1. Nodal Displacements of Rigid Jointed Space Frames
Neglecting Change in Length of Members
Dr. V M Inamdar
Prof. and Head of Structures Dept.
College of Military Engineering
Pune - 31

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1.1 INTRODUCTION
In the prevailing practice of analysing the space frames by matrix methods the change in the length of the members is considered. This makes all the displacement components independent. The analysis requires a stiffness matrix of very large size

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Inverting this matrix and getting the answer require many cumbersome calculations. This procedure is very much advantageous in the solution of space frames since it reduces significantly the size of the stiffness matrix which is otherwise very large. Again, as in the case of plane frames this hardly affects the accuracy of the results.

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1.2 SIMPLE SPACE FRAME:
A simple space frame (Fig.1)is solved first to illustrate the graphical procedure. The figure shows the top view, front view and right side view of the space frame, applied nodal forces and nodal displacements.

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The degree of kinematic indeterminacy is 18. The columns are fixed at the base. Each one of the three joints A, B and C has three degrees of rotational freedom and three degrees of translational freedom.

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Neglecting the change in the length of the members the degree of kinematic indeterminacy is reduced to 12, since six members impose six conditions or interrelationships. Fig.1

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14. 1.3 SPACE FRAME WITH GENERALIZED BEAM MEMBERS

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Here it is tedious to determine the interrelationship by graphical method. Hence the analytical method is opted for.
In this case also the columns supporting the triangle ABC are vertical. Therefore these joints do not move in the direction of Z axis. This makes wA ,wB , wC vanish. Thus displacements of these nodes are limited to XY plane only.
The displacement components at each joint are resolved along the sides of the triangle ABC (Fig5) meeting at that joint. The resolution is

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27. NODAL DISPLACEMENTS-3D CONCLUSION
The individual displacement components of space frame become interrelated when the change in length of the members is neglected. The simple space frames these interrelationships can be derived very easily by using graphical method of relative displacement diagrams.
Also developed an analytical procedure to derive the same interrelationship when the geometry of the space frames is complicated. The results derived here can be used directly for any space frames consisting of six members loosely interpreted as three beam members d three column members.

28. NODAL DISPLACEMENTS-3D CONCLUSION
The procedure to modify the given system of forces at nodal points into a system consisting of unbalanced forces corresponding to the independent displacement components.
Together these two relationships derived make it possible to solve such frames by neglecting the change in the length of the members even when the stiffness method is employed. This reduces the kinematic indeterminacy, when column are fixed at the base. As a result the stiffness matrix is correspondingly smaller and analysis is easier.

29. REFERENCES 1.Gere James M. and Weaver William Jr.
“Analysis of Framed Structures “
An East-West Edition, D.Van Nostrand Company, Inc.
New York and Affiliated East-West Press Pvt. Ltd.
New Delhi 1969
2.Maugh L.C.
“Statically Indeterminate Structures “
Second edition, John Wiley and Sons Inc., New York, 1964

34. THANK YOU