University:
Massachusetts Institute of Technology
Course:
16.01 | Unified Engineering I, II, III, & IV
Academic year:

2021
Views:

299

Pages:

8
Author:

isograda65z

Page 1 M9 Truss Deflections and Statically Indeterminate Trusses TRUSS DEFLECTION EXAMPLE Calculate deflection of loading point E in pin-jointed truss shown below. Bars are at 90° or 45°to each other. All bars have cross sectional area A, Young's modulus E. No temperature change occurs. Draw FBD Â Fy ↑= 0 VA - P = 0 ﬁ VA = P ‹ (1) Æ Â Fx = 0 : H A + HB = 0 H A = - HB Â (2) MA = 0 : HB L - 2LP = 0 HB = 2P ‹ ﬁ HA = -2P ‹ Page 2 Analyze bar forces. Mo J. @B Â Fy ↑= 0 FBA = 0 ‹ Æ Â Fx = 0: FBD + 2P = 0 ﬁ FBD = -2P ‹ @E FECSin45o - P = 0 ﬁ FEC + P 2 ‹ Â Fy ↑= 0 : Æ Â Fx = 0: - FECCos45o - FED = 0 ﬁ FED = - P ‹ HA VA Â MD = 0 : + 2PL - PL - FAC L = 0 ﬁ FAC = + P ‹ V VA Â Fy ↑= 0 : F DC + P = 0 ﬁ FDC = - P ‹ VA Â Fy ↑= 0 : P - FADCos45o = 0 ﬁ FAD = 2P ‹ Page 3 Bar Deflections given by FL AE d Bar Force/P Length/L AB 0 1 0 BD -2 1 -2 AD + 2 2 2 AC +1 1 +1 CD -1 1 -1 DE -1 1 -1 CE + 2 2 + 2 ( FL AE ) Deflection Diagram: 1. Fixed points - 0, A, B 2. Locate D ' via extension/rotations of BD & AD 3. Locate C ' via extensions/rotations of AC & CD 4. Locate E ' via extensions/rotations of CE & DE Page 4 Displacement diagram (to Scale) Horizontal Displacement PL to the left = AE vertical displacement PL = 12.9 AE Page 5 Statically Indeterminate Trusses Can set up problem to yield a set of simultaneous equations with unknown reactions and bar forces but known displacements (at certain points - compatibility) and known constitutive behaviors Can also use superposition and symmetry (two pretty good principles) to simplify seemingly complicated problems. Since trusses are linear (i.e if you double the applied load the internal forces and deflections will also double) we can superimpose the effects of multiple force systems in order to solve a problem. Can extend the idea of deflection diagrams to more complicated trusses - basic principles remain the same: Example: Symmetric 3 bar truss, bars cross sectional area A, Young’s modulus, E Page 6 FBD or go straight to application of method of joints. @D Note: RA = FDA, RB = FDB, RC = FDC Â Fy ↑= 0 FAD cos q + FBD + FCD cos q - P = 0 Â Fx = 0 - FAD sin q + FDC sin q = 0 (1.) Æ FDC = FAD 2 equations; 3 unknowns (symmetry) (2.) Page 7 cannot take moments - all forces pass through D Constitutive behavior. No DT \ d = FL AE Bar Force Length Extension AD FAD L d AD = FAD L / AE (3) BD FBD Lcosq d BD = FBD L cos q / AE (4) CD FCD L dCD = FCDL / AE (5) 5 equations; 6 unknowns. Two equilibrium equations, 3 constitutive relations So must invoke compatibility: bars extend and rotate, but remain attached at D: Displacement diagram Enlarged view of displacement diagram only: Page 8 dAD = dCD = dBD cosq Now have 6 equations, 6 unknowns and can solve. Substitute 3, 4, 5 into 6: FAD L FBD L cos2 q = ﬁ FAD = FBD cos2 q AE AE Substitute into (1) 2FBD cos 3 q + FBD - P = 0 P FBD = (1 + 2cos 3 q ) FAD = FCD = P cos2 q (1 + 2 cos3 q ) (6)