Advanced Methods of Structural Analysis

13.6 Compressed Rods with Lateral Loading 497

SinceQ 0 D0; M 0 D 0 ,then(13.37) become

y.x/Dy 0 C 0 

sinnx

n

C

F

n^3 EI

.nxsinnx/I

.x/D 0 cosnxC

F

n^2 EI

.1cosnx/ :

(13.39)

These equations contain two unknown parameters. They are 0 andy 0. Boundary

conditions are:

1.AtxDl(supportB) the slope of elastic curveD 0 ,so

.l/D 0 cosnlC

F

n^2 EI

.1cosnl/D0;

which leads immediately to the slope at the free end

 0 D

F

n^2 EI

1 cosnl

cosnl

D

Fl^2

2 EI

2.1cos/

^2 cos

;DnlDl

r

P

EI

: (13.40)

2.AtxDlthe vertical displacementyD 0 ,so

y.l/Dy 0 C 0 

sinnl

n

C

F

n^3 EI

.nlsinnl/D0:

Taking into account (13.40), the vertical displacement at the free end becomes

y 0 D

Fl^2

^2 EI

1 cos

cos



sin

n



F

n^3 EI

.sin/

D

Fl^3

^3 EI



1 cos

cos

sin.sin/

D

Fl^3

3 EI

3.tan/

^3

D

Fl^3

3 EI

'y:

(13.41)

If a beam is subjected to lateral forceFonly, then a transversal displacement at

the free end equalsFl^3 =3EI. However, if additional axial forcePacts then the

factor

'yD

3.tan/
^3
must be included.
3.The moment at clamped support equals

M.l/D 0 EInsinnlCMD

Fl^2

^2 EI

1 cos

cos

EInsin

F

n

sinDFl

tan



DFl'M:

(13.42)