A simply-supported beam of span l is discretized into 7

Question

SECTION A-

1. (a) For a structure subjected to arbitrary loading, provide an expression of the Principle of Virtual Work applied for the determination of unknown displacements, in terms of the different component terms which contribute towards the total virtual work done. Explain briefly the methods available for evaluating the resulting integral terms.

(b) A simply-supported beam of span L is discretized into 7 nodes (Node 1 at the left-hand pin support, Node 7 at the right-hand pin support and the other nodes spaced equally at L/6). The stiffness influence coefficient giving the force required at the j^th node to cause a unit displacement at the ith node is defined as kij. Using the Principle of Superposition, obtain an expression for the mid-span deflection if point loads of P₂, P₃ and P₅ are applied respectively at nodes 2, 3 and 5 along the beam.

(c) A translational spring system is composed of three springs in series each having an axial stiffness of k_s N/mm. Another translational spring system is composed of four springs in parallel each having an axial stiffness of k_p N/mm. Obtain an expression of k_s in terms of k_p such that the equivalent spring stiffnesses for both translational spring systems are equal.

SECTION B- General Notes

(i) All Numerical integration is to be carried out using Simpson's Rule.

(ii) The end-moments of a fixed-ended beam AB of span L subjected to a load W at a distance a from the left-hand support A and a distance b from the right-hand support B are given by :

M_A = Wa b²/L²

and

M^B = W a² b/L²

2. With reference to the portal frame shown in Fig.Q2, determine the horizontal sway at the eaves Joint C due to the applied external loading using the Principle of Virtual Work. Assume that all structural members are prismatic and have equal flexural rigidities, EI.

3. Using the Flexibility Method, analyse the portal frame shown in Fig.Q3 and draw the corresponding bending moment and shear force diagrams.

4. Using the Direct Stiffness Method, analyse the continuous beam shown in Fig.Q4 and draw the corresponding bending moment and shear force diagrams.

5. Using the Moment Distribution Method, analyse the continuous beam shown in Fig.Q5 and draw the corresponding bending moment and shear force diagrams.