MAXIMUM BUCKLING LOAD
OF LATTICE SPACEFRAMES
AND TOPICS ON
IMPERFECTIONSENSITIVITY

By PS. CHEN and M.KAWAGUCHI

Peishan Chen Dr.Eng., Hachinohe Tech.

Concept of Buckling

Peishan Chen Dr.Eng., Hachinohe Tech.

Concept of Buckling

0D
Peishan Chen Dr.Eng., Hachinohe Tech.

The Optimization
for Maximum Buckling Load

M axmize BSubject to hX, U   0gX, U   0

B
0D

Peishan Chen Dr.Eng., Hachinohe Tech.

To Find the Bucking LoadKdBKd  fd0TB   uDu 1

Peishan Chen Dr.Eng., Hachinohe Tech.

Bucking Load in Explicit Function

Kd  f0, f i  0bi   1 1
 f i , f i  0Nb f 1T b K du

Tuu

Peishan Chen Dr.Eng., Hachinohe Tech.

Objective Function for Optimization b K duTuuTB   uu 1STuu
bKdi i ;ii

i  1,2,, l u 1

Peishan Chen Dr.Eng., Hachinohe Tech.

The Mathematical ModelM aximize F   w Subject to :h  X, U   0 ; ( j = 1,..., s)
g  X, U   0 , (k = 1,..., t )lii 1ijkSuubKdi i ;iiTi  1,2,, l u 1

Peishan Chen Dr.Eng., Hachinohe Tech.

To Find the Optima
objective function

The concept of
the surface of optima

surface of optima

initial shapeparameter B
path of iteration
parameter A

a plane cutting
line
constraints

Peishan Chen Dr.Eng., Hachinohe Tech.

A Local Optima, OK ?
X1X2

X 1X 2

A Global Optima
A Local Optima
Peishan Chen Dr.Eng., Hachinohe Tech.

Direction to Change the ShapeTTu 1u 1  b T ,i  K u d u  b T  K
 u du,i,iTTTuu b ,i f  b  K ,i d

i  1,2, n  mu 1 ,i  b j ,i f j  b j ,i K jl ,i d l
Peishan Chen Dr.Eng., Hachinohe Tech.

The Process of the Optimization
0D

Peishan Chen Dr.Eng., Hachinohe Tech.

Numerical Analysis ExampleaPeishan Chen Dr.Eng., Hachinohe Tech.

The Analysis CasesCase 1: Full area loading mode;

Change the shape design parameters;
All the member sections are kept constant.

Case 2: Full and Half area loading modes;
Change shape design parameters and themember sections.

Peishan Chen Dr.Eng., Hachinohe Tech.

The Result of Case 1

optimum shapeload parameter

initial shape

8.00

6.08

7.2

6.00
4.00

initial shape

2.00

optimum shape

0.00

The buckling load parameter increase 18.42%
Peishan Chen Dr.Eng., Hachinohe Tech.

Convergence of the Numerical Analysis

objective function

7.0
6.5

Case 16.0
5.5
13579 11 13 15 17 19 21
iteration step

Peishan Chen Dr.Eng., Hachinohe Tech.

load parameter

Load-Displacement Curves

Case 18765a43

init ial shape2

opt imum shape100

2468displacement (cm)The Load-Displacement Curves at Point a
Peishan Chen Dr.Eng., Hachinohe Tech.

load parameter

The Result of Case 2
10.00

8.071
6.08

5.00

initial shape
optimum shape

0.00

load parameter

Full area loading case ;The buckling load parameter increased 32.75%
7.247

8.00
6.00

5.679

4.00

initial shape

2.00

optimum shape

0.00

Half area loading case
The buckling load parameter increased 27.61%
Peishan Chen Dr.Eng., Hachinohe Tech.

Load-Displacement Curves Case 2
load parameter

987654Full area loading case

32optimum shape1

initial shape002468displacement (cm )The Load-Displacement Curves at Point a
Peishan Chen Dr.Eng., Hachinohe Tech.

Load-Displacement Curves Case 2
load parameter

8765Half area loading case432

optimum shape1

initial shape00246810displacement (cm )The Load-Displacement Curves at Point a
Peishan Chen Dr.Eng., Hachinohe Tech.

Convergence of the Numerical Analysis

8Structural Volume (cm3)load-parameter

106
full-loading case4

half-loading case
200

123iteration step

4

The convergence of theoptimization iteration

5670000
650000630000610000590000570000123456iteration step

The change of structural volume

Peishan Chen Dr.Eng., Hachinohe Tech.

ImperfectionBuckling load of aperfect shape may
be more sensitive toits shapeimperfection. ? ? ?

1
2
30

1   2   3
DPeishan Chen Dr.Eng., Hachinohe Tech.

load parameterImperfection for Case 17Λ =5.990
d=6.365

65

Case 1: Change the shapedesign parameters only.

Λ =5.127d=9.374

43

Ratio of the maximum
imperfection point in secondring is R1 :   1 : 0.0005

2
1002

4

68

optimum shape1012 14 16 18
displacement (cm)Full Area Loading

initial shape

Peishan Chen Dr.Eng., Hachinohe Tech.

load parameter

Imperfection for Case 28

Λ=6.905

d=9.462

7

Case 2: Change shapedesign parameters and allof the member sections.

6
Λ=5.846d=6.719

5

Ratio of the maximum
imperfection point in secondring is R1 :   1 : 0.0005

432

Full Area Loading

100246810121416

displacement (cm)optimum shapeinitial shape

The optimum shape withmaximum buckling load
may not be sensitive toits shape imperfection.
Peishan Chen Dr.Eng., Hachinohe Tech.

What is the reason ?

initial shape

optimum shape

Changes in member sections

Peishan Chen Dr.Eng., Hachinohe Tech.

Difference Between
Shells and Space Frames
ShellsKy

Space frames

Kx’

Kx’

Ky
Orthotropic

Kx

Ky’

Kx

Ky’

1. Built by thin continuum in curvature

1. Built by discrete members

2. The stiffness Kx and Ky are continue
respectively and related to each other.

2. The stiffness Kx and Ky are discontinue
respectively and independent with each other.

3. Changes in thickness lead to the changes
in bending stiffness.

3. Changes in member sections cause slightchanges in global bending stiffness.
Peishan Chen Dr.Eng., Hachinohe Tech.

Geometrical Influence
on Buckling Behaviors
PP

No stiffness
reduction

Do notBuckleStiffness reduction:
Limit point
Bifurcation point

BuckleδδPPδδ

Shape and Stiffness  Buckling ?
Peishan Chen Dr.Eng., Hachinohe Tech.

Influence on Shape and Stiffness

GeometricalChanges

Stiffness
Changeby:

Shape & thin
Isotropic
thickness

Shell:
imperfection
Sensitive

Shape & Isotropic
discontinue
member sections

Space frame:
May not
Sensitive toimperfection

Isotropic

Imperfection
Sensitive

orthotropic

May not
Sensitive toimperfection

Mechanical andMaterial Changes

Peishan Chen Dr.Eng., Hachinohe Tech.

An Inference on the Imperfection
Sensitivity
For a lattice space frame as well as ashell with orthotropic material, theoptimum shape with maximum
buckling load may not be sensitive to
its geometrical imperfection.

Peishan Chen Dr.Eng., Hachinohe Tech.

Thank you !

Hachinohe
Hongqiao BridgeJapanKaifeng
China
Peishan Chen Dr.Eng., Hachinohe Tech.