A and B of the structures shown.

Answer 1

The free body diagram of the given beam is shown:

Considering the equilibrium of the free body of the entire beam,

The forces acting along Y-direction,

Taking moment about the point D,

The load intensity of the uniformly varying load at point A and B are shown:

Let the intensity of load at point A be

From properties of similar triangles for triangles DCG and DAF

Let the intensity of load at point B be

From properties of similar triangles for triangles DCG and DBE

The axial force, shear force and bending moment at point A of portion AD of the beam are shown:

A section aa passing through the point A, cutting the beam into portions CA and AD. The portion AD, which is to the right of the section, is used to compute the internal forces.

Axial Force: Considering the external forces acting right as positive,

Therefore, the axial force at point A,

Shear Force: Considering the external forces acting downward as positive,

Therefore, the shear force at point A,

Bending Moment: Considering the counter-clockwise moments of the external forces about point A as positive,

Therefore, bending moment at point A,

The axial force, shear force and bending moment at point B of portion BD of the beam are shown:

A section bb passing through the point B, cutting the beam into portions CB and BD. The portion BD, which is to the right of the section, is used to compute the internal forces.

Axial Force: Considering the external forces acting right as positive,

Therefore, the axial force at point B,

Shear Force: Considering the external forces acting downward as positive,

Therefore, the shear force at point B,

Bending Moment: Considering the counter-clockwise moments of the external forces about point B as positive,

Therefore, bending moment at point B,