Applications of L inear Equations with Constant Coefficients 307c. Both ends are clamped and the load is
i. w(x) =wo ii. w(x) =w 0 sin(7rx/L) iii. w(x) = wacos(7rx/L)Exercise 7. Find the equation for the deflection of a horizontal beam oflength L = 120 inches, assuming y' is small , I(x) = 400 e-x/L in4, E =
15 x 106 lbs/in^2 , and the load is uniform, w(x) = wo where wo is a con-
stant (that is, solve equation (44)). (HINT: Use POLYRTS or your computer
software to find the complementary solution of equation (44).)
A cantilevered beam is one which is clamped at one end and completely
free at the other end. Suppose a cantilevered beam is clamped at x = 0 and
free at x = L. Then two conditions which must be satisfied by the equation
for the deflection of the beam are y(O) = 0 and y' (0) = 0. Suppose further
t hat y' is assumed to be small relative to 1. Then from equations ( 40) and
( 42)- regardless of whether I is constant or variable, we obtain the following
two additional initial conditions
( 45) "(O) = -M(O)
y EI(O)and
( 46 ) (3l(o) = I'(O) "(O) M'(O).
y I(O) y EI(O)The bending moments M(x) for various loads w(x) on a cantilevered beam of
length Lare given in Figure 6.13.
Exerise 8. For each of the loads A, B, C, D, and E shown in Figure 6.13 find
the equation for the deflection for a horizontal cantilevered beam of length
L = 240 inches, assuming y' is small , I(x ) = 500 e-x/L in^4 , P = 500 lbs, d =
50 in, w 0 = 60 lbs/in, and E = 25 x 106 lbs/in^2. (HINT: Solve equation (44)
subject to t he initial condit ions y(O) = 0, y' (0) = 0, and equations ( 45) and
( 46).)
Exercise 9. If y' is small and the moment of area of the cross section, I , is
constant, then the deflection of a horizontal beam resting on an elastic
foundation satisfies the differential equation
(47) EJy<^4 l + k^2 y = w(x)
where Eis Young's modulus, k is the spring constant of the elastic foundation,
and w(x) is the load on the beam. A simple example of such a beam is a single
rail of a railroad track. Find the deflectio n of a horizontal beam resting on an
elastic foundation if the beam is pinned at x = -Land x = L , if k^2 /EI = .09,
and if the load is w(x) = w 0 cos (7rx/ L) where wo is a constant. (HINT: Use
POLYRTS or your computer software to find t he complementary solution of
equation ( 4 7).)