Job No. Sheet 1 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
Worked Example
on the Preliminary Design of a
EUROPEAN PORTAL FRAME to Eurocode 3
by Dr Andrzej Czechowski
Ascot, November 2002
Preliminary Design of a European Portal Frame to Eurocode
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Job No. Sheet 2 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
1. INTRODUCTION
2. DESIGN BRIEF
3. LOADS 3.1 Permanent actions on the roof 3.2 Snow load 3.3 Wind load 3.4 ‘Second order’ sway action 3.5 Combinations of actions
4. STATICS (ELASTIC ANALYSIS) 4.1 UDL over two spans 4.2 Horizontal force case 4.3 Internal forces and reactions
5. MEMBER DESIGN 5.1 Cross-section resistance – definitions 5.2 Columns 5.3 Rafters 5.4 Deformations – serviceability check
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Job No. Sheet 3 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
1. INTRODUCTION
• This is a manual example of the preliminary design of a typical two-span portal frame in steel. The limit state design is carried out in accordance with Eurocodes[1,2]. (U.K. standards [3,4] are used for loading, because of the current unavailability of EC1).
• Elastic analysis of the main frame is performed using the traditional but effective, Prof. Kleinlogel’s Rahmenformeln [5]. In the analysis, initial sway imperfection and second order effects are directly taken into account, so stability of the frame is checked in the non-sway buckling mode.
• The constructional system finally adopted is a very simple and economical solution.
Notation (major symbols) - to Eurocodes[1, 2]
Abbreviations used in the Example:
BMD - Bending Moment Diagram
FB - Flexural Buckling
LTB - Lateral Torsional Buckling
LC - Load Combination
RUR - Resistance Utilization Ratio
SLS - Serviceability Limit State
ULS - Ultimate Limit State
UDL - Uniformly Distributed Load
VA - Variable Action
Preliminary Design of a European Portal Frame to Eurocode
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Job No. Sheet 4 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
2. DESIGN BRIEF Basic data and criteria for structural design
Two-bay portal-frame building – a warehouse (located somewhere in the UK)
• Building size
Width : 2 × 20 = 40 m (column centres)
Length : 60 m (frame centres)
Internal clear height : 6,50 from TOF (± 0,00)
• Loads
Wind : - to BS6399-2 (Ve = 100 km/h)
Snow : - to BS6399-3 (so = sb = 0,75 kN/m2)
Service load : 0,10 kN/m2
• Frame geometry
Span : L = 20 m
Spacing : 6 m
Column base level : –0,20 m
Structural height : h = 7,0 m (base to centre line at knee)
Roof pitch : 5% (~3o)
• Cladding – metal sheeting (panels on purlins/girts)
• Deformation limits for the frame (serviceability)
Vertical deflection : L/200
Horizontal displacement (sway) : H/150
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Job No. Sheet 5 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
General arrangement of main frames
B C
12
34
5
C L
89
10
11 C L
10 x
6.0
m =
60
m
3° 3° 3° 3°
7.0
m
0.2
0 m
20 m
±0.00 (TOF)
CL
8.0
m
PLAN
A
20 m
CROSS SECTION
NOTE: All column bases are to be nominally pinned bases.
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Job No. Sheet 6 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
3. LOADS
3.1 Permanent actions on the roof
• Characteristic values (self-weights):
G1 - rafters & purlins 0,20 kN/m2
G2 - roofing 0,20 kN/m2
G1 - servicess 0,10 kN/m2
Total: Gk = 0,50 kN/m2
• Design values:
Gd,inf (favourable) = γ f,int Gk = 1.00 × 0.50 = 0,50 kN/m2
Gd,sup (unfavourable) = γ f,sup Gk = 1.35 × 0.50 = 0,68 kN/m2
3.2 Snow load
(to BS6399-3, Ref [4])
• Snow load on the roof – characteristic and design values:
Qk = sd = µ1so = 0,8 × 0,75 = 0,60 kN/m2
Qd = γq Qk = 1,50 × 0,60 = 0,90 kN/m2 (leading VA)
• Local drifting in valley (as alternative and exceptional case)
Snow load shape coefficient : µ = 2h/so = 2 × 0,5/0,75 = 1,33
1
1
0.5 L 0.5 L
h=0.5 m3°α≈
' = 1.33
= 0.8 (uniform snow)µ
µ
NOTE: This case will not govern since ½ µ1 < µ.
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Job No. Sheet 7 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
3.3 Wind load
(Standard Method to BS6399-2, Ref [3])
• Basic data:
Dynamic pressure: qs = 45 kN/m2 (Ve ≅ 100 km/h)
Roof pitch angle α ≅ 3o
Building overall dimensions:
W (width) ≅ 40,0 m
L (length) ≅ 60,0 m
H (height) ≅ h + 1,0m = 7,0 + 1,0 = 8,0 m
• Wind cases – pressures coefficients ([3]/Tables 5 & 8):
a) Transverse wind b) Longitudinal wind
b/2 = 8.0 m
BC
D = L
-0.7
+0.6 -0.5
±0.2
b/2 = 8.0 m
-0.7
+0.6
C
-0.5
±0.2
D = L
D/H 4≥
b = 2H = 16.0 m
(C = -0.3)pi (C = -0.3)pi
B = L
B = W
(-0.8)
(-0.8)
w∆
D
(-)
(-)
NOTE: In case D, with regard to the roof, two subcases are to be considered:
pC′ = – 0,2 (suction, Cpi = 0) or pC ′′ = + 0,2 – (–0,3) = 0,5 (pressure & underpressure)
• Tributary areas and size effect coefficients ([3]/Fig. 4: B-line):
Walls A = 6 × 8 = 48 m m1086a 22 =+= Ca = 0,95
Roof As = 6 × 40 = 240 m m40406a 22 ≅+= Ca = 0,86
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Job No. Sheet 8 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
• Wind loads on the roof in Zone B/C:
UDL : ∆w = γ0 γQ qs Cq Cp × 6.0 m [kN/m]
pC′ = + 0,5: ∆w1 = 0,6 × 1,50 × 0,45 × 0,85 × 0,5 × 6,0 ≅ 1,0 kN/m
pC ′′ = – 0,2: ∆w2 = 1,00 × 1,50 × 0,45 × 0,85 × 0,2 × 6,0 ≅ 0,7 kN/m
Coexistent with ∆w2 the extra up-list force in Zone C
Ca = 0,95; Ac = 48m2
Pc = 1,50 × 0,45 × 0,95 × [–0,7 – (–0,2)] × 48m = –15,4 kN
• Horizontal wind forces (for one frame, case a):
Overall surface load on both windward and leeward walls:
P = 0,85 Σqs Cpe Ca A = 0,85 Ca qs A ΣCpe
= 0,85 × 0,95 × 0,45 × 48 × (0,6 + 0,5) = 19,2 kN
NOTE: Factor 0,85 accounts for non-simultaneous action between faces
Frictional force on the roof:
Ps = qs Cf As Ca = 0,45 × 0,01 × 240 × 0,85 = 0,92 kN
Resultant horizontal wind force at rafters (see 4.2):
Hw= kN0,1292,02,190,85,4PP
HH
s ≅+×=+×∆
NOTE: ∆H – upper ‘tributary’ height: ∆ ≅ ½h + 1.0m = 3.5 + 1.0 = 4.5m
Design values of horizontal forces:
Hd1 = ψoγQ Hw = 0,6 × 1,50 × 12,0 ≅ 11,0 kN (accompanying VA)
3.4 ‘Second order’ sway action
(to prEN 1993, Ref [2]/5.2 & 5.3)
• Initial sway imperfection for global analysis:
φ = φo α h αm
φ = 1/200
αh = 75,07/2h/2 ==
αm = 82,0)3/11(5,0)m/11(5,0 =+×=+
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Job No. Sheet 9 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
φ = 2001 × 0,75 × 0,82 = 0,005 × 0,61 = 0,0031
• Equivalent horizontal force due to imperfection:
Hφ = φVEd;
Total max. gravity load (see 3.6/LC1)
VEd = 2w1L = 2 × 9,5 × 20,0 = 380 kN
Hφ = 0,0031 × 380 = 1,2 kN
• Total initial (first order) horizontal action:
Ho = Hd1 + Hφ = 11,0 + 1,2= 12,2 kN
• Critical load factor:
αcrit =
odh
VH
Ed
o
δo = δ(Ho) = Ho/Kh = 12,2/3,2 = 3,8 cm (see 4.2)
αcrit = 8,3
700380
2,12 × = 5,9
NOTE: Since α crit < 10, second order effects (action) cannot be disregarded.
• Total second order (amplified) horizontal action to be used in the global analysis:
19,5
9,52,121
HHcrit
crito
II1 −
×=−
=α
α = 12,2 × 1,20 = 14,7 kN
3.5 Combinations of actions
(to EN 1990, Ref [1])
• The following actions are considered:
G – permanent load (as unfavourable or favourable action)
Q – snow load (as leading variable action)
W – wind load (as accompanying or leading variable action)
• Loading on the frame is represented by two parameters:
- UDL over two spans: w [kN/m]
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Job No. Sheet 10 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
- Horizontal force: H [kN]
• ULS load combinations ([1]/Table A1.2B):
(1)Max. gravity load + wind (1,35G + 1,50Q + 0.6 × 1,50W):
w1 = (Gd,sup + Qd)× 6,0m= (0,68 + 0,90) × 6,0 = 9,5 kN/m
I1w = w1 + ∆w1 = 9,5 + 1,0 = 10,5 kN/m (∆w1 – see 3.3)
II1H = 14,8 kN (see 3.4)
(2)Min. gravity load + wind (1,00G + 1,5W):
w2 = Gd,inf × 6,0m = 0,50 × 6,0 = 3,0 kN/m
12w = w2 - ∆w2= 3,0 – 0,7 = 2,3 kN/m (∆W2 – see 3.3)
H2 = Hd2 = 18,0 kN
• SLS characteristic load combinations ([1]/Table A.1.4):
(3)Max. gravity load (G + Q) – for vertical deflection
w3 = (Gk + Qu) × 6,0m = (0,50 + 0.60) × 6,0 = 6,6 kN/m
(4)Wind load (W) – for horizontal displacement (sway)
H4 = HW = 12,0 kN
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Job No. Sheet 11 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
4. STATICS (ELASTIC ANALYSIS) (From SDM Ref [5])
4.1 UDL over two spans
([5]/Frame VIII)
A+ +
B B' A'
D D'E
IR
IC
C
w (kN/m)
L = 2s L = 2s
h
_ _ _ __ _
HD H 'DV 'VD DVE
Frame data: L = 20,0 m h = 7,0 m f = 0.5 m Ic ≅ 2IR
• Constants:
φ = 07,00,75,0
hf == but φ ≅ 0 accepted for simplification
xI = x = 35,0107
21
sh
II
c
R =⋅=
N1 = 8x + 12 = 14,8; 4N1 = 59.2
• Influence coefficients (for φ = 0 and x = 0.35):
n11 = 1N
2 = 0,1351; n12 = n21 = 1N
1 = 0,0676; n22 = 1N2x + = 0,1588
r1 = n11 + n21 = -0,067; r2 = n12 + n22 = 0,0912
L1 = 1N4
2 = 0,0338; L2 = 1N42x2 + = 0,0456
m1 = L1 – 2r1 = 0,0673; m2 =L2 +
2r2 = 0,0912
c1 = 14
6N
= 0,101; c2 = 1N46x6 + = 0,137
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Job No. Sheet 12 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
• Bending moments:
MA = 4
wLM21
A −= (m1 + 2c1) = –0,067wL2
Mc =4
wL2− (m1 + 2c2) = – 0,091wL2
MB = 2
MM8
wLM cA21
B++−= = 0,046wL2
• Reactions:
V1 = (–Ma + Mc)/L = – 0,024wL; V2 = – V1 = 0,024wL
Vc1 = – V1 + wL/2 = 0,524wL; Vc2 – V2 + wL/2 = 0.524wL
VE = Vc1 + Vc2 = 1,048wL
VD = 1DV = 0,500wL – 0,024wL = 0,476wL
HD = 1DH = MA/h = 0,191wL
• Mid-span deflection:
u(M) = 21M;
EILM125,0
EI8LM 22
== (MA + Mc) = –0,079wL2
u(w) = EI
LM104,0EI384
wL5 2o
4= ; Mo = 0,125wL2
u = u(M) + u(w) = (–0,125 × 0,79 + 0,104 × 0,125)EIwL0031,0
EIwL 44
=
u = ΦwW; where; W = wL, Φw(flexibility) = EI
L0031,0 3
• For I = IR (IPE 360) = 16270cm4:
Φ W = 162700,2100020000031,0 3
×× = 0,073cm/kN; (Kw =
w
1Φ
= 13,7 kN/cm)
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Job No. Sheet 13 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
4.2 Horizontal force case
([5]/Frame II)
B A'
D E
C+
V 'DD
HD D' DH '
VE
B'
__
L LV
h
L
δ
H = 2P A
hI C
IR
w
wh
P
• Bending moments:
MA = Hh21M1
A =− = Ph
• Reactions:
VD = 1DV = Pw/L = 0,35P
HD = H21H1
D = = P
• Horizontal deflection – sway stiffness (to [6]):
δo = δR + δ c = R
2
C
3
r
a
E6HLh
EI3Ph
EI3LhM =+ (1 + k) ⇔ H/Ko
δb = R
2
c
3
c
3
EIkHLh42,1
EI3Ph
EIPh
25 =+ ⇔ H/Kb; k =
LIhI
c
R
NOTE: Kb is additive stiffness component due to the rotational stiffness of the nominally pinned base (= 10% column stiffness)
KH = Ko + Kb =
+
+=+
+ k7,0
k16
LhEI
kLhEI7,0
)k1(LhEI6
2R
2R
2r
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Job No. Sheet 14 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
For k = 207
21 × = 0,175 and IR (IPE360) = 16270cm3:
KH = 9,1 × 27002000162700,21000
×× = 3,2 kN/cm; (Φ H =
HK1 = 0,31 cm/kN)
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Job No. Sheet 15 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
4.3 Internal forces and reactions
Extreme design values
• Bending moments [kNm] at critical sections:
B.M. definition ? Splice at x1 ? Splice at x2 ?
ξ = x/L = 0,00 0,15 0,50 0,80 1,00
Load combination 1 (see 3.5)
M( I1w = 10,5kN/m) – 281 – 29,4 – 193 – 25,2 – 382
M( II1H = 14,7kN) 51,5 44,2 25,8 10,4 0
1a : M( I1w ) + M( II
1H ) – 229 – 14,8 219 14,8 – 382
1b : M( I1w ) – M( II
1H ) – 333 – 44,2 167 – 35,6 – 382
Load combination 2 (see 3.5)
M( I1w = 2,3kN/m) – 61,6 – 6,4 42,3 – 5,5 – 83,7
M( 2H = 18,0kN) 63,0 53,6 31,5 12,6 0
2a : M( I2w ) + M(H2) 1,4 47,2 73,8 7,1 – 83,7
2b : M( I2w ) – M(H2) – 125 – 60,0 10,8 – 181,1 – 83,7
• M(w,x) = MA + Lx (Mc –MA) + 2wL
Lx1
L2x
−
= [– 0,067 – 0,024 ξ + 21 ξ (1 – ξ )] wL2 = [f2(ξ ) + f3(ξ )] wL2, see 4.1
• M(H,x) = MA(H)
−
Lx1 =
21 (1 – ξ ) Hh = f1(ξ ) Hh, see 4.2
f1(ξ ) = 21 (1 – ξ ) 0,500 0,425 0,250 0,100 0
f2(ξ ) = ξ f1(ξ) 0 0,064 0,125 0,080 0
f3(ξ ) = – (0,067 + 0,024 ξ ) – 0,067 – 0,071 – 0,079 – 0,086 – 0,091
f2(ξ ) + f3(ξ ) – 0,067 – 0,007 0,046 – 0,06 – 0,091
NOTE: M(w,x) = 0 for x = x I0 = 0.17L and x = x II
0 = 0,78L
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Job No. Sheet 16 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
• Bending moment envelope:
A
M(-)
M(+)
-60.0
47.2
219
-35.614.8
-333-382
x
BMD (kNm)
x =0.15L
x =0.80L2
L = 20 m
1
CB
NOTE: x1 and x2 are potential splice locations near contraflexure points.
• Reactions:
A'
D E
C
w (kN/m)
V 'VD DVE
DD'
h
(P )C B B'A
H=2P
L L
w =1.2 kN/mo
H '
Leeward suction (C = -0.5)le
(a)‘Windward’ column (LC2a w2 = 2,3 kN/m; p = 21 H2 = 9,0 kN; Pc = 15,4 kN)
min Vd = 0,476 I1w L – 0,35P – Pc = 0,476 × 2,3 × 20 – 0,35 × 9,0 – 15,4 = 3,3 kN
NOTE: There will be no uplift force on footing (min Vd > 0)
(b)Mid column (LC1: w1 = 10,5 kN/m)
max Ve = 1,048 I1w L= 1,048 × 10,5 × 20 = 220 kN
(c) ‘Leeward’ column (LC1b: I1w = 10,5 kN/m); P = 2
1 HD1 = 5,5 KN)
max 1DV = 0,476 I
1w L + 0,35P = 0,476 × 10,5 × 2070,35 × 5,5 = 102 kN
max 1DH = 0,191 I
1w L + P + 0,5woh = 0,191 × 10,5 × 20 + 5,5 + 4,2 = 49.8 kN
NOTE: In above calculations only horizontal force from external (wind) load is used, as contributing to the net reactions
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Job No. Sheet 17 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
5. MEMBER DESIGN (to prEN 1993-1-1, Ref. [2])
5.1 Cross-section resistance – definitions
(Class 1/2 I-sections)
Bending: Mc,Rd = Mpl,Rd = fyWpl/γM1
Wpl – plastic modulus
Shear Vc,Rd = Vpl,Rd = 3yf Av/γM1
Av – shear area; Av = A – 2bftf (for V = Vy)
Compression Nc,Rd = Npl,Rd = fyA/γm1,
where fy = specified yield stress [MPa]
γM1 = partial safety factor: γM1 = 1,00
NOTE 1: In the combined load cases (M, V, N), the design moment resistance need not be reduced, provided VEd ≤ Vo = 0,5Vpl,Rd and NEd ≤ No = min [0,25Npl,Rd, 0,5fyhwtw].
(In the present example, where bending predominates, both above conditions will be satisfied at all sections)
NOTE 2: According to Eurocode convention ‘strong’ axis will be denoted by y – y and ‘weak’ axis by z – z
NOTE 3: Cross-sectional geometrical properties are from Ref [7].
5.2 Columns
5.2.1 External Columns
Try IPE 450 (g = 77,6 kg/m) – steel S235
• Section properties and resistances:
A = 98,8cm2; Av = 98,8 – 2 × 19 × 1,46 = 43,3cm2;
Iy = 33740cm4; Wpl,y = 1702cm3; iy = 18,5 cm; Iz = 1680cm3; iz = 4,12cm
Mpl,Rd = 235 × 1702/103 = 400 kNm
Vpl,Rd = (235/√3) × 43,3/10 = 588 kN
Npl,Rd = 235 × 98,8/10 = 2322 kN
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Job No. Sheet 18 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
• Flexural buckling reduction factors ([2]/Fig. 6.3):
FB about y – y (curve ‘a’)
Lcr = 0,9h = 0,9 × 700 = 630cm (0,9 × system length)
λ y = Lcr/iy = 630/18,5 = 34
λ y = λ y/λ1 = 34/93,9 = 0,36 → χ y = 0,97
FB about z – z (curve ‘b’)
Lcr = l1 = 300cm (wall girt spacing)
λ2 = Lcr/i2 = 300/4,12 = 72,8
λ z = λ z/λ1 = 72,8/93,9 = 0,78 → χ z = 0,73
• Lateral torsional buckling assessment:
l =6.0 mo
MA
N
V
Mo L
a) L - l o o
Lc
c
Lc
xxψ
MLT
c
ψ
MLT
c
c
c
b) L - 0.5l
= 0.5
C =0.8
k =0.86 = 0
C =0.6
k =0.75
x
NOTE: Two cases are assessed: (a) without and (b) with one intermediate lateral restraint to compression flange. (Small local bending Mo = 1,2 × 6,02/8 = 5,4 kNm may be ignored)
Simplified approach ([2]/6.3.2.4)
Af = b × tf = 19 × 1,46 = 27,7cm2 (flange area)
Awc ≅ 21 Av = 2
1 × 43,3 = 21,7cm2 (web compression area)
If,z ≅ 21 Iz = 2
1 × 1680 = 840cm4
if,z = 1,249,34
840AA
I
wc31
f
f ==+
≅ 4,9
Case (a)
λ f = 9,939,4
60075,0xiLk
1z,f
cc
××= = 0,98 → χ ≅ 0,55 (curve ‘c’)
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Job No. Sheet 19 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
χLT ≅ kfL χ (λ f) = 1,1 × 0.55 = 0,61
Mb,Rd = χLTMpl,Rd =0,61 × 400 =244 kNm
MEd = max/MA/ = 333 kNm (see 4.3); MEd > Mb,Rd
NOTE: In this case the buckling moment resistance is not sufficient.
Case (b)
λ f = 9,935
3008,0×× = 0,51 ≅ λ co = λLT,O + 0,1 = 0,5 (Slenderness limit)
NOTE: In this case the moment resistance will not be affected by LTB.
• Interaction factor – case (b) ([2]/Table B.1):
Cmy (x = 0) = 0,6; ny = NEd/χ yNpl,Rd = 0,45 (see below)
kyy = Cmy [1 + (λ y – 0,2)ny] = 0,6 [1 + (0,37 – 0,2) × 0,045] =0,60
NOTE: Since second-order effects have been accounted for on the global analysis, Cmy as for non-sway mode is used.
• ULS/Bending + compression:
VEd = maxH1d = 49,8 kN; (VEd < Vo = 294 kN)
NEd = maxV1d = 102 kN; (NEd < No = 483 kN)
NOTE: For limiting values Vo and No see 5.1
RUR: ]OK[183,0400333
MN
Rd,pl
Ed <==
Member stability check ([2]/6.3.3 – formula 6.6.1
RUR: ]OK[155,040033360,0
232297,0102
MMk
NN
Rd,pl
Edyy
Rd,ply
El <=+×
=+χ
IPE450 can be adopted for external columns, if at least one mid-height lateral restrain to both flanges is provided – see case (b).
5.2.2 Internal column (pin-ended prop)
Try CHS 193,7 × 5,4 (g = 25,1 kg/m) – steel S235
NEd = max 1EV = 220 kN (See 4.3)
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Job No. Sheet 20 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
A =31,9 cm2; i = 6,77
Lcr = h1 = 650 cm; λ = 650/6,66 = 97,6
λ =λ/λ1 = 97,6/93,9 1,04 → χ = 0,65
Nb,Rd = χAfy = 0,65 × 31,9 × 235.10 = 487 kN
ULS/Member stability check
RUR: ]OK[145,0487220
MN
Rd,B
Ed <==
NOTE: This size is a practical minimum for constructional reasons.
5.3 Rafters
5.3.1 Rafter valley segment
Try IPE 450 (g = 77 kg/m) – steel S235
Section properties and resistances – see 5.2
ULS/Section capacity check
VEd = 21 maxVE = 2
1 220 = 110 kN; (VEd < Vo = 294 kN)
MEd = max/Mc/ = 382 kNm (see 4.3)
RUR: 400382
MM
Rd
Ed = = 0,96 < 1 [OK]
NOTE: To prevent this segment from lateral-torsional buckling, adequately located restraints, as in columns (see 5.2), must be provided (Lc ≤ 3,0 m).
5.3.2 Rafter ridge segment
Try IPE360 (g = 57,1 kg/m) – steel S235
• Section properties and resistances:
A = 72,7 cm2; Wpl,y = 1019 cm3; iz = 3,79 cm
Iz = 1043 cm4; IT = 37,4 cm4; Iw = 0,314 cm6
Npl.Rd = 235 × 72,7/10 = 1710 kN
Mpl,Rd = 235 × 1019/103 = 239 kNm
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Job No. Sheet 21 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
• Lateral-torsional buckling assessment ([2]/6.2.2):
M=MB
L c
x x x≈w ≈w
Ridge
k 1.0 k 0.5
L = Lc = 2,0 m (purlin spacing) C1 = 1,0 (quasi-constant moment) k = kz = 1,0; kw = 1,0 (no warping fixity)
Mcr = Ncr,z
z,cr
T
z
w2
w NGI
II
k1 +
, where Ncr,z = 2c
z2
LEIπ
Ncr,z = kN5400200
10430,210002
2=××π (Euler’s critical load)
1043
10315,0II 6
z
w ×= = 302 cm; GIT = 8100,0 × 37,4 = 303000 kN cm2
Mcr 22
10/5400
3030003027,015400 +
= = 1400 kNm
λLT = 1400239
MM
cr
Rd,pl = = 0,48→ χLT = 0,88 curve ‘b’)
NOTE: Since h/b = 360/170 = 2,1 ≅ 2, curve ‘b’ is applied.
• Flexural buckling reduction factor (FB about z – z):
λ z = 85,056,054001710
NN
z,cr
Rd.,p =→== χ = (curve ‘b’)
• ULS/Bending + compression:
Member stability check ([2]/6.3.3 – formula 6.6.2)
NEd = 0,191 w1L = 0,181 × 10,5 × 20 = 38 kN (see 3.5 and 4.3)
MEd = maxMB = 219 kNm (see 4.3)
Interaction factor: kzy ≅ 1,0 (very small compression component)
Preliminary Design of a European Portal Frame to Eurocode
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Job No. Sheet 22 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
RUR: ,107,123988,0
219171085,0
38
,,
>=×
+×
=+RdplLT
Edzy
Rdplz
Ed
MM
kNN
χχ
NOTE: Either accept as preliminary design or reduce purlin centres.
5.4 Deformations – serviceability check
• Deflections of the rafter (L = 20m):
Φw(IPE360) = 0,073 cm/kN (see 4.1)
W = W3L = 6,6 × 20= 132 kN (see 3.5/LC3)
Deflection due to max. gravity load (Gu + Qk);
max u = u (Gk + Qk) = ΦwW = 0,073 × 132 = cm10200Lcm6,9 =< [OK]
Deflection due to imposed load (Qk)
u(Qk) = cm8250L2,56,9
6.05,06,0 =<=×
+ [OK]
• Horizontal displacement of the frame (h = 7,0m):
Kw = 3,2 kN/cm (see 4.2, as assumed: IR/Ic = 0,48 ≅ 21 )
Hw= 12,0 kN (see 3.5/LC4)
maxδ = cm7,4150hcm7,3
2,30,12 =<= [OK]
Preliminary Design of a European Portal Frame to Eurocode
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Job No. Sheet 23 of 23 Rev A Job Title Worked Example
Subject European Portal Frame
Made by AWC Date November 2002 Silwood Park, Ascot, Berks SL5 7QN Telephone: (01344) 623345 Fax: (01344) 622944 CALCULATION SHEET
Client
Checked by JBL Date January 2003
P:\CMP\Cmp657\European Portal Frame Worked Example.Version 3.doc
References
1 EN 1990 : 2002 Eurocode – Basis of structural design 2 prEN 1993-1-1: 2002 Eurocode 3: Design of steel structures Part 1-1: General rules and rules for
Buildings (final draft) 3 BS6399-2: 1997 Loading for buildings. Part 2: Code of practice for wind loads 4 BS6399-3: 1988 Loading for buildings. Part 3: Code of practice for imposed roof loads 5 Steel Designers’ Manual 4 th Edition, SCI, Blackwell’s, Oxford 6 In-plane stability of Portal Frames to BS 5950-1: 2000
SCI Publication P292 7 Structural sections in accordance with European specification. British Steel (1995)
Preliminary Design of a European Portal Frame to Eurocode
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