Sensitivity analysis of fire resistance of a composite floor slab

The paper presents a sensitivity analysis of design bending resistance of a composite slab rib in a fire situation to a change of the value of basic variables. The analysis was carried out for a composite slab being an element of a supporting structure of a reinforced floor referred to in [1, 2]. The calculations were made for a simple calculation model and the standard temperature–time curve according to [6, 7]. The set of basic variables was limited to X1 = {(y, z), h1, fy,a, fy,s, fc}.


Introduction
According to the Regulation of the Minister of Infrastructure [3], every building should meet relevant requirements concerning its fire resistance.Such requirements are determined by defining for a particular building a human risk category (ZL) and the corresponding fire resistance category (OP).For each OP category of a building there is a set of requirements for fire resistance classes (OO) defined for particular elements of a building, such as: primary supporting structure, roof structure, floor, external wall, internal wall, roofing.Fire resistance is expressed by the minimum time tfi, calculated from the flashover, after which a particular element of a building meets the ultimate limit states referred to in [4,7], namely: fire ultimate limit state R, fire insulation limit state I, fire integrity limit state E.
The paper analyzes fire resistance of a reinforced wood floor, being an element of the renovated building referred to in [1].Owing to the expected method of use, the building in which the analyzed floor is located should meet the requirements defined in fire resistance class B, see [1].In a building rated as class B, the floor should be provided with the fire resistance rated as REI 60 meaning, that after 60 minutes upon the flashover, the floor must comply with the requirements of the limit states, R, E, I respectively, see [3,4].
The analysis presented in the paper is limited only to fire resistance R of a separated element of the floor reinforcement, which in this case is a composite steel-concrete slab.The authors assumed the slab fire resistance R as the design sagging moment resistance at time t = 60 minutes Mfi,t,Rd, determined according to [7] for its rib.The change of the moment value Mfi,t,Rd corresponding to the values of basic variables of the assumed calculation model was investigated.

Description of the analyzed element
The elements of the analyzed composite steel-concrete slab are the following [5]: -Cofrastra 40 profiled steel sheet with the thickness of t = 0,75 mm, made of S350 GD steel with the yield strength of fy,a = 350 MPa, -C25/30 concrete with the design strength of fc = 25 MPa, -additional reinforcement bar with the diameter of  = 6 mm, made of A-III N/RB 500W reinforced steel, the yield strength of fy,s = 500 MPa, placed on the steel fold in order to ensure the fire resistance assumed for the floor.The cross-section diagram of the analyzed composite slab is presented in Fig. 1 Fig. 1.The cross-section of the analyzed composite steel-concrete slab Cofrastra 40

Calculation model
The fire resistance of the slab was referred to the design value of the sagging moment resistance in the fire situation at time t = 60 minutes Mfi,t,Rd, determined according to [7] for its rib.The cross-section of the analyzed rib is presented in Fig. 2.
-u1 = u2 = 54,4 [mm] -the shortest distance of the centre of the reinforcement bar to any point of the webs of the steel sheet, -u3 = 23 [mm] -the distance of the reinforcement bar to the lower flange of the steel sheet, -α = 104° -angle of the web.Additional geometry parameters of the rib, calculated basing on the above-mentioned, are as follows: -the rib geometry factor: A/Lr = 22 [mm], -the configuration factor of the upper flange Φ, determined as follows: -and the indication of the position of the reinforcement bar in the rib, which is given by: 4 The design moment resistance of the rib The design moment resistance Mfi,t,Rd was determined according to the simple calculation model, defined in [7].The rules of this model apply to calculating the standard fire resistance of both simple supported and continuous slabs with profiled steel sheets and reinforcement when heated from below according to the standard temperature-time curve, defined in [6].It was assumed that the investigated steel sheet is directly heated and is not protected by any insulation, see Fig. 2. The design bending resistance is determined by plastic theory according to the formula (5):  (5) where : Ai , i = 1, 2, 3-is the elemental steel area, where A1 is the area of the lower flange, A2 is the area of the upper flange and A3 is the area of the web of the steel sheet, As -is the area of the reinforcing bar, Ac = xpl•b -is the area of compression concrete, where xpl is the range of the concrete compression area and b is the width of the analyzed slab rib, αslab = 0,85 -is the coefficient taking into account the assumption of the rectangular stress block when designing slabs, fy,a -is the nominal yield strength for the elemental steel area Ai, fy,s -is the nominal yield strength for the area of reinforcing bar, fc -is the design strength for concrete area, γM,fi -is the partial factor for relevant material property, for the fire situation.For thermal properties of steel and concrete, the recommended value is γM,fi = 1,0, see [6] zi, zs, zc -is the distance from the plastic neutral axis to the centroid of the elemental area Ai, As, Ac, respectively, see Fig. 2. The plastic neutral axis of composite slab is determined basing on the condition for equilibrium of the resultant forces: the compressive one in the concrete Nc = αslab•Ac•fc,Θ, and the tensile force in particular elements of the steel sheet: Np,1 = A1•fy,a,Θ,1, Np,2 = A2•fy,a,Θ,2, Np,3 = A3•fy,a,Θ,3, respectively and in the reinforcement bar Ns = As•fy,s,Θ,.The balance of forces of the cross-section and the geometry parameters of the assumed model are indicated in Fig. 2. The range of the concrete compression area xpl was calculated from the formula (6).
The parameters ky,Θ,i, ky,Θ,s, kc,Θ, in the formulae ( 5) and ( 6) are reduction coefficients allowing calculation of the values reduced due to the increased temperature: the strength yield of the i-th element of the steel sheet fy,a,,i = ky,,i•fy,a, yield strength of the reinforcement bar fy,s, = ky,,s•fy,s, and the compression strength of the concrete fc, = kc,•fc.The values of the coefficients are determined according to item 3.2 in [7] respectively to the determined temperature of particular elements of the analyzed rib cross-section.The temperature of each element is calculated according to the attachment D to the standard [7], respectively to the assumed fire resistance class R60 following the dependences below: -the temperatures a of the lower flange, web, or upper flange: , -the temperature s of the reinforcement bars in the rib: , 1 -the temperature c of the concrete was referred to the temperature of the area distant from the upper slab by h1/2.The temperature of this surface was determined according to the temperature distribution in a solid slab of effective thickness heff composed of normal weight concrete and not insulated, referred to in attachment D to the standard [7].The effective slab thickness was calculated according to the formula: In the formulae (6-9) the parameters: b0, b1, b2, b3, b4 are coefficients for determination of the temperatures of the parts of the steel sheet, the values of which are shown in Table D.2 of the [7]; c0, c1, c2, c3, c4, c5 are coefficients for the determination of the temperatures of the reinforcement bars in the rib, according to Table D.3 of the [7], the importance of the other parameters is as indicated above.
The bending resistance Mfi,t,Rd defined according to the above procedure and the formula (5), depends on the following set of basic variables: X = {l1, l2, l3, h2 t, α, , (y, z), h1, fy,a, fy,s, fc}, where (x, y) are the coordinates of the location of the reinforcement bar in the rib, referred to the coordinate system shown in Fig.    2.
Table 3.Values of parameters for determination of the reference value of Mfi,t,Rd = M0.A subset X1 = {(y, z), h1, fy,a, fy,s, fc} was selected from the set of basic variables X = {l1, l2, l3, h2 t, α, , (y, z), h1, fy,a, fy,s, fc}.The elements of the subset determine the range of the sensitivity analysis.Five analyses that referred to resepctive elements of the set X1 were carried out.In every analysis made, one of the elements of the set X1 was treated as a variable, whereas the other elements of the set were treated as constant with the values as shown in Table 1.For a selected variable, a range of its variability was determined as well as a set of its values comprised by that range.Furthermore, for every value of that set a corresponding value of the design moment resistance Mfi,t,Rd was calculated.The resulting set of values Mfi,t,Rd was then subject to the analysis which meant to determine the range of variability and to compare the extreme values with the reference value M0.
5.1 Mfi,t,Rd value with a varied location (x, y) of the reinforcement bar It was assumed, that the location of the reinforcement bar in the slab rib varies and the values of the other basic variables were as presented in Table 1.The bar location is identified by the coordinates (x, y) of the centre of its cross-section in the assumed coordinate system presented in Fig.The analysis assumed variability of the thickness h1 of the concrete slab above the steel sheet.It was accepted, that h1 changes from 40 mm to 70 mm along with the increment of Δh1 = 0,1h1,0 = 5 mm, where h1,0 = 50 mm is the reference thickness, to which the bending resistance M0 = 1,42 kNm corresponds.Fig. 4a) shows the calculated values of Mfi,t,Rd and their corresponding values of the variable h1.The created graph illustrates a linear dependence of Mfi,t,Rd(h1).The values of h1 from the assumed variability range have their corresponding values of Mfi,t,Rd from the range [Mfi,t,Rd,min = 1,2 kNm = 0,85M0; Mfi,t,Rd,max = 1,86 kNm = 1,31M0].An increase of h1 by 0,1h1,0 leads to an increase of the rib bending resistance by 0,077M0.
5.3 Mfi,t,Rd value with a varied yield strength fy,a fy,a -the nominal yield strength for the elemental steel area Ai of the sheet was taken as a variable.Bending resistance calculations Mfi,t,Rd(fy,a) were made for the value of fy,a from the range of [0,6fy,a,0 = 210; 1,4fy,a,0 = 490 MPa] accepted with the increment of Δfy,a = 0,1fy,a,0, where fy,a,0 = 350 MPa is the reference value.Basing on the obtained results, a dependence graph was made Mfi,t,Rd(fy,a), which is shown in Fig. 4b).As in 5.2, the dependence is linear.In this case, 10% increase of fy,a results in 5,6% increase in the bending resistance, from Mfi,t,Rd,min = 1,1 kNm = 0,77M0 to Mfi,t,Rd,max = 1,74 kNm = 1,23M0.
The graphs in Fig. 6 present the dependence of the nominal value Mfi,t,Rd/M0 on the nominal value of the variable xi/xi0 for which the bending resistance was calculated.The graphs corresponding to the varied position of the reinforcement bar (x, y), refer only to two representative cases, namely:1) the position y = y0 = 23 mm is fixed and the coordinate x varies; 2) the position x = x0 = 62 mm is fixed and y varies.-The bending resistance Mfi,t,Rd shows a similar sensitivity to a change of the value of yield strength of steel sheet fy,a and the yield strength of the reinforcement bar fy,s.The variability ranges fy,a  [0,6fy,a,0; 1,4fy,a,0] and fy,s  [0,6fy,s,0; 1,4fy,s,0] correspond to the values Mfi,t,Rd from the ranges [0,77M0; 1,23M0] and [0,83M0; 1,17M0], respectively.-Three areas of the location of the reinforcement bar can be distinguished, for which sensitivity of bending resistance Mfi,t,Rd to a change of location within the area of each of them is different.These areas are: 1) y  [0,1y0; 0,3y0]; x  [0,2x0; 1,8x0] its corresponding range of variability of bending resistance is : Mfi,t,Rd  [0,65M0; 0,67M0].In this location of the reinforcement bar, the rib has the lowest bending resistance and is not subject to a vivid change.2) y  [0,5y0; 0,8y0]; x  [0,2x0; 0,7x0]  [1,3x0; 1,8x0] within this area the resistance presents a significant sensitivity to a change of the location of the reinforcement bar, the obtained values fit into the range of Mfi,t,Rd  [0,66M0; 0,93M0].3) y  [y0; 1,9y0]; x  [0,7x0; 1,3x0] it is the area that refers to the highest bending resistance of the rib and a change of the bar location within this area slightly affects a change of the bending resistance value Mfi,t,Rd  [0,98M0; 1,06M0].

Fig. 2 .
Fig.2.The cross-section of the analyzed rib of slab Cofrastra 40 The rib geometry parameters and their values indicated in Fig.2 corresponding to the slab assumed in [1] are the following: -l1 = 103,5 [mm] -the distance between the upper corners of the profiled steel sheet, -l2 = 124 [mm] -the width of the lower flange of the steel sheet, -l3 = 46,5 [mm] -the width of the upper flange of the steel sheet, -b = l1 + l3 = 150 [mm] -the width of the slab rib, -h1 = 50 [mm] -the thickness of the concrete slab above the steel sheet, -h2 = 40 [mm]-the height of the steel sheet, the distance x is measured as in Figure in Table

5. 2
Mfi,t,Rd value with a varied thickness of the concrete slab h1

Fig. 6 .
Fig. 6.Dependence between relative increase of the moment resistance Mfi,t,Rd and relative increase of the value of basic variable xi.The sensitivity analysis made leads to the following conclusions: -A varied sensitivity of the calculated bending resistance Mfi,t,Rd to a change of the value of particular basic variables from the set X1 is found; -The biggest changes of Mfi,t,Rd value come along with changes in thickness of the slab above the steel sheet h1, where the variability range Mfi,t,Rd  [0,85M0; 1,31M0] corresponds to the variability range of h1  [0,8h1,0; 1,4h1,0].-The lowest sensitivity can be observed with changes of the value of concrete compression strength fc.In this case, the values fc from the range [0,6fc,0; 1,4fc,0] correspond to the values Mfi,t,Rd from the range [0,986M0; 1,007M0].-Thebending resistance Mfi,t,Rd shows a similar sensitivity to a change of the value of yield strength of steel sheet fy,a and the yield strength of the reinforcement bar fy,s.The variability ranges fy,a  [0,6fy,a,0; 1,4fy,a,0] and fy,s  [0,6fy,s,0; 1,4fy,s,0] correspond to the values Mfi,t,Rd from the ranges [0,77M0; 1,23M0] and [0,83M0; 1,17M0], respectively.-Three areas of the location of the reinforcement bar can be distinguished, for which sensitivity of bending resistance Mfi,t,Rd to a change of location within the area of each of them is different.These areas are: 1) y  [0,1y0; 0,3y0]; x  [0,2x0; 1,8x0] its corresponding range of variability of bending resistance is : Mfi,t,Rd  [0,65M0; 0,67M0].In this location of the reinforcement bar, the rib has the lowest bending resistance and is not subject to a vivid change.2) y  [0,5y0; 0,8y0]; x  [0,2x0; 0,7x0]  [1,3x0; 1,8x0] within this area the resistance presents a significant sensitivity to a change of the location of the reinforcement bar, the obtained values fit into the range of Mfi,t,Rd  [0,66M0; 0,93M0].3) y  [y0; 1,9y0]; x  [0,7x0; 1,3x0] it is the area that refers to the highest bending resistance of the rib and a change of the bar location within this area slightly affects a change of the bending resistance value Mfi,t,Rd  [0,98M0; 1,06M0].

Table 1 .
2. The bending resistance Mfi,t,Rd of the slab rib was calculated with the defined in [1] values of basic variables X, representing the state of reference.It was assumed, that the resulting value of the bending resistance Mfi,t,Rd = M0 = 1,42 kNm is the reference value in the sensitivity analysis conducted hereinafter.The values of basic variables and the values of parameters necessary for determining the bending resistance M0, calculated on the basis of hose basic variables, are presented in Table 1 -3.Values of basic variables according to [1].

Table 2 .
Coefficients for determination of the temperatures of elements of the rib.Calculated temperatures a,i, s, c and reduction factors ky,,i, ky,,s, kc,.

Table 4 .
Results of sensitivity analysis.