Question 3 Figure 2 shows a gantry crane that can be idealised as a simply suppo

Question

Question 3 Figure 2 shows a gantry crane that can be idealised as a simply supported beam designed to carry a maximum load P. The load P can be moved along the beam up to a distance d (but not closer) from the two supports as indicated on the Figure. The largest shear force and the largest bending moment allowed at any section of the beam are Smax and Mmax respectively. You need to determine the maximum length L of the crane while ensuring that the shear force and bending moment requirements are not violated. a. Formulate an optimization problem for maximizing L assuming that the self-weight of the beam is negligible. [18 marks] b. If the shear force requirement is more critical than the bending moment requirement, obtain an expression for the allowable length L. [3 marks) Beam I I LL Figure 2 Gantry crane

Explanation / Answer

a)From maximum shear force:

Maximum shear force in beam = [(L-d)/L]*P=P(1-d/L)=Smax

L =d/[1-(Smax/P)]

From maximum momnet:

PL/4=Mmax

L=4Mmax/P

b) as determined in part a

L=d/[1-(Smax/P)]

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