Advanced Methods of Structural Analysis

(Jacob Rumans) #1
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
Free download pdf