Advanced Methods of Structural Analysis

13.6 Compressed Rods with Lateral Loading 495

Differential equation of elastic curve is

EI

d^2 y

dx^2

DM.x/;

where the bending moment at any sectionxis

M.x/DP.yy 0 /CM 0 CQ 0 x

X

Fi.xai/

qx^2

2

:

Differential equation becomes

d^2 y

dx^2

Cn^2 yD

1

EI



M 0 CQ 0 xPy 0 

X

Fi.xai/

qx^2

2

;nD

r

P

EI

:
(13.36)
Solution of this equation is

yDC 1 cosnxCC 2 sinnxC

Cy 0 

1

n^2 EI



M 0 CQ 0 x



C

1

n^3 EI

P

FiŒn .xai/sinn.xai/



q

n^4 EI



1 

n^2 x^2

2

cosnx
(13.36a)
The first and second terms of this expression are solution of homogeneous equa-
tion (13.36), while other terms are partial solution of nonhomogeneous equation.
For calculation of unknownsC 1 andC 2 , we can use the following boundary con-
ditions: atxD 0 initial displacement and slope areyDy 0 andy^0 D 0 .These
conditions lead to

C 1 D

M 0

n^2 EI

andC 2 D

1

n

 0 C

Q 0

n^2 EI

!

:

Substitution of these constants into expression foryand differentiation with
respect toxleads to the following formulas for displacement, angle of rotation,
bending moment and shear:

y.x/Dy 0 C 0 

sinnx

n



M 0

EI



1 cosnx

n^2



Q 0

EI



nxsinnx

n^3

CyI

.x/D

dy

dx

D 0 cosnx

M 0

EI



sinnx

n



Q 0

EI



1 cosnx

n^2

C:

M.x/DEI

d^2 y

dx^2

D 0 EInsinnxCM 0 cosnxCQ 0 

sinnx

n

CMI

Q.x/D

dM

dx

D 0 EIn^2 cosnxM 0 nsinnxCQ 0 cosnxCQ: (13.37)