LECTURE+35+COE-3001-A

# LECTURE 35 COE-3001-A

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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?...
View Full Document

## 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.

Ask a homework question - tutors are online