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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.
 Spring '09
 ZHU

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