STRUCTURAL ANALYSIS NOTES OF LESSON I UNIT – DEFLECTION OF DETERMINATE STRUCTURES Theorem of minimum Potential Energy Potential energy is the capacity to do work due to the position of body. A body of weight ‗W‘ held at a height ‗h‘ possess energy ‗Wh‘. Theorem of minimum potential energy states that ― Of all the displacements which satisfy the boundary conditions of a structural system, those corresponding to stable equilibrium configuration make the total potential energy a relative minimum‖. This theorem can be used to determine the critical forces causing instability of the structure. Law of Conservation of Energy From physics this law is stated as ―Energy is neither created nor destroyed‖. For the purpose of structural analysis, the law can be stated as ― If a structure and external loads acting on it are isolated, such that it neither receive nor give out energy, then the total energy of the system remain constant‖. With reference to figure 2, internal energy is expressed as in equation (9). External work done We = -0.5 P dL. From law of conservation of energy U i+We =0. From this it is clear that internal energy is equal to external work done. Principle of Virtual Work: Virtual work is the imaginary work done by the true forces moving through imaginary displacements or vice versa. Real work is due to true forces moving through true displacements. According to principle of virtual work ― The total virtual work done by a system of forces during a virtual displacement is zero‖. Theorem of principle of virtual work can be stated as “If a body is in equilibrium under a Virtual force system and remains in equilibrium while it is subjected to a small deformation, the virtual work done by the external forces is equal to the virtual work done by the internal stresses due to these forces”. Use of this theorem for computation of displacement is explained by considering a simply supported bea AB, of span L, subjected to concentrated load P at C, as shown in Fig.6a. To compute deflection at D, a virtual load P‘ is applied at D after removing P at C. Work done is zero a s the load is virtual. The load P is then applied at C, causing deflection C at C and D at D, as shown in Fig. 6b. External work done We by virtual load P‘ is . If the virtual load P‘ produces bending moment M‘, then the internal strain energy stored by M‘ acting on the real deformation d in element dx over the beam equation (14) P' δ D We 2 L M'M dx M'dθ ; Ui dU i 2 2 EI 0 0 0 U L
Where, M= bending moment due to real load P. From principle of conservation of energy We=Wi P' δD L M'M dx 2 2 EI 0 P C A D a B Fig.6a x P P’ D C C A L D B a Fig.6b x L If P‘=1 then M'M dx EI 0 L δD (16) Similarly for deflection in axial loaded trusses it can be shown that n δ 0 P' P dx (17) AE Where, = Deflection in the direction of unit load P‘ = Force in the ith member of truss due to unit load P = Force in the ith member of truss due to real external load n = Number of truss members L = length of ith truss members. Use of virtual load P‘ Unit Load Method Castiglione’s Theorems: = 1 in virtual work theorem for computing displacement is called
Castigliano published two theorems in 1879 to determine deflections in structures and redundant in statically indeterminate structures. These theorems are stated as: 1st Theorem: “If a linearly elastic structure is subjected to a set of loads, the partial derivatives of total strain energy with respect to the deflection at any point is equal to the load applied at that point” U Pj j 1,2, ..... N (18) δ j 2nd Theorem: “If a linearly elastic structure is subjected to a set of loads, the partial derivatives of total strain energy with respect to a load applied at any point is equal to the deflection at that point” U δ j j 1,2,....... N (19) Pj The first theorem is useful in determining the forces at certain chosen coordinates. The conditions of equilibrium of these chosen forces may then be used for the analysis of statically determinate or indeterminate structures. Second theorem is useful in computing the displacements in statically determinate or indeterminate structures. Betti’s Law: It states that If a structure is acted upon by two force systems I and II, in equilibrium separately, the external virtual work done by a system of forces II during the deformations caused by another system of forces I is equal to external work done by I system during the deformations caused by the II system I II Fig. 7 A body subjected to two system of forces is shown in Fig 7. W ij represents work done by ith system of force on displacements caused by jth system at the same point. Betti‘s law can be expressed as Wij = Wji, where Wji represents the work done by jth system on displacement caused by ith system at the same point.
Trusses Two Dimensional Structures ThreeDimensionalStructures Conditions of Equilibrium and Static Indeterminacy A body is said to be under static equilibrium, when it continues to be under rest after application of loads. During motion, the equilibrium condition is called dynamic equilibrium. In two dimensional system, a body is in equilibrium when it satisfies following equation. Fx=0 ; Fy=0 ; Mo=0 ---1.1 To use the equation 1.1, the force components along x and y axes are considered. In three dimensional system equilibrium equations of equilibrium are Fx=0 ; Fy=0 ; Fz=0; Mx=0 ; My=0 ; Mz=0; ----1.2 To use the equations of equilibrium (1.1 or 1.2), a free body diagram of the structure as a whole or of any part of the structure is drawn. Known forces and unknown reactions with assumed direction is shown on the sketch while drawing free body diagram. Unknown forces are computed using either equation 1.1 or 1.2 Before analyzing a structure, the analyst must ascertain whether the reactions can be computed using equations of equilibrium alone. If all unknown reactions can be uniquely determined from the simultaneous solution of the equations of static equilibrium, the reactions of the structure are referred to as statically determinate. If they cannot be determined using equations of equilibrium alone then such structures are called statically indeterminate structures. If the number of unknown reactions are less than the number of equations of equilibrium then the structure is statically unstable. The degree of indeterminacy is always defined as the difference between the number of unknown forces and the number of equilibrium equations available to solve for the unknowns. These extra forces are called redundants.