Advanced Methods of Structural Analysis

13.2 Stability of Structures with Finite Number Degrees of Freedom 455 The total energy of the system is UDU 0 CWDkrot ^2 2 Pl ...

456 13 Stability of Elastic Systems D2CD2l .1cos/Š2l ^2 2 : Strain energy accumulated in elastic support U 0 DR f 2 where r ...

13.2 Stability of Structures with Finite Number Degrees of Freedom 457 subjected to axial static forceP. The structure has one d ...

458 13 Stability of Elastic Systems and for critical force we get the same expression PcrD 2krot l : 13.2.2 Structures with Two ...

13.2 Stability of Structures with Finite Number Degrees of Freedom 459 The trivial solutiona 1 D0; a 2 D 0 corresponds to initia ...

460 13 Stability of Elastic Systems R 1 Dkf 1 Dklˇ 1 andR 2 Dkf 2 Dklˇ 2. The energy accumulated in elastic supports is XRifi 2 ...

13.3 Stability of Columns with Rigid andElastic Supports 461 13.3 Stability of Columns withRigid and Elastic Supports Elastic ba ...

462 13 Stability of Elastic Systems y x P P l x f y 0 Fig. 13.7 Buckling of clamped-free column This equation may be transformed ...

13.3 Stability of Columns with Rigid andElastic Supports 463 13.3.1.2 Uniform Columns with Elastic Supports Now let us consider ...

464 13 Stability of Elastic Systems Unknown parametersC 1 ;C 2 ,andfmay be determined using the following bound- ary conditions: ...

13.3 Stability of Columns with Rigid andElastic Supports 465 presents the columns with specified supports and corresponding stab ...

466 13 Stability of Elastic Systems 13.3.2 Initial Parameters Method.................................... This method may be effe ...

13.3 Stability of Columns with Rigid andElastic Supports 467 P Q 0 φ 0 y 0 M 0 Q 0 Y x Fig. 13.9 Initial parameters of a beam Co ...

468 13 Stability of Elastic Systems The initial parameters equations (13.9) may be presented in equivalent form (sec- ond form), ...

13.3 Stability of Columns with Rigid andElastic Supports 469 For the second portion of the column, the origin 02 is placed at th ...

470 13 Stability of Elastic Systems P 1 crD 2:0678EI 1 l^2 : Summary. Initial parameters method may be applied for stability ana ...

13.4 Stability of Continuous Beams and Frames 471 where dimensionless parameter ̨D EI kl : Equation (c) had been derived earlier ...

472 13 Stability of Elastic Systems As earlier, we need to have the reactions of standard elements, which are subjected to unit ...

13.4 Stability of Continuous Beams and Frames 473 The expressions for unit reactions for standard members subjected to compresse ...

474 13 Stability of Elastic Systems equilibrium. This occurs if the determinant, which is consisting of coefficients of unknowns ...