Advanced Methods of Structural Analysis

13.4 Stability of Continuous Beams and Frames 475 Note that subscript 2 at function'is related to the clamped–clamped member sub ...

476 13 Stability of Elastic Systems The axial compressed forces in both spans are equal, so the dimensionless pa- rameters 1 an ...

13.4 Stability of Continuous Beams and Frames 477 3.Assumel 2 D2l 1 andEI 1 DEI 2 DEI. In this case parameter ̨D l 2 l 1 r EI 1 ...

478 13 Stability of Elastic Systems Stability equation in general and expanded forms are ˇ ˇ ˇ ˇ r 11 r 12 r 21 r 22 ˇ ˇ ˇ ˇD^0 ...

13.4 Stability of Continuous Beams and Frames 479 Solution.The frame has two unknowns of the displacement method. They are the a ...

480 13 Stability of Elastic Systems PP h EI h 3 i h 3 i h j^3 i 1 (u) 3 i h j^1 (u) 3 i h Z 1 =1 r 11 r 11 Q 1 Q 2 Q 3 Q 4 Q 5 3 ...

13.4 Stability of Continuous Beams and Frames 481 Stability equation becomes 1 . /D 0 and the lowest root isD=2.The critical ...

482 13 Stability of Elastic Systems P h l EI kEI a b 12 Primary system c d Elastic curve r 11 3 EI h l 3 kEI Z 1 = (^1) r 21 r 2 ...

13.5 Stability of Arches 483 Nontrivial solution of canonical equations of the displacements method leads to the following stabi ...

484 13 Stability of Elastic Systems constant radiusRof curvature (circular arch), which is subjected to uniformly dis- tributed ...

13.5 Stability of Arches 485 then (13.13) may be presented in polar coordinates in the following form d^2 w d^2 CwD M EI R^2 ; ...

486 13 Stability of Elastic Systems Thus, differential equation (13.13a) becomes d^2 w d^2 C  1 C qR^3 EI  wD k' EI R^2 sin ̨ ...

13.5 Stability of Arches 487 3.Using expression (13.17a), the slope is dw d DBncosnC Ccos n^2  1 : (13.19) The slope at the ...

488 13 Stability of Elastic Systems Solution of this transcendental equation for given ̨and dimensionless parameter k 0 is the c ...

13.5 Stability of Arches 489 Table 13.4Critical parameternfor circular arch with fixed supports ̨ 30 ı 45 ı 60 ı 90 ı N 8:621 5: ...

490 13 Stability of Elastic Systems 13.5.2 Complex Arched Structure: Arch with Elastic Supports In practical engineering, the st ...

13.6 Compressed Rods with Lateral Loading 491 DisplacementZ 1 D 1 and corresponding bending moment diagram is shown in Fig.13.21 ...

492 13 Stability of Elastic Systems 13.6.1 Double Integration Method.................................. Figure13.22shows a simply ...

13.6 Compressed Rods with Lateral Loading 493 In expanded form these conditions are D 1 sinn.lc/DD 2 Œsinn.lc/tannlcosn.lc/ ...

494 13 Stability of Elastic Systems ForPD0:36Pcritwe get #D0:94248and 3.tan##/ #^3 D1:555; i.e., a small compressive loadincrea ...