Advanced Methods of Structural Analysis

494 13 Stability of Elastic Systems

ForPD0:36Pcritwe get

#D0:94248and

3.tan##/

#^3

D1:555;

i.e., a small compressive loadincreases the deflection at forceFby 55%. In this case

tan#

#

D1:4604;

so the maximum bending moment increases by 46%. Thus, compressive force has

unfavorable effect on the state of the beam-column and therefore,P-delta analysis

should not be ignored.

Notes:

1.The displacement andlateral force Faccording to (13.35) are related by linear

law. Since the axial forcePappear in parameternD

q
P
EI, then the displace-
ment andaxial compressedforcePin the same equation are related according
to nonlinear law. It means that superposition principle is applicable only for lat-
eral loads.
2.Equation (13.35a) may be treated as the expression for influence line for bending
moment of a simple-supported compressed beam. For this purpose, we need to
consider the sectionxas a fixed one, while a location of the unit forceFis
defined by a variable parameterc.

13.6.2 Initial Parameters Method....................................

This method is effective forP-delta analysis in case of general case of beam-column

loading. A straight element is subjected to axial compressed forcePas well as

lateral loadsFiand uniformly distributed loadq(Fig.13.23a); dotted line shows

the initial nondeformable position (INDP) of the element; the initial parameters are

y 0 ; 0 ;M 0 ,andQ 0. The shearQ 0 is directed to perpendicular to nondeformed

axis of the beam.

Q 0

x

x

y

y 0

M 0

P

q 0

Fi

q

y

ai

INDP

Fig. 13.23 Loading of the beam-column and positive initial parameters