A structure means a system of connected parts used to support a load. An elastic body is called a structure when a set of loads are applied the elastic body deforms and also the system is set up against the deformation. We can analyze Building frames by various methods. The method of analysis to adopt depends upon the following factors. The types of frame, its configuration in multi-storied frame and degree of indeterminacy. We use building frames as the most common structural form. But the building frames are designed in such a way that the beam-column joints stay rigid. A typical example of a building frame is the reinforced concrete multistory frames.
The two basic methods available for analysis of statically indeterminate structure are, but not limited to these.
1. Compatibility method
This is also known as Flexibility coefficient or Force method. We use this methods to calculate reactions and internal forces in statically indeterminate structures due to loads and imposed deformations.
First step is to select the redundant forces as unknowns. Additional equations are obtained, by considering the geometrical conditions. (imposed on the formations of the structures).
Methods of this category:
- Method of consistent deformation: Developed by Clerk Maxwell (1864), Otto Mohr (1874) and Muller- Breslau (1886).
- Three moment theorem : Claypeyron.
- Column analogy method : Hardy Cross.
- Elastic centre method.
- Maxwell – Mohr equations.
- Castigliano’s theorem of minimum strain energy.
- Flexibility matrix method.
2. Equilibrium or Displacement or Stiffness coefficient method
The displacement-based method is amenable to computer programming and hence we use this in the modern day structural analysis.
- Consider displacements of joints as unknowns, and
- Express equilibrium equations, in terms of these displacement. Then solve the external loads to give the actual joint displacements, from which you have to compute redundant forces.
Methods of this category:
- Slope deflection method : Initially developed by Axel Bendixson (1914) then by G.A.Maney (1915).
- Moment distribution method: Developed by Hardy Cross.
- Rotation contribution method : Kani
- Stiffness matrix method
We use the slope-deflection and moment distribution methods extensively for many years before the computer era. In the displacement method of analysis, primary unknowns are joint displacements which are commonly referred to as the degrees of freedom of the structure.It is necessary to consider all the independent degrees of freedom while writing the equilibrium equations. These degrees of freedom are specified at supports, joints and at the free ends.
Suitability of compatability and equilibrium method
- The choice between which methods to be used largely depends on the type of structure and the type of support.
- If for a beam or a frame: degree of static indeterminacy is more than number of joint displacements, we prefer equilibrium methods.
- Prefer force method : If for a beam or a frame, degree of static indeterminacy is less than kinematic indeterminacy.
- Force approach uses compatability equation whereas for displacement based, we use equilibrium equations.
Lets consider an example of a beam structure. Whose value of static indeterminacy is equal to 3, whereas the value of Dk is 1.
That is Dk < Ds, then we can analyze the above structure by displacement or equilibrium method.
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