Plastic Design of Portal Eurocode 3 - University of · PDF filePlastic Design of Portal ......

Department of Civil and Structural Engineering

Plastic Design of Portal frame to Eurocode 3 Worked Example

University of Sheffield

Contents

1 GEOMETRY ....................................................................................................................................................................... 3

2 DESIGN BRIEF .................................................................................................................................................................. 4

3 DETERMINING LOADING ON FRAME ....................................................................................................................... 5

3.1Combination factors ψ .................................................................................................................................................. 5

3.2Snow loading ................................................................................................................................................................ 6

3.3Self weight of steel members ........................................................................................................................................ 7

4 INITIAL SIZING OF MEMBERS ..................................................................................................................................... 8

5 LOAD COMBINATION (MAX VERTICAL LOAD) (DEAD + SNOW) .................................................................. 10

5.1Frame imperfections equivalent horizontal forces ....................................................................................................... 10

5.2 Partial safety factors and second order effects ........................................................................................................... 11 5.2.1 Sway buckling mode Stability ( αcr,s,est). ............................................................................................................ 14 5.2.2 Snap‐through buckling stability (αcr,r,est ) .......................................................................................................... 16 5.3.2Accounting Second Order effects .......................................................................................................................... 17

6 MEMBER CHECKS ........................................................................................................................................................ 20

6.1 Purlins ....................................................................................................................................................................... 20

6.2 Column (UB 610 x 229 x 101) ...................................................................................................................................... 21 6.2.1Classification .......................................................................................................................................................... 21 6.2.2Cross section resistance ........................................................................................................................................ 21 6.2.3Stability against lateral and torsional buckling (EN 1993‐1‐1: 2005 (E) Sec BB3.2.1): ........................................... 22

6.3Rafter (UB457 x 191 x 89) ............................................................................................................................................ 29 6.3.1Section Classification ............................................................................................................................................. 29 6.3.2 Cross‐section Resistance. ..................................................................................................................................... 29 6.3.3 Check rafter buckling in apex region .................................................................................................................... 31 6.3.4 Stability check for lower bending moments ......................................................................................................... 32

6.4Haunch (UB 457 x 191 x 89) ......................................................................................................................................... 35 6.4.1Classification .......................................................................................................................................................... 35 6.4.2Haunch Stability ..................................................................................................................................................... 36 6.4.3 Cross‐section resistance. ...................................................................................................................................... 41

7. COMPARISON BETWEEN DIFFERENT CODES .................................................................................................... 43

8APPENDIX ...................................................................................................................................................................... 44


Department of Civil Structural Engineering





Date 16/02/2009 Geometry of the Frame Sheet No

2 Reference Calculation

1 Geometry



Revised by: EC3 Plastic Portal Frame Design Prepared by:Cia06mh

Date 16/02/2009 Client brief Sheet No


2 Design Brief A client requires a single-storey building, having a clear floor area 30 m x 80

m, with a clear height to the underside of the roof steelwork of 5 m. The slope of the roof member is to be at least 6o.

Figure 1‐ Plan view of the frame

Figure 2‐ 3 Dimensional view of the building (Plum, 1996)




Date 16/02/2009 Determining load on the frame Sheet No


EN 1991-1-1:2002 (E) Annex A 1 See Supporting Notes Sec 6.4

3 Determining loading on frame

3.1Combination factors ψ The combination ψ must be found from Eurocode 1 (EN1991-1) or relevant NAD. Note that because most portal frame designs are governed by gravity (dead + snow) loading, so in this worked example only maximum vertical load combination is considered. Therefore, the combination factor ψ is never applied in this example, but for full analysis the following load combination should be considered

1) Maximum gravity loads without wind, causing maximum sagging moment in the

rafter and maximum hogging moments in the haunches.

2) Maximum wind loading with minimum gravity loads, causing maximum reversal of moment compared with case 1. The worst wind case might be from either transverse wind or longitudinal wind so both must be checked.

Basic data :

• Total length: b = 72 m • Spacing: s = 7.2 m • Bay width: d = 30 m • Height (max): h = 7.577m • Roof slope: α = 6o

Figure 3‐ Frame spacing (SX016, Matthias Oppe)




Date 16/02/2009 Determining loading on the frame Sheet No


EN 1991-1-3 Sec 5.2.2 Eq.5.1 EN 1991-1-3 Sec 5.3 Table 5.1 See Appendix A Table A1 EN 1991-1-3 Annex C See Appendix A Table A2

3.2Snow loading

General Snow loading in the roof should be determined as follow

µ Where: µ is the roof shape coefficient is the exposure coefficient usually taken as 1 is the thermal coefficient set to 1 for nominal situations Is the characteristic value of ground snow load for

relevant altitude.

Roof shape coefficient Shape coefficients are needed for an adjustment of the ground snow load to a snow

load on the roof taking into account effects caused by non-drifted and drifted snow loading.

The roof shape coefficient depends on the roof angle so

0 30 µ =0.8 Snow load on the ground

For the snow load on the ground; the characteristic value depends on the climatic region; for site in the UK the following expression is relevant

Sk=0.140z-0.1+(A/501) Where:

Z is the( zone number /9 ) depending on the snow load on sea level here in Sheffield z=3

A is the altitude above sea level A=175m




Date 16/02/2009 Determining loading on the frame Sheet No


Self-weight estimated needed to be checked at the end

Snow load on the roof Sk = 0.8 x 1 x 1 0.67 = 0.54 KN/m2

Spacing = 7.2 m For internal frame UDL by snow = 0.54 x 7.2 = 3.89 m

Figure 4‐ Distributed load due to snow per meter span (SX016, Matthias Oppe)

3.3Self weight of steel members Assume the following weight by members,

• Roofing = 0.2 KN/m2 • Services = 0.2 KN/m2 • Rafter and column self weight = 0.25 KN/m2

Total self weight _____________ 0.65 KN/m2




Date 16/02/2009 Initial sizing if members Sheet No


TP/08/43 EC3/08/16 Manual for the design of steelwork building structures to EC3 See Appendix B for the method

4 Initial Sizing of members

Figure 5‐ Dimensions of portal (The institutionof Structural Engineers, TP/08/43 EC3/08/16)

a) L/h = 30/6 = 5

r/L = 1.577/30=0.0526

b) Loading 1) Gravity loading

Snow loading = 0.54 x 7.2 = 3.80KN/m Self weight = 0.65 x 7.2 = 4.68 KN/m

2) Factored load w= (4.68 x 1.35 ) + (3.80 x 1.5 ) = 12.0 KN/m

c) Finding Mp for the sections

1) Total load on the frame (wL)= 12.0 x 30 = 360.5KN

2) Parameter wl2 = 12.0 x 302 = 10816 KNm

3) From Graphs (Figure B2) obtain horizontal force ratio (0.36)

H= 0.36 x 360.5 = 129.8 KN

4) From Graphs (Figure B3) obtain rafter Mp ratio (0.034) Mrafter,,Rd = 0.034 x 10816 = 367.7 KNm




Date 16/02/2009 Initial sizing if members Sheet No


Section Tables of Universal Beams EN 1993-1-1: 2005 (E) Table 3.1

5) From Graphs (Figure B4) obtained column Mp ratio (0.063) Mcolumn, Rd = 0.063 x 10816 = 681.4 KNm.

6) Selecting members a) Wpl (rafter),required = (367.7 x 106) / 275 = 1337 x 103 cm3

Try UB 457x152x74

b) Wpl(column),required = (681.4 x 106)/275= 2478 x 103 cm3

Try UB 533 x 210 x 109

• Properties Rafter Section UB 457x152x74

G=74.2 Kg/m h= 462mm b=154.4mm tw=9.6mm tf=17mm A=94.48 x 102 mm2 d=428mm Iy= 32670 x 104 mm4 Wpl,y=1627 x 103 mm3 iy=186 mm iz = 33,3 mm Iz = 1047 x 104 mm4 Wpl,z = 213.1 x 103 mm3 It = 66.18 x 104 mm4 Iw = 516.3 x 106 mm6

• Properties Column Section UB 533x210x109 G=109 Kg/m h= 539.5mm b=210.8mm tw=11.6mm tf=18.8mm A=138.9 x102 mm2 d=510.9mm Iy= 66820 x 104 mm4 Wpl,y=2828 x 103 mm3 iy=218.7 mm iz = 45.7 mm Iz = 2692 x 104 mm4 Wpl,z = 399.4 x 103 mm3 It = 101.6 x 104 mm4 Iw = 1811 x 106 mm6

Steel grade is S275 Assume Sections Class1, then check




Date 16/02/2009 Ultimate Limit State Analysis Sheet No


EN 1993-1-1:2005 (E) Sec 5.3.2 See Supporting Notes Section 9

5 Load Combination (Max vertical Load) (Dead + Snow)

5.1Frame imperfections equivalent horizontal forces

Ø Ø

Ø = √

0.5 1 .5 Ø = 3.54 x 10-3 The column loads could be calculated by a frame analysis, but a simple calculation based on plan areas is suitable for single storey portals

(i) Permanent loads ( un-factored ): Rafter = (74.5 x 15 x 9.8) / 103 = 11 KN Roofing = (15 x 0.2 x 7.2) = 21.6 KN Services = (15 x 0.2 x 7.2) = 21.6 KN _________ Total = 54.2 KN

(ii) Variable loads ( un-factored ) Snow load = 15 x 0.54 x 7.2 = 58.3 KN

Thus the un-factored equivalent horizontal forces are given by:

(i) Permanent/column = 3.54 x 10-3 x 54.2 = 0.19 KN

(ii) Variable/column = 3.54 x 10-3 x 58.3 = 0.21 KN

Note EC3 requires that all loads that could occur at the same time are considered

together, so the frame imperfection forces and wind loads should be considered as additive to permanent loads and variable loads with the appropriate load factors.






For Second Order effects See Supporting Notes Section 7.1 & Section 7.2

Figure 6‐Frame imperfections equivalent horizontal forces

5.2 Partial safety factors and second order effects Second order effects increases not only the deflections but also the moments and forces beyond those calculated by the first order. Second-order analysis is the term used to describe analysis method in which the effects of increasing deflections under increasing load are considered explicitly in the solution method. The effects of the deformed geometry are assessed in EN 1993-1-1 by calculating alpha crit (αcrit) factor. The limitations to the use of the first-order analysis are defined in EN 1993-1-1 Section 5.2.1 (3) as αcrit 15 for plastic analysis. When a second order analysis is required there are two main methods to proceed: 1) Rigorous 2nd order analysis (i.e. using appropriate second order software). 2) Approximate 2nd order analysis (i.e. hand calculation using first order analysis

with magnification factors). Although the modifications involve approximations, they are sufficiently accurate within the limits given by EN 1993-1-1.






See Supporting Notes Section 7.3

• Carrying first order analysis to obtain first order moments and member forces using partial safety factors (γG=1.35) and (γQ=1.5) with loading calculated above.

• Then Checks if second order effects are relevant by calculating the following αcr,est =min ( αcr,s,est , αcr,r,est ) where αcr,s,est = estimated of αcr for sway buckling mode αcr,r,est =estimated of αcr for rafter snap-through buckling mode.

Figure 7 ‐ Bending moment diagram for first order analysis (Burgess, 20/01/1990)






Load Factor Hinge number Member Hinge status 1.02 1 RHC Formed 1.14 2 LHR Formed

Table 1‐ Position of Hinges and Load factors

Figure 8‐– Member forces (Burgess, 20/01/1990)






See Supporting Notes Section 7.3.2.1 See Figure 9

5.2.1 Sway buckling mode Stability ( αcr,s,est).

αcr,s,est = 0.8 1 ,

, ,

,

,

, is the axial force in rafter {see figure 8 (150.8KN)} , is the Euler load of rafter full span

,

Where is the in-plan second moment of area of rafter L is the full span length.

,

= 752 KN

,

,

, is the minimum value for column 1 to n

, is the horizontal deflection for top of column as indicated in

Appendix

, is the axial force in columns {see figure 8 (207.5KN , 208.1KN)}

• As can be seen that , is the lateral deflection at the top of each column subjected to an arbitrary lateral load HEHF then here an arbitrary load HEHF can be chosen and using analysis software the deflection at top of each column can be obtained.

1) Arbitrary load HEHF=50KN 2) , = 98mm

, = 98mm






So either .

OR .

Min ,

,

, = min(14.75 , 14.75 ) = 14.75

Thus

αcr,s,est = 0.8 1 . 14.75 9.5

Figure 9‐ Sway mode check (Burgess, 20/01/1990)






See Supporting Notes Section 7.3.2.2 & Section 7.2

16 1627 10 275

30 10 238.6

5.2.2 Snapthrough buckling stability (αcr,r,est )

αcr,r,est = .

tan 2

D cross-section depth of rafter (462mm). L span of the bay (30m). h mean height of the column (6m). in-plane second moment of area of column (66820 x 104 mm4) in-plane second moment of area of rafter (32670 x 104 mm4 ) nominal yield strength of the rafter (275 N/mm2) roof slope if roof is symmetrical (6o) Fr/Fo the ratio of the arching effect of the frame where

Fr= factored vertical load on the rafter ( 432 KN see section 3) F0 = maximum uniformly distributed load for plastic failure of the rafter

treated as a fixed end beam of span L , ,

= 1.81

Thus

αcr,r,est = .

.tan 2 6

αcr,r,est = 6.2

• Hence αcr,est =min ( αcr,s,est , αcr,r,est ) = 6.2






See Supporting Notes 7.3.3

• Although the snap-through failure mode is critical mode as shown in calculation

above, but because this example is for designing single bay portal frames, the snap-through mode of failure is irrelevant but included to show complete design steps for simple portal frame design. Snap-through failure mode can be critical mode in three or more spans, as internal bay snap-through may occur because of the spread of the columns inversion of the rafter (The institutionof Structural Engineers, TP/08/43 EC3/08/16) see figure 10.

Figure 10‐ Snap through failure mode critical for 3 bay or more

5.3.2Accounting Second Order effects To account for second order effects the partial safety factors can be modified by the following criteria

1) γG = 1.50

2) γQ = 1.68






See Supporting Notes 7.3.3

• Re-analyze the first order problem with the modified safety factors using same initial sized sections gives the following results,

Load Factor Hinge number Member Hinge status 0.92 1 RHC Formed 1.02 2 LHR Formed

Table 1‐ Hinges obtained from analysis

• It could be seen that using Sections UB 533 x 210 x 109 and UB 457 x 152 x 74 is suitable, although hinge 1 occurs at a load factors ≤ 1 , a mechanism is not formed until the second hinge is formed. Therefore this combination of section sizes is suitable

• Hence size of member initially estimated is suitable and can withstand second-order effects. Note that if the load factors in positions 1 and 2 were less than 1, then the members size needs to be increased to sustain second order effects as the initially sized members cannot sustain second order effects.

Figure 11‐Bending moment diagram for first order analysis (Burgess, 20/01/1990)






Figure 12‐Member forces for first‐order analysis (Burgess, 20/01/1990)




Date 16/02/2009 Member checks – Purlins Sheet No


See Supporting notes section 10.4 Note. Here the safety factors are used as indicated in King span load table See Appendix C

6 Member checks

6.1 Purlins Today the design of the secondary members is dominated by cold formed sections.

The ‘design’ of cold formed members consists of looking up the relevant table for the chosen range of sections. The choice of a particular manufacture’s products is dependent on clients or designer’s experiences and preferences. Table (Appendix C) illustrates a typical purlin load table based on information from manufacture’s catalogue (King span) for double span conditions. As the overall distance between columns is 30 meters, which is assumed to be divided to 18 equal portions would gives purlin centers 1.67 meters (on the slope). The gravity loading (dead (cladding Load plus snow load) is w= (0.1x 1.4) + (0.54 x 1.6) = 1.004 KN/m2. From the Table (Appendix C), knowing the purlin length of 7.2 m, purlin spacing of 1.25m and the gravity load to be supported by purlin 1.004KN/m2, the M175065120 section seems adequate.

Figure 13‐ Connection between rafter section and purlins

Figure 14‐ Purlin cross‐section (Kingspan)

Purlin

Rafter




Date 16/02/2009 Member checks – Column Sheet No


See Figure11 EN 1993-1-1: 2005 (E) Section 5.5 See supporting notes section 12.3

6.2 Column (UB 610 x 229 x 101)

- MEd = 904.7 KNm - VEd = 150.1 KN - NEd = 208.2 KN

6.2.1Classification

Web ( Bending + Axial ) ε = 275/235 =1.08 actual (d/tw ) = .

.44.04 72ε Class 1

Flanges ( Axial Compressive )

actual (c/tf )= . . .

.4.61 9ε Class1

• So the column sections are overall class 1

6.2.2Cross section resistance

The frame analysis assumed that there is no reduction in the plastic moment resistance from interaction with shear force or axial force. This assumption must be checked;

6.2.2.1Shear force effects of Plastic moment resistance (EN 199311: 2005 (E) Sec 6.2.6)

VEd < 0.5 Vpl,Rd

Av = 1.04 h tw = 1.04 x 539.5 x 11.6 = 6508.5 mm2 Vpl,Rd = /√3 /γ Vpl,Rd = 6508.5 275/√3 / 1.1 10 939.4 KN 0.5 Vpl,Rd = 469.7 KN

• VEd < 0.5 Vpl,Rd so the plastic moment of resistance is not reduced by the coexistence of axial force






See supporting notes section 12.4 See supporting notes section 13.4

38 /57.4

756 1 235

6.2.2.2Axial force effects of Plastic moment resistance (EN 199311: 2005 (E) Sec 6.2.9) Check

i. If ,

NEd < .

208.2 < . . .

.

208.2 < 727.8

ii. If

NEd < 0.25 Npl,Rd

NEd < 0.25 plastic tensile resisitance of the section

NEd < .

208.2 < . .

.

208.2 < 868.1

• Therefore, the effect of shear and axial on the plastic moment resistance of the

column sections can be neglected according to EC3 EN1993-1-1: 2005.

6.2.3Stability against lateral and torsional buckling (EN 199311: 2005 (E) Sec BB3.2.1): . The design of the frame assumes hinge forms at the top of the column member, immediately below the haunch level. The plastic hinge position must be torsionally restraint in position by diagonal stays. With the hinge position restraint, the hinge stability is ensured by EC3 by limiting distance between hinge and the next lateral restraint to Lm.




Date 16/02/2009 Member checks - Column Sheet No


38 45.7 / . . .

. .

= 1.53 m

• Thus there must be a lateral restraint at a distance from the hinge not exceeding (1.53m).

• Therefore if 1.5 meters spacing assumed, this would ensure the stability between the intermediate restraints at the top of the column where maximum bending moment occurs, then the spacing of 1.8 meters is OK for sheeting rails below 2.4 meters from the top of the column, where the moment is lower.

Figure 15‐ Column member stability (Plum, 1996)






See supporting notes section 13.4

It must be checked that the column buckling resistance is sufficient, so that the column is stable between the tensional restraint at S2 and the base. This part of the column would be checked using slenderness calculated. Different countries have different procedure to calculate the slenderness of the column and check the susceptibility of this part to lateral tensional buckling. Thus the designer must refer to the national Annex. In this example the procedure used in for assessing the significance of the mode of failure is taken from (King, Technical Report P164).

Figure 16‐ Column between tensional restraints (King, Technical Report P164)

(a) Calculate slenderness λ and λLT Assume side rail depth = 200 mm

Figure 17‐ Column/ Sheeting rails cross‐section (King, Technical Report P164)






(King, Technical Report P164)

Distance from column shear center to center of the side rail, a

a = 607.3/2 + 200/2 = 369.75 mm

is2 = iy

2 + iz2

+ a2

is2 = 218.72 + 45.72 + 369.752 = 186633 mm2

Distance between shear center of flanges

hs = h – tf = 539.5 – 18.8 = 520.7mm

α =

using the simplification for doubly symmetrical I sections

Iw = Iz ( hs / 2 )2

α =

α = . .

= 1.122

The slenderness of the column is given by:

.

.

1 .

. . .

. = 64.35






Appendix D Figure D1 EN1993-1-1:2005 Sec 6.3.2.2 EN1993-1-1:2005 Table 6.3

, /

. ,.

Where:

mt is moment factor obtained from appendix D . Because loads combination considered there is no lateral loads applied to the walls, so there are no intermediate loads

ψ = 0 / 603.1 = 0 y = 82.632 / ( Lt / iz ) = 82.632 / (4000 / 45.7 ) = 0.944

mt = 0.53 c =1 for uniform depth members

0.53 . 1 .

..

64.35 = 42.1

(b) Calculate buckling resistance for axial force

1/ Ф Ф .

Ф 0.5 1 0.2 h/b = 539.5/210.8 = 2.56 curve b for hot rolled I sections α= 0.34

/ 82.8

64.35/82.8 0.78






EN1993-1-1:2005 Sec 6.3.2.2 EN1993-1-1:2005 Table 6.3

Ф 0.5 1 0.34 0.78 0.2 0.78 0.90

, /

Ф 0.5 1 0.2 Ф 0.5 1 0.21 0.485 0.2 0.485

0.65

Xz = 1/ 0.90 0.90 0.78 . = 0.741

= (0.741 x 138.9 x 102 x 275) / (1.1 x 103 )= 2574.13KN

(c) Calculate buckling resistance for bending

Mb,Rd,y = ,

/86.8 = 42.1/ 86.8 = 0.485

1/ Ф Ф .

1/ 0.65 0.65 0.485 . = 0.92

Mb,Rd,y =.

. 650.4







μLT 0.0606

1μLT 1.0

10.0606 208.2 10

0.741 138.9 10 275 1.0

(d) Calculate buckling resistance to combined axial and bending

, ,

,

, ,1

Ψ = 0 βM,LT = 1.8 – 0.7 Ψ = 1.8 – 0.7 (0) = 1.8 μLT 0.15 βM.LT 0.15 but μLT 0.9 μLT 0.15 0.78 1.8 0.15 but μLT 0.9

0.996

.

.. .

. = 0.91

• The column is OK and stable over the section considered (between restraint So and S2).




Date 16/02/2009 Member checks - Rafter Sheet No


EN 1993-1-1: 2005 (E) Section 5.5 See Figure11 (Burgess, 20/01/1990)

6.3Rafter (UB457 x 191 x 89)

6.3.1Section Classification Ensure the section is class 1 to accommodate plastic hinge formation.

ε = 275/235 =1.08

Web ( combined axial and bending )

actual (d/tw ) =

.44.6

44.6 ≤72 ε Class 1

Flanges ( Axial Compressive )


3.66 9ε Class1

• The rafter section is Class 1

6.3.2 Crosssection Resistance. The frame analysis assumed that there is no reduction in the plastic moment resistance from interaction with shear force or axial force. This assumption must be checked because it is more onerous than that the cross-sectional resistance is sufficient.

- Max. shear force VEd = 160.5 KN at haunch tip - Max. axial force NEd = 166.9 KN at haunch tip






See supporting notes section 12.3 See supporting notes section 12.4

6.3.2.1Shear force effects of Plastic moment resistance (EN 199311: 2005 (E) Sec 6.2.6)

VEd < 0.5 Vpl,Rd

Av = 1.04 h tw = 1.04 x 462 x 9.6 = 4613 mm2 Vpl,Rd = /√3 /γ Vpl,Rd = 4613 275/√3 / 1.1 10 666 KN 0.5 Vpl,Rd = 333 KN

• VEd < 0.5 Vpl,Rd so the plastic moment of resistance is not reduced by the coexistence of axial force.

6.3.2.2Axial force effects of Plastic moment resistance (EN 199311: 2005 (E) Sec 6.2.9) Check

i) If ,

NEd < .

166.9 < . . . .

166.9 < 531.4

ii) If NEd < 0.25 Npl,Rd

NEd < 0.25 plastic tensile resistance of the section

NEd < .

231.1 < . .

.

166.9 < 590.5

• Therefore, the effect of shear and axial on the plastic moment resistance of the column sections can be neglected according to EC3 EN1993-1-1: 2005.




Date 16/02/2009 Member checks – Rafter Sheet No


See supporting notes section 13.2 EN 1993-1-1: 2005 (E) Sec BB3.2.2 See section 6.1

38 / 57.4 756 1 235

38 33.3

/166.9 10

57.4 94.48 10 1627 10 275

756 1 94.48 10 66.18 10 235

6.3.3 Check rafter buckling in apex region Another highly stressed region is the length of rafter in which the ‘apex’ hinge occur see fig below. Under (dead + snow) loading, the outstand flange is in tension, while compression flange is restrained by purlin/rafter connection.

Therefore, the buckling resistance of the rafter member between purlins in the

apex region needs to be checked. Because the ‘apex’ hinge is the last to form in order to produce a mechanism (which is true for low pitched portal frame under dead + snow loading), then adequate rotation capacity is not a design requirement, i.e. hinge is required only to develop Mp not to rotate.

It is set by EC3 EN1993-1-1: 2005 that if the value of Lm (as defined in BB.3.1.1) is not exceeded by restraint lateral torsional buckling can be ignored. So assuming that the purlins act as restraint because of their direct attachment to the compression flanges in the apex hinge region, then the purlin spacing should not exceed

Lm = 1221 mm = 1.221m

• As purlin spacing is 1.67m (on slope), thus because Lm has been smaller than 1.67m then the purlin spacing would have to be reduced in the apex region to 1.2m.

Figure 18‐Member stability apex region (Plum, 1996)






See Figure11 (Burgess, 20/01/1990) EN1993-1-1:2005 Sec 6.3.2.2 EN1993-1-1:2005 Table 6.3

6.3.4 Stability check for lower bending moments

Where bending moment is lower, the purlin spacing can be increased:

Figure 19‐ Rafter under lower bending moments

Next critical case is in right hand rafter. Try purlin spacing at 1670 mm centres Check for lateral torsional buckling between purlins:

o MEdmax.y = 345.6 kNm at haunch tip o NEd.max = 166.9 kN at haunch tip

(a) Calculate buckling resistance to axial force L =1670 mm λ z = L/iz = 1670/33.3 = 50.15 λ z = λ z / 86.8 = 50.15 / 86.8 = 0.578 Ф = 0.5 [ 1 + α ( λ – 0.2 ) + λ2 ] =0.5 [ 1 + 0.34 ( 0.578 – 0.2) + (0.578)2 ] = 0.7313

Xz = Ф √Ф

= . √ . .

= 0.8480

Nb,Rd,z = =

. .

= 2003.0 KN






EN1993-1-1:2005 Sec 6.3.2.2 EN1993-1-1:2005 Table 6.3

1 210000 1047 101670 10

516.3 10 1047 10

1670 81000 66.18 10 210000 1047 10

648.1

(b) Calculate buckling resistance to bending moment

•

,

Take C1 = 1 (conservative)

• λ LT = ,

λ LT =

. = 0.831

• ФLT = 0.5 [ 1 + α ( λ – 0.2 ) + λ2 ] ФLT = 0.5 [ 1 + 0.21( 0.831 – 0.2 ) + 0.8312 ] = 0.912

• XLT = Ф √Ф

XLT = . √ . .

= 0.80

• Mb,Rd,y = ,

Mb,Rd,y = .

. = 325.4KNm






(c) Calculate buckling resistance to combined axial and bending

Check that

, , + kLT ,

, 1

Take kLT = 1 ( conservative )

. + 1 x .

. 1

1 1

The value is slightly greater than one but due to the conservative assumption of KLT=1 the rafter can be assumed to be stable between intermediate restraint (purlin/sheeting rails) and purlin spacing could be increased to 1.67m between apex and hunch region shown in figure 19. Otherwise if the value was significantly greater than 1 the purlin spacing (1.67m) should be reduced.






EN 1993-1-1: 2005 (E) Section 5.5

6.4Haunch (UB 457 x 191 x 89) The depth of a haunch is usually made approximately twice depth of the basic rafter sections, as it is the normal practice to use a UB cutting of the same serial size as that of the rafter section for the haunch, which is welded to the underside of the basic rafter (UB 457x191x 89). Therefore it is assumed that the haunch has an overall depth at connection is 0.90 m.

6.4.1Classification

ε = 275/235 =1.08

Web The web can be divided into two, and classified according to stress and geometry of each. actual (d/tw ) =

.44.6

web 1 ( bending ) -------- 44.6≤72 ε Class 1 web 2 ( Compressive) --- 44.6≤38 ε Class 2

Flanges ( Axial Compressive


3.66 9ε Class1

• Thus the haunch section is a class 2.

Figure 20‐ Haunch region cross‐section classification (King, Technical Report P164)




Date 16/02/2009 Member checks - Haunch Sheet No


See Supporting notes section 13.1 & 13.3

6.4.2Haunch Stability First, check the stability of the haunched portion of the rafter ( from eaves connection to the haunch/ rafter intersection) as this represents one of the most highly stressed lengths, and with its outstand flange (inner) in compression, this part of the rafter is the region most likely to fail due to instability. As it has already decided to stay the inside corner of the column/haunch intersection (column hinge position), assume that the haunch/rafter intersection is also effectively torsionally restrained be diagonal braces, giving an effective length of 3m as indicated in Fig below.

Figure 21‐ Member stability haunch‐rafter region (Plum, 1996)




Date 16/02/2009 Member checks – Haunch Sheet No


EN 1993-1-1: 2005 (E) clause BB.3.1.2 (3)B EN 1993-1-1: 2005 (E) section BB.3.1.3 See supporting notes section Appendix B

It would appear that clause BB.3.1.2 (3)B is the most appropriate creation to check the stability of the haunched portion, as there is three flanged haunch, so the distance between rotational restraint should be limited to

Where: Lk is length limit specified where lateral torsional buckling effects can be

ignored where the length L of the segment of a member between restraint section at a plastic hinge location and adjacent torsional restraint.

Lk

.

.

Lk . .

. = 3738 mm

Lk = 3.738m

c is the taper factor (shape factor) which accounts for the haunching of

the restraint length (BB.3.3.3)

1

1

= 1.15






Figure 22‐ Dimensions defining taper factor (BS EN 1993‐1‐1:2005)

Is the modification factor for non-linear moment gradient (BB.3.3.2).

The plastic moduli are determined for five cross-sections indicated on the figure

below, the actual cross-section considered are taken as being normal to the axis of the basic rafter (unhaunched member). The plastic moduli together with the relevant information regarding the evaluation of the ratios Ni/Mi are given in the following table. The worst stress condition at the hunch/rafter intersection (location 5) is taken.




Date 16/02/2009 Member checks – Haunch Sheet No


Position on haunch (FIG ) 1 2 3 4 5

Distance from the eaves (m) 0 0.75 1.5 2.25 3

Depth of bottom web (mm) 428 321 214 107 0

Factored moment (My,Ed) (KNm) 904.5 764.8 625.1 485.3 345.6

Factored axial force (NEd) (KN) 171.1 170.1 169.1 168.1 167.1

Moment capacity (KNm) 1157 1011 784 686 447

Plastic modulus (cm3) 4209 3677 2849 2495 1627

Ratio (N/M) 0.19 0.22 0.27 0.35 0.48

a Value for (R) calculation (mm) 532.5 479.0 425.5 332.0 231.0

R Value 0.86 0.84 0.89 0.79 0.86

Table 2‐ Member forces at locations indicated in figure 18

Figure 23‐ Member stability haunch region (Plum, 1996)






See supporting notes section Appendix A

The modification factor is determined by the form;

in which R1 to R5 are the values of R according to equation below at the ends, quarter points and mid-length ( R values at positions 1 to 5 indicated in Table 2)

In addition, only positive values of ( ) should be included where,

- RE is the greater of R1 and R5 - Rs is the maximum value of R anywhere ( R1 to R5 )

- ,

,

Where (a) is the distance between the centroid of the member and the centroid of

restraining members (such as purlins restraining rafter). Here for simplicity a conservative value of (a) is found by conservative method of ignoring the middle flange as shown if figure (19).

Figure 24‐ Simplification for distance between centriod of rafter and purlin sections







123 4 3 2

120.86 3 0.84 4 0.89 3 0.79 0.86 2 0.89 0.86

1.17

√ . ..

3.4 3

• Thus this portion of the rafter is stable over the assumed restrained length of 3 m

(haunch length), as Ls is around 3m.

• If the value was found to be less than the haunch length then a torsion restraint should be provided in the haunch region as shown below

6.4.3 Crosssection resistance.

6.4.3.1Shear force effects of Plastic moment resistance The shear in the rafter has been checked above, showing that VEd < 0.5 Vpl,Rd . In the haunch, the shear area Av increases more than the applied shear VEd, so the shear force has no effect on the plastic moment capacity of the haunch.







6.4.3.2Axial force effects of Plastic moment resistance, The tables provided below gives the axial and moment resistances of the

haunch section at points 1to 5 shown in figures 18. A series of checks is carried out to determine whether the cross-sectional moment resistance MN,Rd is reduced by coexistence of axial force.

Position Distance (mm)

NEd (KN)

A (mm2)

Npl,Rd (KN)

Aweb (mm2)

(Aweb, fy )/ymo (KN)

1 0 171.1 16092 4023 8216 2054

2 0.75 170.1 15064 3766 7190 1798

3 1.5 169.1 14038 3510 6163 1541

4 2.25 168.1 13010 3253 5136 1284

5 3 167.1 11983 2996 4109 1027

• Npl,Rd = A fy / ymo and fy=275N/mm2 Table 3 – Axial force at positions indicated in figure 18 for haunch

Position Distance (mm)

MEd (KNm)

Is NEd > Does Axial force affect plastic bending

resistance 0.5 Aweb fy

/ymo 0.25 Npl,Rd

1 0 950 No No No

2 0.75 850 No No No

3 1.5 751 No No No

4 2.25 652 No No No

5 3 553 No No No

Table 4‐ Checking the significance of axial force on plastic moment of resistance

• Therefore, the effect of shear and axial on the plastic moment resistance of the column sections can be neglected according to EC3 EN1993-1-1: 2005.




Date 16/02/2009 Comparison between outcomes of different codes Sheet No


7. Comparison between Different Codes

As the dimensions of portal frame designed in this worked example were deliberately chosen to be exactly the same as worked-example in (King, Technical Report P147) a comparison was done between the carried out worked example to (BS EN 1993-1-1:2005), (BS9590-1:2000) and (ENV1993-1-1:1992). The following is a summary of different outcomes and source of design,

Design no

Design Code

Column Section

size

Rafter Section Size

Haunch Length

(m)

Purlin Spacing

(m)

Design Source

1 BS EN 1993-1-1:2005

533x210 UB 109

457x152UB 74 3 1.67 Worked -

example

2 BS9590-1:2000 610x229 UB 113

533x210 UB 82 3 1.85

(King, Technical

Report P164)

3 ENV1993-1-1:1992

610x229 UB 113

457x191UB 74 3 1.85

(King, Technical

Report P164)

Table 5‐ Comparison between different code outcome




Date 16/02/2009 Appendix A Sheet No


EN 1991-1-3:2003 (E) Section 5.3.2 EN 1991-1-3:2003 (E) Table C1

8Appendix

A.1 Roof shape coefficient The values given in table A1 apply when the snow is not prevented from sliding off the

roof. Where the snow fences or other obstruction exists or where the lower edge of the roof is terminated with a parapet, then the snow load shape coefficient should not be reduced below 0.8.

Table A1- Snow load shape coefficients (BS EN 1991-1-3:2003)

A.2 Snow load relationships The snow load on ground; the characteristic value depends on the climatic region; the following table gives different expressions for different regions,

Table A2- Altitude-Snow load relationships (BS EN 1991-1-3:2003)




Date 16/02/2009 Appendix A Sheet No


EN 1991-1-3:2003 (E) Figure C.4

Where: Sk is the characteristic snow load on the ground (KN/m2) A is the site altitude above the sea level (m) Z is the zone number given on the map ( see fig A1 )

The following maps gives the zone number Z for UK and republic of Ireland if other Z values for regions mentioned in Table A2 refer to EN 1991-1-3 Annex C pages ( 41 to 52 ).

Figure A1 UK , Republic of Ireland : snow loads at sea level (BS EN 1991-1-3:2003)




Date 16/02/2009 Appendix B Sheet No


TP/08/43 EC3/08/16 Manual for the design of steelwork building structures to EC3

B1 Initial sizing using Weller’s charts

The method described relies for its simplicity on a series of three charts developed by Alan Weller. The chart has been constructed with the following assumptions and which leads to reasonably economic solution (Note. This is not a rigorous design method; it is a set of rules to arrive at initial size).

1) The rafter depth is approximately span / 55 2) The hunch length is approximately span /10 3) The rafter slope lies between 0o and 20o. 4) The ratio of span to eaves height is between 2 and 5. 5) The hinges in the mechanism are formed at the level of the underside of the

haunch in the column and close to the apex.

Each chart requires a knowledge of the geometry of the frame and the design loading as input data in order to determine approximate sizes for the column and rafter members Using of charts

Figure B1 Dimensions of portal (The institutionof Structural Engineers, TP/08/43

EC3/08/16)

a) Calculate the span/height to eaves ratio = L/h b) Calculate the rise/span ratio = r/L c) Calculate the total design load FL on the frame and then calculate FL2, where F is

the load per unit length on plan of span L (e.g. F =qs, where q is the total factored load per m2 and s is the bay spacing).

d) From figure B2 obtain the horizontal force ratio HFR at the base from r/L and L/h e) Calculate the horizontal force at the base of span H=HFR W L.







f) From figure B3 obtain the rafter Mp ratio MPR from r/L and L/h. g) Calculate the Mp required in the rafter from Mp (rafter) = MPR x W L2. h) From figure B4 obtain the column Mp ratio MPL from r/L and r/h. i) Calculate the Mp required in the rafter from Mp (rafter) = MPL x W L2. j) Determine the plastic moduli for the rafter Wpl,y,R and the column Wpl,y,C from

Wpl,y,R =Mp,(rafter) /fy Wpl,y,C = Mp,(column) /fy Where fy is the yield strength.

Using the plastic moduli, the rafter and column sections may be chosen from the range of plastic sections as so defined in the section books.

0.1 0.2 0.30.15 0.25 0.35 0.450.40

0.05

0.1

0.15

0.2 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Ris

e/sp

an

H/wL

Span to eaves height

Figure B2- Horizontal force at the base (The institutionof Structural Engineers, TP/08/43 EC3/08/16)







2.05.0 4.04.5 3.5

3.02.5

0

0.05

0.1

0.15

0.2

0.02 0.025 0.03 0.035 0.04 0.045


Ris

e/sp

an

M / wL²pr

0

0.05

0.1

0.15

0.2

0.05 0.055 0.06 0.07 0.080.065 0.075

5.0 2.04.5 2.54.0 3.03.5

0.045


Ris

e/sp

an

M / wL²pl

Figure B3- Mp ratio required for the rafter (The institutionof Structural Engineers, TP/08/43 EC3/08/16)

Figure B4- Mp ratio required for the column (The institutionof Structural Engineers, TP/08/43 EC3/08/16)




Date 16/02/2009 Appendix C Sheet No


King Span Website Link http://www.kingspanstructural.com/multibeam/rp/load_tables.htm

C1 King Span Multibeam Purlin (Load tables)

Loading Load Factor Dead load 1.4 Dead load restraining uplift or overturning 1.0 Dead load acting with wind and imposed loads combined 1.2 Imposed load 1.6 Imposed load acting with wind load 1.2 Wind load 1.4 Wind load acting with wind and imposed load 1.2 Forces due to temperature effects 1.2




Date 16/02/2009 Appendix C Sheet No





Date 16/02/2009 Appendix D Sheet No



D1 Equivalent uniform moment factor mt for all other cases

This formula, derived by (Sinhgh, 1969), is applicable in all cases, especially when the bending moment diagram is not a straight line between the tensional restraints defining the ends of the element.

Figure D1- Moment factors (King, Technical Report P164)




Date 16/02/2009 Appendix D Sheet No



MEd1 to MEd5 are the values of the applied moments at the ends, the quarter points and mid- length of the length between effective torsional restraints, as shown in Figure D.2. Only positive values of MEd should be included. MEd is positive when it produces compression in the unrestrained flange.

Figure D2- Intermediate moment (King, Technical Report P164)

D2 Equivalent section factor c

For uniform depth members, c = 1,0.

References BS EN 1991‐1‐3:2003 Eurocode 1 — Actions on structures Part 1‐3: General actions — Snow loads [Book]. ‐ 389 Chiswick High Road, London, W4 4AL : Standards, Institution British.

BS EN 1993‐1‐1:2005 Eurocode 3: Design of steel structures BS EN 1993‐1‐1:2005 [Book]. ‐ 389 Chiswick High Road, London, W4 4AL : Standards, Institution British.

Burgess Ian PLT Portal frame design Software. ‐ 20/01/1990. ‐ Vol. Ver 1.3.

King C M Design of Steel Portal for Europe [Book]. ‐ [s.l.] : The steel construction Institute, Technical Report P164.

Kingspan [Online] // www.kingspanstructural.com. ‐ February 10, 2009. ‐ http://www.kingspanstructural.com/pdf/double_span_tiled_roofs.pdf.

Plum L J Morris & D R Structural Steelwrok Design to BS5950 2nd Edition [Book]. ‐ [s.l.] : Harlow : Longman, 1996.

Sinhgh K.P. Ultimate behaviour of laterally supported beams [Book]. ‐ University of Manchester : [s.n.], 1969.

SX016, Matthias Oppe Determination of loads on a building envelope [Online] // www.access‐steel.com. ‐ Access Steel. ‐ October 20, 2008. ‐ http://www.access‐steel.com/Discovery/ResourcePreview.aspx?ID=J6osLkASHmChe7uBKVEzGw==.

The institutionof Structural Engineers Manual for the design of steelwork building structures to Eurcode 3 [Book]. ‐ TP/08/43 EC3/08/16.



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