calculating beam slope

[ketanco]

hello,

i am studying beam deflection methods such as double integration or moment area method or conjugate beam method etc right now.



how can we know where the slope is 0 for a simply supported beam, with any distributed or point loads on it.



is the beam slope 0 where moment diagram is at its maximum or minimum ? if not is there a shortcut of how to find where slope is 0?

 

[desertfox]

Hi
For a simply supported beam, the beam slope is usually zero at the max deflection point of the beam, if the beam is loaded symmetrically then the maximum deflection occurs at the beam centre, so that the slope is zero at L/2

check out this link:-

http://academic.uprm.edu/pcaceres/Courses/INME4011/MD-3A.pdf

 

[BAretired]

For any load combination on a simply supported beam, bending moment can be determined for every point on the beam. Using conjugate beam method, the area under the M/EI diagram of the real beam is the load on the conjugate beam. The reaction at each end of the conjugate beam is the slope at each end of the real beam.

The slope of the real beam at any point along the span is equal to the shear of the conjugate beam, so the slope of the real beam is zero where the shear of the conjugate beam is zero.

You asked "is the beam slope 0 where moment diagram is at its maximum or minimum ? if not is there a shortcut of how to find where slope is 0?"

A uniformly loaded beam has maximum moment and zero slope at midspan but that is not true of any load distribution. A simply supported beam A-B with an applied moment of M at end B and no other applied loads or moments has a moment diagram varying linearly from 0 at A to M at B. The area under the M/EI curve is ML/2EI. The reactions of the conjugate beam are ML/6EI and ML/3EI, so the slope of the real beam are ML/6EI and ML/3EI for end A and B respectively. Zero slope occurs where the shear in the conjugate beam is zero, namely L/√3 from end A.

BA

 

[desertfox]

page 10 of this link might also be useful:-

http://etidweb.tamu.edu/ftp/ENTC376/Presentations/ENTC 376 Chapter 6 Lecture Notes 1-RK.pdf