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Re: Deflection of Cantilever Beam with Moment at Free End

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Rand:
Really it's twice your deflection (ML^2/2EI) !!
ML^2/EI
Humberto

----- Original Message ----- From: "Rand Holtham, P.E." <rand(--nospam--at)sigmaengineers.com>
To: <seaint(--nospam--at)seaint.org>
Sent: Friday, March 25, 2005 12:00 PM
Subject: Re: Deflection of Cantilever Beam with Moment at Free End


I use the graphical method...draw beam loading,draw shear diagram (it's zero since there is no other load since there's only the external moment), draw moment diagram (area under the shear diag + external applied moments), draw rotation diagram (triangular with max at end of beam = to area under moment diagram {MxL}divided by EI, draw the deflection diagram (first order parab with max equal to area under rotation diag {ML^2/2EI}).

It took me longer to type the procedure than to perform it ;o)
HTH

Rand



----- Original Message ----- From: "Padmanabhan Rajendran" <rakamaka(--nospam--at)yahoo.com>
To: <seaint(--nospam--at)seaint.org>
Sent: Friday, March 25, 2005 9:26 AM
Subject: Re: Deflection of Cantilever Beam with Moment at Free End


Your question has been answered by Jordan. ML^2/(EI)
is correct. Remember, however, that it is easy to
derive formula for deflection of a cantilever using
the conjugate beam method. If you are not familiar,
here are the steps.

(i) Draw the bending moment diagram on a line
represented by the beam span. (ii) Assume that the
free end is fixed and the fixed end is free (iii)
Calculate the bending moment at the "new" fixed end
due to loading represnted by the bending moment
diagram divided by EI (iv) Incidentally, the slope at
any sectiont is given by the shear force at the
section of the conjugate beam. The method is,
particularly, useful for beams with varying moment of
inertia.

For example, with moment M at the free end, the
loading on the conjugate beam is a uniform load
represented by "M/EI". Bending moment at the "new"
fixed end is:
(M/EI)*L*L/2=M*L^2/EI.

Rajendran

--- "M. David Finley, P.E." <pec(--nospam--at)isgroup.net> wrote:
The AISC manual conveniently gives the formula for
the deflection of a cantilever beam for a
concentrated load, a uniformly varying load, and a
concentrated load.

However, it does not provide the case of a
cantilever beam with a moment applied at the free
end of the cantilever.  Can anyone provide me the
formula?  I can get the value from my frame analysis
program, but I want to be able to check it by hand.

Thanks,

David Finley






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