Beam is one of the most important structural components. Beams are usually long, straight, prismatic members and always subjected forces perpendicular to the axis of the beam
Beams are classified based on the type of supports:
Shear force has a tendency to slide the surface, it acts parallel to surface.
∑Fvert = 0
V - qdx - (V+dV) = 0
Only for distributed load not for point load.
Any moment produced by forces acting on the beam must be balance by an equal opposite moment produced by internal forces acting in beam at the section. This moment is called bending moment.
∑M = 0
-M - qdx(dx / z) - (V + dV)dx + M + dm = 0
dM/dx = V ⇒ Ma - MA = ∫V dx
Only for distributed and concentrated load not for couple.
The necessary internal forces to keep the segment of the beam in equilibrium are
∑Fx = 0 ⇒ P
∑Fy = 0 ⇒ V
∑Fz = 0 ⇒ M
[∑Fx = 0 → +]
[∑Fy = 0 ↑ +]
So the differential equations would be:
From equation dV/dx = -P, we can write
From equation dy/dx = -M, we can write
MD - MC = -∫Vdx
Statically Determinate Beam
A beam is said to be statically determinate if all its reaction components can be calculated by applying three conditions of static equilibrium i.e.,
∑ V = 0, ∑ H = 0 and ∑ M = 0
Statically Indeterminate Beam
When the number of unknown reaction components exceeds the static conditions of equilibrium, the beam is said to be statically indeterminate.