Advanced Methods of Structural Analysis

492 13 Stability of Elastic Systems

13.6.1 Double Integration Method..................................

Figure13.22shows a simply supported beam subjected to lateral forceFand com-
pressed forceP. We need to derive the expressions for deflection and internal forces.

Fig. 13.22 Simply supported
beam subjected to
compressive axial loadPand
lateral loadF

P P

l

c

x

y

x

RA RB

ABC

F

y

Differential equation of elastic curve of the beam for left and right parts (portions
1 and 2, respectively) may be written as

EI

d^2 y 1

dx^2

DPy 1 RAxDPy

Fc

l

x; xlc;

EI

d^2 y 2

dx^2

DPy 2 RB.lx/DPy

F.lc/

l

.lx/ ; x > lc:
(13.30)
General solution of these equations is

y 1 DC 1 cosnxCD 1 sinnx

Fc

Pl

x: (13.31)

y 2 DC 2 cosnxCD 2 sinnx

F

Pl

.lc/ .lx/ ; (13.32)

where

nD

r

P

EI

is parameter of compressed loadP.
At the pointsA.x D0/andB.xDl/the displacementyis zero. Equations
(13.31)and(13.32) lead toC 1 D 0 andC 2 DD 2 tannl. Therefore, expressions
for displacements within the left and right portions are

y 1 DD 1 sinnx

Fc

Pl

x;

y 2 DD 2 tannlcosnxCD 2 sinnx

F

Pl

.lc/ .lx/ : (13.33)

For calculation of unknown coefficientsD 1 andD 2 we can use the following
conditions at the pointC:

y 1 Dy 2 and

dy 1

dx

D

dy 2

dx

: