Structural Analysis Under Fire Conditions

The following structural analysis will be based on the temperature of the structural members as estimated in the previous design fire scenario section. At higher temperatures concrete begins to lose strength. One method from EN 1992-1-2 used to evaluate concrete structures is called the 500°C isotherm method. The simplified calculation is based on reducing the effective cross-section of the structural component with respect to a heat damaged zone (above 500°C) at the concrete surface. It is assumed that once when concrete reaches a temperature of 500°C it has zero strength and has no structural benefit. Temperature of the member is lower as the distance from the fire exposed surface increases (i.e at a further depth into the concrete slab). In the 500°C isotherm method, temperatures under 500°C do not experience any strength reduction due to increased temperature and retain 100% of the initial strength.

All of the design fire scenarios discussed above affect reinforced concrete slabs. Thus, the structural analysis will follow the same isotherm method for all three scenarios. This approach follows the steps below. The calculations have been tabulated in Appendix M. It is important to note that this is a simplified hand calculation of the structure and should only be used a first-order approximation of the structural integrity of the building. In order to provide a more accurate analysis of the structure a computer program is required for more intensive calculations.

Structural Analysis Procedure

1. The maximum moment of the slab under fire conditions is calculated as: Mmax,fire =

(1.2Dead + 0.5Live) ∗ 𝐿2

8 Where

L = span of the slab (ft)

0 100 200 300 400 500 600 700 800 900 0 200 400 600 800 1000 1200 1400 1600 Te m p era tu re ( °C) Time (Seconds) Tg x = 0 mm x = 5 mm x = 10 mm

57 Live = Live Load (lbs/ft)

Dead = Superimposed Dead Load + Self Weight of the Slab (lbs/ft) 2. The moment capacity of the slab is calculated as:

Mf= As∗ fy,T(df− af 2) af= As∗ fy,T 0.85 ∗ fc′∗ b Where

As= area of the reinforcing steel (in2)

fy,T= yield stress of the reinforicing steel reduced for temperature (ksi)

fc′ = compressive strength of the concrete (ksi)

df = effective depth of the beam (inches)

b = characteristic width of the slab (3 ft)

3. The comparison Mmax,fire< Mf is checked to ensure that the slab does not fail due to the

induced moment.

4. In addition to verifying that the moment capacity of the beam is not exceeded, it is necessary to ensure that the compression capacity is not reduced so low as to cause a sudden compression failure. This can be ensured by checking that

As∗ fy,T

𝑏𝑑𝑓𝑓𝑐𝑇′

< 0.30 5. Determine the safety factor, Mf/Mmax,fire

Design Fire 1 Structural Analysis:

In the first fire scenario, a mattress is ignited in one of the hotel rooms. The source of the fire is located on the 4th _{floor. The 5}th _{floor above is an 8 inch thick post tensioned concrete slab.}

The temperature reaches 493°C 10 mm into the slab and 577°C 5 mm into the slab. Even though the method calls for the 500°C isotherm to be the cutoff between 100% and 0% strength for concrete, it is assumed that the 500°C isotherm lies 10 mm into the slab as the heat transfer analysis only calculated temperatures at 5 mm increments. This means that the 8 inch slab (approximated from 203.2 mm to 200 mm due to the heat transfer analysis using 5 mm increments) has a new effective depth of 190 mm or 7.48 inches.

Table 39. Isotherms in Concrete Slab based on Design Fire Scenarios Depth of Isotherm Temperature of Isotherm (°C)

5 mm 577

10 mm 493

Assumed 500°C Isotherm Depth 10 mm

58 The temperature of the reinforcing steel bars is approximated as the temperature of the concrete at the location of the steel bar. The steel bars are covered by 1.5 inches (38.1 mm) of concrete which can be approximated as the temperature at 35 mm to be conservative. The tensile steel bars are then estimated to be at 255°C under fire conditions. Steel strength degrades as temperature increases. Based on EN 1992-1-2 Table 3.2a the steel in tension at 255°C has the same yield strength as steel at ambient temperature. The concrete has an ultimate compressive strength of 6 ksi and the steel has a yield strength of 60 ksi under fire conditions.

The live load is estimated at 125 pounds per linear foot and the dead load is estimated as 125 pounds per linear foot. A factored fire design load of 1.2*Dead Load + 0.5*Live Load is used. In order to determine the induced moment on the slab a characteristic width of 3 ft of the slab is taken. This 3 ft width of the slab is assumed to be a one-way spanning element. This is a conservative approach as a two-way spanning slab or beam is provided with additional support. The slab has a span of 21’8” as measured on the architectural set of drawings. The steel reinforcement bars are assumed to be #4 with a diameter of 0.5 inches and are spaced 4 inches apart.

These assumptions and the above analysis procedure produces the following results: 1. The maximum moment of the slab under fire conditions is calculated below.

Mmax,fire=

(1.2Dead + 0.5Live) ∗ 𝐿2

8 = 32,670 lb ∗ ft 2. The moment capacity of the slab is calculated as:

Mf = As∗ fy,T(df−

af

2) = 63,543 lb ∗ ft

3. The comparison Mmax,fire< Mf is checked to ensure that the slab does not fail due to the

induced moment. The inequality is valid; therefore, the slab does not fail.

4. In addition to verifying that the moment capacity of the beam is not exceeded, it is necessary to ensure that the compression capacity is not reduced so low as to cause a sudden compression failure. This can be ensured by checking the below inequality. The inequality is valid;

therefore, the slab does not fail. As∗ fy,T

𝑏𝑑𝑓𝑓𝑐𝑇′

= 0.0656 < 0.30 5. The safety factor is 1.94.

Design Fire 2 Structural Analysis:

In the second fire scenario, a faulty iron in the hotel linen closet on the first floor ignites a stack of hotel linens. The second floor slab is constructed of 10 inch thick cast in place concrete.

The temperature reaches 493°C 25 mm into the slab and 556°C 20 mm into the slab. It is assumed that the 500°C isotherm lies 25 mm into the slab. This means that the 10 inch slab (255 mm) has a new effective depth of 230 mm or 9.055 inches.

59 Table 40. Isotherms in Concrete Slab based on Design Fire Scenarios

Depth of Isotherm Temperature of Isotherm (°C)

5 mm 823

10 mm 720

15 mm 631

20 mm 556

25 mm 493

Assumed 500°C Isotherm Depth 25 mm

Effective Thickness of Slab 230 mm (9.055 inches)

The temperature of the reinforcing steel bars is approximated as the temperature of the concrete at the location of the steel bar. The steel bars are covered by 1.5 inches (38.1 mm) which can be approximated as the temperature at 35 mm. The tensile steel bars are then estimated to be at 393°C under fire conditions. Steel strength degrades as temperature increases. Based on EN 1992-1-2 Table 3.2a the steel in tension at 393°C yield strength degrades by approximately 6%. The concrete has an ultimate compressive strength of 6 ksi and the steel has a yield strength of 60 ksi * 0.94 = 56.4 ksi under fire conditions.

The live load is estimated at 125 pounds per linear foot and the dead load is estimated as 125 pounds per linear foot. A factored fire design load of 1.2*Dead Load + 0.5*Live Load is used. In order to determine the induced moment on the slab a characteristic width of 3 ft of the slab is taken. This 3 ft width of the slab is assumed to be a one-way spanning element. The slab has a span of 28’4” as

measured on the architectural set of drawings. The steel reinforcement bars are assumed to be #5 with a diameter of 0.625 inches and are spaced 5 inches apart.

These assumptions and the above analysis procedure produces the following results: 1. The maximum moment of the slab under fire conditions is calculated below.

Mmax,fire=

(1.2Dead + 0.5Live) ∗ 𝐿2

8 = 64,532 lb ∗ ft 2. The moment capacity of the slab is calculated as:

Mf = As∗ fy,T(df−

af

2) = 88,068 lb ∗ ft

3. The comparison Mmax,fire< Mf is checked to ensure that the slab does not fail due to the

induced moment. The inequality is valid; therefore, the slab does not fail.

4. In addition to verifying that the moment capacity of the beam is not exceeded, it is necessary to ensure that the compression capacity is not reduced so low as to cause a sudden compression failure. This can be ensured by checking that. The inequality is valid; therefore, the slab does not fail.

As∗ fy,T

𝑏𝑑𝑓𝑓𝑐𝑇′

60 5. The safety factor is 1.36.

Design Fire 3 Structural Analysis:

In the third fire scenario, a car in the basement parking garage experiences engine failure and ignites itself. The ground floor slab above the fire is constructed of 10 inch thick cast in place concrete. In this calculation the temperature of the exposed surface of the concrete slab never exceeds 500°C. This means that the concrete is assumed to have 100% of its original strength. The tensile reinforcement steel, covered by 1.5” (38.1 mm), reaches a temperature of 51°C. Per EN 1992-1-2 Section 4.2.4.3 the steel keeps 100% of its original strength. Without facing any strength reduction from this fire scenario, the structural element is not expected to fail.