Advanced Methods of Structural Analysis

6.2 Initial Parameters Method 149

M.x 2 /DP.x 2 aP/;

M.x 3 /DP.x 3 aP/M.x 3 aM/^0 :

4.Integration of differential equation should be performedwithout opening the
parenthesis.
All of these conditions are called Cauchy–Clebsch conditions.
Initial parameters method is based on the equationEI y^00 DM.x/. Integrating
it twice leads to the following expressions for slope and linear displacement

EID

Z

M.x/dxCC 1 ;

EIyD

Z

dx

Z

M.x/dxCC 1 xCD 1 : (6.1)

The transversal displacement and slope atx D 0 areyD y 0 , D  0 .These

displacements are called the initial parameters. Equations (6.1) allow getting the

constants in terms of initial parameters

D 1 DEIy 0 andC 1 DEI 0 :

Finally (6.1) may be rewritten as

EIDEI 0 

Z

M.x/dx;

EIyDEIy 0 CEI 0 x

Z

dx

Z

M.x/dx: (6.2)

These equations are called the initial parameter equations. For practical purposes,

the integrals from (6.2) should be calculated for special types of loads using the

above rules 1–4. These integrals are presented in Table6.1.

Ta b l e 6. 1 Bending moments in unified form for different type of loading

M

aM

x

y

x

aP

P

x

y

aq

q

x

y

ak

k=tanb

x

y

b

M.x/ ̇M.xaM/^0 ̇P.xaP/^1 ̇

q.xaq/^2

2

̇

k.xak/^3

2  3

R

M.x/dx ̇M.xaM/ ̇

P.xaP/^2

2

̇

q.xaq/^3

2  3

̇

k.xak/^4

2  3  4

R

dx

R

M.x/dx ̇

M.xaM/^2

2

̇

P.xaP/^3

2  3

̇

q.xaq/^4

2  3  4

̇

k.xak/^5

2  3  4  5