Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

Evaluate Constants
Substitute the boundary condition x = 0, dv/dx = 0 into Equation (e) to find C 1 = 0. Substitution
of the value of C 1 and the boundary condition x = 0, v = 0 into Equation (f) gives
C 2 = 0. Next, substitute the values of C 1 and C 2 and the boundary condition x = L, v = 0
into Equation (f) to obtain
w
EI
L L A M (0) L L
120 ( ) y
6 ( ) A
=− + +
2 ( )
Solve this equation for M A in terms of the reaction A y :
0 5 3 2
M
A
wL 0 2 AL y
= −
(g)
60 3
From equilibrium Equation (b), M A can also be written as
M
A
wL 0 2 = − AL y
(h)
6
Solve for Reactions
Equate Equations (g) and (h):
wL 0 2 AL y wL 0 2
− = − AL y
60 3 6
Then solve for the vertical reaction force at A:
A = 27 9
y w0L = w0 L
Ans.
120 40
Substitute this result back into either Equation (g) or Equation (h) to determine the
moment M A :
7
MA =− w0L
2 Ans.
120
To determine the reaction force at roller B, substitute the result for A y into Equation (a)
and solve for B y :
B = 33 11
y w0L = w0
L
Ans.
120 40
ExAmpLE 11.2
A beam is loaded and supported as shown. Assume that EI is constant
for the beam. Determine the reactions at supports A and C.
v
w
Plan the Solution
First, draw an FBD of the entire beam and develop two equilibrium
equations in terms of the four unknown reactions A y , C y ,
M A , and M C . Two elastic curve equations must be derived for this
beam and load. One curve applies to the interval 0 ≤ x ≤ L/2, and
the second curve applies to L/2 ≤ x ≤ L. Four constants of integration will result from the
double integration of two equations; therefore, a total of eight unknowns must be determined.
To solve for eight variables, eight equations are required. Four equations are
obtained from the boundary conditions at the beam supports, where the beam deflection
and slope are known. At the junction between the two intervals, two equations can be
A
L—
2
B
L—
2
C
x
449