Moving up 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_023 People profile “Females 30-50 years old [IMO]”

Movement of people flow of “Females 30-50 years old [IMO]” profile up 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	39.6	42.4	-6.6
2	0.05	100	41.2	42.4	-2.8
3	0.1	200	49.4	50.3	-1.8
4	0.15	300	67.2	61.9	8.6
5	0.2	400	89.8	80.3	11.8
6	0.25	500	117.6	112.4	4.6
7	0.3	600	160	187.2	-14.5
8	0.35	700	186.4	510.2	-63.5
9	0.4	800	228.6	510.2	-55.2
10	0.45	900	252.2	510.2	-50.6
11	0.5	1000	297	510.2	-41.8
12	0.6	1200	314	510.2	-38.5
13	0.7	1400	442	510.2	-13.4
14	0.8	1600	474.2	510.2	-7.1
15	0.9	1800	505.6	510.2	-0.9

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

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

Test_024 People profile “Females older than 50 years [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	46.8	51.0	-8.2
2	0.05	100	50.6	51.0	-0.8
3	0.1	200	68.8	60.6	13.5
4	0.15	300	80.8	74.5	8.5
5	0.2	400	97.0	96.4	0.6
6	0.25	500	143.0	134.9	6.0
7	0.3	600	177.8	224.1	-20.7
8	0.35	700	222.8	609.8	-63.5
9	0.4	800	270.8	609.8	-55.6
10	0.45	900	307.6	609.8	-49.6
11	0.5	1000	347.8	609.8	-43.0
12	0.6	1200	379.8	609.8	-37.7
13	0.7	1400	529.8	609.8	-13.1
14	0.8	1600	568.6	609.8	-6.8
15	0.9	1800	604.2	609.8	-0.9

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

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

Test_025 People profile “Females older than 50, mobility impaired (1) [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	59.0	67.6	-12.7
2	0.05	100	70.0	67.6	3.6
3	0.1	200	89.0	80.2	11.0
4	0.15	300	102.2	98.6	3.7
5	0.2	400	129.4	127.7	1.3
6	0.25	500	180.8	178.5	1.3
7	0.3	600	238.8	296.5	-19.5
8	0.35	700	294.6	806.5	-63.5
9	0.4	800	344.0	806.5	-57.3
10	0.45	900	400.0	806.5	-50.4
11	0.5	1000	465.6	806.5	-42.3
12	0.6	1200	502.2	806.5	-37.7
13	0.7	1400	701.0	806.5	-13.1
14	0.8	1600	751.0	806.5	-6.9
15	0.9	1800	799.2	806.5	-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 up the stairs.

Test_026 People profile “Females older than 50, mobility impaired (2) [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	79.8	80.6	-1.0
2	0.05	100	79.2	80.6	-1.7
3	0.1	200	91.8	95.8	-4.2
4	0.15	300	127.8	117.8	8.5
5	0.2	400	159.4	152.7	4.4
6	0.25	500	224.8	213.4	5.3
7	0.3	600	289.8	354.0	-18.1
8	0.35	700	353.0	961.5	-63.3
9	0.4	800	433.8	961.5	-54.9
10	0.45	900	501.2	961.5	-47.9
11	0.5	1000	544.0	961.5	-43.4
12	0.6	1200	599.4	961.5	-37.7
13	0.7	1400	837.4	961.5	-12.9
14	0.8	1600	894.6	961.5	-7.0
15	0.9	1800	952.8	961.5	-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 up the stairs

Test_027 People profile “Females younger than 30 years [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	39.2	39.7	-1.3
2	0.05	100	40.8	39.7	2.8
3	0.1	200	51.4	47.1	9.1
4	0.15	300	58.4	57.9	0.9
5	0.2	400	81.0	75.1	7.9
6	0.25	500	107.0	105.0	1.9
7	0.3	600	144.0	174.8	-17.6
8	0.35	700	174.8	471.7	-62.9
9	0.4	800	207.4	471.7	-56.0
10	0.45	900	244.0	471.7	-48.3
11	0.5	1000	270.6	471.7	-42.6
12	0.6	1200	296.4	471.7	-37.2
13	0.7	1400	413.0	471.7	-12.4
14	0.8	1600	441.8	471.7	-6.3
15	0.9	1800	467.4	471.7	-0.9

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

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

Test_028 People profile “Males 30-50 years old [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	30.4	39.7	-23.4
2	0.05	100	40.0	39.7	0.8
3	0.1	200	49.4	47.1	4.9
4	0.15	300	57.6	57.9	-0.5
5	0.2	400	78.2	75.1	4.1
6	0.25	500	104.6	105.0	-0.4
7	0.3	600	140.8	174.8	-19.5
8	0.35	700	173.0	471.7	-63.3
9	0.4	800	208.4	471.7	-55.8
10	0.45	900	242.2	471.7	-48.7
11	0.5	1000	270.4	471.7	-42.7
12	0.6	1200	295.8	471.7	-37.3
13	0.7	1400	413.0	471.7	-12.4
14	0.8	1600	439.4	471.7	-6.8
15	0.9	1800	467.4	471.7	-0.9

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

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

Test_029 People profile “Males older than 50 years [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	47.4	49.0	-3.3
2	0.05	100	49.4	49.0	0.8
3	0.1	200	62.8	58.2	7.9
4	0.15	300	81.8	71.6	14.2
5	0.2	400	98.8	92.7	6.6
6	0.25	500	139.8	129.7	7.8
7	0.3	600	180.8	215.9	-16.3
8	0.35	700	219.4	581.4	-62.3
9	0.4	800	254.0	581.4	-56.3
10	0.45	900	296.2	581.4	-49.1
11	0.5	1000	338.4	581.4	-41.8
12	0.6	1200	363.6	581.4	-37.5
13	0.7	1400	509.4	581.4	-12.4
14	0.8	1600	546.0	581.4	-6.1
15	0.9	1800	576.2	581.4	-0.9

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

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

Test_030 People profile “Males older than 50, mobility impaired (1) [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	59.8	64.1	-6.7
2	0.05	100	64.4	64.1	0.5
3	0.1	200	77.6	76.1	2.0
4	0.15	300	91.4	93.6	-2.4
5	0.2	400	135.0	121.2	11.4
6	0.25	500	177.4	169.6	4.6
7	0.3	600	223.6	282.4	-20.8
8	0.35	700	279.6	757.6	-63.1
9	0.4	800	338.6	757.6	-55.3
10	0.45	900	391.2	757.6	-48.4
11	0.5	1000	434.2	757.6	-42.7
12	0.6	1200	477.8	757.6	-36.9
13	0.7	1400	665.2	757.6	-12.2
14	0.8	1600	713.4	757.6	-5.8
15	0.9	1800	750.8	757.6	-0.9

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

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

Test_031 People profile “Males older than 50, mobility impaired (2) [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	63.2	75.8	-16.6
2	0.05	100	74.0	75.8	-2.4
3	0.1	200	96.6	90.0	7.3
4	0.15	300	108.6	110.8	-2.0
5	0.2	400	148.4	143.6	3.3
6	0.25	500	214.2	200.9	6.6
7	0.3	600	265.0	334.0	-20.7
8	0.35	700	336.4	892.9	-62.3
9	0.4	800	399.4	892.9	-55.3
10	0.45	900	454.6	892.9	-49.1
11	0.5	1000	532.0	892.9	-40.4
12	0.6	1200	567.4	892.9	-36.5
13	0.7	1400	787.6	892.9	-11.8
14	0.8	1600	838.0	892.9	-6.1
15	0.9	1800	884.8	892.9	-0.9

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

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

Test_032 People profile “Males younger than 30 years [IMO]”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	34.2	37.3	-8.3
2	0.05	100	38.4	37.3	2.9
3	0.1	200	44.2	44.3	-0.2
4	0.15	300	53.8	54.5	-1.3
5	0.2	400	76.2	70.7	7.8
6	0.25	500	100.4	98.8	1.6
7	0.3	600	134.8	164.1	-17.9
8	0.35	700	165.2	446.4	-63.0
9	0.4	800	191.4	446.4	-57.1
10	0.45	900	234.4	446.4	-47.5
11	0.5	1000	254.2	446.4	-43.1
12	0.6	1200	276.4	446.4	-38.1
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 10. Movement of people flow of “Males younger than 30 years [IMO]” profile up the stairs

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

Test_033 People profile “SFPE contingent”

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

№	Flow density, m²/m²	Number of people	Movement time, s (Fenix+ 3)	Movement time, s (Expected)	Deviation, %
1	0.01	20	23.8	26.1	-8.8
2	0.05	100	25.0	26.1	-4.2
3	0.1	200	31.4	30.4	3.3
4	0.15	300	38.2	37.2	2.7
5	0.2	400	54.2	47.8	13.4
6	0.25	500	70.6	66.8	5.7
7	0.3	600	87.6	111.1	-21.2
8	0.35	700	110.0	329.7	-66.6
9	0.4	800	131.4	20000	-99.3
10	0.45	900	153.2	20000	-99.2
11	0.5	1000	171.2	20000	-99.1
12	0.6	1200	190.2	20000	-99.0
13	0.7	1400	269.4	20000	-98.7
14	0.8	1600	284.6	20000	-98.6
15	0.9	1800	330.0	20000	-98.4

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

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

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