Advanced Methods of Structural Analysis

498 13 Stability of Elastic Systems

Let’s the compressive forcePand critical forcePcrare related as follows:

PDkPcrDk

2 EI

4l^2

;

wherekis any positive number,k 1. In this case the dimensionless parameter is

Dl

r

P

EI

D

2

p

k:

Coefficients'yand'Mfor different parameterskare presented in Table13.8.

Table 13.8 Coefficients'yand'Mfor different parameterskDP=Pcr

Parameterk

0.0 0.2 0.4 0.6 0.8 1.0

'y 1.0 1.2467 1.6576 2.4792 4.9434 1

'M 1.0 1.2051 1.5453 2.2234 4.2562^1

Bolded data corresponds to casePD 0 .EvenifkDP=PcrD0:2, the transver-
sal displacement at free end and bending moment at the fixed support are 24.6% and
20.5% higher than for same beam without compressive load.
Some important cases in the expanded form are presented in Table A.23; note
that this table contains formulas in terms of parameter

DnlDl

r

P

EI

:

This table allows performing the analysis for limiting cases.PD0/. For example,
for pinned-clamped beam subjected to lateral concentrated loadPonly we get

lim

! 0

M.l/DFllim

! 0

sin



2



1 cos



2



sincos

D

3

16

Fl:

This result is presented in Table A.3, pos. 3 foruD0:5.

Summary. Beam-column analysis allows determining the real deflections and in-
ternal forces, which are much higher than those, which would be determined without
taking into account compressive load. This is an essence of theP-delta analysis. Ne-
glecting of the compressive forces may lead to failure of a structure; therefore, for
high and multileveled structures, analysis on the basis of deformable design diagram
(P-delta analysis) is necessary.
ForP-delta analysis we can use two different approaches. In the first approach,
we need to complete and solve the second-order differential equation of the beam for
each specified loading. In the second approach (initial parameters method), we use
the once derived formulas for displacements, slope, bending moments, and shear.
These formulas can be used for any loads and procedure integration do not required.