Strength of Materials Design of steel beam under high temperature.

Poznan University of Technology
Institute of Structural Engineering
Strength of Materials
Design of steel beam under
high temperature.
Jakub Turbakiewicz, SE
2014/2015

I. Task information
The lowest, right sided beam was selected for calculations:
-length of the beam: 6.5 m
-type of the beam: simply supported beam
Assumed steel: S355:
-yield strength: f y = 355 MPa
-modulus of elasticity: E = 210 GPa
-shear modulus: G = 81 GPa
-Poisson’s ratio in elastic stage: ν = 0,3
Figure 1. Frame scheme
-co-efficient of linear thermal expansion: α = 1,2 ∙ 10 −5 1
°C
for T < 100°C
II.
Chosen cross-section:
IPE 550 was chosen:
h = 550 mm I y = 67120 cm 4
b f = 210 mm I z = 2668 cm 4
t f = 17,2 mm W el,y = 2440 cm 3
t w = 11,1 mm W pl,y = 2787 cm 3
r = 24 mm I ω = 1884000 cm 6
A = 134,0 cm 2 I t = 123 cm 4
m = 106 kg/m
Figure 2. Cross-section scheme

Cross-section class:
ε = √ 235
f y
= √ 235
355 = 0,814
-web
-flange
c
t = h − 2 ∙ (t f + r) 550 − 2 ∙ (17,2 + 24)
= = 42,13
t w 11,1
42,13 < 72ε
42,13 < 58,32 → class 1
c
t = 0,5 ∙ (b f − t w − r) 0,5 ∙ (210 − 11,1 − 24)
= = 5,08
t f 17,2
5,08 < 9ε
5,08 < 7,29 → class 1
The whole cross-section if a class 1 cross-section.
III.
Design in ambient temperature
Characteristic loading:
-dead load: beam weight,
-dead load: ceiling dead load (3,5 kN/m 2 , with frame spacing of 6,2 m equal to 21,7 kN/m),
-live load (2,0 kN/m 2 , with frame spacing of 6,2 m equal to 12,4 kN/m).
Figure 3. Beam loading scheme
Figure 4. Moments in the designed beam - ULS loading combination
Figure 5. Moments in the designed beam - SLS loading combination

Beam capacity due to bending:
W pl,y = 2787 cm 3
M c,Rd = W pl ∙ f y 2787 ∙ 35,5
= = 98938 kNcm = 989,38 kNm
γ M0 1,0
M Ed
≤ 1,0 → 258,40 = 0,261 < 1,0 → conditions met
M c,Rd 989,38
Beam capacity due to shear force:
h w
t w
< 72 ∗ ε η
h w
= h − 2 ∗ (t f + r) 550 − 2 ∙ (17,2 + 24)
= = 42,13
t w t w 11,1
72 ∗ ε 0,814
= 72 ∗
η 1,0 = 58,32
h w
< 72 ∗ ε → 42,13 < 58,32 → conditions met
t w η
Beam capacity due to shearing stress:
Figure 6. Shear force in the designed beam - ULS loading combination
V Ed = 114,18 kN
A w = t w ∗ (h − 2 ∗ t f ) = 1,11 ∗ (55 − 2 ∗ 1,72) = 57,23 cm 2
τ Ed = V Ed
= 114,18
A w 57,23
= 2,00
kN
cm 2
τ Ed
f y /(√3 ∙ γ M0 ) < 1,0 → 2,00
< 1,0 → 0,10 < 1,0 − conditions met
35,5/(√3 ∙ 1,0)
Beam capacity due to the elastic critical moment:
π 2 EI z
M cr = C 1
(kL) 2 { √( k 2 I w
) + (kL)2 GI t
k w I z π 2 + (C
EI 2 z g ) 2 − C 2 z g }
z
k = 1,0; k w = 1,0; C 1 = 1,127; C 2 = 0,454

M cr = 1,127 ∗ 3,142 ∗ 21000 ∗ 2668
(1 ∗ 650) 2 ∗
∗ {√( 1 1 )2 1884000
2668 + (1 ∗ 650)2 ∗ 8100 ∗ 123
3,14 2 ∗ 21000 ∗ 2668 + (0,454 ∗ 27,5)2 − 0,454 ∗ 27,5}
= 45670,19 kNcm
λ LT = √ W pl,yf y 2787 ∗ 35,5
= √
M cr 45670,19 = 1,472
α LT = 0,34; (buckling curve b: h b > 2)
Φ LT = 0,5 [1 + α LT (λ LT − 0,2) + λ 2 LT] = 0,5 ∗ [1 + 0,34 ∗ (1,472 − 0,2) + 1,472 2 ] = 1,799
χ LT =
1
Φ LT + √ Φ 2 LT − λ 2 LT
1
1,0
=
= 0,353 ≤
1,799 + √1,799 2 − 1,4722 1,472 2 = 0,46
M b,Rd = χ LT ∗ W y ∗ f y 0,353 ∗ 2787 ∗ 35,5
= = 34904,85 kNcm = 349,05 kNm
γ M1 1,0
M Ed
M = 258,40 = 0,74 → as close to requested design ratio as possible
b,Rd 349,05
IV.
Design of steel beam at high temperature
Figure 6. Moments in the designed beam – accidental loading combination
Beam bending capacity:
M fi,Ed
M b,fi,t,Rd
≤ 1,0
M b,fi,t,Rd = χ LT,fi ∙ W pl,y ∙ k y,θ ∙ f y
γ M,fi
Assumed critical temperature: Θ = 550℃
Max bending moment: M fi,Ed = 120,05 kNm
The elastic critical moment: M cr = 34904,85 kNcm

Ambient temperature slenderness: λ LT = 1,472
Reduction factors for steel parameters in high temperature (Θ = 550℃):
k y,Θ = 0,625; k E,Θ = 0,455
λ LT,Θ = λ LT √ k y,Θ
k E,Θ
= 1,472√ 0,625
0,455 = 1,725
α = 0,65 √ 235
f y
= 0,65 √ 235
355 = 0,53
2
Φ LT,Θ = 0,5 ∗ [1 + α ∗ λ LT,Θ + λ LT,Θ ] = 0,5 ∗ [1 + 0,53 ∗ 1,725 + 1,725 2 ] = 2,445
χ LT,fi =
1
2
Φ LT,Θ + √Φ LT,Θ
2
− λ LT,Θ
=
1
2,445 + √2,445 2 − 1,725 2 = 0,239
The design buckling resistance moment in high temperature:
M b,fi,t,Rd = χ LT,fi ∗ W y,pl ∗ k y,Θ ∗ f y
γ M,fi
= 148,01 kNm
=
0,239 ∗ 2787 ∗ 0,625 ∗ 35,5
1,0
= 14801 kNcm
M fi,Ed
= 120,05 = 0,811 = 81,1 % → Θ = 550℃ is the critical temperature
M b,fi,t,Rd 148,01
V. Emphasizing the critical temperature and the fire resistance
Design assumptions:
-partial exposure
-density of steel: ρ a = 7850 kg/m 3
-increment of temperature rise: ∆t = 4s
-heat transfer coefficient: α c = 25 W
m 2 K
-configuration factor: Φ = 1,0
-emissivity of the steel surface: ε m = 0,7
-fire emissivity: ε f = 1,0
-Boltzmann’s constant: σ = 5,67 ∗ 10 −8
Section factors:
⌈ A m
V ⌉ b
= 113 1 m ; A m
V = 140 1 m
W
m 2 K
k sh = 0,9 ∙
⌈ A m
V
⌉
b
A mV
= 0,9 ∙ 113
140 = 0,726

Temperature [°C]
Functions used in MS Excel calculations
Specific heat of steel for 20 0 C ≤ θ a < 600 0 C: c a = 425 + 7,73 ∙ 10 −1 ∙ θ a − 1,69 ∙ 10 −3 ∙ θ a 2 +
2,22 ∙ 10 −6 ∙ θ a
3 J
kgK
Net design heat flux: ḣ
net,d = ḣ
conv + ḣ
rad
Convection flux: ḣ
conv = α c (Θ g − Θ m )
Radiation heat flux: ḣ
net,r = Φ ∙ ε m ∙ ε f ∙ σ ∙ [(Θ r + 273) 4 − (Θ m + 273) 4 ]
Gas temperature: Θ g = 20 + 345 ∙ log 10 (8 ∙ t + 1)
Temperature increase: ΔΘ a,t = k sh ∙
Am
V
c a ∙ρ a
∙ ḣ
net,d ∙ Δ
Results of steel temperature calculations:
Air and element's temperature
steel
air
900,0
800,0
700,0
600,0
500,0
400,0
300,0
200,0
100,0
0,0
0,0 10,0 20,0 30,0
Time [min]
Graph 1. Increase in steel temperature in time
The critical steel temperature is reached after 816 seconds (13,6 min).

VI.
Fire paint design
Promapaint SC4 was chosen as fire protection paint. Three different assumptions were set:
-fire resistance R15
Figure 7. Fire paint specification – R15
-fire resistance R30
Figure 8. Fire paint specification – R30

-fire resistance R60
Figure 9. Fire paint specification – R60
R15 R30 R60
Layer thickness [μm] 188 191 1094
Amount of pain [l/m 2 ] 0,31 0,32 1,82
Table 1. Fire paint specifications
VII. Results
Utilisation factor for
normal conditions
Design
section
Critical moment in
ambient temp.
Med/Med,fire
Critical
temperature
Utilistaion for
fire conditions
Fire
resistance
time
74 % IPE 550 349,05 kNm 2,36 550⁰C 81,1 % 816 s
Table 2. Final results of the calculations