LECTURE 35 COE-3001-A

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Unformatted text preview: 2 ]3/2 •  As we have seen, in the case of the small beam rota?ons, the curvature of the beam is given by: 1 d2 v = = 2 (8- 5) ⇢ dx •  The assump?on of small beam rota?ons is therefore equivalent to assuming that (ν’)2 <<< 1 in (8- 13). 4/9/13 M. Mello/Georgia Tech Aerospace 15 DEFLECTIONS BY INTEGRATION OF THE BENDING- MONENT EQUATION 00 EIv (x) = M (x) (8- 12a) •  GENERAL STRATEGY FOR SOLVING (8- 12a) AND OBTAINING DEFLECTIONS OF BEAMS ☞  Determine the equa?on for the bending moment (or moments) in a beam ☞  These are usually obtained from free- body diagrams and equa?ons of equilibrium ☞  Some?mes a single bending- moment formula applies for en?re length of the beam ☞  Some?mes bending- moment changes abruptly and so bending- moment in each region between points where the abrupt changes occur. ☞ For each region, subs?tute bending moment expression M=M(x) (a < x < b) into (8- 12a) and integrate twice to obtain the deflec?on curve rela?on v(x). ☞ Two integra?ons yields two arbitrary constants, which must be determined 4/9/13 M. Mello/Georgia Tech Aerospace 16 Determining the constants of integraBon 00 EIv (x) = M (x) (8- 12a) 0 EIv (x) = Z x M ( ⇠ ) d⇠ + C 1 •  “Method of successive integra?ons” •  Determine C1 from known condi?ons “Known condi?ons” fall into 3 categories: 1.  Boundary condiBons: Pertain to deflecBons and slopes at the supports ☞ at a simple support (pin or roller), the beam deflecBon is zero ☞ at a fixed (clamped) support, both deflecBon and slope are zero 4/9/13 M. Mello/Georgia Tech Aerospace 17 Determining the constants of integraBon (cont.) 2.  ConBnuity condiBons: Occur at points where regions of integraBon meet ☞ such as point C in the diagram below ☞ DeflecBon must be conBnuous across C i.e., displacement from the lel and right regions must be the same at point C ☞ Slopes from lel and right regions must be the same at point C 3.  Symmetry condiBons: may someBmes be invoked as well ☞ example: for case of a simply supported beam with uniform load along its length, we know that v’(L/2) = 0 (slope = zero at midpoint) 4/9/13 M. Mello/Georgia Tech Aerospace 18 EXAMPLE 8- 1 Problem: Simply supported beam with uniform load of intensity q ac?...
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This note was uploaded on 09/19/2013 for the course CEE 3001 taught by Professor Zhu during the Spring '09 term at Georgia Institute of Technology.

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