Statically indeterminate

Template:Short description In statics and structural mechanics, a structure is statically indeterminate when the equilibrium equationsTemplate:Snd force and moment equilibrium conditionsTemplate:Snd are insufficient for determining the internal forces and reactions on that structure.<ref>Template:Cite book</ref><ref name=":0">Template:Cite book</ref>

MathematicsEdit

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

File:Statically Indeterminate Beam.svg

Free body diagram of a statically indeterminate beam

In the beam construction on the right, the four unknown reactions are Template:Math, Template:Math, Template:Math, and Template:Math. 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 variables) (Template:Math, Template:Math, Template:Math, and Template:Math) 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 deformations.

Statically determinateEdit

If the support at Template:Mvar is removed, the reaction Template:Math cannot occur, and the system becomes statically determinate (or isostatic).<ref>Template:Cite book</ref> Note that the system is completely constrained here. The system becomes an exact constraint kinematic coupling. The solution to the problem is:<ref name=":0" />

<math>\begin{align}

 \mathbf H_A &= \mathbf F_h \\
 \mathbf V_C &= \frac{\mathbf F_v \cdot a}{a + b + c} \\
 \mathbf V_A &= \mathbf F_v - \mathbf V_C

\end{align}</math>

If, in addition, the support at Template:Mvar is changed to a roller support, the number of reactions are reduced to three (without Template:Math), but the beam can now be moved horizontally; the system becomes unstable or partly constrained—a mechanism rather than a structure. In order to distinguish between this and the situation when a system under equilibrium is perturbed and becomes unstable, it is preferable to use the phrase partly constrained here. In this case, the two unknowns Template:Math and Template:Math can be determined by resolving the vertical force equation and the moment equation simultaneously. The solution yields the same results as previously obtained. However, it is not possible to satisfy the horizontal force equation unless Template:Math.<ref name=":0" />

Statical determinacyEdit

Descriptively, a statically determinate structure can be defined as a structure where, if it is possible to find internal actions in equilibrium with external loads, those internal actions are unique. The structure has no possible states of self-stress, i.e. internal forces in equilibrium with zero external loads are not possible. Statical indeterminacy, however, is the existence of a non-trivial (non-zero) solution to the homogeneous system of equilibrium equations. It indicates the possibility of self-stress (stress in the absence of an external load) that may be induced by mechanical or thermal action.

Mathematically, this requires a stiffness matrix to have full rank.

A statically indeterminate structure can only be analyzed by including further information like material properties and deflections. Numerically, this can be achieved by using matrix structural analyses, finite element method (FEM) or the moment distribution method (Hardy Cross) .

Practically, a structure is called 'statically overdetermined' when it comprises more mechanical constraintsTemplate:Snd like walls, columns or boltsTemplate:Snd than absolutely necessary for stability.

ReferencesEdit

Template:Reflist

External linksEdit

Beam calculation online (Statically indeterminate)