Principale of super position and maxwell reciprocal therom

Principale of super position and maxwell reciprocal therom

1. Subject:- SA-1 Branch :-CIVIL SEM :- 4 Na me :- Mistry Kevin H. Enrollment No.:- 180140106027

2. The principle ef superposition is stated as follows : "The displacements resulting from each of a number of forces may be added to obtain the displacements resulting from the sum of forces . A ( 1 + 2) = A1 + A2 "The forces that correspond to a number of displacements may be added to yield the force that corresponds to the sum of displacement." OR PRINCIPLE OF SUPERPOSITION :

3. Consider a cantilever beam as shown in figur e The deflections caused by the three seprate loads are shown in figure. The same final deflections would result if all the three loads are applied together as shown in figur e . This is even true if the sequence of loading is altered. It is important to note that this useful result would not occur if the deflection was not a liear elastic function of load. Superposition thus allows us to separate the loads in any desired way, analyse the structure for a separate set of loads and find the result for the sum of loads by adding individual load effects. Superposition applies equally to forces, stresses, strains and displacements.

4. (1) (2) (3) (4)

5. The requirements must be imposed for the principle of superposition to apply : The material must behave in a linear elastic manner, so that Hooke's law is valid and therefore the load will be proportional to displacement. The geometry of the structure must not undergo significant change when the loads are applied. i.e. small displacement theory applies. Large displacements will significantly change the position i an d orientation of the loads. 3. The supports are unyielding. It is not applicable for slender columns. It is also can not be applied in plastic design as in plastic range load-deformation relationship is not linear.

6. James clerk Maxwell, a Cavendtsh professor at Cambridge universily offered this law in 1964. The Maxwell's theorem of reciprocal dell.ctions can be stated by three ways as follows (1) The delfection at A due to unit load at B is equal to. dellection at B due to unit load at A. Fig 1.15 (al). ( 2) The slope at A due to unit couple at B is equal to, the slope at B due to unit couple at A. (Fig. 1.15. (bl. E Pan =OuA (3) The slope at A riue to tunit load al B is equal to, deflection at B due to unit couple at A. 'AD = ‘DA Proof : As per ullit load method. general cquation of deleetion is, Where M = Bending moment at any point X due to external load. m = Bending moment at any point X due to unit load applied at the point where deler is required. MAXWELL'S RECIPROCAL THEOREM :

7. Let, mxA = B.M. at any point X due to unit load at A mxB = B.M. at any point X duc to unit load at B When unit load (external load) is applied at A, M = mxA. To find deflection at B due to unit load at A. apply unit load at B.

8. Similarly, When unit load is applied at B, M = mxB To find deflection at A due to unit load at B, apply unit load at A. Then, m= mxA

9. Betti's Law : In any structure, if the material is elastic and follows Hooke's law and in which the supports are unyielding and the temperature is constant, the virtual work done by a system of forces P 1 , P2, P 3. .. during the displacements causes by a system of forces Q 1. Q 2, Q 3 … is equal to the virtual work done by the system of forces Q 1,Q 2, Q3. .. during the displacement caused by the system of force P, P2, P3... As per Betti's law, Virtual work done by P systen of loads Virtual work done by 9 system of loads

10. Betti's Law

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