Guide on Second-order and Advanced analysis of structures

Nonlinear Integrated Design and Analysis software for structures www.nida-naf.com

Ir Prof. SL Chan [email protected]

Guide on

Second-order and Advanced analysis of structures

Version 2 - 2009


Ir Prof. SL Chan [email protected]

CONTENT

1 Methods of analysis.......................................................................... - 1 -

2 Linear analysis and the effective length method .......................... - 5 -

Important remark .................................................................................... - 5 - 2.1 Non-sway frame ........................................................................ - 6 - 2.2 Design chart method .................................................................. - 6 - 2.3 Sway-sensitive frames ............................................................... - 8 - 3.4 Sway ultra-sensitive frames ....................................................... - 9 -

3 Design by second-order analysis .................................................. - 10 -

3.1 Software .......................................................................................... - 10 -

3.2 Imperfections .................................................................................. - 10 - 3.2.1 Frame imperfection .......................................................... - 10 -

3.2.1.1 Using eigen-buckling mode as imperfection mode .... - 10 -

4 Examples ......................................................................................... - 16 - Example 1 Simple benchmark example for testing of software ..... - 16 - Example 2 Portal frame .................................................................. - 17 - Example 3 Design of a 3-Dimensional regular steel building ........ - 21 - Example 4 Slender frame in practice ............................................. - 27 -

5 Application of nonlinear finite shell element method ................ - 28 - 5.1 Design of beams for lateral-torsional buckling ............... - 28 - 5.2 Large deflection nonlinear analysis for column buckling ....... - 31 - 5.3 Buckling analysis of hinged cylindrical shell .......................... - 33 -

6 References ....................................................................................... - 36 -

CHECK LIST ......................................................................................... - 37 -


Prof. SL Chan [email protected]

- 1 -

1 Methods of analysis

In the Hong Kong steel Code, both the first-order linear and second-order non-linear analysis methods can be used. But we need to ensure the effects of change of geometry and buckling effects are correctly considered with λcr not less than 5 otherwise the second-order analysis must be used. Also, the design of irregular and slender structures should adopt the second-order analysis with appropriate values of initial imperfections from code (see Table 5.1 of the Hong Kong Steel Code or Table 6.1 of Eurocode-3). Software NIDA is used for the present analysis and design. Load factor λ in Figure 1 represents a scalar multiplied to the set of design load in a particular combined load case. To understand the method, we must first appreciate the behaviour of a structure under an increasing load. Various methods provide an answer of the collapse load under its assumptions, such as plastic collapse load which does not consider any buckling effect and P- Δ-only second-order analysis does not consider member imperfection and member buckling. The results of these methods are compared with the true collapse or ultimate load of a structure, λu in the Figure 1 below.

Deflection

Load factor λ

Second-order P elastic analysis

First-order Linear Analysis

Elastic critical load= 2EI/L

2cr

P

Design resistance or collapse load by Second-order Plastic Analysis

Design resistance by "First-plastic-hinge"

p=1.2

cr=2.2

U=0.85

y=0.76 λPwhere P=1000kN

Braced & hori. members 152x152x37

Others : 203x203x60

Yield stress : 355MPa

4m4m3m

4m

λ

λ

λ

λ

=λπ cr

pyRigid plastic load=p S= λ

Δ

Δ

−Δ−δ

Second-order P only elastic analysis−Δ−

cr

Figure 1 Design methods



- 2 -

Types of analyses can be explained as follows. Elastic Critical Load Factor λcr λcr a factor multiplied to the design load to cause the structure to buckle elastically. The deflection before buckling, large deflection and material yielding effects are not considered here and the factor is an upper bound solution that cannot be used directly for design. λcr can be used to measure the instability stage of a frame against sway and buckling. Plastic collapse load factor, λp. λp is a load factor multiplied to the design load to cause the structure to collapse plastically but buckling and second-order effects are not considered. Because of the ignorance of buckling effects, λp cannot be used for direct design and it is an upper bound solution to the true collapse load of the structure. This load factor was widely used in the past for plastic design because of its simplicity to determine. P-delta effects : refer to the second-order effect. There are two types, being P-Δ and P-δ. P-Δ effect : second-order effect due to change of geometry of the structure P-δ effect : second-order effect due to member curvature and change of member stiffness under load. A member under tension is stiffer than under compression. Second-order analysis and design method is a better design method than the effective length method. The method determines the P-Δ effect and the P-δ effect with initial imperfections so that effective length needs not assumed. Note that use of second-order analysis without consideration of initial imperfection should be avoided as it under-designs a structure since all structures contain imperfections.



- 3 -

Figure 2 The P-Δ and P-δ effects

Linear analysis or First-order linear analysis : an analysis assuming the deflection and stress are proportional to load. It does not consider buckling nor material yielding which is then considered in the design stage using code. Notional Force : a small force applied horizontally to a structure to simulate lack of verticality and imperfection. It can also be used to measure the lateral stiffness so that the elastic critical factor can be determined.

Figure 3 Simulation of out-of-plumbness by the notional force

P P

δ

Δ

φP P P P

P

φ



- 4 -

Second-order P-Δ-only analysis for plotting bending moment: an analysis used to plot the bending moment and force diagrams based on the deformed nodal coordinates. It does not consider member curvature or the P-δ effect. This method is commonly used in software because of its simplicity. In fact, most software can only do this P-Δ-only analysis which is not qualified for a full second-order analysis accounting for P-Δ and P-δ effects with imperfections at frame and member levels. Second-order analysis with section capacity check : an analysis allows for P-Δ effect and the P-δ effect and stops at first plastic hinge. It needs not assume an effective length for the buckling strength check, but imperfection must be allowed for.



- 5 -

2 Linear analysis and the effective length method

In the first-order linear analysis, the analysis finds the load in the columns and the buckling strength is unknown. Unlike the second-order P-Δ−δ analysis which consider the increase in stress due to the second-order and buckling effect, the first-order linear analysis needs to reduce the resistance of the columns when taking the load without considering second-order moment. To calculate the effective length other than making an assumption, we have the following method. 1. Calculate λcr by one of the following methods 1.1 Application of notional force. λcr can be determined in Eqt. 2. 1.2 Use computer to find λcr

δλ h

FF

V

Ncr = (Eqt.2)

For multi-storey frames, the maximum λcr among all stories should be used to obtain the minimum critical load factor. λcr is defined as the factor multiplied to the design load causing the frame to buckle elastically. Notional force is (1) to simulate lack of verticality of frames and taken as 0.5% of the factored dead and imposed loads applied horizontally to the structure and (2) to calculate the elastic critical load factor λcr to Equation 4. This percentage of notional force may vary for other types of structures like scaffolding where imperfections are expected to be more serious. (3) to classify the frame as non-sway, sway-sensitive and sway-ultra sensitive frames.

Important remark

Elastic Critical Load Factor λcr for assessing a sway and a non-sway frames should only be used for rectilinear frames under gravitational loads and it should not be used in irregular frames or structures dominated by lateral loads such as wind.



- 6 -

The following section describes the method of using chart to determine the effective length of a member in a regular frame. 2.1 Non-sway frame When λcr ≥10, the frame is considered as non-sway. P-Δ effect can be ignored here and only P-δ effect is needed. The effective length of members in frames can be designed by chart in Figure 6.4b of Hong Kong Steel Code. 2.2 Design chart method For a sub-frame in a multi-storey frame in Figure 4, the distribution factors, k1 and k2, are required to be determined as,

Determine k1 and k2 as joint at the members theall of stiffness Total

joint at the columns theof stiffness Total=k

12111

11 KKKK

KKk

c

c

++++

=

22212

22 KKKK

KKkc

c

++++

=

To calculate the load resistance of the column cgc pAP = , its effective length is

needed to be determined from k1and k2 above with the chart in Figure 5.



- 7 -

Figure 4 Restraint coefficients in a sub-frame

Figure 5 Effective length chart for non-sway frame

column-length being designed

K11

K21

K12

K22

K1

Kc

K2

k1

k2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pinned

k 1

Fixed

k 2 PinnedFixed



- 8 -

2.3 Sway-sensitive frames

When 510 ≥> λ , it is a sway sensitive frame.

We need to use the following chart to find the effective length factor ⎟⎠⎞

⎜⎝⎛

LLe to eqt.

8.79 in HK steel Code (2005). k1 and k2 are calculated similarly to the above but the stiffness coefficients for beams should follow Table 6.2 of Hong Kong Steel Code.

Figure 6 Effective length chart for sway frame

Member buckling check to Eqt. 8.79 in HK Steel Code Using effective length greater than member length to find the Pc is first carried out and the moment is not required to be amplified here because we concentrate on member buckling check.

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF (Eqt.3)

Amplification moment check to Egt.8.80 in HK Steel Code Additionally, moment amplification factor should be used to enlarge the moment due

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pinned

k 1

Fixed

k 2 PinnedFixed



- 9 -

to sway effect as the second checking as

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF

(Eqt.4)

Here the buckling resistance cP (note the bar at top while Eqt.8.79 has no bar at P)

is determined using effective length equal to member length (i.e. 1=LLE ), but Mx and

My are needed to be amplified as follows.

Also, the amplified moment should be used for connection design.

1−

=cr

crMMλ

λ (Eqt. 5)

As the above considers the P-Δ effect such that the effective length of the column is then taken as its member length or from Figure 6. When the beam may experience beam lateral-torsional buckling, eqt. 8.81 in HK Steel Code should also be checked. 3.4 Sway ultra-sensitive frames When λcr < 5, only second-order analysis method can be used.



- 10 -

3 Design by second-order analysis

We need to use proper software for second-order analysis indicated in Check List in the end of this note. With the qualified software, we need to model imperfections as follows.

3.1 Software

Proper software is essential and many programs are not for second-order analysis and we must be careful on using suitable software. Refer to checklist for some benchmark problems.

3.2 Imperfections

Unlike the first-order linear analysis, imperfections must be considered in any second-order analysis since no real structure is perfect and possesses no residual stress and initial crookedness. In Hong Kong Steel Code and Eurocode-3 (2005), a special feature is about the consideration of frame and member imperfections which are not so explicitly expressed in most other codes like BS5950(2000). The effects of imperfections shall be taken into account for two conditions. Global analysis : P-Δ effect Member design : P-δ effect

3.2.1 Frame imperfection We need to model global or frame imperfection due to unavoidable construction tolerance. This can be done by one of the following two methods.

3.2.1.1 Using eigen-buckling mode as imperfection mode

The effects of imperfections for typical structures shall be incorporated in frame



- 11 -

analysis using an equivalent geometric imperfection as an alternative to the notional horizontal force as,

200h

=Δ (Eqt. 6)

where h is the storey height ; 　is the initial deformation or out-of-plumbness deflection.

The shape of imperfection may be determined using the notional horizontal force for a regular frame or from the elastic critical mode. For regular multi-floor building frames, the shape may be simply taken as an inclined straight line. In many structures, the buckling mode shape is not obvious and we need to use computer program to determine the buckling mode. We can use the buckling mode as imperfection mode as the worst scenario as follows. In software, we can use specify this eigen-buckling mode option and a magnitude equal to 0.5% multiplied by the height or the longest span or an expected value of imperfection for a particular type of structures. 1% imperfection deflection or notional force is needed for temporary structures and 3% may be needed for structures under demolition.



- 12 -

Figure 7 Use of buckling mode as imperfection mode

These initial sway imperfections should be applied in all unfavourable horizontal directions, but need only be considered in one direction at a time. Temporary structures and structures under demolition require greater imperfections.



- 13 -

3.2.1.2 Method of Notional force For regular frames where the buckling mode is in a sway mode and obvious to engineers, a 0.5% of the vertical load should be applied horizontally to a frame which is basically regular in order to simulate imperfection. For structures used for other functions and durations, a varied value of notional force is used.

Figure 8 Horizontal notional force as 0.5% of factored vertical loads



- 14 -

The simulation of out-of-plumbness with notional horizontal force is indicated in Figure 3. 3.2.2 Member imperfections For practical members, initial bow and residual stress are unavoidable and must be considered in the buckling strength determination. Table 6.1 in the HKSC are the equivalent imperfection for these two sources of imperfections and they are the equivalent imperfection. The value of these imperfections cannot be measured from the initial bow or crookedness of the member but it can be determined by a curve-fitting procedure against the buckling strength vs. slenderness curve. In other words, we can try different values of imperfections to obtain a curve giving a 5% lower bound curve to the experimental curve. Alternatively or more directly, we can calculate the imperfection using the available Perry Robertson constants (Cho and Chan, 2002). For a compression member, the equivalent initial bow imperfection specified in Table 6.1 of Hong Kong Steel Code may be used in a second order analysis of the member.

Input of member initial imperfections in NIDA Figure 9 Member imperfection in NIDA



- 15 -

The effects of imperfections can be considered in member design when using the effective length method and the moment amplification method. This is the reason we have different buckling curves in Hong Kong Steel Code. Instead of using different curves, the second-order analysis uses different imperfections in Table 6.1 of HK Steel Code (2005) or Table 5.1 of Eurocode-3 (2005). For P-Δ-only analysis, we still need to curves a to d for member bucking check. Note that we cannot directly measure these imperfections since they are “equivalent “ imperfections considering geometrical initial curvature and residual stress. The geometrical initial curvature is measured to be around L/1000 to L/1500 which is much less than the equivalent initial imperfection allowing for residual stress and geometrical crookedness.

Figure 10 Buckling curves for sections with different imperfections (cr

y

PAp

=λ )

As can be seen above, a curved member is needed for the simulation. Although the Eurocode-3 (2005) suggests to use equivalent load along a member to simulate the effect, there will be an additional stress induced by this fictitious load which should not exist and it is therefore not allowed in the HKSC which uses the direct method of curved member.



- 16 -

4 Examples

Example 1 Simple benchmark example for testing of software Determine the elastic buckling load Pcr and design strength Py (design load resistance) of a fix-pin (prop-cantilever) column by 1 element/member. Do not input effective length which is generally unknown in a complex frame. Methods Elastic buckling

load (kN) Design Load

Capacity (kN) NIDA 131 98.5

Analytical1 130.8 - HKSteelCode2 - 108.6

The small difference between HK Steel Code and program is due to conservative use of initial imperfection of L/500. As seen here, the second-order analysis is in fact conservative. If the software being used by yourself cannot give you a right answer for a simple column, how can you expect it to give you a correct answer for real

structures?

Figure 11 Buckling of a pin-fix column by 1 element/member

Note :

1 Analytical 2

2

)7.0( LEIπ

= = 130.8kN

2 HKSteelCode(2005), Figure 6.2, k1=0, k2=1, LE/L =0.7, L/r = 0.7x5000/30.3=115.5, pc = 126 N/mm2, Pc = 108.6 kN



- 17 -

Example 2 Portal frame Check the structural adequacy of the following portal. The section is 686x254x140 UB (I = 1.363 x 10-3 m4) of grade S355 steel. The frame is rigid-jointed and pin-supported with dimensions shown in Figure 12 below. Properties of 140254686 ××UB : B=253.7mm, D=683.5mm, T=19mm, t=12.4mm A=17800mm2, I=1.363x109mm4, r=277mm, Sx=4.56x106mm3

k1

k2 30m

10mk1

k2

100kN

1000kN

Figure 12 Portal frame under point load on one column Deflection due to horizontal force =154 mm (Note that it is unnecessary to use 0.5% for notional force as the same elastic critical load factor λcr will be obtained because the deflection δN below will be changed proportionally and the final answer will be the same.)

49.6154.010

1000100

===NV

Ncr

hFF

δλ

This is a sway frame. Design Section classification Section Classification : ( Rolled ; I/H Section ) For web shear buckling check :



- 18 -

Web : d/t = 52.0565 =< 70e O.K. This member is Not susceptible to shear buckling. For section classification : Mat. Coeff. e = 0.8928 Web : d/t = 52.0565 <= 100e/(1+1.5r1) = 57.0807 Flange : b/T = 6.6763 <= 9e = 8.0352 Section Class = [ 2 ] "Compact" Material Reduction Factor for Slender Section = 1.0000 We can use plastic modulus in design. For 16<T<40, py=345N/mm2

Plastic moment = Mcx=Sx py = 4.56x106x345/106= 1573 kN-m Base shear =100/2=50 kN Moment at top of column = 50x10 =500 kN-m Section capacity check to Equation 8.78

1≤++cy

y

cx

x

yg

c

MM

MM

pAF

py=345N/mm2

..,1481.01573500

10345178001000

3 KOxx

<=+−

Buckling check using effective length under sway mode to Equation 8.79

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF

Relative stiffness of beam in sway mode should be LI

×5.1

check to Egt.8.79 in HK Steel Code

67.030/5.10010/

010/

12111

11 =

×++++

=+++

+=

III

KKKKKK

kc

c

112111

12 =

++++

=KKKK

KKk

c

c

9.2=LLe

Le=2.9x10=29m



- 19 -

7.104277.0

29000==

rLe

From Table 8.8(a), pc=162.3 N/mm2, Pc=162.3x17800/103= 2888.9 kN

From Table 8.9, β=0, mx=0.6

Therefore, additional moment at joint B and C due to Δ−P effect,

=++cy

yy

cx

xx

c

c

MMm

MMm

PF 537.0

15735006.0

28891000

=×

+ <1, O.K.

check to Egt.8.80 in HK Steel Code

Member buckling check to non-sway mode effective length under amplified moment to

equation 8.80 in HK Steel Code

For cP , effective length factor (Le/L) =1

0.1=LLe

Le=10m

1.36277.0

10000==

rLe

From Table 8.8(a), pc=327.9 N/mm2, Pc=327.9x17800/103= 5837 kN Therefore, additional moment at joint B and C due to Δ−P effect,

mkNMMcr

cr −=−

×=−

= 591149.6

49.65001λ

λ

=++cy

yy

cx

xx

c

c

MMm

MMm

PF

397.01573

5916.058371000

=×

+ <1, O.K.

Note : cP can be found from Table 6 for greater accuracy and assuming effective

length factor 1=LLe is only for conservative design.

No beam buckling check to equation 8.81 in HK Steel Code is needed here as the out-of-plane deflection is restrained. By second-order analysis and design We can see NIDA first use the sway initial buckling mode shape as imperfection mode shape. When using second-order analysis for design in NIDA, the section capacity factor is 0.562. This is close to the most critical value of 0.537 above.



- 20 -

Top restraint case When the top of the portal is restrained laterally, the frame becomes non-sway and the application of notional force becomes useless.

In NIDA, the buckling mode shape can be adjusted automatically and used for design check.



- 21 -

Example 3 Design of a 3-Dimensional regular steel building This example is for the extension to a three-dimensional structure and to demonstrate the application of a second-order analysis dealing on a three-dimensional problem. Figure 13 show the three-dimensional four-storey frame with steel grade S355. The loadings on the floors and the roof are 30kN/m2 and 10kN/m2 respectively and the self-weight of the material is included. The sections used are shown in Figure 13. In this example, column C1 (see Figure 13) is to be designed. The frame is 4-storey high of dimensions 12m(H)x12m(L)x8m(W) with story height of 3m. All base connections are assumed pinned and beam to column connections are assumed rigid. Concrete floor is assumed as rigid diaphragm and the stiffness in the Z-axis is strengthened by cross bracings at the two end bays while the sway stiffness in the X-axis is provided by moment frame action. The major axis of the columns is located about the Z-axis in order to provide a larger resistance against sway in the X-axis for which bracings are not provided. Select member “C1 of 254x254x73 UC” for demonstration. Other members follow the same procedural check.



- 22 -

Figure 13 The 4-storey steel frame

C1

Section properties:



- 23 -

The elastic buckling load factor for the unbraced plane is 5.07 and for the braced plane is greater than 10 so that bucking about member major Z-axis is classified as sway and about member minor Y-axis as non-sway. For major Z-axis For minor Y-axis For finding of k1

12111

11 KKKK

KKk

C

C

++++

=

53

1 1052.13

10568.4 −=×

=== xLIKKC

For finding of k1

12111

11 KKKK

KKk

C

C

++++

=

65

1 1016.53

10548.1 −=×

=== xLIKKC

54

1112 1083.66

10102.4 −−

=×

=== xLIKK 5

4

1112 10865.44

10946.1 −−

=×

=== xLIKK

Thus,

182.083.6252.12

52.121 =

×+××

=k

For finding of k2

175.06.480.116.52

16.521 =

×+××

=k

For finding of k2

22212

22 KKKK

KKk

C

C

++++

=

44

2 1038.03

10141.1 −−

×=×

=K

22212

22 KKKK

KKk

C

C

++++

=

55

2 103.13

10908.3 −−

×=×

=K

280.083.6252.18.3

52.18.32 =

×+++

=k 117.065.483.116.5

3.116.52 =

+++

=k

From Table 6.4a

From Table 6.4b

16.1=L

LE

LE = 1.16x3=3.48 ; rZ = 0.088m

kNPmmNp

rL

CZ

CZ

z

e

1872/319

5.39

2

==

=

545.0=L

LE

LE = 0.545x3=1.635 ; rY = 0.051m

kNPmmNp

rL

CY

CY

Y

e

1947/7.331

06.32

2

==

=

Smaller PC=1872 kN



- 24 -

For CP with effective length = member length

LE =3.48 ; rz = 0.088m

kNP

mmNp

rL

C

C

z

e

7.1927

/4.328

1.34

2

=

=

=

Design moment Mymax = 17.12 kN-M, Mzmax=5.1 kN-m, F=900.8 kN Section Classification

2355 mmNpy = for mmT 16≤

(Table 3.2)

88.0275==

ypε (Table 7.1 Note b)

4.111317.85.12

23.204=≤== ε

Tb

∴ flange is semi-compact (Table 7.1)

173.13559.72.181

10880 3

1 >=××

×==

y

c

dtpFr Equation (7.1)

∴ 11 =r

2.351809.22

9.72.181

1

=+

≤==rt

d ε

∴ web is plastic (Table 7.1)

∴Section is Class 3 “Semi-compact”, use elastic modulus Z only. Moment and Axial force Capacity MCZ=Zz py = 159.6 kN-m > Mymax = 17.12 kN-m, O.K. MCY=Zypy=0.1825x10-3x355 = 54 kN-m > Mzmax=5.1 kN-m, O.K. Py= Axpy= 0.587x10-2x0.355x106 = 2083 kN > F=900.8 kN, O.K.



- 25 -

Moment equivalent factors From Table 8.4b, mLT= 0.44 From Table 8.9, my = 0.407, mz = 0.40 Lateral-torsional buckling check of beam For lateral-torsional buckling of beam in a steel building, the effective length factor

can be taken as 5.0=L

LE

4.29051.0

35.0=

×=

y

E

rL

2/355

80.234.299702.08345.0

mmNp

uv

b

LT

=

=××== λλ

Mb = 159.6 kN-m > Mzmax = 5.1 kN-m, O.K.

Cross-Section Capacity

1≤++cy

y

cx

x

yg

c

MM

MM

pAF (8.78)

..,1781.054

12.176.159

1.52083

8.900 KO<=++

Using sway effective length without moment amplification

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF (8.79)

..,1622.054

12.17407.06.159

1.54.05.18728.900 KO<=

×+

×+

Using non-sway effective length with amplified moments

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF (8.80 HKSC)

..,1641.054

12.17)14.5(

4.5407.0

6.159

1.5)14.5(

4.54.0

7.19278.900 KO<=

×−

×+

×−

×+

Beam-column interactive check against beam lateral-torsional buckling

1≤++cy

yy

b

LTLT

cy

c

MMm

MMm

PF (8.81 HKSC)



- 26 -

..,1606.054

12.17407.06.159

1.5)14.5(

4.544.0

19478.900 KO<=

×+

×−

×+

We follow a logic of using either the sway effective length (which is greater than member length) OR the amplified moment in a single equation check, but not both at a time. In the last equation, as Pcy is obtained from sway effective length, we need not amplified the My which is about the same y-axis. However, MLT is about the major axis and the effective length about this major axis has not been considered and thus we need to amplify MLT. Using Second-order analysis Mzmax = 5.56 kN-m Mymax = 16.52 kN-m F = 901.1 kN Section capacity factor φ = 0.773 Output results Linear analysis Second-order analysis % Difference Max. Mz 5.10 kN-m 5.56 kN +9% Max. My 17.12 kN-m 16.52 kN -4% Axial Force F 900.8 kN 901.1 kN-m 0% Section capacity factor (1/factor of safety)

0.781 0.773 1%

For a typical frame designed by the two methods, we can see the difference is small because the effective length can be quite accurately determined and the frame is not too slender with λcr greater than 5. For slender frames where effective length cannot be determined by the charts, the error by the linear analysis method can be very large because the effective length is unknown. Nevertheless, the first-order analysis method has under-estimated the moment about the major axis by 9% which can be compensated by the use of effective length or amplified moment in element design, but NOT in connection design where the second-order moment has not been determined and used.



- 27 -

Example 4 Slender frame in practice The example below demonstrates the design of a space frame without assuming any effective length which should not be used here as the structure is classified as “irregular”. The space frame shown below has been designed without assuming any effective length. All expected loadings have been allowed for in the analysis and design. The figure shows the structure near completion and the computer model.

Figure 14 Computer model



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5 Application of nonlinear finite shell element method

5.1 Design of beams for lateral-torsional buckling The concept is elegant as it can also be applied to design of structures using the finite plate and shell elements. Example in 5.1 is a solution based on finding of λcr and example in 5.2 is based on the concept of second-order analysis. Here we need not assume an effective length whereas the beam-column design for “lateral-torsional” beam buckling in previous examples requires an effective length for beam buckling check because this effective length does not require frame classification, which is required for flexural (column) effective length, and thus it is much simple and a second-order analysis to simulate beam buckling is practically not needed. Eigen-buckling approach In addition to the use of the method for design of frames, the concept can also be applied to design of beams against lateral-torsional buckling as shown in Figure 15 below. According to the Hong Kong Code, we can use the elastic critical moment Mcr to find the buckling resistance of a beam Mb under virtually all conditions. From equation 8.17 in the Hong Kong Code, the equivalent slender ratio can be found as,

ycr

pLT pM

EM 2πλ =

in place of wLT uv βλλ = in which we need to consider effective length which may

not be obvious such as warping restrained and load height etc. With this value of λLT we can find the buckling resistance stress pb and resistance moment Mb. The approach can be applied to beams with opening or with unequal flanges etc.



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Figure 15 Buckling of a cantilever I-beam in laboratory and by NIDA-8. (Left or right direction of buckling is arbitrary as the beam is symmetrical)

For typical cases we obtain results compared very well with the code results shown below.



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Figure 16 Comparison between code and computer results



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5.2 Large deflection nonlinear analysis for column buckling Alternatively we can use a more refined method to determine the load causing yielding in a section by the geometrically nonlinear analysis.

Figure 17 Large deflection nonlinear analysis of a column allowing for local and flexural buckling



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In some structures like glass panels not controlled by buckling, the design by eigen-buckling analysis cannot be used and the large deflection nonlinear analysis provides a reliable and accurate solution and in many standards like ASTM-1300E, the non-linear analysis method is used.

Figure 18 Large deflection nonlinear analysis of glass panel under wind pressure



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5.3 Buckling analysis of hinged cylindrical shell

This problem demonstrates the accuracy of the program NIDA against large

deflection and buckling problems. Figure 19 shows a circular cylindrical shell of

thickness 12.7mm subjected to a concentrated central load at the convex side. The

length of the shell is 508mm, the Young’s modulus of elasticity is 3.10275kN/mm2

and the Poisson’s ratio is 0.3. The longitudinal boundaries are hinged and immovable

while curved edges are free. A four by four mesh for one-quarter of the panel has been

considered. The problem has been used extensively for testing of a new element and

the nonlinear solution method for snap-through problems.

Figure 19 Hinged cylindrical shell

R = 2540 mm

L = 254 mm

θ= 0.1 rad

E= 3.10275 kN/mm2

v = 0.3

Case 1: h = 12.7 mm

Case 2: h = 6.35 mm



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Figure 20 One quarter of the panel considered The results of applied load against the central defections are shown in Figure 21. The results agree very well with those by Surana 4 and Sabir and Lock which has a similar result.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 5 10 15 20 25 30 35

Displacement (mm)

Load

P (k

N)

Vertical deflection at point C Vertical deflection at point ESurana at Point C Surana at Point E

Figure 21 Load Deflection curve for Cylindrical shell

(Thickness = 12.7mm)

When the thickness of the cylindrical shell is reduced to 6.35mm and the



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geometric and physical data remains unchanged, the structural response of the

structure is varied from the previous example. Snap-back in addition to snap-through

buckling is observed. The load-deflection curves for central point and mid-point of

curved edge are plotted in Figure 22. Again, the results are in very good agreement

with the reference in the figure.

-0.45

-0.35

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.75

0 5 10 15 20 25 30

Displacement (mm)

Load

P (k

N)

Vertical deflection at Point C Vertical deflection at Point E Surana at Point CSurana at Point E Sablr and Lock

Figure 22 Load Deflection curve for Cylindrical shell

(Thickness = 6.35mm)



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6 References

1 BD, Code of practice for structural design of steel structures – A limit state approach, Buildings Department, Hong Kong, 2005. (http://www.bd.gov.hk/english/documents/index_crlist.html)

2 CEN, Eurocode 3, Part 1-1: General Rules and Rules for Building, EN 1993-1-1, British Standards Institute, London, 2005.

3 Surana, K.S., Geometrically Nonlinear Formulation for Curved Shell Elements, International Journal for Numerical Methods in Engineering, Vol.19, No.4, pp.581-615, 1983

4 Chan, S.L. and Zhou, Z.H., “Elastoplastic and large deflection analysis of steel frames by one element per member. Parts 1 and 2, Journal of Structural Engineering, ASCE, April, 2004, vol. 130, No.4, pp.538-553.



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CHECK LIST

The use of the formulaδ

λ hFF

V

Ncr = is only applicable to regular frames of rectilinear

geometry with dominant gravitational loads such as steel buildings. Its use to irregular structures such as space frames and sign board supporting structures with dominant load as wind is unsuitable. Step 1 Elastic critical load factor λcr.

Use notional force for regular structures or use computer program for regular and irregular structures. Elastic critical load factor λcr δ

λ hFF

V

Ncr = or by computer program

λcr ≥ 10 If λcr ≥ 10, assume member length as effective length or use Table 6.4b of HKSteelCode to find the effective length LE. Then use eqt 8.791. If beam buckling may occur, use eqt. 8.81 of HKSteel Code. Use second-order analysis with P-δ imperfection and P-Δ sway effect here is unimportant. Note again P-δeffect and its imperfection are still important here.

5 ≤ λcr < 10 Use effective length greater than member length found in Table 6.4a. (Eqt. 8.79 of HK Steel Code) + moment amplification method (Eqt.8.80 of HK Steel Code). If beam buckling may occur, use eqt. 8.81 of HK Steel Code Use second-order analysis with frame and member imperfections.

λcr <5 Use second-order analysis with frame and member imperfections only.

Note:



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(1) Pc is the same as cP . Pc is the buckling resistance of the member using the

Table 6.4a or 6.4b. cP is the buckling resistance using member length as effective

length. A bar represents a unit value of L

LE when finding cP ).

Important remark

Elastic Critical Load Factor λcr for assessing a sway and a non-sway frames should only be used for rectilinear frames under gravitational loads and it should not be used in irregular frames or structures dominated by lateral loads such as wind.



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Step 2 Imperfections Both direction and magnitude are critical for computing the design load resistance. Frame imperfection is taken as span, height or maximum dimension of a structure

divided by 200 for permanent structures (200Hor

200L ). Member imperfections are

found in Table 6.1. The direction should be chosen as the one same as the buckling mode shape. For regular structures, notional force may be used. Step 3 Quality assurance checks – very important to make sure the design is proper 3.1 Randomly select 2 or 3 members in a particular load case out of several or hundreds to ask the design engineer to see whether the imperfections are put in the direction as the buckling mode shape. 3.2 Check the geometry of the structure before loading to have an imperfection shape the same as buckling mode shape with imperfection equal to L/200 (i.e. maximum initial deflection like out-of-plumbness).



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Comparison between First-order and Second-order analysis First-order analysis assumes effective length and uses the following code equations. This method cannot be used for long span, irregular and slender structures and the following code method is only for regular building frames. This limitation is adopted in both the Hong Kong Steel Code 2005 and Eurocode-3. 8.9.1 Cross-section capacity Except for Class 4 slender cross-sections, the cross-section capacity can be checked as,

1≤++cy

y

cx

x

yg

c

MM

MM

pAF (8.78)

in which xM and yM are the design moments about the x- and y-axes. cxM and

cyM are the moment capacities about the x- and y-axes. For Class 4 slender cross-sections, the effective area effA should be used in place of the gross sectional area. Alternatively, for members of class 1 plastic or class 2 compact and symmetrical I and rectangular hollow cross-sections, a more exact check can be carried out by formulae in literature or the sectional strength analysis described in clause 8.8. For more exact design of asymmetric and mono-symmetrical sections, clause 8.8 should be referred. 8.9.2 Member buckling resistance For Class 4 slender section, the effective moduli effxZ , and effyZ , should be used in

place of the elastic moduli Zx and Zy in the following cases. When using the P-Δ-δ analysis allowing for initial imperfection and beam buckling in clause 6.8.3 a check for member buckling resistance in this clause is not required. The resistance of the member of a sway frame as defined in clause 6.3 can be checked using,

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF (8.79)

1≤++cy

yy

cx

xx

c

c

MMm

MMm

PF (8.80)

1≤++cy

yy

b

LTLT

cy

c

MMm

MMm

PF (8.81)



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Second-order analysis does not assume effective length factor (LE/L) and buckling effects and second-order effects are considered by P-Δ and P　 -δ moments through 　

the following code equation. 6.8.3 Applications and limitations The method is limited to the capacity at the load level where the first plastic hinge is formed. Except for Class 4 slender sections, the sectional capacity should be checked using the following equation allowing for the P-Δ and P-δ effects.

1)()(

≤+Δ+

++Δ+

+=++cy

yycy

cx

xxcx

yg

c

cy

y

cx

x

yg

c

MFM

MFM

pAF

MM

MM

pAF δδ (6.12)

in which yx δδ , are the member deflections due to member initial bow and loads on the

member and about x- and y-axes; For Class 4 slender cross-sections, the effective area effA should be used in place of the gross sectional area.

Guide on Second-order and Advanced analysis of structures

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