Moving down the stairs

Test group description

The scenario is a staircase with a width of 4 m, a height of 30 m and a projection length of 40 m. Thus, the length of the staircase is 50 m (since the speed of movement does not depend on the staircase inclination angle, the above parameters are chosen only to simplify the calculations). At the bottom level of the staircase there is an exit and a registering device. There is such a number of people on the stairs to provide the necessary density of placement. The start time of movement of all people is zero. The time of the last person passing through the middle of the staircase, calculated using the Fenix+ 3 program, is compared with the time determined by the formula:

$$ t(D) = \frac{0.5 \cdot L}{V(D)}(1) $$

where L is the length of the stair and V is the velocity. The flow density is determined by the formula:

$$ D = \frac{Nf}{S} (2)$$

where N is the number of people on the stairs, S is the area of the stairs, and f is the area of the horizontal projection of a person.

Projects were developed for each contingent of people. Each design consists of several scenarios (each scenario corresponds to a certain flow density).

Expected results

The time for the last person to pass through the middle of the stairs coincides with the time calculated by formula (1).

Modeling results

Test_012 People profile “Females 30-50 years old [IMO]”

Movement of people flow of “Females 30-50 years old [IMO]” profile down the stairs

(a) Flow density 0,01 m²⁄m² (b) Flow density 0,05 m²⁄m² (c) Flow density 0,1 m²⁄m² (d) Flow density 0,15 m²⁄m² (e) Flow density 0,20 m²⁄m² (f) Flow density 0,25 m²⁄m² (g) Flow density 0,3 m²⁄m² (h) Flow density 0,35 m²⁄m² (i) Flow density 0,4 m²⁄m² (j) Flow density 0,45 m²⁄m² (k) Flow density 0,5 m²⁄m² (l) Flow density 0,6 m²⁄m² (m) Flow density 0,7 m²⁄m² (n) Flow density 0,8 m²⁄m² (o) Flow density 0,9 m²⁄m²

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	34.2	38.5	-11.2
2	0.05	100	37.6	38.5	-2.3
3	0.1	200	47.0	45.7	2.8
4	0.15	300	59.2	56.2	5.3
5	0.2	400	78.4	72.8	7.7
6	0.25	500	104.4	101.9	2.5
7	0.3	600	134.4	169.8	-20.8
8	0.35	700	170.8	463.0	-63.1
9	0.4	800	198.4	463.0	-57.1
10	0.45	900	234.0	463.0	-49.5
11	0.5	1000	266.6	463.0	-42.4
12	0.6	1200	287.2	463.0	-38.0
13	0.7	1400	400.8	463.0	-13.4
14	0.8	1600	430.6	463.0	-7.0
15	0.9	1800	458.8	463.0	-0.9

Table 1. Movement of people flow of “Females 30-50 years old [IMO]” profile down the stairs

Figure 1. Movement of people flow of “Females 30-50 years old [IMO]” profile down the stairs

Test_013 People profile “Females older than 50 years [IMO]”

Movement of people flow of “Females older than 50 years [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	33.8	41.7	-18.9
2	0.05	100	43.8	41.7	5.0
3	0.1	200	47.2	49.5	-4.6
4	0.15	300	60.8	60.9	-0.2
5	0.2	400	81.4	78.9	3.2
6	0.25	500	120.6	110.4	9.2
7	0.3	600	148.2	183.6	-19.3
8	0.35	700	186.6	500.0	-62.7
9	0.4	800	223.6	500.0	-55.3
10	0.45	900	248.2	500.0	-50.4
11	0.5	1000	281.6	500.0	-43.7
12	0.6	1200	309.8	500.0	-38.0
13	0.7	1400	433.8	500.0	-13.2
14	0.8	1600	465.6	500.0	-6.9
15	0.9	1800	495.6	500.0	-0.9

Table 2. Movement of people flow of “Females older than 50 years [IMO]” profile down the stairs

Figure 2. Movement of people flow of “Females older than 50 years [IMO]” profile down the stairs

Test_014 People profile “Females older than 50, mobility impaired (1) [IMO]”

Movement of people flow of “Females older than 50, mobility impaired (1) [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	51.8	55.6	-6.8
2	0.05	100	54.0	55.6	-2.9
3	0.1	200	69.6	66.0	5.5
4	0.15	300	81.6	81.2	0.5
5	0.2	400	114.4	105.3	8.6
6	0.25	500	153.6	147.3	4.3
7	0.3	600	198.2	244.9	-19.1
8	0.35	700	245.0	657.9	-62.8
9	0.4	800	295.6	657.9	-55.1
10	0.45	900	339.8	657.9	-48.4
11	0.5	1000	383.6	657.9	-41.7
12	0.6	1200	413.0	657.9	-37.2
13	0.7	1400	577.8	657.9	-12.2
14	0.8	1600	620.6	657.9	-5.7
15	0.9	1800	652.0	657.9	-0.9

Table 3. Movement of people flow of “Females older than 50, mobility impaired (1) [IMO]” profile

Figure 3. Movement of people flow of “Females older than 50, mobility impaired (1) [IMO]” profile down the stairs

Test_015 People profile “Females older than 50, mobility impaired (2) [IMO]”

Movement of people flow of “Females older than 50, mobility impaired (2) [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	48.2	64.1	-24.8
2	0.05	100	64.2	64.1	0.2
3	0.1	200	71.4	76.1	-6.2
4	0.15	300	95.4	93.6	1.9
5	0.2	400	129.0	121.2	6.4
6	0.25	500	178.0	169.6	5.0
7	0.3	600	217.0	282.4	-23.2
8	0.35	700	292.6	757.6	-61.4
9	0.4	800	336.6	757.6	-55.6
10	0.45	900	376.0	757.6	-50.4
11	0.5	1000	438.2	757.6	-42.2
12	0.6	1200	477.0	757.6	-37.0
13	0.7	1400	665.2	757.6	-12.2
14	0.8	1600	714.0	757.6	-5.8
15	0.9	1800	750.8	757.6	-0.9

Table 4. Movement of people flow of “Females older than 50, mobility impaired (2) [IMO]” profile.

Figure 4. Movement of people flow of “Females older than 50, mobility impaired (2) [IMO]” profile down the stairs

Test_016 People profile “Females younger than 30 years [IMO]”

Movement of people flow of “Females younger than 30 years [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	29.6	33.3	-11.1
2	0.05	100	33.2	33.3	-0.3
3	0.1	200	41.4	39.6	4.5
4	0.15	300	55.8	48.7	14.6
5	0.2	400	75.8	63.1	20.1
6	0.25	500	93.4	88.2	5.9
7	0.3	600	114.6	146.9	-22.0
8	0.35	700	150.4	396.8	-62.1
9	0.4	800	175.2	396.8	-55.8
10	0.45	900	201.4	396.8	-49.2
11	0.5	1000	228.4	396.8	-42.4
12	0.6	1200	247.0	396.8	-37.8
13	0.7	1400	347.4	396.8	-12.4
14	0.8	1600	371.8	396.8	-6.3
15	0.9	1800	393.4	396.8	-0.9

Table 5. Movement of people flow of “Females younger than 30 years [IMO]” profile down the

Figure 5. Movement of people flow of “Females younger than 30 years [IMO]” profile down the stairs.

Test_017 People profile “Males 30-50 years old [IMO]”

Movement of people flow of “Males 30-50 years old [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	29.0	29.2	-0.7
2	0.05	100	28.6	29.2	-2.1
3	0.1	200	32.8	34.7	-5.5
4	0.15	300	42.6	42.7	-0.2
5	0.2	400	60.8	55.4	9.7
6	0.25	500	75.2	77.4	-2.8
7	0.3	600	108.8	128.6	-15.4
8	0.35	700	128.6	352.1	-63.5
9	0.4	800	155.4	352.1	-55.9
10	0.45	900	175.8	352.1	-50.1
11	0.5	1000	202.6	352.1	-42.5
12	0.6	1200	218.8	352.1	-37.9
13	0.7	1400	305.2	352.1	-13.3
14	0.8	1600	327.6	352.1	-7.0
15	0.9	1800	349.0	352.1	-0.9

Table 6. Movement of people flow of “Males 30-50 years old [IMO]” profile down the stairs

Figure 6. Movement of people flow of “Males 30-50 years old [IMO]” profile down the stairs

Test_018 People profile “Males older than 50 years [IMO]”

Movement of people flow of “Males older than 50 years [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	36.4	37.3	-2.4
2	0.05	100	37.8	37.3	1.3
3	0.1	200	48.6	44.3	9.7
4	0.15	300	59.4	54.5	9.0
5	0.2	400	75.4	70.7	6.6
6	0.25	500	101.4	98.8	2.6
7	0.3	600	132.6	164.1	-19.2
8	0.35	700	172.0	446.4	-61.5
9	0.4	800	197.2	446.4	-55.8
10	0.45	900	223.4	446.4	-50.0
11	0.5	1000	254.0	446.4	-43.1
12	0.6	1200	275.6	446.4	-38.3
13	0.7	1400	388.6	446.4	-12.9
14	0.8	1600	417.4	446.4	-6.5
15	0.9	1800	442.4	446.4	-0.9

Table 7. Movement of people flow of “Males older than 50 years [IMO]” profile down the stairs

Figure 7. Movement of people flow of “Males older than 50 years [IMO]” profile down the stairs

Test_019 People profile “Males older than 50, mobility impaired (1) [IMO]”

Movement of people flow of “Males older than 50, mobility impaired (1) [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	47.2	49.0	-3.7
2	0.05	100	50.0	49.0	2.0
3	0.1	200	58.4	58.2	0.3
4	0.15	300	68.2	71.6	-4.7
5	0.2	400	100.4	92.7	8.3
6	0.25	500	139.4	129.7	7.5
7	0.3	600	176.4	215.9	-18.3
8	0.35	700	221.0	581.4	-62.0
9	0.4	800	252.8	581.4	-56.5
10	0.45	900	296.4	581.4	-49.0
11	0.5	1000	331.6	581.4	-43.0
12	0.6	1200	362.4	581.4	-37.7
13	0.7	1400	509.4	581.4	-12.4
14	0.8	1600	547.2	581.4	-5.9
15	0.9	1800	576.2	581.4	-0.9

Table 8. Movement of people flow of “Males older than 50, mobility impaired (1) [IMO]” profile

Figure 8. Movement of people flow of “Males older than 50, mobility impaired (1) [IMO]” profile down the stairs

Test_020 People profile “Males older than 50, mobility impaired (2) [IMO]”

Movement of people flow of “Males older than 50, mobility impaired (2) [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	52.6	56.8	-7.4
2	0.05	100	56.0	56.8	-1.4
3	0.1	200	65.4	67.4	-3.0
4	0.15	300	84.4	82.9	1.8
5	0.2	400	112.4	107.4	4.7
6	0.25	500	151.0	150.5	0.3
7	0.3	600	200.8	251.0	-20.0
8	0.35	700	249.0	675.7	-63.1
9	0.4	800	295.4	675.7	-56.3
10	0.45	900	347.0	675.7	-48.6
11	0.5	1000	383.8	675.7	-43.2
12	0.6	1200	424.4	675.7	-37.2
13	0.7	1400	590.2	675.7	-12.7
14	0.8	1600	625.8	675.7	-7.4
15	0.9	1800	669.6	675.7	-0.9

Table 9. Movement of people flow of “Males older than 50, mobility impaired (2) [IMO]” profile.

Figure 9. Movement of people flow of “Males older than 50, mobility impaired (2) [IMO]” profile down the stairs

Test_021 People profile “Males younger than 30 years [IMO]”

Movement of people flow of “Males younger than 30 years [IMO]” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	23.2	24.8	-6.5
2	0.05	100	25.6	24.8	3.2
3	0.1	200	27.4	29.4	-6.8
4	0.15	300	37.2	36.2	2.8
5	0.2	400	50.0	46.9	6.6
6	0.25	500	68.8	65.6	4.9
7	0.3	600	89.6	109.2	-17.9
8	0.35	700	107.8	297.6	-63.8
9	0.4	800	130.4	297.6	-56.2
10	0.45	900	150.6	297.6	-49.4
11	0.5	1000	170.0	297.6	-42.9
12	0.6	1200	185.2	297.6	-37.8
13	0.7	1400	258.8	297.6	-13.0
14	0.8	1600	277.0	297.6	-6.9
15	0.9	1800	295.0	297.6	-0.9

Table 10. Movement of people flow of “Males younger than 30 years [IMO]” profile down the stairs

Figure 10. Movement of people flow of “Males younger than 30 years [IMO]” profile down the stairs.

Test_022 People profile “SFPE contingent”

Movement of people flow of “SFPE contingent” profile down the stairs

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	24.2	26.1	-7.3
2	0.05	100	27.6	26.1	5.7
3	0.1	200	33.6	30.4	10.5
4	0.15	300	37.8	37.2	1.6
5	0.2	400	58.6	47.8	22.6
6	0.25	500	72.8	66.8	9.0
7	0.3	600	87.4	111.1	-21.3
8	0.35	700	112.4	329.7	-65.9
9	0.4	800	135.2	20000	-99.3
10	0.45	900	153.6	20000	-99.2
11	0.5	1000	171.8	20000	-99.1
12	0.6	1200	190.0	20000	-99.1
13	0.7	1400	269.4	20000	-98.7
14	0.8	1600	284.4	20000	-98.6
15	0.9	1800	330.8	20000	-98.3

Table 11. Movement of people flow of “SFPE contingent” profile down the stairs

Figure 11. Movement of people flow of “SFPE contingent” profile down the stairs

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