PSI - Issue 6

L.M. Kagan-Rosenzweig / Procedia Structural Integrity 6 (2017) 216–223 L.M. Kagan-Rosenzweig / Structural Integrity Procedia 00 (2017) 0 0–000

221

6

0

0

/ 2 2

M =

=

ql

is the moment's module at support in the absence of compression, deflection

. Constants are

w

0

0

0

as follows: 0 0 = b . For the load in Fig. 2, a), Eq. (14) and Eq. (19) coincide, move to a known result 0 1 0 a w = − ,

/ P P P

0

0

0

= +

w w −

.

(20)

(

)

M M

1

−

1

c

When compressive force amounts to 90% of a critical one, exact and approximate moments at support differ in 3.5%. The load in Fig. 2, b) is not covered by the traditional Eq. (20), we must apply Eq. (19). Let's take the sum of compressive forces as the load parameter K , parameter m sets the forces ratio. In Eq. (17.1) 2 = n , 1 0 = p , p m = 1 , p m = − 1 2 , l x = α 1 , l x = 2 , 0 = r . The error of the approximate solution and the critical K value c K depend on m and α . When K amounts to 90% of c K , the error of the moment calculation is 1-3%. For the load in Fig. 2, c), the load parameter is Rl K = . In Eq. (17.2) it is necessary to put l r 1/ = , 0 = i p . For c K K 0.9 = , the difference between exact and approximate moments at support is 1.8%.

P

q

l

R

a )

b )

Fig. 3. Pinned-fixed rod

4.2. Statically indeterminate rod of constant cross-section

( ) 0 f w is

Eq. of type (1) are not applicable to the rod in Fig. 3. We are to use Eq. (14), (19). In Eq. (19) function

0 0 w w ∆ = . In the absence of compression,

0

/ 8 2

M =

ql

taken in the form Eq. (17.2);

is the moment's module at the

0

0 0 , a b are calculated according to Eq. (15). For load in Fig. 3, a), we obtain

support. Constants

b w x l x dx l ) ( )( 0 0 −

3

240 1

l

= 

=

.

3

EI

0

If c P P 0.9 = , the moment at support increases in 6.5 times, but its exact and approximate values differ by 6.7%. If c P P 0.8 = , the difference is reduced to 4.8%. Pay attention that calculations are made for a very high level of compressive load not found in structural design. For the load in Fig. 3, b), load parameter is taken as Rl K = , so that 0 = i p , l r 1/ = . Eq. (17.2) takes the form