PSI - Issue 6

L.M. Kagan-Rosenzweig / Procedia Structural Integrity 6 (2017) 216–223 L.M. Kagan-Rosenzweig / Structural Integrity Procedia 00 (2017) 000–000

219

4

P P P −

M ∆ =

∆

.

(13)

M

2

1

c

Combining the above equations, we get the required formula:

/ P P P

0 0 0 0 w w w ∆ = − .

0

0 0 0 w a b x

= +

( ∆ + +

(14)

,

)

M M

−

1

c

In what follows the lower support is assumed to be immovable, that is 0 0 , a b of this formula are calculated from the boundary conditions, have the meaning of specific (divided by the force P ) support reactions at the lower end, arising due to deflection 0 w . For the cantilever rod it turns out that 0 1 0 a w = − , 0 0 = b . 0 0 0 = w and 0 0 w w ∆ = . Constants

0 0 = a ,

0 0 = b .

For the hinged rod

In these two cases Eq. (14) converts into known approximate formulas. When the rod is fixed at the bottom and hinged at its upper end, constants

0 0 , a b provide the zero moment and

deflection at the upper end. Applying the Maxwell-More formula, we obtain

l

l

0

2

− EI w x l x ( )(

− EI l x

)

(

)

 0

 0

0

0

0

a = −

=

lb

.

(15)

,

b

dx

dx

For the rod with fixed both ends, linear and angular deflections of the lower end with fixed upper one are equal zero, constants 0 0 , a b are the solution of the system

l

l

l

l

l

l

0

2 EI x

0

1

( )

( )

EI x

EI w x

EI x

EI xw x

0 

0 

0 

0 

0 

0 

0

0

0

0

⋅ dx a

+

⋅ dx b

+

=

⋅ dx a

+

⋅ dx b

+

=

,

.

(16)

0

0

dx

dx

EI

3. The rod compressed by a system of forces

The system of concentrated and distributed compressive forces is specified by the load parameter K . Concen trated forces i i P Kp = , n i 1,2..., = are applied in sections with coordinates i x x = ; ( ) ( ) Kr x R x = is distributed compressive force. 0 0 P Kp = is the vertical reaction at lower support. c K is the critical value of K . Let 0 w be the deflection in the absence of compressive load. We divide the rod into two pieces and calculate the moment ( ) 0 m Kf w = in the section of curved rod due to vertical load only. Such calculation of m does not provide

( ) 0 f w can be written in two different forms, depending we consider the upper or the

equilibrium, so the function

lower part of the rod:

x  0

n

i  = 1

0

0

0

0

0

0

0

f w p w w ( ( ) =

) − −

p w w ( −

−

−

(17.1)

)

r y w x w y dy ( )] ( )[ ( )

0

0

i

i

x x i >

l

n

i  = 1



0

0

0

0

0

=

p w w ( −

+

−

(17.2)

( )

)

r y w x w y dy ( )] ( )[ ( )

f w

i

i

x x i <

x

( ) 0 f w affects

Eq. (17.1) and (17.2) differ linearly with respect to x , so that the choice of particular expression for