Page 269 - Excel for Scientists and Engineers: Numerical Methods

246 EXCEL: NUMERICAL METHODS

y', we calculate y for a suitable range of x values from xo to x,, and compare the
calculated value of y at x, with the known value. If the calculated value does not
agree with the known value, we repeat the calculations with a different trial value
of y', until we calculate a value of y at the other boundary, x,,, that agrees with the
boundary value, hence the name "shooting method."

An Example: Deflection of a Simply Supported Beam
A simply supported beam (a beam supported at the ends) is bent downwards
by the applied load, consisting of the weight of the beam itself plus any other
loads.

Figure 11-1. Diagram of a simply supported beam.
The simply supported steel beam shown in Figure 11-1 supports a uniformly
distributed load of 2000 lblft. The length L of the span is 30 feet. The deflection
(downward bending displacement) y of the beam as a function of distance x along
the span of the beam is given by the second-order differential equation 1 1-1,
known as the general equation of the elastic curve of a deflected beam.
d2y - M (11-1)
dx2 EI
M, the bending moment at distance x, is given by equation 1 1-2
M= (WLXB) - (WX2/2) (1 1-2)
where L is the length of the beam and w is the weight of the beam per unit length.
E is the modulus of elasticity of the beam material; for carbon steel, E = 2.9 x 10'
psi, and I is the moment of inertia of the cross section of the beam, given by
equation 1 1-3.
I= bh3/12 (1 1-3)

where b is the width and h the height of the beam cross section. In this example,
for a beam 6 in wide x 16 in deep, I = 2048 in4.
Equation 1 1 - 1 can be transformed into the two equations

dY
- (1 1-4)
=z
dx
dz A4
ans -=- (1 1-5)
dx EI
where z is the slope of the beam.