Advanced Methods of Structural Analysis

13.6 Compressed Rods with Lateral Loading 493

In expanded form these conditions are

D 1 sinn.lc/DD 2 Œsinn.lc/tannlcosn.lc/ ;

D 1 ncosn.lc/DD 2 nŒcosn.lc/Ctannlsinn.lc/C

F

P

:(13.34)

Solution of these equations is

D 1 D

Fsinnc

Pnsinnl

;D 2 D

Fsinn.lc/

Pntannl

:

Displacements at any point of the left part of the beam

y 1 D

Fsinnc

Pnsinnl

sinnx

Fc

Pl

x: (13.35)

Knowing (13.35) we may construct the equations for slope and internal forces.
In particularly, the bending moment for left part of the beam

M.x/DEI

d^2 y

dx^2

DEI

Fnsinnc

Psinnl

sinnx: (13.35a)

Limiting case

If a forceFis placed at the middle span.cD0:5l/, then for sectionxD0:5l
from (13.35)and(13.35a), we get the maximum deflection and bending moment

y



l

2



D

Fl^3

48 EI

3.tan##/

#^3

;#D

l

2

r

P

EI

D

nl

2

;

M



l

2



D

Fl

4

tan#

#

:

If a compressive forceP! 0 ,then

3.tan##/

#^3

!1;

tan#

#

!1;soy



l

2



!

Fl^3

48 EI

andM



l

2



!

Fl

4

:

Let us evaluate numerically effect of the compressive forceP. Since a critical
force for simply supported column is

PcrD

2 EI

l^2

;

then the dimensionless parameter#may be rewritten as

#D

2

s

P

Pcrit

: