Example 1 1 Calculate The Lateral Stiffness For The Frame Shown In Fig Ella Assuming The Elements To Be Axially Rigid 1 (53.81 KiB) Viewed 17 times
Example 1 1 Calculate The Lateral Stiffness For The Frame Shown In Fig Ella Assuming The Elements To Be Axially Rigid 2 (53.81 KiB) Viewed 17 times
Example 1.1 Calculate the lateral stiffness for the frame shown in Fig. Ella, assuming the elements to be axially rigid. Ry 4E1/h + 4EI, IL :6E 16, ko 2/2 ET _201201 6E1 ko ET EI Ir T w = 1 (c) L = 2 () (b) Solution This structure can be analyzed by any of the standard methods, including moment distribution. Here we use the definition of stiffness influence coefficients to solve the problem. The system has the three DOFs shown in Fig. Ella. To obtain the first column of the 3 x 3 stiffness matrix, we impose unit displacement in DOF With M2 = 13 = 0. The forces kat required to maintain this deflected shape are shown in Fig. E11b. These are determined using the stiffness coefficients for a uniform flexural element presented in Appendix 1. The elements ka in the second column of the stiffness matrix are determined by imposing M2 = 1 with = 3 = 0; see Fig. Elle. Similarly, the elements kig in the third column of the stiffness matrix cap be determined by imposing displacements #3 = 1 with uy = 42 = 0. Thus the 3 x 3 stiffness matrix of the structure is known and the equilibrium equations can be written. For a frame with Is = le subjected to lateral force fs, they are 24 61 6h ΕΙ, 6h 6h? (a) h L6h h? 612 From the second and third equations, the joint rotations can be expressed in terms of lateral displacement as follows: 6)--[ (b) 12 6h Substituting Eq. (b) into the first of three equations in Eq. (a) gives 24 ETC ΕΙ, 6 Is = 96 ET (6 6h) 137h 13 Thus the lateral stiffness of the frame is 96 ET k= (d) 713 This procedure to eliminate joint rotations, known as the statie condensation method, is presented in textbooks on static analysis of structures. We return to this topic in Chapter 9. 12 16}-{8 672 1 [S] C (oor » []). = 7131 (c)
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