Structures for which equilibrium equations are sufficient to obtain the solution are classified as statically determinate. But for some combination of members subjected to axial loads, the solution cannot be obtained by merely using equilibrium equations. The structural problems with number of unknowns greater than the number independent equilibrium equations are called statically indeterminate.
The following equations are required to solve the problems on statically indeterminate structure.
1)Equilibrium equations based on free body diagram of the structure or part of the structure.
2) Equations based on geometric relations regarding elastic deformations, produced by the loads.
A compound bar is one which is made of two or more than two materials rigidly fixed, so that they sustain together an externally applied load. In such cases
(i)Changes in length in all the materials are same. (ii) Applied load is equal to sum of the loads carried by
Any material is capable of expanding or contracting freely due to rise or fall in temperature. If it is subjected to rise in temperature of T°C, it expands freely by an amount ‘αTL’ as shown in figure. Where αis the coefficient of linear expansion, T°C = rise in temperature and L = original length.
From the above figure it is seen that ‘B’shifts to B’ by an amount ‘αTL’. If this expansion is to be prevented a compressive force is required at B’.
Temperature strain = αTL/(L + αTL) ≈αTL/L= αT
Temperature stress = αTE
Hence the temperature strain is the ratio of expansion or contraction prevented to its original length.
If a gap δis provided for expansion then
Temperature strain = (αTL –δ)/ L
Temperature stress = [(αTL –δ)/L]
A beam with a built in support at one side (i.e no rotation about or translation in the x, y, z direction) and a point support at the other i.e no translation in the x, y, z direction but rotation.
The fixed end moments are reaction moments developed in a beam member under certain load conditions with both ends fixed. A beam with both ends fixed is statically indeterminate to the 3rd degree, and any structural analysis method applicable on statically indeterminate beams can be used to calculate the fixed end moments.
Many structural analysis methods including the moment distribution method, slope deflection method and the matrix method make use of the fixed end moments.
Moment distribution method
Moment distribution method is basically a displacement method of analysis. But this method side steps the calculation of the displacement and instead makes it possible to apply a series of converging corrections that allow direct calculation of the end moments.
This method of consists of solving slope deflection equations by successive approximation that may be carried out to any desired degree of accuracy. Essentially, the method begins by assuming each joint of a structure is fixed. Then by unlocking and locking each joint in succession, the internal moments at the joints are distributed and balanced until the joints have rotated to their final or nearly final positions. This method of analysis is both repetitive and easy to apply. Before explaining the moment distribution method certain definitions and concepts must be understood.
Sign convention: In the moment distribution table clockwise moments will be treated +ve and anti-clockwise moments will be treated –ve. But for drawing BMD moments causing concavity upwards (sagging) will be treated +ve and moments causing convexity upwards (hogging) will be treated –ve.
Fixed end moments: The moments at the fixed joints of loaded member are called fixed end moment. FEM for few standards cases are given below:
Member stiffness factor:
a) Consider a beam fixed at one end and hinged at other as shown in figure 3 subjected to a clockwise couple M at end B. The deflected shape is shown by dotted line
BM at any section xx at a distance x from ‘B’ is given by