The deflection at the midpoint of a uniform beam of length l, flexural rigidity EI and weight

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The deflection at the midpoint of a uniform beam of length l, flexural rigidity EI and weight per unit length w, subject to an axial force P, is

W d = (sec-ml - 1) - mP wl 8P

where m2 = P/EI. On making the substitution θ = 1/2ml, show that

d = wl 2 sec 02-0 32EI 04

As the force P is relaxed, the deflection should reduce to that of a beam sagging under its own weight. By first representing sec θ by its Maclaurin series expansion, show that

lim d = 00 5w14 384EI

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