13.6 Compressed Rods with Lateral Loading 499
13.6.3 P-Delta Analysis of the Frames..............................
Analysis of a structure on the basis of deformable design diagram (P-delta analysis)
contains two steps. The first step presents the classical analysis of structure on the
basis of nondeformable scheme. If a frame is subjected to arbitrary set of external
forces, including compressive forces at the joints, then on the first stage of analysis
these compressive forces should be omitted. This classical analysis is performed by
any appropriate methods, which were discussed previously. For each member of the
frame, the axial forces must be calculated.The second step begins from calculation
of axial forces with taking into account external axial compressive forces. For each
member, dimensionless parameter of compressive load
Dlr
P
EIshould be calculated. After that, the internal force diagrams of the first step should
be reconstructed taking into account compressive load. The displacement method
on this stage is more preferable. Unit reactions depend on parameter; the unit
reactions are presented in Table A.22. The free terms of canonical equations depends
on lateral loads. The following example demonstrates application of a beam-column
theory for detailed analysis of two-span sidesway frame.
Example 13.7.The frame in Fig.13.25a is subjected to horizontal forceFat the
level of the cross bar and compressed forcesP. The cross bar of the frame is con-
nected with vertical members by means of the hinges. The stiffnesses of the columns
and cross bar areEIand 2EI. Construct the internal force diagrams. Cross section
for vertical members is W15037.ID22:2 106 mm^4 /, the modulus of elasticity
ED 2 105 .N=mm^2 /.
Solution.The primary system of the displacement method is shown in Fig.13.25b.
The introduced constraints are 1 and 2. The primary unknowns are angular dis-
placementZ 1 of constraint 1 and linear displacementZ 2 of constraint 2. The
bending stiffness per unit length for members areivert D EI=5 D 0:2EIand
ihorD 2 EI=6D0:333EI.
Stage 1. Analysis of the frame on the basis of nondeformable design diagram. In
this step, we need to provide analysiswithoutthe axial compressive forcesN.This
analysis is presented in Chap. 8, Example 8.1. The internal forces are the following:
the bending moments at the fixed supports are
MADMBDMCD 50 kNm;the axial forces at the columns areNADNBDNCD 0 and axial forces at the left
and right cross bar are
NleftD 20 kN;NrightD 10 kNStage 2. Analysis of the frame on the basis of deformable design diagram. This step
should be performed taking into account axial forcesN. Axial forces that arise in