Statically Indeterminate Structures

Statically indeterminate structures are those which can not be solved by using the equations of static equilibrium alone. Additional compatibility equations are required to solve indeterminate structures e.g.  fixed ended beam, continuous beam, propped cantilever etc.

It is very important to understand the behaviour of different types of supports. In general every hinged support will have two reaction components (along x and y axes) . Roller support will have one reaction in the direction perpendicular to the surface on which the rollers are kept. Fixed support will have three reactions ; two in x and y directions and the third is moment about z-axis at the support.

Let's consider a beam which  has roller supports at A and B and a hinged support at C. Its free body diagram is given below which shows that the given beam has 4 support reactions (C_x along x-axis and A_y, B_y, C_y along y-axis).

 

Every member of a structure will provide three equations of static equilibrium (ΣF_x = 0, ΣF_y = 0 and ΣM_z = 0).

(i) If the number of unknown reaction components are equal to the number of equations of equilibrium, the structure is known as statically determinate.

(ii) If the unknowns are more than the number of equations of equilibrium the structure is known as statically indeterminate. The difference between unknown reaction components and the equations of equilibrium is known as degree of indeterminacy. Therefore the beam shown in the above figure is statically indeterminate of degree one as it has 4 unknown reactions and only 3 equations of static equilibrium equilibrium..

 (iii)  If the unknown reaction components are less than the number of equilibrium equation, the structure is known as unstable. A structure is also defined as unstable if all the reaction components are concurrent or parallel (even if reactions are equal or more than the number of equations). This is evident from the beam shown in the above figure that if we change the hinged support at C to a roller support, all the reactions will be vertical and hence parallel to each other and this beam will not be able to support any horizontal force applied on it.

Methods for Solving Indeterminate Structures

Most of the structures being used in construction are indeterminate in nature. Force method of analysis or displacement method of analysis could be used to solve for indeterminate structures.

Some of the common methods to solve for indeterminate structures are;

Method of consistent deformation (compatibility method also known as flexibility method),

Moment distribution method, (Watch our Video Lecture on Moment distribution method for solving indeterminate structures This video is presenting different steps of moment distribution method with the help of an illustrated solved example.)

Slope-deflection equations

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three-moment theorem

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Energy theorems

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Stiffness method etc.

You can also use our Moment Distribution Calculator for solving indeterminate beams

Use our Fixed Beam Calculator to determine fixed-end moments for different loading cases.