MT-1620 al.2002 Unit 17 The beam-Column Readings Theory of Elastic Stability Timoshenko(and Gere), McGraw-Hill, 1961(2nd edition), Ch. 1 Paul A Lagace, Ph. D Professor of aeronautics Astronautics and Engineering Systems Paul A Lagace @2001
MIT - 16.20 Fall, 2002 Unit 17 The Beam-Column Readings: Theory of Elastic Stability, Timoshenko (and Gere), McGraw-Hill, 1961 (2nd edition), Ch. 1 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
MT-1620 al.2002 Thus far have considered separately beam - takes bending loads column -takes axial loads Now combine the two and look at the beam-column (Note: same geometrical restrictions as on others l > cross-sectional dimensions) Consider a beam with an axial load (general case) Figure 17.1 Representation of beam-column could also have p for bending in y direction) X Consider 2-D case: Paul A Lagace @2001 Unit 17-2
MIT - 16.20 Thus far have considered separately: • beam -- takes bending loads • column -- takes axial loads Now combine the two and look at the “beam-column” (Note: same geometrical restrictions as on others: l >> cross- sectional dimensions) Consider a beam with an axial load (general case): Figure 17.1 Representation of beam-column Fall, 2002 (could also have py for bending in y direction) Consider 2-D case: Paul A. Lagace © 2001 Unit 17 - 2
MT-1620 al.2002 Cut out a deformed element d Figure 17.2 Loads and moment acting on deformed infinitesimal element of beam-column M+rdx dx 召 dx dx 积 2 X Assume small angles such that sIn ax dw cOS d Paul A Lagace @2001 Unit 17-3
MIT - 16.20 Fall, 2002 Cut out a deformed element dx: Figure 17.2 Loads and moment acting on deformed infinitesimal element of beam-column Assume small angles such that: sin dw ≈ dw dx dx dw cos ≈ 1 dx Paul A. Lagace © 2001 Unit 17 - 3
MT-1620 al.2002 Sum forces and moments dF F+F+dx+p, dx d1 ds, dw d S-+s+adx dx|=0 ax This leaves (dx) dF dsdw dw dx+p, dx S- ldx+HOT=0 dx dF → p S 17-1) ax new term Paul A Lagace @2001 Unit 17-4
MIT - 16.20 Fall, 2002 Sum forces and moments: + • ∑Fx = 0 : dF − F F + + dx + px dx dx 2 − S dw + S + dS dx dw + d w dx = 0 dx dx dx d x2 This leaves: 2 dF d S dw S dw dx + px dx + + 2 dx + H O T. = 0 (dx)2 .. dx d x dx dx ⇒ dF dx p d dx S dw dx = − x − (17-1) new term Paul A. Lagace © 2001 Unit 17 - 4
MT-16.20 al.2002 F=01+ dF, d Fa+F+=dx dx ax S-S+dx +pdx=0 This results in d/dw d x dx、dx (17-2) new term ∑M=0 M+m dM x dx+p dx dw dx dS x S+dx dx=0 dx 2 (using the previous equations)this results in Paul A Lagace @2001 Unit 17-5
MIT - 16.20 Fall, 2002 • ∑Fz = 0 + : 2 − F dw + F + dF dx dw + d w dx dx dx dx d x2 dS + S − S + dx + pz dx = 0 dx This results in: dS dx p d dx F dw dx = z + (17-2) new term • ∑ My = 0 + : dM dx − M + M + dx + pz dx dx 2 dw dx dS − p dx − S + dx dx = 0 x dx 2 dx (using the previous equations) this results in: Paul A. Lagace © 2001 Unit 17 - 5