Statically Indeterminate Structures

A structure is statically indeterminate when the static equilibrium equations are insufficient for determining the internal forces and reactions on that structure.

• :: the vectorial sum of the forces acting on the body equals zero. This translates to

Σ H = 0: the sum of the horizontal components of the forces equals zero;

Σ V = 0: the sum of the vertical components of forces equals zero;

• : the sum of the moments (about an arbitrary point) of all forces equals zero.

Procedures:

• Obtain equation from static equilibrium

• Compatibility: Obtain equation from deformation

• Apply Hooke's Law

• Solve the unknown

In the beam construction, the four unknown reactions are V_A,V_B, V_C and H_A. The equilibrium equations are:

Σ V = 0:

V_A − F_v + V_B + V_C = 0

Σ H = 0:

H_A − F_h = 0

Σ M_A = 0:

F_v · a − V_B · (a + b) - V_C · (a + b + c) = 0.

The degree of indeterminacy is taken as the difference between the umber of reactions to the number of equations in static equilibrium that can be applied. In the case of the propped beam shown, there are three reactions R₁, R₂, and M and only two equations (ΣM = 0 and ΣF_v = 0) can be applied, thus the beam is indeterminate to the first degree (3 - 2 = 1).