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A Note on the Principle of Uniqueness

A Note on the Principle of Uniqueness STRESSING A Note on the Principle of and GC as tensions and compress­ Uniqueness ion (since the triangle BGC is the triangle of forces for the joint G). A Principle that can be Used to Simplify the When the load W is removed and the tension Stress Analysis where some Members of a in the brace is T, the brace is equivalent to two Structure are Unloaded applied loads T acting at E, G. If these are re- By H. Roberts, B.Sc., A.F.R.Ae.S., A.M.I.Mech.E. Introduction (a) with the brace removed and the load W applied; HE stress analysis of perfect frameworks, in (6) with the load W removed and an unknown either two or three dimensions, is possible by a large number of standard methods. load T in the brace. The case of structures which under a prescribed The strain energy of the combined systems is loading system have some of the members un­ then evaluated and minimized to evaluate T. loaded deserves a little consideration, since it is Finally the deflection with the brace present is often possible to use the principle of superposition found from the equations to break down a given external loading into loads each of which give only a limited number of stressed members. When this is possible, the technique of stress analysis can be radically sim­ solved into components along the sides of the plified by consideration of the principle of where U =the strain energy with the brace panel EFGH, we have four loads each of magni­ uniqueness as stated below. 1 absent, tude T/Ö2 acting on a perfect framework as shown (FIG. 3). U =the strain energy with the brace Principle of Uniqueness present, A simple 'path' for the horizontal load at E In any perfect framework subjected to external A =the deflection with the brace present. is to load AE, EB only. A simple 'path' for the forces, the number of members is such that any It is the first step in this process, the stress vertical load at E is to load EH, HA, HD only. method of solution must give one, and only one, A 'path' for the vertical load at G is to load GB, set of internal resisting loads. If it were possible GC only, while a 'path' for the horizontal load to obtain more than one set of internal loads, at G, is to load GH, HD, HC only. In each case then the structure is inherently redundant, and the loads in the members can be written down the true internal loads correspond to minimum without calculation, since all four side bays are strain energy. similar (if they were not the calculations would It follows that if on inspection of a structure it is possible to find a simple 'path' for the loads, be very limited) and so the loads are that 'path' is the only one possible, since other­ wise there would be at least two sets of internal loads corresponding to the one set of external times those in GB, GC due to W—except for EH, loads, which is clearly not permissible. The find­ ing of an obvious 'path' for the loads will in GH which have loads of by inspection. general simplify the stress analysis and save a great deal of labour. These loads, by the uniqueness theorem must be the only set of loads which can exist so the stress Example analysis under conditions (a), (b), that concerns A good example of the use of the principle us here. If the method of tension coefficients is of uniqueness is given by a problem set in a used, a system of twelve simultaneous equations recent examination. The problem is as follows: TABLE I results, and although this is not too unmanage­ 'The pin jointed structure shown (FIG. 1) is Load due to tension able for (a) it is far too lengthy for (b). A simpler connected to a rigid vertical structure at A, B, Member Load due to \V T in EG analysis is as follows: C, D. The members AE, BF, CG, DH have areas AE A , the members EF, FG, GH, HE have areas A , 1 2 Considering the case when the brace is re­ BF 0 and the members BE, BG, CH, AH have areas A . moved, the only applied load to the perfect frame- CG -W Ö11/ 4 The brace GE has an area A . If the vertical DH 0 work is W, and by inspection it is possible to deflection of G when the brace is removed is 8, AH 0 balance this load by internal loads in GB and GC what would its deflection be when the brace is BE 0 only. Thus it follows that all other members are reinserted, all members being of the same ma­ 0 CH unstressed since the loads in GB and GC provide BG terial, the applied load being W at G.' + WÖ19/4 the unique system of internal balancing loads. EF 0 The problem reduces to the stress analysis of A glance at the true view of the side panel 0 FG GH 0 the framework BFGC (FIG. 2) gives the value of the loads in GB HE EG 0 analysis is complete. TABLE I gives the loads in the various members. The remainder of the solution to the problem is purely formal so it will not be completed. Conclusion The principle to which attention is drawn here can be used to simplify the derivation of the loads in certain kinds of structures. The possible scope of the technique illustrated in the particular example discussed is limited, but the technique is none the less one which is well worth knowing. 20 Aircraft Engineering http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Aircraft Engineering and Aerospace Technology Emerald Publishing

A Note on the Principle of Uniqueness

Aircraft Engineering and Aerospace Technology , Volume 22 (1): 1 – Jan 1, 1950

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Publisher
Emerald Publishing
Copyright
Copyright © Emerald Group Publishing Limited
ISSN
0002-2667
DOI
10.1108/eb031850
Publisher site
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Abstract

STRESSING A Note on the Principle of and GC as tensions and compress­ Uniqueness ion (since the triangle BGC is the triangle of forces for the joint G). A Principle that can be Used to Simplify the When the load W is removed and the tension Stress Analysis where some Members of a in the brace is T, the brace is equivalent to two Structure are Unloaded applied loads T acting at E, G. If these are re- By H. Roberts, B.Sc., A.F.R.Ae.S., A.M.I.Mech.E. Introduction (a) with the brace removed and the load W applied; HE stress analysis of perfect frameworks, in (6) with the load W removed and an unknown either two or three dimensions, is possible by a large number of standard methods. load T in the brace. The case of structures which under a prescribed The strain energy of the combined systems is loading system have some of the members un­ then evaluated and minimized to evaluate T. loaded deserves a little consideration, since it is Finally the deflection with the brace present is often possible to use the principle of superposition found from the equations to break down a given external loading into loads each of which give only a limited number of stressed members. When this is possible, the technique of stress analysis can be radically sim­ solved into components along the sides of the plified by consideration of the principle of where U =the strain energy with the brace panel EFGH, we have four loads each of magni­ uniqueness as stated below. 1 absent, tude T/Ö2 acting on a perfect framework as shown (FIG. 3). U =the strain energy with the brace Principle of Uniqueness present, A simple 'path' for the horizontal load at E In any perfect framework subjected to external A =the deflection with the brace present. is to load AE, EB only. A simple 'path' for the forces, the number of members is such that any It is the first step in this process, the stress vertical load at E is to load EH, HA, HD only. method of solution must give one, and only one, A 'path' for the vertical load at G is to load GB, set of internal resisting loads. If it were possible GC only, while a 'path' for the horizontal load to obtain more than one set of internal loads, at G, is to load GH, HD, HC only. In each case then the structure is inherently redundant, and the loads in the members can be written down the true internal loads correspond to minimum without calculation, since all four side bays are strain energy. similar (if they were not the calculations would It follows that if on inspection of a structure it is possible to find a simple 'path' for the loads, be very limited) and so the loads are that 'path' is the only one possible, since other­ wise there would be at least two sets of internal loads corresponding to the one set of external times those in GB, GC due to W—except for EH, loads, which is clearly not permissible. The find­ ing of an obvious 'path' for the loads will in GH which have loads of by inspection. general simplify the stress analysis and save a great deal of labour. These loads, by the uniqueness theorem must be the only set of loads which can exist so the stress Example analysis under conditions (a), (b), that concerns A good example of the use of the principle us here. If the method of tension coefficients is of uniqueness is given by a problem set in a used, a system of twelve simultaneous equations recent examination. The problem is as follows: TABLE I results, and although this is not too unmanage­ 'The pin jointed structure shown (FIG. 1) is Load due to tension able for (a) it is far too lengthy for (b). A simpler connected to a rigid vertical structure at A, B, Member Load due to \V T in EG analysis is as follows: C, D. The members AE, BF, CG, DH have areas AE A , the members EF, FG, GH, HE have areas A , 1 2 Considering the case when the brace is re­ BF 0 and the members BE, BG, CH, AH have areas A . moved, the only applied load to the perfect frame- CG -W Ö11/ 4 The brace GE has an area A . If the vertical DH 0 work is W, and by inspection it is possible to deflection of G when the brace is removed is 8, AH 0 balance this load by internal loads in GB and GC what would its deflection be when the brace is BE 0 only. Thus it follows that all other members are reinserted, all members being of the same ma­ 0 CH unstressed since the loads in GB and GC provide BG terial, the applied load being W at G.' + WÖ19/4 the unique system of internal balancing loads. EF 0 The problem reduces to the stress analysis of A glance at the true view of the side panel 0 FG GH 0 the framework BFGC (FIG. 2) gives the value of the loads in GB HE EG 0 analysis is complete. TABLE I gives the loads in the various members. The remainder of the solution to the problem is purely formal so it will not be completed. Conclusion The principle to which attention is drawn here can be used to simplify the derivation of the loads in certain kinds of structures. The possible scope of the technique illustrated in the particular example discussed is limited, but the technique is none the less one which is well worth knowing. 20 Aircraft Engineering

Journal

Aircraft Engineering and Aerospace TechnologyEmerald Publishing

Published: Jan 1, 1950

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