Editing Statically indeterminate (section)

== Mathematics ==
Based on [[Newton's laws of motion]], the equilibrium equations available for a two-dimensional body are:<ref name=":0" />

:<math> \sum \mathbf F = 0 :</math> the [[Euclidean vector|vectorial]] sum of the [[force]]s acting on the body equals zero. This translates to:
::<math> \sum \mathbf H = 0 :</math> the sum of the horizontal components of the forces equals zero;
::<math> \sum \mathbf V = 0 :</math> the sum of the vertical components of forces equals zero;
:<math> \sum \mathbf M = 0 :</math> the sum of the [[moment (physics)|moment]]s (about an arbitrary point) of all forces equals zero.

[[File:Statically Indeterminate Beam.svg|thumb|350px|right|[[Free body diagram]] of a statically indeterminate [[beam (structure)|beam]]]]
In the [[beam (structure)|beam]] construction on the right, the four unknown reactions are {{math|'''V'''{{sub|''A''}}}}, {{math|'''V'''{{sub|''B''}}}}, {{math|'''V'''{{sub|''C''}}}}, and {{math|'''H'''{{sub|''A''}}}}. The equilibrium equations are:<ref name=":0" />

: <math>\begin{align}
\sum \mathbf V = 0 \quad & \implies \quad \mathbf V_A - \mathbf F_v + \mathbf V_B + \mathbf V_C = 0 \\
\sum \mathbf H = 0 \quad & \implies \quad \mathbf H_A = 0 \\
 \sum \mathbf M_A = 0 \quad & \implies \quad \mathbf F_v \cdot a - \mathbf V_B \cdot (a + b) - \mathbf V_C \cdot (a + b + c) = 0 
\end{align}</math>

Since there are four unknown forces (or [[Variable (mathematics)|variables]]) ({{math|'''V'''{{sub|''A''}}}}, {{math|'''V'''{{sub|''B''}}}}, {{math|'''V'''{{sub|''C''}}}}, and {{math|'''H'''{{sub|''A''}}}}) but only three equilibrium equations, this system of [[simultaneous equations]] does not have a unique solution. The structure is therefore classified as ''statically indeterminate''. 

To solve statically indeterminate systems (determine the various moment and force reactions within it), one considers the material properties and compatibility in [[deformation (engineering)|deformation]]s.