StiFF Meeting
September 23, 2003
Analysis of Catenary Action in Steel Beams under Fire Conditions Yin ingz gzh hi Yin Manchester Manchest er Centre for Civil and Construction Engineering University of Manchester & UMIST
OVERVIEW
●
●
Introduction Numerical Simulations by ABAQUS
●
Hand Calculation Method
●
Conclusions
Introduction
Objectives
●
Investigate the large deflection behaviour of steel beams at elevated temperatures
●
Develop a simple hand calculation method for predicting the deflec def lectio tions ns and and catena catenary ry for force cess in steel steel bea beams ms at elev elevat ated ed temperatures
Numerical Simulations by ABAQUS ●
Shell element S4R
●
Boundary conditions:
Stiff end plates applied to beam end sections
Spring elements applied at centre of beam end sections sections
Spring couple elements applied to the top and bottom flanges of beam end sections relative to section centre
Model validation
0.6m
0.8m
0.6m
178x102x19UB
1 . 5 m 1 . 5 m
o
Bott Bottom flange temperat temp erature ure ( C) -10
0
100
2 00
30 0
400
500
600
70 0
800
900
8 00
9 00
) -30 m m-50 ( δ
n-70 o i t c -90 e l f e -110 D
simulation, LR=0.5 simulation, LR=0.7 test, LR=0.5 test, LR=0.7
-130
δ
-150 200 )150 N k (100 R e 50 c r o 0 f n-50 o i t -100 c a e -150 r l a i -200 x A
-250
simulation, LR=0.5 simulation, LR=0.7 test, LR=0.5 test, LR=0.7 0
100
20 0
3 00
R
R
o
Bottom Bottom flange flang e temperature temp erature ( C) 4 00
50 0
60 0
700
Fully axially restrained beams with lateral restraints Temperature distributions h/8
0.5T
h
T Uniform
T on-uniform
Uniform temperature distribution 1000
Temperautre (oC)
0 0
100
200
300
-100 -200
400
500
600
700
800
900 10 1000
600
200
) -300 m m ( n-400 o i t c e l -500 f e D
) N k ( -200 n o i t c a e-600 R
-600
-1000
Temperautre (oC) 0
100
200
300
400
500
600
700
800
900 10 1000
-700 -1400 -800 -900 -1000
8m, 5m, 8m, 5m,
LR=0.7 LR=0.7 LR=0.4 LR=0.4
-1800
-2200
8m, 5m, 8m, 5m, Axial
LR=0.7 LR=0.7 LR=0.4 LR=0.4 capacity
Non-uniform temperature temperature distribution distribution 1000
Maximum temperautre temperautre (oC)
0 0
100
200
300
400
500
600
700
800
900 10 1000
-100
600
-200
200
) -300 m m (
) N k ( -200 n o i t c a e-600 R
-600
-1000
n-400 o i t c e l -500 f e D
Maximum temperautre (oC) 0
100
200
300
400
500
600
700
800
900 10 1000
-700 -1400 -800 -900 -1000
8m, 5m, 8m, 5m,
LR=0.7 LR=0.7 LR=0.4 LR=0.4
-1800
-2200
8m, 5m, 8m, 5m, Axial
LR=0.7 LR=0.7 LR=0.4 LR=0.4 capacity
Laterally restrained beams with different levels of axial restraint 400
Maximum temperature (oC) 0 0
100
200
300
-100
400
500
600
700
800
900 10 1000
200
-200
Maximum temperature ( oC)
0 ) N k ( n o-200 i t c a e R
-300 ) m-400 m ( n o -500 i t c e l f e -600 D
-700 -800
KA=0.02EA/L KA=0.05EA/L
-600
200
300
400
500
600
700
800
900 10 1000
KA=0.02EA/L KA=0.05EA/L KA=0.15EA/L
KA=0.15EA/L
-800
KA=EA/L Fully restrained
-1000
100
-400
KA=0.30EA/L -900
0
KA=0.30EA/L KA=EA/L Fully restrained restrained
-1000
Laterally restrained beams with different levels of rotational restraint 400
Maximum temperature temperature (oC) 0 0
100
200
300
400
500
600
700
800
900
10 1000
200
-100
Maximum temperature temperature (oC)
0 -200
0
-1000
400
500
600
700
800
900
10 1000
-800
-600
-900
300
) N k -400 ( n o i t c -600 a e R
) m m-400 ( n o i t c -500 e l f e D
-800
200
-200
-300
-700
100
Fully restrained KR=2EI/L KR=EI/L KR=0.6EI/L KR=0.1EI/L Free rotation
-1000 -1200 -1400 -1600
Fully restrained restrained KR=2EI/L KR=EI/L KR=0.6EI/L KR=0.1EI/L Free rotation
Effect Effe ct of late lateral ral tor torsio sional nal buc buckli kling ng
Maximum temperature ( oC) 0 0
100
200
300
400
-100 -200
) m m (
-300
n o i t c e l f-400 e d e n -500 a l p n I
-600 -700
Uniform, with L&A restraints Non-uniform, Non-un iform, with L&A L &A restr res traints aints Uniform Uni form,, with only on ly A res traints Non-uniform, Non-un iform, with only A rest res traints Uniform, Uniform , without L&A restr res traint aint Non-uniform, Non-un iform, without L&A restr res traint aint
-800
500
600
700
800
900
1000
Maximum temperature ( oC) 200 0 -200 ) -400 N k ( -600 n o i t c a-800 e R
0
100
200
300
400
500
600
700
800
900
-1000
Uniform, with L&A restraints
-1200
Non-uniform, Non-uniform , with L&A restraints
-1400
Uniform Uni form,, with only A res traints
-1600
Non-uni Non -uniform, form, with only onl y A res traints
-1800
1000
Hand Calculation Method P
P q K A
K A
K R
K R L M P
F T T
F T T δ m + δ t
M R z
M T T
M T T
M R
x
Equilibrium equations F M T + M M T (δ m +δ t ) + R + P =0 F T = K ε = K ' A m
' A
∆ Lm
1
L
K A'
=
1
+
L
+
1
K A E T A K A
M T = E T I y ϕ m x= L 2
M R = K R' θ x=0
1 '
K R
=
1
+
L
+
1
K R E T I y K R
M P :the externally applied free bending moment
Deflection profiles z ( x) = z m ( x) + z t ( x)
1/ 2
L
∆ L = ∫
∆ Lm = ∆ L − ∆ Lt
0
ϕ m x = L = 2
d 2 z m dx 2
L x = 2
dz 2 1 + dx − L dx
θ x =0 =
dz dx x =0
∆ Lt = α TL
Uniform temperature distribution z t = 0
Zero end rotational restraint Under UDL Under CPL
z m =
16δ 16δ m ,max x 4 5 L
3 − 2 + x L L 2 x 3
z m=free bend moment diagram
Complete end rotational restraint z m =
16δ m,max x 4 L2
2 − + x 2 L L 2 x 3
Non-uniform temperature temperature distribution distribution
Zero end rotational restraint z t = −
2h
( x 2 − Lx )
z m = z UDL ,UT
Under UDL Under CPL
α ∆T
z m =
z UDL ,UT + z CPL ,UT 2
Complete end rotational restraint z t = 0
z m =
16δ m,max x 4 L2
2 − + x 2 L L 2 x 3
M t =
E T I yα ∆T h
Axial load & bending moment interaction 1
1 − γ
0.9
M
(1 + α )2 γ 2 M p 1− α [2(1 + β ) + α ]
0.8
+
F F p
=1
0.7 0.6 p F / F
0.5 0.4 0.3
α =
0.2
β =
0.1
γ =
0 0
Aw
PNA at web/flange junction (1 + α ) 2 γ 2 1 − , γ α [2(1 + β ) + α ]
2 A f t h0
Aw 2 A f + Aw
0. 1
2
F (1 + α ) M =1 + F p ( ) M p α [2 1 + β + α ] 2
0. 2
0. 3
0. 4
0. 5 M/Mp
0. 6
0. 7
0. 8
0. 9
1
Validation ●
Complete axial restraint, zero rotational restraint uniform temperature, UDL 1000
Temperautre (oC)
Limiting temperature of BS5950 Part8
0 0
100
200
300
400
500
600
700
800
900 10 1000
LR=0.4 0.7 500
-200 ) N k ( n o i t c a e R
) -400 m m ( n o i t c -600 e l f e D
Temperautre (oC)
0 0
100
200
300
400
500
600
700
800
900 10 1000
-500
-800 ABAQUS,LR= 0.4 ABAQUS,LR=0.4 -1000
-1000
HCM,LR= HCM,LR= 0.4 0. 4
HCM,LR=0.4
HCM,LR=0.7
HCM,LR=0.7 -1200
ABAQUS,LR= 0.7
ABAQUS,LR=0.7
Tensile capa c apacit city y -1500
●
Complete axial restraint, zero rotational restraint non-uniform temperature, central point load 1000
Maximum temperautre (oC) 0 0
100
200
300
400
500
600
700
800
900 10 1000 500
-200 ) N k ( n o i t c a e R
) -400 m m ( n o i t c -600 e l f e D
Maximum temperautre (oC)
0 0
100
200
300
400
500
600
700
800
900 10 1000
-500
-800 ABAQUS,LR= 0.4 ABAQUS,LR=0.4 -1000
-1200
-1000
ABAQUS,LR= 0.7
ABAQUS,LR=0.7
HCM,LR= 0.4
HCM,LR=0.4
HCM,LR=0.7
HCM,LR=0.7
Tensile capa c apacit city y -1500
●
Different levels of axial restraint, zero rotational restraint uniform temperature, central point load, load ratio = 0.7 Temperature (oC)
0 0
100
200
300
400
500
-100
600
400 700
800
900 10 1000 200
-200
Temperature (oC)
0 -300
0
) m-400 m ( n o -500 i t c e l f e -600 D
-700 -800 -900 -1000
100
200
300
400
500
600
700
800
900 10 1000
) N-200 k ( n o i t c a -400 e R
ABAQUS,KA= ABAQUS, KA=0.05EA/L 0.05EA/L ABAQUS,KA= ABAQUS, KA=0.15EA/L 0.15EA/L ABAQUS,KA= ABAQUS, KA=0.30EA/L 0.30EA/L ABAQUS,KA= ABAQUS, KA=EA/ EA/L L HCM,KA=0.05EA/L HCM,KA=0.15EA/L HCM,KA=0.30EA/L HCM,KA=EA/L
-600
-800
-1000
ABAQUS,KA=0. ABAQUS,K A=0.05EA/L 05EA/L ABAQUS,KA=0. ABAQUS,K A=0.15EA/L 15EA/L ABAQUS,KA=0. ABAQUS,K A=0.30EA/L 30EA/L ABAQUS,KA=E ABAQUS,K A=EA/L A/L HCM,KA=0.05EA/L HCM,KA=0.15EA/L HCM,KA=0.30EA/L HCM,KA=EA/L
●
Different levels of axial restraint, zero rotational restraint non-uniform temperature, temperature, central point load, load ratio = 0.7 Maximum temperature temperature (oC)
0 0
100
200
300
400
500
-100
600
700
400 800
900 10 1000 200
-200
Maximum temperature (oC)
0 -300
0
) m-400 m (
-800 -900 -1000
200
300
400
500
600
700
800
900 10 1000
) N-200 k ( n o i t c a -400 e R
n o -500 i t c e l f e -600 D
-700
100
ABAQUS,KA=0. ABAQUS,K A=0.05EA/L 05EA/L ABAQUS,KA=0. ABAQUS,K A=0.15EA/L 15EA/L ABAQUS,KA=0. ABAQUS,K A=0.30EA/L 30EA/L ABAQUS,KA=EA ABAQUS,K A=EA/L /L HCM,KA=0.05EA/L HCM,KA=0.15EA/L HCM,KA=0.30EA/L HCM,KA=EA/L
-600
-800
-1000
ABAQUS,KA=0.05EA/L ABAQUS,KA =0.05EA/L ABAQUS,KA=0.15EA/L ABAQUS,KA =0.15EA/L ABAQUS,KA=0.30EA/L ABAQUS,KA =0.30EA/L ABAQUS,KA=EA/ ABAQUS,KA =EA/L L HCM,KA=0.05EA/L HCM,KA=0.15EA/L HCM,KA=0.30EA/L HCM,KA=EA/L
●
Different levels of rotational restraint, complete axial restraint uniform temperature, UDL, load ratio = 0.7 500
Temperature (oC)
0 0
100
200
300
400
500
-100
600
700
800
900 10 1000
250 Temperature (oC)
0
-200
0 -300
-250
) m-400 m (
) N k -500 ( n o i t c a -750 e R
n o -500 i t c e l f e -600 D
-700 -800 -900 -1000
ABAQUS,KR= ∞ ABAQUS,KR=EI/L ABAQUS,KR=0.1EI/L ABAQUS,KR=0 HCM,KR= ∞ HCM,KR=EI/L HCM,KR=0.1EI/L HCM,KR=0
-1000 -1250 -1500 -1750
100
200
300
400
500
600
700
800
900 10 1000
ABAQUS,KR= ∞ ABAQUS,KR=EI/L ABAQUS,KR=0.1EI/L ABAQUS,KR=0 HCM,KR=∞ HCM,KR=EI/L HCM,KR=0.1EI/L HCM,KR=0
●
Different levels of rotational restraint, complete axial restraint uniform temperature, central point load, load ratio = 0.7 600
Maximum temperature (oC)
0 0
100
200
300
400
500
-100
600
700
800
900 10 1000
300 Maximum temperature (oC)
0
-200
0 -300
-300
) m-400 m ( n o -500 i t c e l f e -600 D
) N k -600 ( n o i t c a -900 e R
-700 -800 -900 -1000
ABAQUS,KR= ABA QUS,KR=∞ ABAQUS,KR=E ABA QUS,KR=EI/L I/L ABAQUS,KR=0.1E ABA QUS,KR=0.1EI/L I/L ABAQUS,KR=0 ABA QUS,KR=0 HCM,KR=∞ HCM,KR=EI/L HCM,KR=0.1EI/L ∞ HCM,KR=0
-1200 -1500 -1800 -2100
100
200
300
400
500
600
700
800
900 10 1000
ABAQUS,KR= ABA QUS,KR=∞ ABAQUS,KR=EI/L ABA QUS,KR=EI/L ABAQUS,KR=0.1EI/L ABA QUS,KR=0.1EI/L ABAQUS,KR=0 ABA QUS,KR=0 HCM,KR=∞ HCM,KR=EI/L HCM,KR=0.1EI/L HCM,KR=0
Conclusions ●
If a steel beam is reliably provided with some axial restraints, catenary cate nary acti action on will occur occur and will will enable enable the beam beam to survive survive very high temperatures without a collapse
●
Whether a beam will exper Whether experienc iencee lateral lateral torsi torsional onal buck buckling ling or or not will will only only have have some some minor minor effects effects on its its large large deflect deflection ion behaviour
●
A simplified hand calculation method is developed to predict the maxim maximum um deflec deflection tion and and caten catenary ary forc forcee in a steel steel beam, beam, which can be used in design applications
StiFF Meeting
September 23, 2003
Thank you!
