effective buckling length of columns in sw … · effective buckling length of columns in sway...

EFFECTIVE BUCKLING LENGTH OF COLUMNS IN SWAY

FRAMEWORKS: COMPARISONS

4.1 Objectives

In the present context, two different approaches are employed to determine the value the

effective buckling length effX mem

cn,L of a column mem

cn about the local axis (X). The first is

the simplified approach to BS 5950, while the second is a more accurate method. This

chapter answers two questions;

(i) when does the value of the effective buckling length factor memc

memc

Codeeff,

X nn,LL

determined by the BS 5950 approach differ from the effective buckling length factor

memc

memc

DMeff,

X nn,LL calculated by the direct method?

(ii) what are reasons for the different results of memc

memc

effX nn,

LL when applying the

different approaches?

The elastic stability of multi–bay single–storey frameworks permitted to sway

are investigated. Here, comparisons have been made between memc

memc

Codeeff,

X nn,LL and

memc

memc

DMeff,

X nn,LL where mem

cn refers to the column member under consideration.

IV

Effective Buckling Length of Columns in Sway Frameworks: Comparisons

101

The developed FORTRAN program for the stability analysis has been used and

verification of results has been performed using ANSYS.

4.2 Compar isons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL

The stability analysis of structural steel frameworks is based on the following

assumptions:

1. The framework material is perfectly elastic.

2. Buckling out plane of the framework is prevented.

3. The effect of the axial deformation of members produced by second order forces is

neglected.

4. The effect of the second order variations in the axial forces of framework members

due to sidesway is neglected when determining the stability functions.

5. The deflections of columns and beams are small, thus the relationship between the

bending moment and the curvature can be approximately expressed by the second

order differential equation discussed in Section 3.5.1.

4.2.1 Example 1: Single–bay single–storey framework

Consider the rectangular fixed base framework ABCD shown in Figure 4.1a which is

permitted to sway laterally. The framework is subjected to two vertical loads P at it

corners (B and C) and horizontal force αP where α equals 0.01. The columns of the

framework have different types of cross sections. The stability functions of left hand

side (LHS) column, column 1, are different from those of right hand side (RHS)

column, column 2. When the vertical loads P reach their critical values Pcr, any small

sidesway can take place. This results in two opposite horizontal forces (H1 and H2)

when using the finite element analysis. The difference between (H1 and H2) is however


102

neglected when using the direct method. The operations of sway and rotations are then

built–up for every member of the framework separately as shown in Figure 4.1c

corresponding to the distorted configuration assumed in Figure 4.1b of the framework.

Since at the critical load there are no external moments set up at the corners to

keep the framework in the distorted position, the sum of moments

0BCBA =+∴ MM ,

0242

2b1b1

111 =+++∴ θθθ KKLHm

Kn .

2b4 θK

222 θKn – LHm

22

222 θKO− – LHm

22

2b2 θK

CD∆BA∆

111 θKO− + LHm

21

111 θKn + LHm

21

PP

PP

HH

2θ1θB C

DA

αPbK

2K1K

bL

L

P P

1b4 θK

(b) Distorted configuration(a) Loading pattern

(c) Operations of rotation and sway

Figure 4.1. Single–bay single–storey framework

1b2 θK


103

Dividing this equation by 1K :

02

24

1

12

1

b1

1

b1 =��

��+��+��

�� +∴K

LHm

K

K

K

Kn θθ and (4.1)

0CBCD =+∴ MM ,

024H2

1b2b2

222 =++−∴ θθθ KKLm

Kn .

Dividing this equation by 1K :

02

42

1

22

1

b

1

221

1

b =��−��

�� ++��∴

K

LHm

K

K

K

Kn

K

Kθθ . (4.2)

Since the sidesways at joints B and C are equal, the sum of the corresponding

components of pure–shear and no–shear sway of the two columns can be expressed as

CDAB ∆=∆ ,

222

22

2

111

11

1

)(122)(122 K

LH

CS

mm

K

LH

CS

mm ��

++��

��=��

+−��

��∴ θθ ,

0)(12)(1222 12

1

22

2

11

12

21

1 =��

++

+−��

��−��∴

K

LH

K

K

CS

m

CS

mmmθθ . (4.3)

Eliminating the three unknowns 121 and, KLHθθ from (4.1), (4.2) and (4.3), the

general form (4.4) of the critical buckling load is obtained:

0

)(12)(1222

2

42

2

24

det

2

1

22

2

11

121

2

1

b

1

22

1

b

1

1

b

1

b1

=

��

�

�

�

��

�+

++

−��

�−��

��

��

�−��

��

+��

��

��

��

��

+

K

K

CS

m

CS

mmm

m

K

K

KKn

K

K

m

K

K

K

Kn

. (4.4)


104

The solution of this form is obtained following the flow chart given in Figure 3.13

by the trial and error method, where the variables are the height of the columns L, the

span of the beam bL and the properties of the cross section of each member of the

framework. These properties can be taken from Steel Construction Institute (1985).

To verify the results obtained from equation (4.4), the elastic critical load of two

single–bay single–storey frameworks, having the same dimensions of L= 6 m, bL = 24

m and the sections described in Table 4.1, have been compared to the results obtained

using ANSYS, where E = 205 kN/mm2.

Table 4.1. Dimensions and cross sections of single–bay single–storey framworks

Case study LHS column RHS column Beam

Framework 1 305 × 305 × 97 UC 305 × 305 × 97 UC 457 × 191 × 67 UB

Framework 2 203 × 203 × 46 UC 305 × 305 × 97 UC 457 × 191 × 67 UB

The ANSYS data file is prepared in a sequence of three steps: preprossesing,

solution and postprocessing. In the preprocessing stage, nodes and members of the

framework are defined, where each member is divided into several elements to get

adequate accuracy. Four different divisions into elements for each member are studied:

• Case 1: five elements for each member,

• Case 2: ten elements for each member,

• Case 3: twenty elements for each member and

• Case 4: one hunderd elements for each member.

In the solution stage, the type of analysis required as well as the end conditions

and loading patteren are defined. Finally, ANSYS Parametric Design Language (APDL)

is used in the postprocessing stage to obtain the results.


105

Table 4.2 describes the comparisons between the critical load FEcrP , obtained using

ANSYS, and that DMcrP by using the direct method.

Table 4.2. Single–bay single–storey framework: verification of obtained results

Cr itical loads

FEcrP (kN) Case study

Case 1 Case 2 Case 3 Case 4

DMcrP

(kN)

Framework 1 6594.59 6594.36 6594.351 6594.346 6598.02

Framework 2 4081.257 4080.577 4080.534 4080.531 4087.6

To compare the effective length factors memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL , the

parameters L, bL and cross section of the beam are fixed while the cross sections of

columns are varied, thus, two sets of parameters are considered:

Set 1: L= 6 m, bL = 18 m and the cross section of the beam is 406 × 178 × 54 UB,

Set 2: L= 6 m, bL = 24 m and the cross section of the beam is 457 × 191 × 67 UB.

The value of memc

memc

DMeff,

X nn,LL is computed by

memc

memc

memc

1DMeff,

X

nn

n,

L

L

ρ= (4.5)

where memcn

ρ = memc

memc E n,n

PF , memcn

F is the axial force at the critical buckling load DMcrP

and 22E mem

cmemc

memc nnn,

LIEP π= .

Comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL are listed in Tables 4.3

and 4.4 for the framework having parameter sets 1 and 2 respectively. From these tables,

It can be observed that when the sections of columns are the same, the values of

memc

memc

Codeeff,

X nn,LL equal to their corresponding values of mem

cmemc

DMeff,

X nn,LL .

Table 4.3. Single–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL when using parameter set 1

LHS column RHS column

Direct method BS 5950 Direct method BS 5950 Case study

Cross section

1ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section

2ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 203 × 203 × 46 UC 0.7999 1.118 1.17 203 × 203 × 46 UC 0.7999 1.118 1.17

2 254 × 254 × 73 UC 0.6281 1.261 1.36 254 × 254 × 73 UC 0.6281 1.261 1.36

3 254 × 254 × 89 UC 0.5821 1.3106 1.32 254 × 254 × 89 UC 0.5821 1.3106 1.32

4 305 × 305 × 97 UC 0.4983 1.416 1.52 305 × 305 × 97 UC 0.4983 1.416 1.52

5 203 × 203 × 46 UC 1.1525 0.9315 1.17 254 × 254 × 73 UC 0.461 1.4728 1.36

6 203 × 203 × 46 UC 1.264 0.8894 1.17 254 × 254 × 89 UC 0.4031 1.575 1.32

7 203 × 203 × 46 UC 1.53 0.8084 1.17 305 × 305 × 97 UC 0.3143 1.783 1.52

8 254 × 254 × 73 UC 0.6786 1.213 1.36 254 × 254 × 89 UC 0.5410 1.359 1.32

9 254 × 254 × 73 UC 0.7960 1.120 1.36 305 × 305 × 97 UC 0.4087 1.564 1.52

10 254 × 254 × 89 UC 0.6770 1.215 1.32 305 × 305 × 97 UC 0.4361 1.514 1.52

Table 4.4. Single–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL when using parameter set 2

LHS column RHS column


Cross section

1ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section

2ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 203 × 203 × 46 UC 0.825 1.10 1.15 203 × 203 × 46 UC 0.825 1.10 1.15

2 254 × 254 × 73 UC 0.662 1.228 1.32 254 × 254 × 73 UC 0.662 1.228 1.32

3 254 × 254 × 89 UC 0.616 1.273 1.36 254 × 254 × 89 UC 0.616 1.273 1.36

4 305 × 305 × 97 UC 0.529 1.374 1.48 305 × 305 × 97 UC 0.529 1.374 1.48

5 203 × 203 × 46 UC 1.201 0.912 1.15 254 × 254 × 73 UC 0.4807 1.44 1.32

6 203 × 203 × 46 UC 1.319 0.87 1.15 254 × 254 × 89 UC 0.4207 1.541 1.36

7 203 × 203 × 46 UC 1.595 0.791 1.15 305 × 305 × 97 UC 0.327 1.747 1.48

8 254 × 254 × 73 UC 0.717 1.18 1.32 254 × 254 × 89 UC 0.571 1.32 1.36

9 254 × 254 × 73 UC 0.8418 1.089 1.32 305 × 305 × 97 UC 0.432 1.521 1.48

10 254 × 254 × 89 UC 0.7173 1.18 1.36 305 × 305 × 97 UC 0.4624 1.47 1.48


108

4.2.2 Example 2: Two–bay single–storey framework

In this section, the two–bay single–storey framework shown in Figure 4.2a is studied.

Following the same procedure described in Section 4.2.1, equations (4.6–4.10) can be

deduced, where the distorted configuration as well as the operations of rotations of the

framework are given in Figure 4.2b and 4.2c respectively:

�= 0BM ,

02

24

1

112

11

1

b1 =��

��+��+��

�� +∴K

LHm

K

K

K

Kn b θθ , (4.6)

�= 0CM ,

02

282

1

223

1

b2

1

b

1

221

1

b =��+��

��+�� ++��

��∴K

LHm

K

K

K

K

K

Kn

K

Kθθθ , (4.7)

�= 0EM ,

022

42

1

23

1

133

1

b

1

332

1

b =��−��

��−�� ++��

��∴K

LHm

K

LHm

K

K

K

Kn

K

Kθθ , (4.8)

EFAB ∆=∆ ,

0)(12

)(12)(1222

1

2

3

1

33

3

1

1

3

1

33

3

11

13

31

1

=��

+

−��

++

+−��

�� −�� ∴

K

LH

K

K

CS

m

K

LH

K

K

CS

m

CS

mmmθθ

(4.9)

and EFCD ∆=∆

.K

LH

K

K

CS

m

CS

m

K

LH

K

K

CS

mmm

0)(12)(12

)(1222

1

2

3

1

33

3

22

2

1

1

3

1

33

33

32

2

=��

++

+

−��

+−��

��−��∴ θθ

(4.10)


109

2b2 θK

EF∆CD∆BA∆

333 θKO− ( )LHHm

213

2+−

333 θKn ( )LHHm

213

2+−

3b2 θK

2b4 θK

2P P

E

F

3K

bK

3θ

21 HH +

2P

111 θKO− + LHm

11

2

111 θKn + LHm

11

2

PP

2PP

2H1H

2θ1θ

B C

DA

bK

2K1K

bL

L

1b4 θK

2b4 θK

1b2 θK

2b2 θK

222 θKn + LHm

22

2

222 θKO− + LHm

22

2

P

P

αP

3b4 θK

(a) Loading pattern

(b) Distorted configuration


Figure 4.2. Two–bay single–storey framework

bL


110

Eliminating the unknowns 1211321 and,,, KLHKLHθθθ from (4.6)–(4.10), the

critical buckling load (4.11) is derived:

0det

55

5444

534333

52423222

5141312111

=

��

�

�

��

�

�

.

..

...

....

.....

a

aa

aaa

aaaa

aaaaa

(4.11)

where 0425131 === ... aaa , 1

b11.1

4

K

Kna += ,

1

b3.22.1

2

K

Kaa == ,

21

4.1m

a = ,

1

b

1

222.2

8

K

K

K

Kna += ,

22

4.2m

a = , 1

b

1

333.3

4

K

K

K

Kna += ,

23

5.34.3m

aa −== ,

3

1

33

3

11

144

)(12)(12 K

K

CS

m

CS

ma . +

++

= , 3

1

33

354

)(12 K

K

CS

ma . +

= , and

3

1

33

3

2

1

22

255

)(12)(12 K

K

CS

m

K

K

CS

ma . +

++

= .

This equation (4.11) is used to obtain DMcrP when dimensions of the framework are

L = 8 m, bL = 20 m. The cross sections of members, given in Table 4.5, are assumed.

Table 4.5. Two–bay single–storey framework: assumed cross sections

Memeber Cross section

Outer columns 356 × 356 × 129 UC

Inner Column 254 × 254 × 89 UC

Beams 533 × 210 × 82 UB

The finite element model of the framework was built as illustrated in Section

4.2.1, then FEcrP was obtained to verify the results. Table 4.6 shows the comparison

between FEcrP and DM

crP assuming E = 205 kN/mm2 .

Symmetric


111

Table 4.6. Two–bay single–storey framework: verification of obtained results

Cr itical loads

FEcrP (kN)


DMcrP

(kN)

4312.172 4311.113 4311.045 4311.041 4317.13

Using (4.11) of the critical buckling load, the effective length factors

memc

memc

DMeff,

X nn,LL of columns of frameworks (8 cases) are computed assuming L = 8 m,

bL = 20 m and the cross section of beam is 533 × 210 × 82 UB. Then, the effective

length factors memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL for each member are compared as

given in Table 4.7. It is also assumed that the two outer columns, columns 1 and 3, have

the same type of cross section.

From this table, it can be deduced that the value of the effective length factor

memc

memc

Codeeff,

X nn,LL can be either greater or smaller than the value of the effective length

factor memc

memc

DMeff,

X nn,LL and this depends on the section properties of the framework

members, i.e. the bending stiffness of the members.

Table 4.7. Two–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL

Outer columns Inner column


Cross section

1ρ = 3ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section

2ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 254 × 254 × 89 UC 0.6049 1.285 1.18 254 × 254 × 89 UC 1.209 0.909 1.09

2 305 × 305 × 97 UC 0.5559 1.341 1.25 305 × 305 × 97 UC 1.1119 0.948 1.14

3 305 × 305 × 97 UC 0.602 1.28 1.25 305 × 305 × 118 UC 0.9697 1.015 1.18

4 305 × 305 × 118 UC 0.5287 1.375 1.3 305 × 305 × 118 UC 1.057 0.9726 1.18

5 356 × 356 × 129 UC 0.4782 1.446 1.4 356 × 356 × 129 UC 0.9564 1.022 1.23

6 356 × 356 × 129 UC 0.3397 1.715 1.4 254 × 254 × 89 UC 1.9103 0.7235 1.09

7 356 × 356 × 129 UC 0.3928 1.5955 1.4 305 × 305 × 97 UC 1.4225 0.838 1.14

8 356 × 356 × 129 UC 0.4223 1.538 1.4 305 × 305 × 118 UC 1.2303 0.9015 1.18

where 3

DMeff,3X

L

L ,=

1

DMeff,1X

L

L ,,

3

Codeeff,3X

L

L ,=

1

Codeeff,1X

L

L , and 321 LLL ==


113

4.2.3 Example 3: Three–bay single–storey framework

The effective length factors memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL of the columns of the

three–bay single–storey framework shown in Figure 4.3a are compared following the

same procedure described in Section 4.2.1. The general equation of the elastic critical

load is obtained:

0

00

000

0000

det

77

7666

756555

74645444

734333

623222

512111

=

��

�

�

��

�

�

.

..

...

....

...

...

...

a

aa

aaa

aaaa

aaa

aaa

aaa

(4.12)

where 1

b4.33.22.1

2

K

Kaaa === ,

21

5.1m

a = , 22

6.2m

a = , 23

7.3m

a = ,

24

7.46.45.4m

aaa −=== , 1

b11.1

4

K

Kna += ,

1

b

1

222.2

8

K

K

K

Kna += ,

1

b

1

333.3

8

K

K

K

Kna += ,

1

b

1

444.4

4

K

K

K

Kna += ,

4

1

44

4767565

)(12

-

K

K

CS

maaa ... +

=== ,

��

++

+−=

4

1

44

4

11

155

)(12)(12 K

K

CS

m

CS

ma . ,

��

++

+−=

4

1

44

4

2

1

22

266

)(12)(12 K

K

CS

m

K

K

CS

ma . and

��

++

+−=

4

1

44

4

3

1

33

377

)(12)(12 K

K

CS

m

K

K

CS

ma . .

The unknowns of this form are and, 12114321 KLH,KLH,,, θθθθ 13 KLH .

Symmetric

(c) Operations of rotation

and sway

2P 2P P P

3b4 θK 3b2 θK

BA∆ GJ∆

333 θKn + LHm

33

2

333 θKO− + LHm

33

2

2P

2P

3H

4θ

EF∆ CD∆

−− 444 θKO ( )LHHHm

3214

2++

444 θKn ( )LHHHm

3214

2++−

3b2 θK

2b4 θK

3θ

321 HHH ++

2P

111 θKO− + LHm

11

2

111 θKn + LHm

11

2

P P

P

2H 1H

2θ 1θ

1b4 θK

2b4 θK

1b2 θK

2b2 θK

222 θKn + LHm

22

2

222 θKO− + LHm

22

2

P

3b4 θK

2b2 θK

2P

4b2 θK 4b4 θK

J

G

4K

bKE

F

3K

bK B C

D A

bK

2K1K L

αP

bL bL bL

(a) Loading pattern


Figure 4.3. Three–bay single–storey framework


115

To verify the results when using the developed program, FEcrP and DM

crP , the

results are compared assuming dimensions of the framework L = 6 m, bL = 10 m and

the cross sections of its members are as given in Table 4.8.

Table 4.8. Three–bay single–storey framework: used cross sections


Outer columns 305 × 305 × 97 UC

Inner Columns 203 × 203 × 46 UC

Beams 457 × 191 × 67 UB

The finite element model of the framework was built as demonstrated in Section

4.2.1. The obtained results are then compared as shown in Table 4.9 assuming E = 205

kN/mm2.

Table 4.9. Three–bay single–storey framework: verification of obtained results

Cr itical loads

FEcrP (kN)


DMcrP

(kN)

3254.258 3252.923 3252.838 3252.832 3253.6

Applying (4.12) of the critical buckling load, 8 cases (Table 4.10) are investigated.

In this investigation, the effective length factors memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL are

obtained considering different types of cross sections of columns. The framework

dimensions L = 6 m, bL = 10 m and the cross sections of beams of 457 × 191 × 67 UB

are assumed. It is also assumed that outer columns, columns 1 and 4, have the same

cross section as well as inner columns, column 2 and 3.

Table 4.10. Three–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL

Outer columns Inner columns


Cross section

1ρ = 4ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section

2ρ = 3ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 152 × 152 × 37 UC 0.64145 1.248 1.03 152 × 152 × 37 UC 1.2829 0.8829 1.02

2 203 × 203 × 46 UC 0.6189 1.271 1.065 203 × 203 × 46 UC 1.237 0.8991 1.04

3 203 × 203 × 46 UC 1.056 0.9731 1.065 254 × 254 × 73 UC 0.8448 1.087 1.08

4 203 × 203 × 46 UC 0.9404 1.031 1.065 203 × 203 × 86 UC 0.9066 1.050 1.07

5 203 × 203 × 46 UC 1.608 0.788 1.065 305 × 305 × 97 UC 0.6606 1.23 1.15

6 305 × 305 × 97 UC 0.2615 1.955 1.27 203 × 203 × 46 UC 2.546 06267 1.04

7 305 × 305 × 97 UC 0.3411 1.712 1.27 203 × 203 × 86 UC 1.6009 07903 1.07

8 305 × 305 × 97 UC 0.3692 1.645 1.27 254 × 254 × 73 UC 1.437 0834 1.08

where 4

DMeff,4X

L

L ,=

1

DMeff,1X

L

L ,,

3

DMeff,3X

L

L ,=

2

DMeff,2X

L

L ,,

4

Codeeff,4X

L

L ,=

1

Codeeff,1X

L

L ,,

3

Codeeff,3X

L

L ,=

2

Codeeff,2X

L

L , and 4321 LLLL ===


117

4.2.4 Example 4: Four–bay single–storey framework

The next example to investigate is the four–bay single–storey framework shown in

Figure 4.4a. Following the same procedure described in Section 4.2.1, the general

equation (4.13) of the critical buckling load is obtained:

0

000

0000

00000

000000

det

99

9888

978777

96867666

9585756555

945444

834333

723222

612111

=

��

�

�

��

�

�

.

..

...

....

.....

...

...

...

...

a

aa

aaa

aaaa

aaaaa

aaa

aaa

aaa

aaa

(4.13)

where 1

b54433221

2

K

Kaaaa .... ==== ,

21

6.1m

a = , 22

7.2m

a = , 23

8.3m

a = , 24

9.4m

a = ,

25

9.48.47.46.4m

aaaa −==== , 1

b11.1

4

K

Kna += ,

1

b

1

222.2

8

K

K

K

Kna += ,

1

b

1

333.3

8

K

K

K

Kna += ,

1

b

1

444.4

4

K

K

K

Kna += ,

1

b

1

555.5

4

K

K

K

Kna += ,

5

1

55

5989787968676

)(12

-

K

K

CS

maaaaaa ...... +

====== , 1

b11.1

4

K

Kna +=

��

++

+−=

5

1

55

5

11

166

)(12)(12 K

K

CS

m

CS

ma . ,

��

++

+−=

5

1

55

5

2

1

22

277

)(12)(12 K

K

CS

m

K

K

CS

ma . ,

��

++

+−=

5

1

55

5

3

1

33

388

)(12)(12 K

K

CS

m

K

K

CS

ma . and

Symmetric

2b4 θK2b4 θK2b2 θK 2b2 θK 3b4 θK 3b2 θK 4b2 θK 4b4 θK

5θ

444 θKn + LHm

44

2


555 θKn ( ) LHHHHm

43215

2+++−

444 θKO− + LHm

44

2

2P

UV∆GJ∆

4H

2P

V

U G

J

2P

5K

bK

2P

333 θKn + LHm

33

2

333 θKO− + LHm

33

2

2P

2P

3H

4K

4θ

bK

2P

EF∆CD∆BA∆

555 θKO− ( ) LHHHHm

43215

2+++−

3b2 θK

2P P

E

F

3K

bK

3θ

4321 HHHH +++

2P

111 θKO− + LHm

11

2

111 θKn + LHm

11

2

P P

P

2H1H

2θ1θ

B C

D A

bK

2K1K L

1b4 θK 1b2 θK

222 θKn + LHm

22

2

222 θKO− + LHm

22

2

P

P

αP

bL

3b4 θK 4b2 θK 4b4 θK 5b2 θK 5b4 θK


(a) Loading pattern

Figure 4.4. Four–bay single–storey framework

bL bL bL


119

��

++

+−=

5

1

55

5

4

1

44

499

)(12)(12 K

K

CS

m

K

K

CS

ma . .

The unknowns of this form are ,,, 121154321 KLH,KLH,,, θθθθθ

13 KLH and 14 KLH .

To verify the results obtained from (4.13), the framework is modelled using

ANSYS and a stability analysis is carried out where dimensions of the framework are

L = 8 m, bL = 16 m and cross sections of members are assumed as given in Table 4.11.

Table 4.11. Four–bay single–storey framework: used cross sections


Columns 356 × 368 × 129 UC

Beams 457 × 191 × 82 UB

The finite element model was built as demonstrated in Section 4.2.1. The obtained

results are then compared as shown in Table 4.12 assuming E = 205 kN/mm2.

Table 4.12. Four–bay single–storey framework: verification of obtained results

Cr itical loads

FEcrP (kN)


DMcrP

(kN)

5347.766 5347.346 5347.320 5347.318 5351.361

Eight cases (Table 4.13) are studied where the framework dimensions of L = 8 m,

bL = 16 m as well as the beams sections of 457 × 191 × 82 UB are assumed. It is also

assumed that the two outer columns, columns 1 and 5, have the same section

as well as the inner columns, column 2, 3 and 4. .

Table 4.13. Four–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL



Cross section

1ρ = 5ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section 2ρ = 3ρ

= 4ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 203 × 203 × 86 UC 0.5514 1.346 1.12 203 × 203 × 86 UC 1.102 0.9525 1.07

2 254 × 254 × 107 UC 0.5066 1.404 1.21 254 × 254 × 107 UC 1.013 0.993 1.12

3 305 × 305 × 118 UC 0.4629 1.469 1.3 305 × 305 × 118 UC 0.9258 1.039 1.17

4 305 × 305 × 158 UC 0.42549 1.533 1.39 305 × 305 × 158 UC 0.8509 1.084 1.23

5 356 × 368 × 129 UC 0.42108 1.541 1.4 356 × 368 × 129 UC 0.8421 1.089 1.24

6 305 × 305 × 158 UC 0.43371 1.518 1.39 356 × 368 × 129 UC 0.8352 1.094 1.24

7 305 × 305 × 158 UC 0.358 1.671 1.39 305 × 305 × 118 UC 1.004 0.998 1.17

8 305 × 305 × 158 UC 0.2844 1.875 1.39 254 × 254 × 107 UC 1.258 0.891 1.12

where 5

DMeff,5X

L

L ,=

1

DMeff,1X

L

L ,,

3

DMeff,3X

L

L ,=

4

DMeff,4X

L

L ,=

2

DMeff,2X

L

L ,,

5

Codeeff,5X

L

L ,=

1

Codeeff,1X

L

L ,,

3

Codeeff,3X

L

L ,=

4

Codeeff,4X

L

L ,=

2

Codeeff,2X

L

L , and 54321 LLLLL ====


121

4.2.5 Example 5: Five–bay single–storey framework

Following the procedure described in Section 4.2.1, the critical buckling load equation

(4.14) of the five–bay single–storey framework shown in Figure 4.5 was obtained:

0

0000

00000

000000

0000000

00000000

det

1111

11101010

11910999

1181089888

117107978777

11610696867666

1156555

1045444

934333

823222

712111

=

��

�

�

��

�

�

.

..

...

....

.....

......

...

...

...

...

...

a

aa

aaa

aaaa

aaaaa

aaaaaa

aaa

aaa

aaa

aaa

aaa

(4.14)

where 1

b6554433221

2

K

Kaaaaa ..... ===== ,

21

7.1m

a = , 22

8.2m

a = , 23

9.3m

a = ,

24

10.4m

a = , 25

11.5m

a = , 2

- 611.610.69.68.67.6

maaaaa ===== ,

1

b111

4

K

Kna . += ,

1

b

1

2222

8

K

K

K

Kna . += ,

1

b

1

333.3

8

K

K

K

Kna += ,

1

b

1

444.4

4

K

K

K

Kna += ,

11

555.5

b8

K

K

K

Kna += ,

1

b

1

666.6

4

K

K

K

Kna += , ��

�

�

++

+−=

6

1

6

6

11

177

)(12)(12 K

K

CS

m

CS

ma

6. ,

6

1

66

6109118108981171079787

)(12

-

K

K

CS

maaaaaaaa ........ +

======== ,

6

1

66

61110119

)(12

-

K

K

CS

maa .. +

== ,

Symmetric

2P 2P 2P 2P P P

111 θKO− + LHm

22

2

4H 3H 2H1H

2P WY∆

Y

W

2b4 θK

6θ

555 θKO− + LHm

55

2

111 θKn + LHm

11

2 444 θKn + LH

m4

4

2

444 θKO− + LHm

44

2

555 θKn + LHm

55

2

6b2 θK

333 θKO− + LHm

33

2

222 θKn + LHm

22

2 333 θKn + LH

m3

3

2

5H

CD∆

6K

bK

2P


−666 θKn ( ) LHHHHHm

543216

2++++

UV∆GJ∆ 2P

V

U G

J

2P

5K

bK

2P 2P

4K

4θ

bK

2P

EF∆ BA∆

−− 666 θKO ( ) LHHHHHm

543216

2++++

3b2 θK

2P P

E

F

3K

bK

3θ

54321 HHHHH ++++

111 θKO− + LHm

11

2

P

2θ 1θ

B C

D A

bK

2K 1K L

1b4 θK

2b4 θK

1b2 θK

2b2 θK

P

P

αP

bL

3b4 θK

2b2 θK

4b2 θK

3b4 θK

4b4 θK

3b2 θK

5θ

4b2 θK 4b4 θK

5b2 θK 5b4 θK

5b4 θK 5b2 θK

6b4 θK

bL bL bL bL


(a) Loading pattern

Figure 4.5. Five–bay single–storey framework


123

��

�

�

++

+−=

6

1

6

6

2

1

22

288

)(12)(12 K

K

CS

m

K

K

CS

ma

6. ,

��

�

�

++

+−=

6

1

66

6

3

1

33

399

)(12)(12 K

K

CS

m

K

K

CS

ma . ,

��

�

�

++

+−=

6

1

66

6

4

1

44

41010

)(12)(12 K

K

CS

m

K

K

CS

ma . and

��

�

�

++

+−=

6

1

66

6

5

1

55

51111

)(12)(12 K

K

CS

m

K

K

CS

ma . .

The unknowns of this form are ,,,, 1211654321 KLH,KLH,,, θθθθθθ

13 KLH , 14 KLH and 15 KLH .

To validate the results obtained from (4.14), the framework is modelled using

ANSYS and a stability analysis is carried out considering dimensions of the framework

of L = 8 m, bL = 16 m and cross sections of members as given in Table 4.14.

Table 4.14. Five–bay single–storey framework: used cross sections


Columns 305 × 305 × 118 UC

Beams 457 × 191 × 82 UB

Following the same technique illustrated in Section 4.2.1 for building the

framework model (Appendix B), the obtained results FEcrP and DM

crP are compared as

given in Table 4.15 assuming E = 205 kN/mm2.


124

Table 4.15. Five–bay single–storey framework: verification of obtained results

Cr itical loads

FEcrP (kN)


DMcrP

(kN)

4198.16 4197.708 4197.680 4197.678 4201.118

Using (4.14), 8 cases (Table 4.16) are investigated to compare the effective factors

memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL where dimensions of the framework are L = 8 m,

bL = 16 m and cross section of beams of 457 × 191 × 82 UB are assumed. It is also

assumed that the two outer columns, columns 1 and 6, have the same of cross section as

well as the inner columns, columns 2, 3, 4 and 5. From this table, it can be extrated that

the value of the effective length factor memc

memc

Codeeff,

X nn,LL can be either greater or less than

the value of the effective length factor memc

memc

DMeff,

X nn,LL and this depends on the section of

the framework members, i.e. the bending stiffness of the members.

Table 4.16. Five–bay single–storey frame: comparisons between memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL



Cross section

1ρ = 6ρ 1

DMeff,1X

L

L , 1

Codeeff,1X

L

L ,

Cross section 2ρ = 3ρ

= 4ρ = 5ρ 2

DMeff,2X

L

L , 2

Codeeff,2X

L

L ,

1 203 × 203 × 86 UC 0.53321 1.369 1.12 203 × 203 × 86 UC 1.0664 0.9683 1.07

2 254 × 254 × 107 UC 0.4915 1.426 1.21 254 × 254 × 107 UC 0.983 1.008 1.12

3 305 × 305 × 118 UC 0.4501 1.49 1.3 305 × 305 × 118 UC 0.9003 1.053 1.17

4 305 × 305 × 158 UC 0.4144 1.553 1.39 305 × 305 × 158 UC 0.848 1.098 1.23

5 356 × 368 × 129 UC 0.4101 1.561 1.4 356 × 368 × 129 UC 0.8203 1.104 1.24

6 305 × 305 × 158 UC 0.4231 1.537 1.39 356 × 368 × 129 UC 0.8147 1.108 1.24

7 305 × 305 × 158 UC 0.3434 1.706 1.39 305 × 305 × 118 UC 0.963 1.019 1.17

8 305 × 305 × 158 UC 0.2666 1.936 1.39 254 × 254 × 107 UC 1.179 0.921 1.12

where 6

DMeff,6X

L

L ,=

1

DMeff,1X

L

L ,,

3

DMeff,3X

L

L ,=

4

DMeff,4X

L

L ,=

5

DMeff,5X

L

L ,=

2

DMeff,2X

L

L ,,

6

Codeeff,6X

L

L ,=

1

Codeeff,1X

L

L ,,

3

Codeeff,3X

L

L ,=

4

Codeeff,4X

L

L ,=

5

Codeeff,5X

L

L ,=

2

Codeeff,2X

L

L , and

654321 LLLLLL =====


126

4.3 Concluding remarks

In this chapter, the stability concept of steelwork has been investigated. The critical

buckling load equations of five multi–bay single–storey frameworks were obtained

using the direct method of analysis. To verify the results obtained using these equations,

the finite element package, ANSYS, was used. From the results presented in this

chapter, the observations can be summarized as follows:

1. It can be observed that there is a difference between the effective length factors

memc

memc

Codeeff,

X nn,LL determined by the BS 5950 approach and the effective buckling

length factor memc

memc

DMeff,

X nn,LL calculated by the direct method. This difference varies

according to the geometric dimensions and section properties of each member of the

framework under consideration. This difference is obtained because the simplified

method adapted by Wood (1974a) and presented in BS 5950 mainly depend on two

assumptions:

a) involving the no–shear stability function only, while the general case of sway is a

superposition of no–shear sway and pure–shear sway as discussed in Chapter 3.

b) the consideration of a limited framework which contains the column under

consideration plus all members, in the framework in at either end. This means, the

effect of other members not included in the limited framework were neglected.

2. When the single–bay single–storey framework is symmetric and symmetrically

loaded, the value memcn

ρ for both columns becomes identical because the effect of

the pure–shear stability functions is very small. Consequently, the values of

memc

memc

Codeeff,

X nn,LL and mem

cmemc

DMeff,

X nn,LL are almost equal. This indicates that when the


127

assumptions made by Wood (1974a) are met, the simplified approach gives the same

results as those by more accurate approach.

3. The comparison shows that the effective length factor memc

memc

Codeeff,

X nn,LL can be either

greater or smaller than memc

memc

DMeff,

X nn,LL and this depends on the geometric

dimensions and section properties of each member of the framework under

consideration.

Four questions now arise. First, what is the maximum difference that could occur

between the effective length factor when using the code approach and the more accurate

approach? Second, what is the position of the column member where the maximum

difference is obtained? Third, what are the cross sections of the framework members at

the maximum difference position? Fourth, how does the method of the determination of

the effective length factor influence the optimum design? To answer these questions,

application of optimization techniques is needed. Consequently, the aim of next chapter

is to introduce an algorithm, tune its parameters, suggest new modifications and finally

test the algorithm.

effective buckling length of columns in sw … · effective buckling length of columns in sway...

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