Fire Resistance of Materials and Structures - Heat Transfer and Thermal Analysis

Fire Resistance of Materials & Structures
Heat Transfer & Thermal Analysis
Date of Submission
15 Dec 2015
Submitted by
Seyed Mohammad Sadegh Mousavi
836 154
Submitted to
Prof. R. Felicetti
Dr. P. Bamonte
Structural Assessment & Residual Bearing
Capacity, Fire & Blast Safety
Civil Engineering for Risk Mitigation
Politecnico di Milano
[ 1 s t H o m e w o r k - H e a t T r a n s f e r & T h e r m a l A n a l y s i s ]

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Politecnico di Milano – Lecco Campus
Prof. R. Felicetti & Dr. P. Bamonte
Seyed Mohammad Sadegh Mousavi (836154)
Fire Resistance of Materials and Structures
Prof. R. Felicetti and Dr. P. Bamonte
1st
Homework – Heat Transfer and Thermal Analysis
The figure below shows a composite section (steel beam HE360A + concrete slab) subjected to a
standard fire (ISO834) at the intrados of the steel beam.
Figure 1 – Exercise Scheme
With reference to the provisions of EC1 (for the boundary conditions of convection and radiation), and
EC4 (for the thermal properties of concrete and steel) determine:
1. The temperature distribution along axis AB at different time steps;
2. The temperature distribution along CD at different time steps;
3. The temperature at points A and M as a function of the fire duration.
Make your comments on the above results, with special attention to the following points:
a) Temperature differences in the steel profile: more massive zones (web-flange intersection) vs thinner
plates (web) an their evolution in time (initial fast heating vs smoother final stage);
b) Effect of the heat sink on the top flange of the steel beam;
c) Shadow effect on the web and internal face of flanges;
d) Comparison at points A and M with the heating curves of a steel plate (th = 17.5 mm) exposed on one
or two sides (see thermal analysis of steel structures and the related spreadsheet file);
e) Concrete response and its progression with or without the flange.

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The composite section of steel IPE beam and concrete slab was modeled in ABAQUS application, to
evaluate its thermal response when exposed to standard fire ISO834. Also for avoid the repetitive
information and useless pages some of the theories and formulas are neglected but the Heat Transfer
and Steel pdfs were used as references for this issue.
Steps in ABAQUS are as following
1. Edit Model Attributes
Inserting the values of absolute zero temperature (-273.15) and Stefan-Boltzmann constant (5.67E-008)
Figure 2 – Model Attributes window
2. Module-Creating Parts
In this Part the information of geometry of the model was defined, two parts (concrete slab and steel
beam) need to be created. For this issue for the both parts are created as shell elements, deformable and
2D planar. Due to the restrictions in number of mesh elements in student’s version of ABAQUS, choice
was to take advantage of symmetry of the section and model only half of the section.
Figure 3 – Create Cross Sections (Steel & Concrete) window

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3. Module-Property
In Property module, materials such as concrete and steel are to be defined, by defining their thermal
properties such as density (ρ), conductivity (λ) and Specific Heat (c).
Figure 4 – Materials Manager window
Concrete
Figure 5 – Concrete Properties window (Conductivity, Density and Specific Heat)
 Density (ρ) - weight per unit mass of the material (kg/m3)
According to the Eurocode 2 Part 1&2 (2004)
𝜌(𝑡) = 𝜌(20 ℃) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 ≤ 115 ℃
𝜌(𝑡) = 𝜌(20 ℃) ∙ (1 −
0.02(𝑡 − 115)
85
) 𝐹𝑜𝑟 115 ℃ < 𝑡 ≤ 200 ℃
𝜌(𝑡) = 𝜌(20 ℃) ∙ (0.98 −
0.03(𝑡 − 200)
200
) 𝐹𝑜𝑟 200 ℃ < 𝑡 ≤ 400 ℃
𝜌(𝑡) = 𝜌(20 ℃) ∙ (0.95 −
0.07(𝑡 − 400)
800
) 𝐹𝑜𝑟 400 ℃ < 𝑡 ≤ 1200 ℃

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The following graph represent the concrete density with respect to the equations for different
temperatures that mentioned in the previous page and plotted in excel code.
Figure 6 – Concrete Density graph
 Specific Heat (c) - amount of heat required to heat unit mass of the material by one degree (J/kg.K).
According to the Eurocode 2 Part 1&2 (2004)
𝑐 𝑝(𝑡) = 900 (
𝐽
𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 ≤ 100 ℃
𝑐 𝑝(𝑡) = 900 + (𝑡 − 100) (
𝐽
𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 100 ℃ < 𝑡 ≤ 200 ℃
𝑐 𝑝(𝑡) = 1000 +
𝑡 − 200
2
(
𝐽
𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 200 ℃ < 𝑡 ≤ 400 ℃
𝑐 𝑝(𝑡) = 1100 (
𝐽
𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 400 ℃ < 𝑡 ≤ 1200 ℃
Concrete is assumed with 0% of moisture. Because according to the discussion during the class, if there
is moist inside the concrete the graph of the temperature versus time is different. But in the following
graph the increasing in the trend happened due to vaporization of the water that called latent heat.
2075
2125
2175
2225
2275
2325
2375
2425
20 150 280 410 540 670 800 930 1060 1190
DENSITY(KG/M3)
TEMPERATURE (℃)
CONCRETE - DENSITY

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Figure 7 – Concrete Specific Heat graph
 Conductivity (λ) - rate of heat transferred per unit thickness of material per unit temperature difference
(W/m.K). EN1992-1-2 proposes lower and upper limit of thermal conductivity. Regarding the
maximum and minimum formulas for thermal conductivity in the Eurocode 2, the average value of these
formulas considered as a reference value for the concrete thermal conductivity in the ABAQUS code.
According to the Eurocode 2 Part 1-2 (2004) – MAX & MIN Conductivity
𝜆 𝑐
𝑚𝑎𝑥(𝑡) = 2 − 0.2451 (
𝑡
100
) + 0.0107 (
𝑡
100
)
2
𝜆 𝑐
𝑚𝑖𝑛(𝑡) = 1.36 − 0.136 (
𝑡
100
) + 0.0057 (
𝑡
100
)
2
Figure 8 – Concrete Conductivity graph (Average Value)
850
900
950
1000
1050
1100
1150
20 150 280 410 540 670 800 930 1060 1190
SPECIFICHEAT(J/KGK)
TEMPERATURE (℃)
CONCRETE - SPECIFIC HEAT
0.2
0.6
1.0
1.4
1.8
20 145 270 395 520 645 770 895 1020 1145
CONDUCTIVITY(W/MK)
TEMPERATURE (C)
CONCRETE - THERMAL CONDUCTIVITY (AVERAGE VALUE)

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Steel
In this case again thermal properties such as density (ρ), conductivity (λ) and Specific Heat (c) have
been defined. Specific heat and conductivity are temperature dependent, while density is not.
Figure 9 – Steel Properties Window (Conductivity, Density and Specific Heat)
 Density (ρ): constant, doesn’t change with increased temperature. Assigned value is ρ=7850kg/m3.
 Conductivity (λ): According to the Eurocode 3 – Part 1&2 (2005)
𝜆 𝑎(𝑡) = 54 − 0.0333 ∙ 𝑡 ( 𝑊
𝑚 ℃⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 ≤ 800 ℃
𝜆 𝑎(𝑡) = 27.3 ( 𝑊
𝑚 ℃⁄ ) 𝐹𝑜𝑟 800 ℃ ≤ 𝑡 < 1200 ℃
Figure 10 – Steel Thermal Conductivity
20
25
30
35
40
45
50
55
60
20 150 280 410 540 670 800 930 1060 1190
CONDUCTIVITY(𝑊⁄𝑚℃)
TEMPERATURE (℃)
STEEL - THERMAL CONDUCTIVITY

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 Specific Heat (c)
According to the Eurocode 3 – Part 1&2 (2005)
𝑐 𝑎 = 425 − 0.773 𝑡 − 1.69 × 10−3
𝑡2
+ 2.22 × 10−6
𝑡3
(
𝐽
𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 < 600 ℃
𝑐 𝑎 = 666 +
13002
738 − 𝑡
(
𝐽
𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 600 ℃ ≤ 𝑡 < 735 ℃
𝑐 𝑎 = 545 +
17820
𝑡 − 731
(
𝐽
𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 735 ℃ ≤ 𝑡 < 900 ℃
𝑐 𝑎 = 650 (
𝐽
𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 900 ℃ ≤ 𝑡 < 1200 ℃
Figure 11 – Steel Specific Heat
Endothermic Reaction: a process or reaction in which the system absorbs energy from its
surroundings; usually, but not always, in the form of heat. So in this graph as you can see, marked with
red dash line, at around 735 °C (Critical Temperature) with providing the energy to the steel the trend of
the graph increased sharply and then return to the previous one due to phase change of material. Also
the peak value represented the Thermos-Physical Transformation of material.
Time-Temperature curve used: ISO834
For the translation the real fire to an equivalent ISO fire with the same severity in corresponding
time, the formula was defined in excel code with respect to this equation T= 20 + 345log (8t+ 1)
where t is time (min) and so the following graph is represented in temperature (°C) versus time (min).
0
1000
2000
3000
4000
5000
20 120 220 320 420 520 620 720 820 920 1020 1120
SPECIFICHEAT
TEMPERATURE (℃)
STEEL - SPECIFIC HEAT

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Figure 12 – Standard Fire ISO834
Assign a Section
Then, in section manager, materials that we defined with their properties have been assigned to the
relevant cross sections.
Figure 13 – Section Manager
4. Module-Assembly
Concrete Slab and Steel Beam sections are then assembled into one composite section.
0
200
400
600
800
1000
1200
0 1200 2400 3600 4800 6000 7200 8400 9600 10800
TEMPERATURE(°C)
TIME (SECONDS)
FIRE CURVE - ISO 834

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5. Module-Step
After we assembled the sections, the step was created in this part, type Heat transfer for transient
response. Time period was set for 3 hours but converted to the 10800 sec which is the duration of
standard fire curve, ISO834, also some changes in incrementation part were defined.
In general the value of emissivity is equal to 𝜀𝑓𝑖𝑟𝑒 = 0.8 in order to Eurocode 1 but in Italian code
(UNI9502) this value is 0.56.
Figure 14– Step Window
Another change in this step was in field output manager, that Thermal output variables have been
chosen.
Figure 15 – Field Output Manager windows

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6. Module-Interaction
In this part, boundary conditions are defined and assigned to the section. Boundary conditions that
govern the heat transmission on the fire side have been chosen with respect to the convection
( 𝜶 𝒄 = 𝟐𝟓 𝑾
𝒎 𝟐℃⁄ , fire side) and radiation(𝜺 𝑹𝑬𝑺 = 𝟎. 𝟕). Internal heat transfer is governed by conductivity
(in porous materials such as concrete the conduction is happening)
In assignment of boundary condition, we can consider shadow effect or not. The concept of shadow
effect was introduced by Wickstrom, to consider the fact that the incident heat radiation received by an
open steel section, such as IPE-section, is not the same as what is received by a so called boxed (closed)
section. For the shadow effects two methods have been modeled and results will be presented in the following.
Method 1
In this method, no shadow effect taken into account.
Figure 16 – Radiation & Convection in Method 1
In case of no shadow effect the value of radiation is chosen 0.7 for inside the steel section. And also for
the cold side (top) of the concrete, the value of convection on cold environment (Still Air) including
radiation in linearized form is 9 W/m2 °C.

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Figure 17 – Radiation & Convection in Method 1
Method 2
Apply a different correction for the emissivity of each side in the concave region according to the
Hottel’s rule. So, in this case due to the shadow effect the resultant emissivity between the combusted
surface and member surface ( 𝜀 𝑟) can be reduced by multiplication with 𝑘 𝑠ℎ (Correction factor for
shadow effect).
Figure 18 – Correction Factor Scheme
𝑘 𝑠ℎ =
ℎ1
ℎ2
=
0.315
0.5818
≅ 𝟎. 𝟓𝟒𝟏
𝜀 𝑟 = 0.7 ∗ 𝑘 𝑠ℎ ≅ 𝟎. 𝟑𝟕𝟗

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Modified Boundary Conditions are represented in the following:
Figure 19 – Shadow Effect
7. Module-Mesh
For this part some definition were defined:
Figure 20 – Mesh window

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Figure 21 – Local Seeds & Mesh Controls
Figure 22 – Element Type & Global Seeds
8. Module-Job
In this part we define the commands for the analysis, in the job manager window we select the
Submit to start the analysis:
Figure 23 – Job Module Section

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9. Module-Visualization
After the previous module, the results will be represent in the cross section:
Figure 24 – Temperature in the cross section at around 30 min (1800 sec) – Max temperature is around 830 °C

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Part 1 – Distribution along A-B (With & Without Shadow Effect) and C-D at different time steps
So, the behavior of steel and concrete sections are entirely as expected from theory. In the steel
section, we can neglect the internal resistance due to very high thermal conductivity and Biot’s
number is less than 0.1. In addition, an important property of steel is specific heat that is lower than
the concrete, so steel requires less amount of heat with respect to the unit weight to be heated for one
degree.
Temperature along A to B at different time steps represented in the following graph. This graph
evaluate different temperature without considering the effect of shadow effect. At the beginning when
steel heating fast, there is dynamic revolution of temperature, so it is dynamic situation makes
highlights the differences between curves that the thinner (less massive) heating faster and the thicker
(more massive) heating slower. The web is thinner than the other parts, so it is hotter than the other
parts. As you can see on the Fig.24 at page 14 in 30 min the lower flange that is exposed to the fire
and almost whole the web already reached to the 830 °C. So, on the bottom flange, there is no shadow
effect, so, it is larger with small difference. But in the contrary, in the top flange the temperature is
dropped because it is much colder than the remaining parts of the cross section. These differences get
flatter and smaller as far as heating rate decreases. Subsequently, from 30 min to 180 min the
temperature of steel is increased just around 300 °C and achieved 1100 °C in the whole steel section
after 180 min. It can be described with the relationship of specific heat and conductivity with
temperature. As you can see on Fig.10 at page 6, the conductivity of steel is decreasing with
temperature and it is the highest in room temperature and then decrease linearly until 800 °C and
remain constant. Also, the trend of specific heat between 600 to 735 °C has increased sharply and then
dropped immediately to its initial value at 800 °C due to Endothermic reaction (Phase change of
material) that is already discussed in page 7.
On the other hand, concrete has low thermal conductivity and its heating much slower and not
uniform as in the steel section. As you can see on fig.30 at page 17, the temperatures of point B are
greatly lower than at point M that is located at the beam-slab interaction. Also due to standard ISO
fire is logarithmic plot and the slope is very steep in 1 minute, so the reduces get start in the concrete
deck.
When the heating is very slow, the differences are reduce as you can see on the graph, it is more
obvious for the trends after 90 min that these are almost flatten because at these different durations
there is slower heating. However, in order for the whole steel section to be uniformly heated and for
point M to reached the same temperature as the other points on the web, it takes more time,
approximately 150 min.
However, it is a dynamic process, if the dynamic component reduces, there is a kind of quasi-static
heating which gradually drive to all the section to reach an equilibrium with the compartment. So the
revolution of temperature of steel in time under the assumption temperature is almost uniform in the
cross section, so there is no difference on the bottom and top flanges and single temperature
considered for whole cross section.

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Figure 30 – Temperature along AMB without shadow effect at different time steps
Due to the C-D surface is located on the concrete part, it is not affected by shadow effect and so the
temperature progress will remain unchanged for both methods. But along A-M surface has a
difference between with or without shadow effect that we will discuss it in the following and in the
Fig. 32.
Figure 31 – Temperature along C-D at different time steps
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
TEMPERATURE(°C)
DISTANCE ALONG AMB (METERS)
Initial
1800s
3600s
5400s
7200s
9000s
10800s
A M B
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
TEMPERATURE(°C)
DISTANCE ALONG C-D (METERS)
Initial
1800s
3600s
5400s
7200s
9000s
10800s

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Temperature along A-M-B surface has a difference between with or without shadow effect as you
can see on the Fig. 32 but it is more obvious in just 30 min of fire and for the 60, 90 and 120 min fire
duration it is dominant only close to the flange-slab interaction area (Point M), while for the later on
fire duration when the temperature is more than 1000 °C, it is negligible.
Figure 32 – Temperature along A-M-B With Shadow Effect (SE-Solid line) and Without Shadow Effect (WSE-Dash
line) at different time steps
Figure 33 – The temperature at points A & M as a function of the fire duration
0
200
400
600
800
1000
1200
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
TEMPERATURE(°C)
DISTANCE ALONG A-B
INITIAL
1800s-SE
1800s-WSE
3600s-SE
3600s-WSE
5400s-SE
5400s-WSE
7200s-SE
7200s-WSE
9000s-SE
9000s-WSE
10800s-SE
10800s-WSE
0
200
400
600
800
1000
1200
0 1200 2400 3600 4800 6000 7200 8400 9600 10800
TEMPERATURE[°C]
TIME [S]
A Without Shadow
A With Shadow
M Without Shadow
M With Shadow

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The result of the temperature at points A & M as a function of fire duration is represented in the
previous graph (Fig.33). Because of Endothermic process of steel at around 735 °C that discussed in
page 7, results of the numerical analysis of points A and M with or without shadow effect have a kind of
dropped in the heating rate. Due to, in the Critical Temperature (at around 735 °C) with providing the
energy to the steel the temperature is not rising much and most of the heat providing is taking with the
material not to rise the temperature but to switch from some phase in the metal to the another phase
(phase change). In addition, considering the dash line in figure 33 that represents the effect of shadow
effect, takes longer time to reach the 600 °C due to less effective radiation and convection on the
assumption in the Eurocode and so the shadow effect decreases the temperature in the steel section.
Part e - Concrete response and its propagation with or without the flange
Figure 34 – Concrete Response along M-B with and without Flange
Flange behaves like some kind of protection for concrete section, as the external resistance of concrete
to the fire is lower than the internal. As a result, when the concrete is not protected by the flange, it is
directly exposed to the fire and temperatures are extremely higher especially for the initial times (30 and
60 min). However, due to lower conductivity of concrete, the internal resistance of concrete is much
better than the steel, so the temperature will decrease in case of depth of cross section. As a
consequence, the effect of flange will be remarkable for short duration of fires at lower depth.
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
TEMPERATURE(°C)
DISTANCE ALONG C-D (METERS)
Initial
1800s-F
1800s-WF
3600s-F
3600s-WF
5400s-F
5400s-WF
7200s-F
7200s-WF
Dashed Line = Without Flange
Continuous Line = With Flange

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