- Email: [email protected]
The distribution of fire load
Fire Safety Journal 20 (1993) 83-88
The Distribution of Fire Load
S. J. M e l i n e k Building Research Establishment, Fire Research Station, Borehamwood, Hertfordshire, WD6 2BL, UK (Received 20 September 1990; revised version received 12 March 1991; accepted 15 November 1991)
ABSTRACT Data are presented for fire load densities in office buildings. It is shown that the distribution of fire load density is approximately log normal. NOTATION In n
P s.d. t tc V
#, Ol O,
Natural logarithm (i.e. log to the base 'e') N u m b e r of rooms Probability of load density exceeding tc Standard deviation Fire load density Critical value of t Coefficient of variation ( s . d . / m e a n ) M e a n value of In (t) M e a n value of t Standard deviation of In (t) Standard deviation of t Cumulative probability for a normally distributed variable, x, with zero m e a n and unit variance
INTRODUCTION W h e t h e r structural failure occurs in a fire depends largely on the fire severity. For given wall properties, r o o m or c o m p a r t m e n t g e o m e t r y and 83 © 1992 Crown Copyright. Published by permission of the Controller of HMSO.
84
S. J. Melinek
ventilation, the fire severity is approximately proportional to the fire load density (the load per unit floor area).~ H e n c e , the fire load density will be one of the main variables determining w h e t h e r failure occurs. Since failure d e p e n d s on extreme values of the fire load, the probability of failure will d e p e n d on the frequency distribution of the fire load density rather than just on the m e a n and standard deviation. This paper presents data for live loads (i.e. total movable load) and fire loads (i.e. combustible load) in offices. It is shown that the distribution of the load per unit floor area is approximately log normal. This paper is concerned only with the nature of the fire load distribution. A n extensive s u m m a r y of fire load data is given in reference 2. THEORY It can be shown that if the distribution of a r a n d o m variable, t, is log normal, then tr~ = In (1 + (tr,//~,) 2) = In (1 + V 2)
(la)
/~, = In (~,) - ½o~
(lb)
where/~t and ot are the m e a n and standard deviation of t, /~, and crj are the m e a n and standard deviation of In (t), V is the coefficient of variation of t, and In signifies natural logarithm (i.e. log to the base 'e'). One way to judge w h e t h e r the distribution of a variable is log normal is to c o m p a r e the observed values of/~, and or, with the values given by eqn (1). A n o t h e r way is to c o m p a r e observed percentile values (values of t with a stated probability of being exceeded) with predicted values. For a log normal distribution the probability, p, of t exceeding a value tc is given by p = ~(x) (2) where x = {In (to) - / ~ , } / o , , and ~ ( x ) is the cumulative probability for a normally distributed variable with zero m e a n and unit variance. Data are often represented by their means, their standard deviations and sometimes percentiles (values with a stated probability of being exceeded). E q u a t i o n (1) can be used to derive /~l and o, from these quantities. E q u a t i o n (2) can then be used to estimate the failure probability at any value of the variable. RESULTS Tables 1 to 3 show load data from two surveys of office buildings. T h e British data consist of m e a s u r e m e n t s on nine office buildings selected at
The distribution of fire load
85
TABLE 1 L o a d D e n s i t i e s in Office Buildings
British data
n
American data
Office rooms
All rooms
(office rooms)
801
1143
453
Live load density (kg/m 2 of floor area) Mean s.v. V
44 24 0.56
54 66 1.21
53 24 0.45
~uIa /,1 h
3-65 3-64
3.53 3-64
3-88 3.87
ol ~
0.52
O'1b
0"55
0.95 0"82
0-43 0"46
Fire load density (kg/m 2 of floor area) Mean s.d. V
25 18 0.72
29 43 1-48
21 15 0.72
a D e r i v e d f r o m e q n (1) a s s u m i n g the distribution of t to b e log n o r m a l . b O b s e r v e d values.
r a n d o m from a survey of 32 office buildings in L o n d o n c o n d u c t e d during the period 1965-67. 3 The A m e r i c a n data (Gross, D., 1971, pers. c o m m . ) 4 are taken from a survey carried out by the National B u r e a u of Standards. 5 The data in Tables 1 to 3 are mostly for the total live (movable) load density, including steel furniture and its contents, since steel items may, in other buildings or at other times, be m a d e of wood or plastic or open metalwork. Table 1 also gives data for fire load density, which is defined for the British data as all u n p r o t e c t e d items of wood or other combustible material. For the A m e r i c a n data, it includes a proportion of the contents of enclosed steel items. It can be seen that fire load is on average about half the live load. It can be seen from Table 1 that the observed values of/u~ and ti~ are close to the values derived, assuming the distribution of the load density to be log normal. It can be seen from Table 2 that a log normal distribution gives a better fit to the data than a normal distribution. D a t a for fire load only
S. J. Melinek
86
TABLE 2 Observed and Predicted Distributions of Live (Movable) Load Density
Probability (%)
Load density (kg/m ~ of floor area) Observed
Predicted a
b
116
200
100 83 75 64 44
137 94 77 61 38
260
505
207 162 138 109 54
260 148 110 76 38
American data, office rooms(n =453) Highest value 169 122
180
Brit~hdata, office rooms(n = 801) Highest value 241 99 95 90 80 50
130 87 74 59 39
Brit~hdam, aH rooms (n = 1 143) Highest value 1096 99 95 90 80 50
99 95 90 80 50
307 144 92 67 39
135 104 84 67 51
110 93 84 74 53
141 103 87 71 48
a Assuming distribution of load density to be normal. b Assuming distribution of load density to be log normal.
(not shown) reveal a similar pattern, though less clear cut. These findings are in accordance with other work. 6 Table 3 shows that, contrary to earlier theories, 7 there is no systematic decrease in or, with room size. On the contrary, a, tends to be greater for large rooms than for m e d i u m size rooms. This appears to be due to the fact that s o m e large rooms have very low loads, probably because they are used as reception or conference areas.
The distribution of fire load
87
TABLE 3 Variation of Live Load Density with Room Size
Room area
No. of rooms
(m 2)
Iz,a
o, a
#~
o~
45 41 46
24 18 28
3-65 3.62 3-66
0-60 0.41 0-59
58 49 53
82 39 62
3.65 3.69 3-61
0.88 0-62 0.87
54 57 49
22 24 25
3.91 3-95 3.76
0.43 0.42 0-50
British data(office rooms) <15 15-20 >20
244 223 334
British data(all rooms) <15 15-20 >20
401 269 473
Ame~candata(office rooms) <15 15-20 >20
80 206 167
a In kg/m 2 of floor area.
It can be seen from Table 3 that there is little variation in #, or /~1 with room size. The distribution of the highest load densities (not shown) also did not vary significantly with room size. Table 4 shows fire loads per unit enclosing area for fire compartments. 8 It can be seen that the coefficients of variation, V, are somewhat less than the values given in Table 1. That is to be expected since fire compartments will normally consist of several rooms. TABLE 4 Fire Loads in Fire Compartments (Swedish data) Occupancy
Fire load density (M J i m 2 of enclosing area) Mean
s.d.
V
150 139
25 20
0.16 0-14
Offices
114
39
0-35
Schools
80
23
0.29
116
36
0-31
67
19
0.29
Dwellings (2 rooms + kitchen) (3 rooms + kitchen)
Hospitals Hotels
88
S. J. Melinek
It is of interest to note in Table 4 that the m e a n fire load density is higher in small dwellings (two rooms and kitchen) than in larger dwellings (three rooms and kitchen).
CONCLUSIONS 1. The log normal distribution gives a better fit to fire load data than does a normal distribution. 2. Some values for m e a n fire load densities and their standard deviations are presented. 3. The effect of r o o m size is considered.
REFERENCES 1. Law, M., Prediction of Fire Resistance. JFRO Symposium No. 5, HM Stationery Office, London, 1973, pp. 16-29. 2. Fire Load density. Appendix 1 of Design Guide, Structural Fire Safety, (CIB W14 Workshop), Fire Safety J., 10 (2), (1986), 101-18. 3. Mitchell, G. R. & Woodgate, R. W., Floor Loadings in Office Buildings-the Results of a Survey. Building Research Station Current Paper CP3/71, Garston, UK, 1971. 4. Gross, D. Private communication (1971). 5. Bryson, J. O. & Gross, D., Techniques for the survey and evaluation of live floor loads and fire loads in modern office buildings. Building Science Series 16, US Department of Commerce, National Bureau of Standards, Washington, DC, 1967. 6. Investigation on the statistical interpretation of fire loads in buildings. Cornell, A. C., Supplement to Report No. 1 on Fire load study for the European Convention for Constructional Steelwork, March 1971. 7. Home, M. R., The variation of mean floor loads with area. Engineering, London, 171(Feb.) (1951) 179-82. 8. CIB W14 Workshop report. Fire Safety J., 6 (1983) 24-55.
The Distribution of Fire Load
S. J. M e l i n e k Building Research Establishment, Fire Research Station, Borehamwood, Hertfordshire, WD6 2BL, UK (Received 20 September 1990; revised version received 12 March 1991; accepted 15 November 1991)
ABSTRACT Data are presented for fire load densities in office buildings. It is shown that the distribution of fire load density is approximately log normal. NOTATION In n
P s.d. t tc V
#, Ol O,
Natural logarithm (i.e. log to the base 'e') N u m b e r of rooms Probability of load density exceeding tc Standard deviation Fire load density Critical value of t Coefficient of variation ( s . d . / m e a n ) M e a n value of In (t) M e a n value of t Standard deviation of In (t) Standard deviation of t Cumulative probability for a normally distributed variable, x, with zero m e a n and unit variance
INTRODUCTION W h e t h e r structural failure occurs in a fire depends largely on the fire severity. For given wall properties, r o o m or c o m p a r t m e n t g e o m e t r y and 83 © 1992 Crown Copyright. Published by permission of the Controller of HMSO.
84
S. J. Melinek
ventilation, the fire severity is approximately proportional to the fire load density (the load per unit floor area).~ H e n c e , the fire load density will be one of the main variables determining w h e t h e r failure occurs. Since failure d e p e n d s on extreme values of the fire load, the probability of failure will d e p e n d on the frequency distribution of the fire load density rather than just on the m e a n and standard deviation. This paper presents data for live loads (i.e. total movable load) and fire loads (i.e. combustible load) in offices. It is shown that the distribution of the load per unit floor area is approximately log normal. This paper is concerned only with the nature of the fire load distribution. A n extensive s u m m a r y of fire load data is given in reference 2. THEORY It can be shown that if the distribution of a r a n d o m variable, t, is log normal, then tr~ = In (1 + (tr,//~,) 2) = In (1 + V 2)
(la)
/~, = In (~,) - ½o~
(lb)
where/~t and ot are the m e a n and standard deviation of t, /~, and crj are the m e a n and standard deviation of In (t), V is the coefficient of variation of t, and In signifies natural logarithm (i.e. log to the base 'e'). One way to judge w h e t h e r the distribution of a variable is log normal is to c o m p a r e the observed values of/~, and or, with the values given by eqn (1). A n o t h e r way is to c o m p a r e observed percentile values (values of t with a stated probability of being exceeded) with predicted values. For a log normal distribution the probability, p, of t exceeding a value tc is given by p = ~(x) (2) where x = {In (to) - / ~ , } / o , , and ~ ( x ) is the cumulative probability for a normally distributed variable with zero m e a n and unit variance. Data are often represented by their means, their standard deviations and sometimes percentiles (values with a stated probability of being exceeded). E q u a t i o n (1) can be used to derive /~l and o, from these quantities. E q u a t i o n (2) can then be used to estimate the failure probability at any value of the variable. RESULTS Tables 1 to 3 show load data from two surveys of office buildings. T h e British data consist of m e a s u r e m e n t s on nine office buildings selected at
The distribution of fire load
85
TABLE 1 L o a d D e n s i t i e s in Office Buildings
British data
n
American data
Office rooms
All rooms
(office rooms)
801
1143
453
Live load density (kg/m 2 of floor area) Mean s.v. V
44 24 0.56
54 66 1.21
53 24 0.45
~uIa /,1 h
3-65 3-64
3.53 3-64
3-88 3.87
ol ~
0.52
O'1b
0"55
0.95 0"82
0-43 0"46
Fire load density (kg/m 2 of floor area) Mean s.d. V
25 18 0.72
29 43 1-48
21 15 0.72
a D e r i v e d f r o m e q n (1) a s s u m i n g the distribution of t to b e log n o r m a l . b O b s e r v e d values.
r a n d o m from a survey of 32 office buildings in L o n d o n c o n d u c t e d during the period 1965-67. 3 The A m e r i c a n data (Gross, D., 1971, pers. c o m m . ) 4 are taken from a survey carried out by the National B u r e a u of Standards. 5 The data in Tables 1 to 3 are mostly for the total live (movable) load density, including steel furniture and its contents, since steel items may, in other buildings or at other times, be m a d e of wood or plastic or open metalwork. Table 1 also gives data for fire load density, which is defined for the British data as all u n p r o t e c t e d items of wood or other combustible material. For the A m e r i c a n data, it includes a proportion of the contents of enclosed steel items. It can be seen that fire load is on average about half the live load. It can be seen from Table 1 that the observed values of/u~ and ti~ are close to the values derived, assuming the distribution of the load density to be log normal. It can be seen from Table 2 that a log normal distribution gives a better fit to the data than a normal distribution. D a t a for fire load only
S. J. Melinek
86
TABLE 2 Observed and Predicted Distributions of Live (Movable) Load Density
Probability (%)
Load density (kg/m ~ of floor area) Observed
Predicted a
b
116
200
100 83 75 64 44
137 94 77 61 38
260
505
207 162 138 109 54
260 148 110 76 38
American data, office rooms(n =453) Highest value 169 122
180
Brit~hdata, office rooms(n = 801) Highest value 241 99 95 90 80 50
130 87 74 59 39
Brit~hdam, aH rooms (n = 1 143) Highest value 1096 99 95 90 80 50
99 95 90 80 50
307 144 92 67 39
135 104 84 67 51
110 93 84 74 53
141 103 87 71 48
a Assuming distribution of load density to be normal. b Assuming distribution of load density to be log normal.
(not shown) reveal a similar pattern, though less clear cut. These findings are in accordance with other work. 6 Table 3 shows that, contrary to earlier theories, 7 there is no systematic decrease in or, with room size. On the contrary, a, tends to be greater for large rooms than for m e d i u m size rooms. This appears to be due to the fact that s o m e large rooms have very low loads, probably because they are used as reception or conference areas.
The distribution of fire load
87
TABLE 3 Variation of Live Load Density with Room Size
Room area
No. of rooms
(m 2)
Iz,a
o, a
#~
o~
45 41 46
24 18 28
3-65 3.62 3-66
0-60 0.41 0-59
58 49 53
82 39 62
3.65 3.69 3-61
0.88 0-62 0.87
54 57 49
22 24 25
3.91 3-95 3.76
0.43 0.42 0-50
British data(office rooms) <15 15-20 >20
244 223 334
British data(all rooms) <15 15-20 >20
401 269 473
Ame~candata(office rooms) <15 15-20 >20
80 206 167
a In kg/m 2 of floor area.
It can be seen from Table 3 that there is little variation in #, or /~1 with room size. The distribution of the highest load densities (not shown) also did not vary significantly with room size. Table 4 shows fire loads per unit enclosing area for fire compartments. 8 It can be seen that the coefficients of variation, V, are somewhat less than the values given in Table 1. That is to be expected since fire compartments will normally consist of several rooms. TABLE 4 Fire Loads in Fire Compartments (Swedish data) Occupancy
Fire load density (M J i m 2 of enclosing area) Mean
s.d.
V
150 139
25 20
0.16 0-14
Offices
114
39
0-35
Schools
80
23
0.29
116
36
0-31
67
19
0.29
Dwellings (2 rooms + kitchen) (3 rooms + kitchen)
Hospitals Hotels
88
S. J. Melinek
It is of interest to note in Table 4 that the m e a n fire load density is higher in small dwellings (two rooms and kitchen) than in larger dwellings (three rooms and kitchen).
CONCLUSIONS 1. The log normal distribution gives a better fit to fire load data than does a normal distribution. 2. Some values for m e a n fire load densities and their standard deviations are presented. 3. The effect of r o o m size is considered.
REFERENCES 1. Law, M., Prediction of Fire Resistance. JFRO Symposium No. 5, HM Stationery Office, London, 1973, pp. 16-29. 2. Fire Load density. Appendix 1 of Design Guide, Structural Fire Safety, (CIB W14 Workshop), Fire Safety J., 10 (2), (1986), 101-18. 3. Mitchell, G. R. & Woodgate, R. W., Floor Loadings in Office Buildings-the Results of a Survey. Building Research Station Current Paper CP3/71, Garston, UK, 1971. 4. Gross, D. Private communication (1971). 5. Bryson, J. O. & Gross, D., Techniques for the survey and evaluation of live floor loads and fire loads in modern office buildings. Building Science Series 16, US Department of Commerce, National Bureau of Standards, Washington, DC, 1967. 6. Investigation on the statistical interpretation of fire loads in buildings. Cornell, A. C., Supplement to Report No. 1 on Fire load study for the European Convention for Constructional Steelwork, March 1971. 7. Home, M. R., The variation of mean floor loads with area. Engineering, London, 171(Feb.) (1951) 179-82. 8. CIB W14 Workshop report. Fire Safety J., 6 (1983) 24-55.