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 ]

Page 1 of 19

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.

Page 2 of 19





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

Page 3 of 19





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 ℃

Page 4 of 19





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)

𝑐𝑝(𝑡) = 900 (𝐽

𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 ≤ 100 ℃

𝑐𝑝(𝑡) = 900 + (𝑡 − 100) (𝐽

𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 100 ℃ < 𝑡 ≤ 200 ℃

𝑐𝑝(𝑡) = 1000 +𝑡 − 200

2 (

𝐽𝐾𝑔 𝐾⁄ ) 𝐹𝑜𝑟 200 ℃ < 𝑡 ≤ 400 ℃

𝑐𝑝(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

DEN

SITY

(K

G/M

3)

TEMPERATURE (℃)

C O N C R E T E - D E N S I T Y

Page 5 of 19





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

𝜆𝑐𝑚𝑎𝑥(𝑡) = 2 − 0.2451 (

𝑡

100) + 0.0107 (

𝑡

100)

2

𝜆𝑐𝑚𝑖𝑛(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

SPEC

IFIC

HEA

T (J

/KG

K)

TEMPERATURE (℃)

C O N C R E T E - S P E C I F I C H E A T

0.2

0.6

1.0

1.4

1.8

20 145 270 395 520 645 770 895 1020 1145

CO

ND

UC

TIV

ITY

(W/M

K)

TEMPERATURE (C)

C O N C R E T E - T H E R M A L C O N D U C T I V I T Y ( A V E R A G E V A L U E )

Page 6 of 19





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)

𝜆𝑎(𝑡) = 54 − 0.0333 ∙ 𝑡 (𝑊𝑚 ℃⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 ≤ 800 ℃

𝜆𝑎(𝑡) = 27.3 (𝑊𝑚 ℃⁄ ) 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

CO

ND

UC

TIV

ITY

(𝑊⁄𝑚

℃)

TEMPERATURE (℃)

S T E E L - T H E R M A L C O N D U C T I V I T Y

Page 7 of 19





Specific Heat (c)

According to the Eurocode 3 – Part 1&2 (2005)

𝑐𝑎 = 425 − 0.773 𝑡 − 1.69 × 10−3𝑡2 + 2.22 × 10−6𝑡3 (𝐽

𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 20 ℃ ≤ 𝑡 < 600 ℃

𝑐𝑎 = 666 +13002

738 − 𝑡 (

𝐽𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 600 ℃ ≤ 𝑡 < 735 ℃

𝑐𝑎 = 545 +17820

𝑡 − 731 (

𝐽𝑘𝑔 ℃⁄ ) 𝐹𝑜𝑟 735 ℃ ≤ 𝑡 < 900 ℃

𝑐𝑎 = 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

SPEC

IFIC

HEA

T

TEMPERATURE (℃)

S T E E L - S P EC I F I C H E AT

Page 8 of 19





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

TEM

PER

ATU

RE

(°C

)

TIME (SECONDS)

FIRE CURVE - ISO 834

Page 9 of 19





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

Page 10 of 19





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

(𝜶𝒄 = 𝟐𝟓 𝑾𝒎𝟐℃⁄ , fire side) and radiation(𝜺𝑹𝑬𝑺 = 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.

Page 11 of 19





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 (𝜀𝑟) can be reduced by multiplication with 𝑘𝑠ℎ (Correction factor for

shadow effect).

Figure 18 – Correction Factor Scheme

𝑘𝑠ℎ = ℎ1

ℎ2

= 0.315

0.5818≅ 𝟎. 𝟓𝟒𝟏

𝜀𝑟 = 0.7 ∗ 𝑘𝑠ℎ ≅ 𝟎. 𝟑𝟕𝟗

Page 12 of 19





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

Page 13 of 19





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

Page 14 of 19





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



Page 15 of 19








Page 16 of 19





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.

Page 17 of 19





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

TEM

PER

ATU

RE

(°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

TEM

PER

ATU

RE

(°C

)

DISTANCE ALONG C-D (METERS)

Initial

1800s

3600s

5400s

7200s

9000s

10800s

Page 18 of 19





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

TEM

PER

ATU

RE

(°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

TEM

PER

ATU

RE

[°C

]

TIME [S]

A Without Shadow

A With Shadow

M Without Shadow

M With Shadow

Page 19 of 19





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

TEM

PER

ATU

RE

(°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

Education

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