Ideal minimum curve radii and bevel angles for American standard-gauge mainlines.

JonMyrlennBailey

Active member
I'm thinking about building a new model route in the future in TANE or newer.

What are the ideal minimum curve radii and super-elevation bank angles for the following train speeds?

10 mph
15 mph
20 mph
25 mph
30 mph
35 mph
40 mph
45 mph
50 mph
55 mph
60 mph
65 mph
70 mph
 
That data doesn't consider tightness of the curves. The faster the train goes, the tighter the turns, the more it must lean just like riding a motorcycle.
 
While riding in a Trainz engine, one can "feel" hard or or gentle the curves are when hit. On my latest model layout construction, I'm making my curves stupid large. 1,500 foot radius in a 50 mph zone and 2,000 foot radius in a 70 mph zone. In 1/10 scale, that 150 actual foot radius and 200 actual foot radius respectively. I'm sure civil engineers while designing track geometry, place a coffee cup on a dining car table and observe whether the cup slides on turns at a given speed. Coffee cup physics to indicate G force.
 
I'm thinking about building a new model route in the future in TANE or newer.

What are the ideal minimum curve radii and super-elevation bank angles for the following train speeds?

10 mph
15 mph
20 mph
25 mph
30 mph
35 mph
40 mph
45 mph
50 mph
55 mph
60 mph
65 mph
70 mph
The following is from a spreadsheet I developed for my own route development in Trainz. It is based primarily on requirements of US Fed Regulations 49CFR213 but includes other sources. Optimum minimun radius - what's that? The minimum radius indicated in the table (Rf or Rm) is the minimum that can be assigned the indicated allowed speed based on the superelevation used. Max superelevation permitted for any track above class 2 in US is 7 inches. The table is based on a superelevation of 5 inches and a cant deficiency of 3 inches. The choice of what superelevation limit (set by P2) to use depends among other things on whether both passenger and freight traffic use the track and whether they are assigned different allowed speeds based on the class of track.

Code:
Superelevation	Actual 	Ea	5	in	Maximum Ea = 7.0 inches		
	Cant deficiency	Eu	3	in	Applicable to all railcars		
	Track gauge	g	56.5	in (standard gauge)			
	Cant angle	A = Ea/g	0.0885	rad			
			Easement Lengths				
V	V	P1	L1	L2	Rf	Rm	D
km/h	mph 	Rad-m	ft	ft	ft	m	deg
							
24.1	15	3.04	73	55	113	34	50.79
32.2	20	5.41	98	73	201	61	28.57
40.2	25	8.45	122	92	313	96	18.29
48.3	30	12.17	147	110	451	138	12.70
56.3	35	16.57	171	128	614	187	9.33
64.4	40	21.64	196	146	802	245	7.14
72.4	45	27.39	220	165	1015	309	5.64
80.5	50	33.81	245	183	1253	382	4.57
88.5	55	40.91	269	201	1517	462	3.78
96.6	60	48.69	293	220	1805	550	3.17
104.6	65	57.14	318	238	2118	646	2.70
112.7	70	66.27	342	256	2457	749	2.33
120.7	75	76.07	367	275	2820	860	2.03
128.7	80	86.56	391	293	3209	978	1.79
136.8	85	97.71	416	311	3623	1104	1.58
144.8	90	109.55	440	329	4061	1238	1.41
152.9	95	122.06	465	348	4525	1379	1.27
160.9	100	135.24	489	366	5014	1528	1.14
169.0	105	149.11	513	384	5528	1685	1.04
177.0	110	163.64	538	403	6067	1849	0.94
185.1	115	178.86	562	421	6631	2021	0.86
193.1	120	194.75	587	439	7220	2201	0.79
201.2	125	211.32	611	458	7834	2388	0.73
209.2	130	228.56	636	476	8474	2583	0.68
217.3	135	246.48	660	494	9138	2785	0.63
225.3	140	265.08	685	512	9827	2995	0.58
233.4	145	284.35	709	531	10542	3213	0.54
241.4	150	304.30	734	549	11281	3439	0.51
249.4	155	324.92	758	567	12046	3672	0.48
257.5	160	346.22	782	586	12836	3912	0.45
265.5	165	368.20	807	604	13650	4161	0.42
273.6	170	390.85	831	622	14490	4417	0.40
281.6	175	414.18	856	641	15355	4680	0.37
289.7	180	438.19	880	659	16245	4952	0.35
							
V = ((Ea + Eu)/(0.0007D))^0.5		 Rm = 1.2226V^2/(Ea+Eu)			
D = (Ea+Eu)/(0.0007V^2)			 P2 = Ea/g			
Rf = 50.0/(sin(D/2)) = 5730/D (approx)   P1 = RmP2  P1 = [1.2226V^2/(Ea+Eu)](Ea/g)		
Rf = 4.011V^2/(Ea+Eu)						
				                               		
The above table is based on:   	          	
Ea = 5.0 in		                           		
Eu = 3.0 in		                            		
g = 56.5 in		                          		
P2 = 0.0885 radians			                    		
P1 = 0.0135243V^2 m-radians         			
Rf = 0.501375V^2 ft           
Rm = 0.152825V^2 ft

The following is the supplementary table from the spreadsheet to cover superelevations from 0.5 to 7 inches and coefficients to use to correct P1 and R values given in the main table if Ea other than 5 is used. L3 is an overall minimum easement length to use based only on Ea value:
Code:
			Easement	
Ea	P2	Ke	L3	Fe
in	rad		ft 					
0.5	0.0088	0.2286	31	2.2857
1.0	0.0177	0.4000	62	2.0000
1.5	0.0265	0.5333	93	1.7778
2.0	0.0354	0.6400	124	1.6000
2.5	0.0442	0.7273	155	1.4545
3.0	0.0531	0.8000	186	1.3333
3.5	0.0619	0.8615	217	1.2308
4.0	0.0708	0.9143	248	1.1429
4.5	0.0796	0.9600	279	1.0667
5.0	0.0885	1.0000	310	1.0000
5.5	0.0973	1.0353	341	0.9412
6.0	0.1062	1.0667	372	0.8889
6.5	0.1150	1.0947	403	0.8421
7.0	0.1239	1.1200	434	0.8000
				
P1e = KeP1
Re = FeR

The numbers I use as noted are based on 49CFR 213 which governs minimum standards for US railroads. I include minimum radii and suggestions for easement lengths in the table if you want to take it that far. Be advised that the Trainz track spline is a 3D cubic Bezier spline and can only approximate circular arcs and cubic spiral easements. Spacing of vertices for circular curves has to be uniform (equal with central angles not exceeding 30 deg) with straightened end tangent track at the correct angles to get really good approximations. Fitting in easements is difficult and requires some planning to simulate real ones.

Geophil provided some info for superelevation back in 2015 based on 1st principles and German regulations and the above link (post #2) is to a recent table provided by Anathoth71 which appears to be based on Geophil's info. [EDIT]My tables agree with Geophil's for the Trainz vertex input parameter P1 - following Geophil's original work - if I use the following input values: actual superelevation Ea = 118mm (4.65 in), cant deficiency Eu = 81.28mm (3.2 in), cant width g = 1500mm (standard gauge + 2.55 in). In the above table I use the track gauge 1435mm (56.5 in) for the cant width assuming wheel contact is on the inside rail edges and 76.2mm (3 in) for the cant deficiency because in the US all equipment is qualified to use that value. The superelevation used in the above table Ea = 5 inches is a nominal value applicable to track used by both freight and passenger trains.

I have a metric version of the table I'll post later.

The parameter P2 is used by Trainz to set the maximum cant angle/superelevation that it will apply to the track. Trainz probably calculates the cant angle to apply by multiplying the track horizontal curvature (1/radius in meters) by the parameter P1. If that result is greater than P2, then P2 is used instead.

Bob Pearson

[EDIT]
PS some notes and examples that go with the tables.

Code:
Note1: Linear interpolation within the tables is only approximate but should be OK for Trainz								
Note2: For the above Table the Trainz inputs, P1 and P2, result in superelevations calculated by Trainz equal to:								
          a) 5.0 inches when the radius (R) of the curve is equal to or less than the Rf value indicated
          b) a hyperbolic variation between 5 inches and 0 inches when the radius (R) of the curve is greater than Rf
             where Ea = 3.28084*P1*g/R (g in inches and R in ft)

Note 3: Any curve to which the inequality in 2a applies will have an allowed speed of less than that used to enter the table								
	Allowed speed can be solved for using V = ((Ea + Eu)*R/4.011)^0.5						

Examples:								
Trainz vertex inputs for an allowed speed of 70 mph based on Ea of 3, 5 and 7 inches. 								
								
Ea =	3 in		         Ea =	5 in		Ea =	7 in	
 V = 	70 mph	                  V = 	70 mph	         V = 	70 mph
from the main table:
P1 =    66.27     	         P1 =   66.27	        P1 =    66.27 
Rf = 	2457 ft		         Rf = 	2457 ft		Rf = 	2457 ft
from the supplementary table:
P2 =    0.0531	                 P2 =   0.0885          P2 =    0.1239
Ke = 	0.8000		         Ke = 	1.00		Ke = 	1.1200	
Fe = 	1.3333		         Fe = 	1.00		Fe = 	0.8000	
				
Trainz vertex inputs:	         Trainz vertex inputs:  Trainz vertex inputs:		
P1 = 0.80x66.27 = 53.02	         P1 = 	66.27		P1 = 	1.12x66.27 =	74.70
P2 = 	0.0531		         P2 = 	0.0885		P2 = 	0.1239

Min radius =	                 Min radius = 	        Min radius = 
1.3333x2457 = 3276 ft		 2457 ft		0.80x2457 =	1966 ft
Any radius equal to or larger than this will have an allowed speed of 70 mph with the Ea calculated by Trainz.
Any radius smaller than this will have an allowed speed of less than 70 mph with the Ea set to P2 by Trainz.
Allowed speed can be calculated using formula in Note 3.
										
Length of easement curves leading 	
into and out of the circular portion:	
from the main table:
L1 = 	342 ft		         L1 = 	342 ft		L1 = 	342 ft	
L2 = 	256 ft		         L2 = 	256 ft		L2 = 	256 ft	
from the supplementary table:
L3 = 	186 ft		         L3 = 	310 ft		L3 = 	434 ft	
L1 is recommended based on speed								
L2, L3  larger is the minimum to use
 
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This is the metric version of my spreadsheet:
Code:
Superelevation	Actual 	Ea	120	mm	Maximum Ea = 177.8 mm	
	Cant deficiency	Eu	76.2	mm	Applicable to all railcars	
	Track gauge	g	1435	mm   (standard gauge)		
	Cant angle	A = Ea/g	0.0836	rad		
			Easement Lengths			
V	V	P1	L1	L2	Rm	Rf
mph 	km/h	Rad-m	m	m	m	ft
						
12.4	20	2.04	18.5	14.9	24	80
18.6	30	4.60	27.8	22.3	55	180
24.9	40	8.18	37.0	29.8	98	321
31.1	50	12.78	46.3	37.2	153	501
37.3	60	18.40	55.6	44.6	220	722
43.5	70	25.04	64.8	52.1	299	982
49.7	80	32.70	74.1	59.5	391	1283
55.9	90	41.39	83.4	67.0	495	1624
62.1	100	51.10	92.6	74.4	611	2005
68.4	110	61.83	101.9	81.8	739	2426
74.6	120	73.59	111.1	89.3	880	2887
80.8	130	86.36	120.4	96.7	1033	3388
87.0	140	100.16	129.7	104.2	1198	3930
93.2	150	114.98	138.9	111.6	1375	4511
99.4	160	130.82	148.2	119.0	1564	5132
105.6	170	147.68	157.4	126.5	1766	5794
111.8	180	165.57	166.7	133.9	1980	6496
118.1	190	184.48	176.0	141.4	2206	7238
124.3	200	204.41	185.2	148.8	2444	8020
130.5	210	225.36	194.5	156.2	2695	8842
136.7	220	247.33	203.8	163.7	2958	9704
142.9	230	270.33	213.0	171.1	3233	10606
149.1	240	294.34	222.3	178.6	3520	11548
155.3	250	319.38	231.5	186.0	3819	12531
161.6	260	345.45	240.8	193.4	4131	13553
167.8	270	372.53	250.1	200.9	4455	14616
174.0	280	400.64	259.3	208.3	4791	15718
180.2	290	429.76	268.6	215.8	5139	16861
186.4	300	459.91	277.8	223.2	5500	18044

V = 12.0693((Ea + Eu)/D)^0.5		Rf = 39.3359V^2/(Ea+Eu)	
D = 145.668(Ea+Eu)/V^2			P2 = Ea/g	
Rm = 15.24/(sin(D/2)) = 1746.5/D (approx)	
Rm = 11.9895V^2/(Ea+Eu)		P1 = RmP2  P1 = [11.9895V^2/(Ea+Eu)](Ea/g)	
				
The above table based on
Ea = 120 mm
Eu = 76.2 mm	
 g = 1435 mm
P2 = 0.0836 radians
P1 =  0.00511014V^2 m-radians
Rm = 0.061109V^2 m

The supplementary table:
Code:
			Easement	
Ea	P2	Ke	L3	Fe
mm	rad		m	
				
10	0.0070	0.1897	7.4	2.2761
20	0.0139	0.3399	14.9	2.0395
30	0.0209	0.4619	22.3	1.8475
40	0.0279	0.5628	29.8	1.6885
50	0.0348	0.6478	37.2	1.5547
60	0.0418	0.7203	44.6	1.4405
70	0.0488	0.7828	52.1	1.3420
80	0.0557	0.8374	59.5	1.2561
90	0.0627	0.8854	67.0	1.1805
100	0.0697	0.9279	74.4	1.1135
110	0.0767	0.9659	81.8	1.0537
120	0.0836	1.0000	89.3	1.0000
130	0.0906	1.0308	96.7	0.9515
140	0.0976	1.0587	104.2	0.9075
150	0.1045	1.0842	111.6	0.8674
160	0.1115	1.1075	119.0	0.8307
170	0.1185	1.1290	126.5	0.7969
180	0.1254	1.1487	133.9	0.7658
				
P1e = KeP1
Re = FeR

Bob Pearson

[EDIT]
PS
Some notes and examples that go with the tables:

Code:
Note1: Linear interpolation within the tables is only approximate but should be OK for Trainz								
Note2: For the above Table the Trainz inputs, P1 and P2, result in superelevations calculated by Trainz equal to:								
          a) 120.0 mm when the radius (R) of the curve is equal to or less than the Rm value indicated
          b) a hyperbolic variation between 120 mm and 0 mm when the radius (R) of the curve is greater than Rm
             where Ea = P1*g/R (g in mm and R in meters)

Note 3: Any curve to which the inequality in 2a applies will have an allowed speed of less than that used to enter the table								
	Allowed speed can be solved for using: V = ((Ea + Eu)*R/11.9895)^0.5
						
Examples:								
Trainz vertex inputs for an allowed speed 110 km/h based on Ea of 70, 120 and 170 mm. 								
								
Ea =	70 mm	                 Ea =    120 mm		Ea =	170 mm	
 V = 	110 km/h	          V =    110 km/h        V = 	110 km/h	
from the main table:
P1 =    61.83     	         P1 =   61.83	        P1 =    61.83 
Rm = 	739 m 	                 Rm = 	 739 m 	        Rm = 	739 m 	
from the supplementary table:
P2 =    0.0488	                 P2 =   0.0836          P2 =    0.1185
Ke = 	0.7828	                 Ke = 	 1.00		Ke = 	1.1290	
Fe = 	1.3420	                 Fe = 	 1.00		Fe = 	0.7969	
				
Trainz vertex inputs:	         Trainz vertex inputs:  Trainz vertex inputs:		
P1 = 0.7828x61.83 = 48.40	 P1 = 	61.83		P1 = 	1.1290x61.83 =	69.81
P2 = 	0.0488		         P2 = 	0.0836		P2 = 	0.1185
	
Min radius =	                 Min radius = 	        Min radius = 
1.3420x739 = 992 m		 739 m		        0.7969x739 = 589 m	
Any radius equal to or larger than this will have an allowed speed of 110 km/h with the Ea calculated by Trainz.
Any radius smaller than this will have an allowed speed of less than 110 km/h with the Ea set to P2 by Trainz.
Allowed speed can be calculated using formula in Note 3.
								
Length of easement curves leading 	
into and out of the circular portion:	
from the main table:
L1 = 	101.9 m	                 L1 = 	101.9 m		L1 = 	101.9 m	
L2 = 	81.8 m		         L2 = 	81.8 m		L2 = 	81.8 m	
from the supplementary table:
L3 = 	52.1 m		         L3 = 	89.3 m		L3 = 	126.5 m	
L1 is recommended based on speed								
L2, L3  larger is the minimum to use

[EDIT]
PPS
Cant angle is the ratio of the superelevation to the distance between the contact point of the wheels on each rail. Because the rail heads are slightly rounded and the wheel treads form a conic surface that point will lie somewhere between the center of the top and the inside edge. Track gauge is the distance between the inside edges of the rail heads. For any track gauge the rail width will vary based on rail size used and varies say between 2.31 and 3 inches for rails used with std gauge. If the contact point is 1/6 the width from the center, then for standard gauge the contact point distance is between 58.04 and 58.5 in. They differ from the gauge width by 2.7 and 3.5% resp. For narrow gauges the % increases though smaller rail widths offset that a bit. While I suggest using the gauge width to set the parameter P2, it results in slightly larger values of cant angle and corresponding superelevations, if you prefer you can use g in the above formulas as the width between contact points.
 
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I got a PM recently asking if the above tables were finalized. Actually I had planned to post about an alternate way to calculate the parameters P1 & P2. The formulas are a lot simplier because we don't have to calculate the minimum radius that is associated with a given allowed track speed and superelevation/cant angle. You will have to determine that some other way if you are following prototype design and operation practices.

The following is the post that wasn't included when I posted the 2 main tables. This approach ignores any reference to track speed, though allowed track speed is the main reason for adding superelevation. The accelerations that affect safety and comfort as a train travels around a curve are directly related to train speed and track curvature. But if you just want to apply a specific superelevation to a curve, which in Trainz is expressed as a cant angle, it is possible.

I'll add my definition of the 2 parameters for what it's worth:

P1 is the ratio of the cant angle to the horizontal curvature the track. Units are radians-meters
P2 is the upper limit for the cant angle that Trainz will apply to a local track section. Units are radians.

If we know a curve has a specific value of superelevation based on track data of the prototype railroad or we just want to set a specific superelevation value without any regard to prototype speed on the curve, we can set the P1 and P2 parameters using the following:

Start by calculating the P2 parameter:
Code:
P2 = Ed/g       Ed is the desired superelevation on the curve
                g is the track gauge
Use the same units for both Ed and g: in and in or mm and mm. The result is a ratio that approximates the cant angle in radians. Strictly speaking P2 is the arcsin(Ed/g) but for small angles measured in radians the approximation is accurate enough. The supplementary tables in the above 2 posts show values of P2 for various superelevations on standard gauge track (see last comment in post above for info on distance between contact points vs track gauge).

Next calculate parameter P1 by multiplying P2 by the radius of the curve (in meters) that we want to have the cant angle P2. In Trainz Surveyor module we can measure the radius of the curve in question at a few points and pick a radius, Rmax, that is somewhat larger than the largest one we measure:
Code:
P1 = P2xRmax       P2 is an angle in radians 
                   Rmax is a radius in meters

The following is provided for information but is not necessary to calculate the 2 parameters.

Spline track in Trainz doesn't create constant radius curves- mathematically it can't. It can approximate them but the radius will vary in some fashion along the curve. Hence the suggestion to measure the radius at a few points and pick a max value. Trainz calculates the cant angle to apply by multiplying parameter P1 by the curvature of the track at that point. Since curvature is equal to 1/R (in meters), for any point where the radius is less than the one used to calculate P1 the calculated cant angle will be greater than P2 and Trainz will therefore use the limiting value, P2 for the cant angle at this point.

Using P1 & P2 as described above will assign a cant angle P2 to any curve that has a radius equal to or less than Rmax. Using these P1 & P2 values on curves with radii larger than Rmax will result in superelavations that range from Ed at Rmax down to 0 on tangent (straight) track. That might be ok for your needs or on larger radius curves you can just recalculate appropriate P1 & P2 values to apply and Trainz will calculate the cant angle you want it to use.

Bob Pearson


PS. An example: I have a curve in my route that I know the prototype used a superelevation value of 5.0 inches and I want to have Trainz use that same value on the curve.

I measure a few values of the radius and get the following values: 225, 250, 260, 240 & 220 m. I set Rmax to 270 m.

P2 = 5.0 in / 56.5 in = 0.0885 radians
P1 = 0.0885 x 270 m = 23.90 radian-meter

Checking the tables in post 1 above I'll note that the allowed speed on a curve with 270m radius and Ea = 5.0 in would be approx 42 mph.

Example 2: I have a curve in my route that I want to have a superelevation value of 180 mm.

I measure a few values of the radius and get the following values: 1975, 2000, 2015, 2005 & 1988 m. I set Rmax to 2020 m.

P2 = 180 mm / 1435 mm = 0.1254 radians
P1 = 0.1254 x 2020 m = 253.30 radian-meter

Checking the tables in post 2 above I'll note that the allowed speed on a curve with 2020m radius and Ea = 180mm would be approx 208 km/h.
(P2 =0.1254 -> Ke = 1.1487 -> P1 = 253.3/1.1487 = 220.51 -> V = 208 or by formula in note3)
 
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