PSI - Issue 6

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

220

5

only 0 0 , a b values, but does not affect the result. In the equations above, if condition i x x > (

i x x < ) written below

the line is violated, the corresponding term is to be omitted. When solving a particular problem, we choose a convenient expression for

( ) 0 f w , then write the first-order cor

rection

] 0 1 M K f w a b x + + ∆ = . [ ( ) 0 0

(18)

0 0 , a b keep the meaning of the specific support reactions at the rod's end (at the bottom or top end ( ) 0 f w choice), they are found by formulas for the rod with a force at its top, replacing in these

Here, constants

depending on the

( ) 0 f w .

0 w ∆ with function

formulas deflection

Eq. (14) takes the form

/ K K K

0

] [ ( ) f w a b x + + 0 0 0

= +

.

(19)

M M

−

1

c

Eq. (19) is deduced the same way as the Eq. (14). Now, however, the following more complicated equation is used instead of Eq. (2):

EI N

′ +

M w R q − ′ = −

.

M

In it N is the projection of forces on one side of the section onto the axis of the stainless rod. Note that both Eq. (14), (19) relate to the rod of variable cross-section. Below are examples of beam-columns, bent by uniformly distributed transverse load q , illustrating the accuracy of approximate formulas.

4. Examples

4.1. Cantilever rod

P

K (1- m )

q

R

Km

l

α l

c )

a )

b )

Fig. 2. Cantilever rod

The rod of length l is loaded by transverse load q and by three different systems of vertical forces (Fig. 2).