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Structural and Nonstructural Panels

Posted by admin on 12/ 11/ 15

Plywood and OSB are the structural panels most often specified to sheathe wood framing and increase its shear strength. For example, a 20-ft. wall sheathed with %6-in. plywood can withstand more than a ton of lateral force pushing against the top of the wall.

PLYWOOD

Structural plywood is made by laminating softwood plies. Each panel is stamped to indicate veneer grade, species group or span rating, thickness, exposure durability, mill number, and certifying agency.

Veneer grades. Veneer grades range from A to D, with letters appearing in pairs to indicate the front and back veneers of the panel. "A/B Exterior,” for example, has a grade A front veneer, a grade B back veneer, and grade C inner plies. When you buy CDX (C/D exterior-grade), it’s advisable to place the grade C side toward the weather—or up, if used as subflooring.

Most roof and wall sheathing and subflooring is CDX. If a panel is also stamped PTS, its imperfections have been plugged and touch sanded. Lower veneer grades have more plugs and bigger knots.

Grade D is the lowest grade of interior plywood panels; it should not be exposed to weather.

Species grade or span rating. Plywood’s strength may be indicated by two marks. One is a species group number (1-5). Group 1 is the strongest and often contains Douglas fir or southern yellow pine.

The second mark, a span rating, is more common. The two-digit rating looks like a fraction, but it’s not. Rather, a rating of 24/16 indicates that a panel can sheathe rafters spaced 24 in. on center and studs spaced 16 in. on center.

Another common stamp is Struc I, which stands for Structural I sheathing, a five-ply CDX that’s tested and guaranteed for a given shear value. If an engineer specifies Struc I, it must be used. Note: Plywood used for structural sheath-

image120" Подпись: ee PANELS ing must have a minimum of five plies. Avoid three-ply, h-in. CDX. Although it is widely available and cheaper than five-ply, it’s vastly inferior.

Thickness and length. APA (American Plywood Association) panels rated for Struc I wall sheathing, roof sheathing, and subflooring range from 3з8 in. to 23/з2 in. thick. Although 4×8 panels are the most common, 4×9 or 4×10 sheets enable you to run panels vertically from mudsills to the rim joists atop the first floor, thereby reducing the shear-wall blocking you might need behind panel edges and greatly improving the shear strength of the wall. (Shear walls are specially engineered walls that brace a building against lateral seismic and wind forces.) Although the square-foot price of 4×9 and 4×10 panels is higher than that of 4x8s, the larger panels enable you to work more quickly.

Exposure durability. How much weather and moisture a wood-based panel can take is largely a function of the glues used. Exterior-grade panels can be exposed repeatedly to moisture or used in damp climates because their plies are bonded with waterproof adhesives. Exposure 1 is suitable if there’s limited exposure to moisture—say, if construction gets delayed and the house doesn’t get closed in for a while. Exposure 2 panels are okay for protected applications and moderate construction delays. Interior-grade panels will deteriorate if they get wet; use them only in dry, protected applications.

OSB PANELS

OSB and plywood have almost exactly the same strength, stiffness, and span ratings. Both are fabricated in layers, and they weigh roughly the same. Both can sheathe roofs, walls, and floors. Their installation is almost identical, down to the blocking behind subfloor edges and need for H-clips between the unsupported edges of roof sheathing. Exposure ratings and grade stamps

tongue and groove good two sides good one side

000: performance-rated panel (number follows)

select tight face uniform surface, acceptable for underlayment sheathing

are also very similar. But in some respects, OSB is superior to plywood. It rarely delaminates, it holds screws and nails better, and it has roughly twice the shear values. (That’s why I-joists have OSB webs.) So given OSB’s lower cost (10 percent to 15 percent cheaper, on average), it’s not surprising that OSB grabs an increasing market share every year.

But OSB has one persistent and irreversible shortcoming: Its edges swell when they get wet and appear as raised lines (ghost lines) through roofing. To mitigate this swelling, OSB makers seal the panel edges; but when builders saw panels, the new (unsealed) edges swell when wet. Buildings under construction get rained on, so edge swelling is a real problem. Swollen edges can also raise hell in OSB subflooring or under – layment if it absorbs moisture, as commonly occurs over unfinished basements and uncovered crawl spaces. Thus many tile and resilient-flooring manufacturers insist on plywood underlayment.

Given the huge market for OSB, however, count on solutions before long. At this writing, J. M. Huber AdvanTech®, Louisiana-Pacific Top Notch®, and Weyerhaeuser Structurwood® are all tongue-and-groove-edged OSB panels purported to lie flat, install fast, and have minimal "edge swell.” Stay tuned.

Vertical Alignment

Posted by admin on 12/ 11/ 15

The design of the vertical alignment of a roadway also has a direct effect on the safety and comfort of the driver. Steep grades can slow down large, heavy vehicles in the traffic stream in the uphill direction and can adversely affect stopping ability in the downhill direction. Grades that are flat or nearly flat over extended distances will slow down the rate at which the pavement surface drains. Vertical curves provide a smooth change between two tangent grades, but must be designed to provide adequate stopping sight distance.

Tangent Grades. The maximum percent grade for a given roadway is determined by its functional classification, surrounding terrain, and design speed. Table 2.16 shows how the maximum grade can vary under different circumstances. Note that relatively flat grade limits are recommended for higher functional class roadways and at higher design speeds, whereas steeper grade limits are permitted for local roads and at lower design speeds.

Concerning minimum grades, flat and level grades may be used on uncurbed roadways without objection, as long as the pavement is adequately crowned to drain the surface laterally. The preferred minimum grade for curbed pavements is 0.5 percent, but a grade of 0.3 percent may be used where there is a high-type pavement accurately crowned and supported on firm subgrade.

Critical Length of Grade. Freedom and safety of movement on two-lane highways are adversely affected by heavily loaded vehicles operating on upgrades of sufficient lengths to result in speeds that could impede following vehicles. The term critical length of grade is defined as the length of a particular upgrade which reduces the operating speed of a truck with a weight-to-horsepower ratio of 200 lb/hp (0.122 kg/W) to 10 mi/h (1.6 km/h) below the operating speed of the remaining traffic. Figure 2.11 provides the amount of speed reduction for these trucks given a range of percent upgrades and length of grades. The entering speed is assumed to be 70 mi/h (113 km/h). The curve representing a 10-mi/h (1.6-km/h) reduction is the design guideline to be used in determining the critical length of grade.

Design speed, mi/h

TABLE 2.16 Maximum Grades as Determined by Function, Terrain, and Speed, %

Functional

classification	Terrain	25	30	35	40	45	50	55	60	65	70	75—
Urban:
Interstate,* other
freeways, and
expressways	Level						4	4	3	3	3	3
	Rolling						5	5	4	4	4	4
	Hilly						6	6	6	5	5
Arterial street^	Level		8	7	7	6	6	5	5
	Rolling		9	8	8	7	7	6	6
	Hilly		11	10	10	9	9	8	8
Collector streets^	Level	9	9	9	9	8	7	7	6
	Rolling	12	11	10	10	9	8	8	7
	Hilly	13	12	12	12	11	10	10	9
Local streets^	Level	7	7	7	7	7	6	6	5
	Rolling	11	10	10	10	9	8	7	6
	Hilly	15	14	13	13	12	10	10
Rural:
Interstate,* other
freeways, and
expressways	Level						4	4	3	3	3	3
	Rolling						5	5	4	4	4	4
	Hilly						6	6	6	5	5
Arterials^	Level				5	5	4	4	3	3	3	3
	Rolling				6	6	5	5	4	4	4	4
	Hilly				8	7	7	6	6	5	5	5
Collectors^	Level	7	7	7	7	7	6	6	5
	Rolling	10	9	9	8	8	7	7	6
	Hilly	11	10	10	10	10	9	9	8
Local roads’^	Level	7	7	7	7	7	6	6	5
	Rolling	11	10	10	10	9	8	7	6
	Hilly	15	14	13	13	12	10	10

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

*Grades 1% steeper may be used for extreme cases where development in urban areas precludes the use of flatter grades. Grades 1% steeper may also be used for one-way down-grades except in hilly terrain.

^Grades 1% steeper may be used for short lengths (less than 500 ft) and on one-way down-grades. For rural highways with current ADT less than 400, grades may be 2% steeper.

Source: Location and Design Manual, Vol. 1, Roadway Design. Ohio Department of Transportation, with

permission.

If after an investigation of the project grade line, it is found that the critical length of grade must be exceeded, an analysis of the effect of the long grades on the level of service of the roadway should be made. Where speeds resulting from trucks climbing up long grades are calculated to fall within the range of service level D or lower, consideration should be given to constructing added uphill lanes on critical lengths of grade. Refer to the “Highway Capacity Manual” (Ref. 10) for methodology in determining level of service. Where the length of added lanes needed to preserve the recommended level of service on sections with long grades exceeds 10 percent of the total distance between major termini, consideration should be given to the ultimate construction of a divided multilane facility.

Vertical Alignment

FIGURE 2.11 Critical lengths of grade based on typical heavy truck of 200 lb/hp (0.122 kg/W) at entering speed of 70 mi/h (113 km/h). Notes: (1) This graph can also be used to compute the critical length of grade for grade combinations. For example, find the critical length of grade for a 4 percent upgrade preceded by 2000 ft (610 m) of 2 percent upgrade and a tolerable speed reduction of 15 mi/h (24 km/h). From the graph, 2000 ft (610 m) of 2 percent upgrade results in a speed reduction of 7 mi/h (11 km/h). Subtracting 7 mi/h (11.2 km/h) from the tolerable speed reduction of 15 mi/h (24 km/h) gives the remaining tolerable speed reduction of 8 mi/h (12.8 km/h). The graph shows that the remaining tolerable speed reduction would occur on 1000 ft (305 m) of the 4 percent upgrade. (2) The critical length of grade is the length of tangent grade. When a vertical curve is part of the critical length of grade, an approximate equivalent tangent grade should be used. Where A < 3 percent, the vertical tangent lengths can be used (VPI to VPI). Where A > 3 percent, about V4 of the vertical curve length should be used as part of the tangent grade. Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m. (From Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission)

Vertical Curves. A vertical curve is used to provide a smooth transition between vertical tangents of different grades. It is a parabolic curve and is usually centered on the intersection point of the vertical tangents. One of the principles of parabolic curves is that the rate of change of slope is a constant throughout the curve. For a vertical curve, this rate is equal to the length of the curve divided by the algebraic difference of the grades. This value is called the K value and represents the distance required for the vertical tangent to change by 1 percent. The K value is useful in design to determine the minimum length of vertical curve necessary to provide minimum stopping sight distance given two vertical grades.

Allowable Grade Breaks. There are situations where it is not necessary to provide a vertical curve at the intersection of two vertical grades because the difference in grades is not large enough to provide any discomfort to the driver. The difference

TABLE 2.17 Maximum Change in Vertical Alignment Not Requiring a Vertical Curve

Design speed, mi/h	Design speed, km/h	Maximum grade change, %*
25	40	1.85
30	48	1.30
35	56	0.95
40	64	0.75
45	72	0.55
50	80	0.45
55	88	0.40
60	96	0.30
65	105	0.30
70	113	0.25

Based on the following equation:

. = 46.5L = 1162.5

A = V2 = V2

where A = maximum grade change, %

L = length of vertical curve, ft; assume 25 V = design speed, mi/h

Note: The recommended minimum distance between consecutive deflections is 100 ft (30 m) where design speed > 40 mi/h (64 km/h) and 50 ft (15 m) where design speed < 40 mi/h.

*Rounded to nearest 0.05%.

Source: Location and Design Manual, Vol. 1, Roadway

Design, Ohio Department of Transportation, with permission.

varies with the design speed of the roadway. At 25 mi/h (40 km/h), a grade break of 1.85 percent without a curve may be permitted, while at 55 mi/h (88 km/h) the allowable difference is only 0.40 percent. Table 2.17 lists the maximum grade break permitted without using a vertical curve for various design speeds. The equation used to develop the distances is indicated as well as a recommended minimum distance between consecutive grade breaks. Where consecutive grade breaks occur within 100 ft (30 m) for design speeds over 40 mi/h (64 km/h), or within 50 ft (15 m) for design speeds at 40 mi/h (64 km/h) and under, this indicates that a vertical curve may be a better solution than not providing one.

Crest Vertical Curves. The major design consideration for crest vertical curves is the provision of ample stopping sight distance for the design speed. Calculations of available stopping sight distance are based on the driver’s eye 3.5 ft (1.07 m) above the roadway surface with the ability to see an object 2 ft (0.61 m) high on the roadway ahead over the top of the pavement. Table 2.18 lists the calculated design stopping sight distance values and the corresponding K values for design speeds from 20 to 70 mi/h (32 to 113 km/h). The values shown are based on the assumption that the curve is longer than the sight distance. In those cases where the sight distance exceeds the vertical curve length, a different equation is used to calculate the stopping sight distance provided. The equations are shown in the table.

Another consideration in designing crest vertical curves is passing sight distance, especially when dealing with two-lane roadways. This has already been discussed

Height of eye, 3.50 ft; height of object, 2.00 ft

TABLE 2.18 Stopping Sight Distance (SSD) for Crest Vertical Curves at Design Speeds from 20 to 70 mi/h (32 to 113 km/h)

Design speed, mi/h	Design SSD, ft	Design K, ft/%	Design speed, mi/h	Design SSD, ft	Design K, ft/%
20	115	7	46	375	66
21	120	7	47	385	69
22	130	8	48	400	75
23	140	10	49	415	80
24	145	10	50	425	84
25	155	12	51	440	90
26	165	13	52	455	96
27	170	14	53	465	101
28	180	15	54	480	107
29	190	17	55	495	114
30	200	19	56	510	121
31	210	21	57	525	128
32	220	23	58	540	136
33	230	25	59	555	143
34	240	27	60	570	151
35	250	29	61	585	159
36	260	32	62	600	167
37	270	34	63	615	176
38	280	37	64	630	184
39	290	39	65	645	193
40	305	44	66	665	205
41	315	46	67	680	215
42	325	49	68	695	224
43	340	54	69	715	237
44	350	57	70	730	247
45	360	61

Using S = stopping sight distance, ft

L = length of crest vertical curve, ft A = algebraic difference in grades, %, absolute value K = rate of vertical curvature, ft per % change

• For a given design speed and A value, the calculated length L = KA.

• To determine S with a given L and A, use the following:

For S < L: S = 46.45VK where K = L/A

For S > L: S = 1079/A + L/2

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Note: For design criteria pertaining to collectors and local roads wih ADT less than 400, please refer to the AASHTO publication, Guidelines for Geometric Design of Very Low-Volume Local Roads (ADT < 400).

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio

Department of Transportation, with permission.

under “Passing Sight Distance” earlier in this chapter. Also, in addition to being designed for safe stopping sight distance, crest vertical curves should be designed for comfortable operation and a pleasing appearance whenever possible. To accomplish this, the length of a crest curve in feet should be, as a minimum, 3 times the design speed in miles per hour.

Sag Vertical Curves. The main factor affecting the design of a sag vertical curve is headlight sight distance. When a vehicle traverses an unlighted sag vertical curve at night, the portion of highway lighted ahead is dependent on the position of the headlights and the direction of the light beam. For design purposes, the length of roadway lighted ahead is assumed to be the available stopping sight distance for the curve. In calculating the distances for a given set of grades and a length of curve, the height of the headlight is assumed to be 2 ft (0.61 m) and the upward divergence of the light beam is considered to be 1°. Table 2.19 lists the calculated design stopping sight distance values and the corresponding K values for design speeds from 20 to 70 mi/h (32 to 113 km/h). As was the case with crest curves, the values shown are based on the assumption that the curve is longer than the sight distance. In those cases where the sight distance exceeds the vertical curve length, a different equation is used to calculate the actual stopping sight distance provided as indicated in the table.

Note for sag curves, when the algebraic difference of grades is 1.75 percent or less, stopping sight distance is not restricted by the curve. In these cases, the equations in Table 2.19 will not provide meaningful answers. Minimum lengths of sag vertical curves are necessary to provide a pleasing general appearance of the highway. To accomplish this, the minimum length of a sag curve in feet should be equal to 3 times the design speed in miles per hour.

Vertical Alignment Considerations. The following items should be considered when establishing new vertical alignment:

• The profile should be smooth with gradual changes consistent with the type of facility and the character of the surrounding terrain.

• A “roller-coaster” or “hidden dip” profile should be avoided.

• Undulating grade lines involving substantial lengths of steeper grades should be appraised for their effect on traffic operation, since they may encourage excessive truck speeds.

• Broken-back grade lines (two vertical curves—a pair of either crest curves or sag curves—separated by a short tangent grade) should generally be avoided.

• Special attention should be given to drainage on curbed roadways where vertical curves have a K value of 167 or greater, since these areas are very flat.

• It is preferable to avoid long, sustained grades by breaking them into shorter intervals with steeper grades at the bottom.

Horizontal Alignment and Superelevation

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The horizontal alignment of a roadway should be designed to provide motorists with a facility for driving in a safe and comfortable manner. Adequate stopping sight distance should be furnished. Also, changes in direction should be accompanied by the use of curves and superelevation when appropriate in accordance with established guidelines. Some changes in alignment are slight and may not require curvature. Table 2.5 lists the maximum deflection angle which may be permitted without the use of a horizontal curve for each design speed shown. It is assumed that a motorist can easily negotiate the change in direction and maintain control over the vehicle without leaving the lane.

TABLE 2.4 Decision Sight Distance (DSD) for Design Speeds from 30 to 70 mi/h (48 to 113 km/h)

Decision sight distance, ft
Avoidance maneuver

Design speed, mi/h	A	B	C	D	E
30	220	490	450	535	620
35	275	590	525	625	720
40	330	690	600	715	825
45	395	800	675	800	930
50	465	910	750	890	1030
55	535	1030	865	980	1135
60	610	1150	990	1125	1280
65	695	1275	1050	1220	1365
70	780	1410	1105	1275	1445

• The avoidance maneuvers are as follows: A—rural stop; B—urban stop; C—rural speed/path/direction change; D—suburban speed/path/direction change; E—urban speed/path/direction change

• Decision sight distance (DSD) is calculated or measured using the same criteria as stopping sight distance: 3.50 ft (1.07 m) eye height and 2.00 ft (0.61 m) object height.

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of

Transportation, with permission.

TABLE 2.5 Maximum Centerline Deflection Not Requiring a Horizontal Curve

Design speed, mi/h Maximum deflection*

25	5°30′
30	3°45′
35	2°45′
40	2°15′
45	1°45′
50	1°15′
55	1°00′
60	1°00′
65	0°45′
70	0°45′
Based on the following formulas: Design speed 50 mi/h or over: tan A = 1.0/V Design speed under 50 mi/h: tan A = 60/V2

where V = design speed, mi/h A = deflection angle

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Note: The recommended minimum distance between consecutive horizontal deflections is:

200 ft where design speed > 45 mi/h 100 ft where design speed < 45 mi/h *Rounded to nearest 15 min.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission.

When centerline deflections exceed the values in Table 2.5, it is necessary to introduce a horizontal curve to assist the driver. Curves are usually accompanied by superelevation, which is a banking of the roadway to help counteract the effect of centrifugal force on the vehicle as it moves through the curve. In addition to superelevation, centrifugal force is also offset by the side friction developed between the tires of the vehicle and the pavement surface. The relationship of the two factors when considering curvature for a particular design speed is expressed by the following equation:

U. S. units: e + f =

V2 15V

(2.1a)

SI units: e + f =

127R

(2.1b)

where e = superelevation rate, ft per ft (m per m) of pavement width f = side friction factor V = design speed, mi/h (km/h)

R = radius of curve, ft (m)

In developing superelevation guidelines for use in designing roadways, it is necessary to establish practical limits for both superelevation and side friction factors. Several factors affect the selection of a maximum superelevation rate for a given highway. Climate must be considered. Regions subject to snow and ice should not be superelevated too sharply, because the presence of these adverse conditions causes motorists to drive slower, and side friction is greatly reduced. Consequently, vehicles tend to slide to the low side of the roadway. Terrain conditions are another factor. Flat areas tend to have relatively flat grades, and such conditions have little effect on superelevation and side friction factors. However, mountainous regions have steeper grades, which combine with superelevation rates to produce steeper cross slopes on the pavement than may be apparent to the designer. Rural and urban areas require different maximum superelevation rates, because urban areas are more frequently subjected to congestion and slower-moving traffic. Vehicles operating at significantly less than design speeds necessitate a flatter maximum rate. Given the above considerations, a range of maximum values has been adopted for use in design. A maximum rate of 0.12 or 0.10 may be used in flat areas not subject to ice or snow. Rural areas where these conditions exist usually have a maximum rate of 0.08. A maximum rate of 0.06 is recommended for urban high-speed roadways, 50 mi/h (80 km/h) or greater, while 0.04 is used on low-speed urban roadways and temporary roads.

Various factors affect the side friction factors used in design. Among these are pavement texture, weather conditions, and tire condition. The upper limit of the side friction factor is when the tires begin to skid. Highway curves must be designed to avoid skidding conditions with a margin of safety. Side friction factors also vary with design speed. Higher speeds tend to have lower side friction factors. The result of various studies leads to the values listed in Table 2.6, which shows the side friction factors by design speed generally used in developing superelevation tables (Ref. 1).

Taking into account the above limits on superelevation rates and side friction factors, and rewriting Eq. (2.1), it follows that for a given design speed and maximum superelevation rate, there exists a minimum radius of curvature that should be allowed for design purposes:

R. = ————— (2.2)

mn 15(e + f) v ‘

To allow a lesser radius for the design speed would require the superelevation rate or the friction factor to be increased beyond the recommended limit.

Подпись: Design speed, mi/h Side friction factor f 20 0.27 30 0.20 40 0.16 50 0.14 55 0.13 60 0.12 65 0.11 70 0.10 Source: Adapted from Ref. 1.

Highway design using U. S. Customary units defines horizontal curvature in terms of degree of curve as well as radius. Under this definition, the degree of curve is defined as the central angle of a 100-ft (30-m) arc using a fixed radius. This results in the following equation relating R (radius, ft) to D (degree of curve, degrees):

Подпись: (2.3) 5729.6

Substituting in Eq. (2.2) gives the maximum degree of curvature for a given design speed and maximum superelevation rate:

Before presenting the superelevation tables, one final consideration must be addressed. Because for any curve, superelevation and side friction combine to offset the effects of centrifugal force, the question arises how much superelevation should be provided for curves flatter than the “maximum” allowed for a given design speed. The following five methods have been used over the years (Ref. 1):

Method 1. Superelevation and side friction are directly proportional to the degree of curve or the inverse of the radius.

Method 2. Side friction is used to offset centrifugal force in direct proportion to the degree of curve, for curves up to the point where fmax is required. For sharper curves, fmax remains constant and e is increased in direct proportion to the increasing degree of curvature until e is reached.

Method 3. Superelevation is used to offset centrifugal force in direct proportion to the degree of curve for curves up to the point where emax is required. For sharper curves, emax remains constant and f is increased in direct proportion to the increasing degree of curvature until f is reached.

Method 4. Method 4 is similar to method 3, except that it is based on average running speed instead of design speed.

Method 5. Superelevation and side friction are in a curvilinear relationship with the degree of curve (inverse of radius), with resulting values between those of method 1 and method 3.

Figure 2.8 shows a graphic comparison of the various methods. Method 5 is most commonly used on rural and high-speed [50 mi/h (80 km/h) or higher] urban highways. Method 2 is used on low-speed urban streets and temporary roadways.

Recommended minimum radii for a given range of design speeds and incremental superelevation rates are given in Tables 2.7 through 2.11, where each table represents

Horizontal Alignment and Superelevation

FIGURE 2.8 Methods of distributing superelevation and side friction. (a) Superelevation. (b) Corresponding friction factor at design speed. (c) Corresponding friction factor at running speed. (From A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission)

TABLE 2.7 Minimum Radii for Design Speeds from 15 to 60 mi/ti (24 to 97 km/h) and Superelevation Rates to 4 Percent

e (%)	V, = 15 mi/h a R (ft)	V. = 20 mi/h a R (ft)	V, = 25 mi/h a R (ft)	V, = 30 mi/h a R (ft)	V. = 35 mi/h a R (ft)	V. = 40 mi/h a Я (ft)	V. = 45 mi/h a R (ft)	V, = 50 mi/h a R (ft)	V. = 55 mi/h a R (ft)	V. = 60 mi/h a R (ft)
1.5	796	1410	2050	2830	3730	4770	5930	7220	8650	10300
2.0	506	902	1340	1880	2490	3220	4040	4940	5950	7080
2.2	399	723	1110	1580	2120	2760	3480	4280	5180	6190
2.4	271	513	838	1270	1760	2340	2980	3690	4500	5410
2.6	201	388	650	1000	1420	1930	2490	3130	3870	4700
2.8	157	308	524	817	1170	1620	2100	2660	3310	4060
3.0	127	251	433	681	982	1370	1800	2290	2860	3530
3.2	105	209	363	576	835	1180	1550	1980	2490	3090
3.4	88	175	307	490	714	1010	1340	1720	2170	2700
3.6	73	147	259	416	610	865	1150	1480	1880	2350
3.8	61	122	215	348	512	730	970	1260	1600	2010
4.0	42	86	154	250	371	533	711	926	1190	1500

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

R = radius of curve Vd = design speed e = rate of superelevation

Note: Use of emax = 4 percent should be limited to urban conditions.

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

TABLE 2.8 Minimum Radii for Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 6 Percent

e (%)	Vd = 15 mi/h R (ft)	Vd = 20 mi/h R (ft)	Vd = 25 mi/h R (ft)	Vd = 30 mi/h R (ft)	Vd = 35 mi/h R (ft)	Vd = 40 mi/h R (ft)	Vd = 45 mi/h R (ft)
1.5	868	1580	2290	3130	4100	5230	6480
2.0	614	1120	1630	2240	2950	3770	4680
2.2	543	991	1450	2000	2630	3370	4190
2.4	482	884	1300	1790	2360	3030	3770
2.6	430	791	1170	1610	2130	2740	3420
2.8	384	709	1050	1460	1930	2490	3110
3.0	341	635	944	1320	1760	2270	2840
3.2	300	566	850	1200	1600	2080	2600
3.4	256	498	761	1080	1460	1900	2390
3.6	209	422	673	972	1320	1740	2190
3.8	176	358	583	864	1190	1590	2010
4.0	151	309	511	766	1070	1440	1840
4.2	131	270	452	684	960	1310	1680
4.4	116	238	402	615	868	1190	1540
4.6	102	212	360	555	788	1090	1410
4.8	91	189	324	502	718	995	1300
5.0	82	169	292	456	654	911	1190
5.2	73	152	264	413	595	833	1090
5.4	65	136	237	373	540	759	995
5.6	58	121	212	335	487	687	903
5.8	51	106	186	296	431	611	806
6.0	39	81	144	231	340	485	643

(Continued)

e	Vd = 50 mi/h	Vd = 55 mi/h	II o о 3 &	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	7870	9410	11100	12600	14100	15700	17400
2.0	5700	6820	8060	9130	10300	11500	12900
2.2	5100	6110	7230	8200	9240	10400	11600
2.4	4600	5520	6540	7430	8380	9420	10600
2.6	4170	5020	5950	6770	7660	8620	9670
2.8	3800	4580	5440	6200	7030	7930	8910
3.0	3480	4200	4990	5710	6490	7330	8260
3.2	3200	3860	4600	5280	6010	6810	7680
3.4	2940	3560	4250	4890	5580	6340	7180
3.6	2710	3290	3940	4540	5210	5930	6720
3.8	2490	3040	3650	4230	4860	5560	6320
4.0	2300	2810	3390	3950	4550	5220	5950
4.2	2110	2590	3140	3680	4270	4910	5620
4.4	1940	2400	2920	3440	4010	4630	5320
4.6	1780	2210	2710	3220	3770	4380	5040
4.8	1640	2050	2510	3000	3550	4140	4790
5.0	1510	1890	2330	2800	3330	3910	4550
5.2	1390	1750	2160	2610	3120	3690	4320
5.4	1280	1610	1990	2420	2910	3460	4090
5.6	1160	1470	1830	2230	2700	3230	3840
5.8	1040	1320	1650	2020	2460	2970	3560
6.0	833	1060	1330	1660	2040	2500	3050

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

R = radius of curve Vd = design speed e = rate of superelevation

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

TABLE 2.9 Minimum Radii for Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 8 Percent

e	Vd = 15 mi/h Vd	= 20 mi/h	Vd = 25 mi/h Vd	= 30 mi/h	Vd = 35 mi/h	II 4^ О 3 &	Vd = 45 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	932	1640	2370	3240	4260	5410	6710
2.0	676	1190	1720	2370	3120	3970	4930
2.2	605	1070	1550	2130	2800	3570	4440
2.4	546	959	1400	1930	2540	3240	4030
2.6	496	872	1280	1760	2320	2960	3690
2.8	453	796	1170	1610	2130	2720	3390
3.0	415	730	1070	1480	1960	2510	3130
3.2	382	672	985	1370	1820	2330	2900
3.4	352	620	911	1270	1690	2170	2700
3.6	324	572	845	1180	1570	2020	2520
3.8	300	530	784	1100	1470	1890	2360
4.0	277	490	729	1030	1370	1770	2220
4.2	255	453	678	955	1280	1660	2080
4.4	235	418	630	893	1200	1560	1960
4.6	215	384	585	834	1130	1470	1850
4.8	193	349	542	779	1060	1390	1750
5.0	172	314	499	727	991	1310	1650
5.2	154	284	457	676	929	1230	1560
5.4	139	258	420	627	870	1160	1480
5.6	126	236	387	582	813	1090	1390
5.8	115	216	358	542	761	1030	1320
6.0	105	199	332	506	713	965	1250
6.2	97	184	308	472	669	909	1180
6.4	89	170	287	442	628	857	1110
6.6	82	157	267	413	590	808	1050
6.8	76	146	248	386	553	761	990
7.0	70	135	231	360	518	716	933
7.2	64	125	214	336	485	672	878
7.4	59	115	198	312	451	628	822
7.6	54	105	182	287	417	583	765
7.8	48	94	164	261	380	533	701
8.0	38	76	134	214	314	444	587

e	Vd = 50 mi/h	Vd = 55 mi/h	II C О 3 &	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	8150	9720	11500	12900	14500	16100	17800
2.0	5990	7150	8440	9510	10700	12000	13300
2.2	5400	6450	7620	8600	9660	10800	12000
2.4	4910	5870	6930	7830	8810	9850	11000
2.6	4490	5370	6350	7180	8090	9050	10100
2.8	4130	4950	5850	6630	7470	8370	9340
3.0	3820	4580	5420	6140	6930	7780	8700
3.2	3550	4250	5040	5720	6460	7260	8130
3.4	3300	3970	4700	5350	6050	6800	7620
3.6	3090	3710	4400	5010	5680	6400	7180
3.8	2890	3480	4140	4710	5350	6030	6780
4.0	2720	3270	3890	4450	5050	5710	6420
4.2	2560	3080	3670	4200	4780	5410	6090
4.4	2410	2910	3470	3980	4540	5140	5800
4.6	2280	2750	3290	3770	4310	4890	5530
4.8	2160	2610	3120	3590	4100	4670	5280
5.0	2040	2470	2960	3410	3910	4460	5050
5.2	1930	2350	2820	3250	3740	4260	4840
5.4	1830	2230	2680	3110	3570	4090	4640
5.6	1740	2120	2550	2970	3420	3920	4460
5.8	1650	2010	2430	2840	3280	3760	4290
6.0	1560	1920	2320	2710	3150	3620	4140
6.2	1480	1820	2210	2600	3020	3480	3990
6.4	1400	1730	2110	2490	2910	3360	3850
6.6	1330	1650	2010	2380	2790	3240	3720
6.8	1260	1560	1910	2280	2690	3120	3600
7.0	1190	1480	1820	2180	2580	3010	3480
7.2	1120	1400	1720	2070	2470	2900	3370
7.4	1060	1320	1630	1970	2350	2780	3250
7.6	980	1230	1530	1850	2230	2650	3120
7.8	901	1140	1410	1720	2090	2500	2970
8.0	758	960	1200	1480	1810	2210	2670

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

R = radius of curve Vd = design speed e = rate of superelevation

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

TABLE 2.10 Minimum Radii for Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 10 Percent

e Vd	= 15 mi/h Vd	= 20 mi/h	Vd = 25 mi/h Vd	= 30 mi/h	Vd = 35 mi/h	II 4^ О 3 &	Vd = 45 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	947	1680	2420	3320	4350	5520	6830
2.0	694	1230	1780	2440	3210	4080	5050
2.2	625	1110	1600	2200	2900	3680	4570
2.4	567	1010	1460	2000	2640	3350	4160
2.6	517	916	1330	1840	2420	3080	3820
2.8	475	841	1230	1690	2230	2840	3520
3.0	438	777	1140	1570	2060	2630	3270
3.2	406	720	1050	1450	1920	2450	3040
3.4	377	670	978	1360	1790	2290	2850
3.6	352	625	913	1270	1680	2150	2670
3.8	329	584	856	1190	1580	2020	2510
4.0	308	547	804	1120	1490	1900	2370
4.2	289	514	756	1060	1400	1800	2240
4.4	271	483	713	994	1330	1700	2120
4.6	255	455	673	940	1260	1610	2020
4.8	240	429	636	890	1190	1530	1920
5.0	226	404	601	844	1130	1460	1830
5.2	213	381	569	802	1080	1390	1740
5.4	200	359	539	762	1030	1330	1660
5.6	188	339	511	724	974	1270	1590
5.8	176	319	484	689	929	1210	1520
6.0	164	299	458	656	886	1160	1460
6.2	152	280	433	624	846	1110	1400
6.4	140	260	409	594	808	1060	1340
6.6	130	242	386	564	772	1020	1290
6.8	120	226	363	536	737	971	1230
7.0	112	212	343	509	704	931	1190
7.2	105	199	324	483	671	892	1140
7.4	98	187	306	460	641	855	1100
7.6	92	176	290	437	612	820	1050
7.8	86	165	274	416	585	786	1010
8.0	81	156	260	396	558	754	968
8.2	76	147	246	377	533	722	930
8.4	72	139	234	359	509	692	893
8.6	68	131	221	341	486	662	856
8.8	64	124	209	324	463	633	820
9.0	60	116	198	307	440	604	784
9.2	56	109	186	291	418	574	748
9.4	52	102	175	274	395	545	710
9.6	48	95	163	256	370	513	671
9.8	44	87	150	236	343	477	625
10.0	36	72	126	200	292	410	540

(Continued)

e Vd	= 50 mi/h	Vd = 55 mi/h	II o О 3 &	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	8280	9890	11700	13100	14700	16300	18000
2.0	6130	7330	8630	9720	10900	12200	13500
2.2	5540	6630	7810	8800	9860	11000	12200
2.4	5050	6050	7130	8040	9010	10100	11200
2.6	4640	5550	6550	7390	8290	9260	10300
2.8	4280	5130	6050	6840	7680	8580	9550
3.0	3970	4760	5620	6360	7140	7990	8900
3.2	3700	4440	5250	5930	6680	7480	8330
3.4	3470	4160	4910	5560	6260	7020	7830
3.6	3250	3900	4620	5230	5900	6620	7390
3.8	3060	3680	4350	4940	5570	6260	6990
4.0	2890	3470	4110	4670	5270	5930	6630
4.2	2740	3290	3900	4430	5010	5630	6300
4.4	2590	3120	3700	4210	4760	5370	6010
4.6	2460	2970	3520	4010	4540	5120	5740
4.8	2340	2830	3360	3830	4340	4900	5490
5.0	2240	2700	3200	3660	4150	4690	5270
5.2	2130	2580	3060	3500	3980	4500	5060
5.4	2040	2460	2930	3360	3820	4320	4860
5.6	1950	2360	2810	3220	3670	4160	4680
5.8	1870	2260	2700	3090	3530	4000	4510
6.0	1790	2170	2590	2980	3400	3860	4360
6.2	1720	2090	2490	2870	3280	3730	4210
6.4	1650	2010	2400	2760	3160	3600	4070
6.6	1590	1930	2310	2670	3060	3480	3940
6.8	1530	1860	2230	2570	2960	3370	3820
7.0	1470	1790	2150	2490	2860	3270	3710
7.2	1410	1730	2070	2410	2770	3170	3600
7.4	1360	1670	2000	2330	2680	3070	3500
7.6	1310	1610	1940	2250	2600	2990	3400
7.8	1260	1550	1870	2180	2530	2900	3310
8.0	1220	1500	1810	2120	2450	2820	3220
8.2	1170	1440	1750	2050	2380	2750	3140
8.4	1130	1390	1690	1990	2320	2670	3060
8.6	1080	1340	1630	1930	2250	2600	2980
8.8	1040	1290	1570	1870	2190	2540	2910
9.0	992	1240	1520	1810	2130	2470	2840
9.2	948	1190	1460	1740	2060	2410	2770
9.4	903	1130	1390	1670	1990	2340	2710
9.6	854	1080	1320	1600	1910	2260	2640
9.8	798	1010	1250	1510	1820	2160	2550
10.0	694	877	1090	1340	1630	1970	2370

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

R = radius of curve Vd = design speed e = rate of Superelevation

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

TABLE 2.11 Minimum Radii for Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 12 Percent

e	Vd = 15 mi/h Vd	= 20 mi/h	Vd = 25 mi/h Vd	= 30 mi/h	Vd = 35 mi/h	II О 3	Vd = 45 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	950	1690	2460	3370	4390	5580	6910
2.0	700	1250	1820	2490	3260	4140	5130
2.2	631	1130	1640	2250	2950	3750	4640
2.4	574	1030	1500	2060	2690	3420	4240
2.6	526	936	1370	1890	2470	3140	3900
2.8	484	863	1270	1740	2280	2910	3600
3.0	448	799	1170	1620	2120	2700	3350
3.2	417	743	1090	1510	1970	2520	3130
3.4	389	693	1020	1410	1850	2360	2930
3.6	364	649	953	1320	1730	2220	2750
3.8	341	610	896	1250	1630	2090	2600
4.0	321	574	845	1180	1540	1980	2460
4.2	303	542	798	1110	1460	1870	2330
4.4	286	512	756	1050	1390	1780	2210
4.6	271	485	717	997	1320	1690	2110
4.8	257	460	681	948	1260	1610	2010
5.0	243	437	648	904	1200	1540	1920
5.2	231	415	618	862	1140	1470	1840
5.4	220	395	589	824	1090	1410	1760
5.6	209	377	563	788	1050	1350	1690
5.8	199	359	538	754	1000	1300	1620
6.0	190	343	514	723	960	1250	1560
6.2	181	327	492	694	922	1200	1500
6.4	172	312	471	666	886	1150	1440
6.6	164	298	452	639	852	1110	1390
6.8	156	284	433	615	820	1070	1340
7.0	148	271	415	591	790	1030	1300
7.2	140	258	398	568	762	994	1250
7.4	133	246	382	547	734	960	1210
7.6	125	234	366	527	708	928	1170
7.8	118	222	351	507	684	897	1130
8.0	111	210	336	488	660	868	1100

e Vd	= 15 mi/h	Vd = 20 mi/h	Vd = 25 mi/h	Vd = 30 mi/h	Vd = 35 mi/h	II 4^ О 3 &	Vd = 45 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
8.2	105	199	321	470	637	840	1070
8.4	100	190	307	452	615	813	1030
8.6	95	180	294	435	594	787	997
8.8	90	172	281	418	574	762	967
9.0	85	164	270	403	554	738	938
9.2	81	156	259	388	535	715	910
9.4	77	149	248	373	516	693	883
9.6	74	142	238	359	499	671	857
9.8	70	136	228	346	481	650	832
10.0	67	130	219	333	465	629	806
10.2	64	124	210	320	448	608	781
10.4	61	118	201	308	432	588	757
10.6	58	113	192	296	416	568	732
10.8	55	108	184	284	400	548	707
11.0	52	102	175	272	384	527	682
11.2	49	97	167	259	368	506	656
11.4	47	92	158	247	351	485	629
11.6	44	86	149	233	333	461	600
11.8	40	80	139	218	312	434	566
12.0	34	68	119	188	272	381	500

(Continued)

TABLE 2.11 Minimum Radii for Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 12 Percent (Continued)

e	Vd = 50 mi/h Vd	= 55 mi/h	Vd = 60 mi/h	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
1.5	8370	9990	11800	13200	14800	16400	18100
2.0	6220	7430	8740	9840	11000	12300	13600
2.2	5640	6730	7930	8920	9980	11200	12400
2.4	5150	6150	7240	8160	9130	10200	11300
2.6	4730	5660	6670	7510	8420	9380	10500
2.8	4380	5240	6170	6960	7800	8700	9660
3.0	4070	4870	5740	6480	7270	8110	9010
3.2	3800	4550	5370	6060	6800	7600	8440
3.4	3560	4270	5030	5690	6390	7140	7940
3.6	3350	4020	4740	5360	6020	6740	7500
3.8	3160	3790	4470	5060	5700	6380	7100
4.0	2990	3590	4240	4800	5400	6050	6740
4.2	2840	3400	4020	4560	5130	5750	6420
4.4	2700	3240	3830	4340	4890	5490	6120
4.6	2570	3080	3650	4140	4670	5240	5850
4.8	2450	2940	3480	3960	4470	5020	5610
5.0	2340	2810	3330	3790	4280	4810	5380
5.2	2240	2700	3190	3630	4110	4620	5170
5.4	2150	2590	3060	3490	3950	4440	4980
5.6	2060	2480	2940	3360	3800	4280	4800
5.8	1980	2390	2830	3230	3660	4130	4630
6.0	1910	2300	2730	3110	3530	3990	4470
6.2	1840	2210	2630	3010	3410	3850	4330
6.4	1770	2140	2540	2900	3300	3730	4190
6.6	1710	2060	2450	2810	3190	3610	4060
6.8	1650	1990	2370	2720	3090	3500	3940
7.0	1590	1930	2290	2630	3000	3400	3820
7.2	1540	1860	2220	2550	2910	3300	3720
7.4	1490	1810	2150	2470	2820	3200	3610
7.6	1440	1750	2090	2400	2740	3120	3520
7.8	1400	1700	2020	2330	2670	3030	3430
8.0	1360	1650	1970	2270	2600	2950	3340

e Vd	= 50 mi/h	Vd = 55 mi/h	Vd = 60 mi/h	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)	R (ft)
8.2	1320	1600	1910	2210	2530	2880	3260
8.4	1280	1550	1860	2150	2460	2800	3180
8.6	1240	1510	1810	2090	2400	2740	3100
8.8	1200	1470	1760	2040	2340	2670	3030
9.0	1170	1430	1710	1980	2280	2610	2960
9.2	1140	1390	1660	1940	2230	2550	2890
9.4	1100	1350	1620	1890	2180	2490	2830
9.6	1070	1310	1580	1840	2130	2440	2770
9.8	1040	1280	1540	1800	2080	2380	2710
10.0	1010	1250	1500	1760	2030	2330	2660
10.2	980	1210	1460	1720	1990	2280	2600
10.4	951	1180	1430	1680	1940	2240	2550
10.6	922	1140	1390	1640	1900	2190	2500
10.8	892	1110	1350	1600	1860	2150	2460
11.0	862	1070	1310	1560	1820	2110	2410
11.2	831	1040	1270	1510	1780	2070	2370
11.4	799	995	1220	1470	1730	2020	2320
11.6	763	953	1170	1410	1680	1970	2280
11.8	722	904	1120	1350	1620	1910	2230
12.0	641	807	1000	1220	1480	1790	2130

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

R = radius of curve Vd = design speed e = rate of superelevation

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

a different maximum superelevation rate. Table 2.7 shows values for a maximum rate of 0.04; Table 2.8, for 0.06; Table 2.9, for 0.08; Table 2.10, for 0.10; and Table 2.11, for 0.12. Method 5 was used to calculate the minimum radius for each superelevation rate less than the maximum rate in each design speed column in the tables.

The superelevation rates on low-speed urban streets are set using method 2 described above, in which side friction is used to offset the effect of centrifugal force up to the maximum friction value allowed for the design speed. Superelevation is then introduced for sharper curves. The design data in Table 2.12, based on method 2 and a maximum superelevation rate of 0.04, can be used for low-speed urban streets and temporary roads. The design data in Table 2.13 can be used for a wider range of design speeds and superelevation rates.

In attempting to apply the recommended superelevation rates for low-speed urban roadways, various factors may combine to make these rates impractical to obtain. These factors include wide pavements, adjacent development, drainage conditions, and frequent access points. In such cases, curves may be designed with reduced or no superelevation, although crown removal is the recommended minimum.

Effect of Grades on Superelevation. On long and fairly steep grades, drivers tend to travel somewhat slower in the upgrade direction and somewhat faster in the downgrade direction than on level roadways. In the case of divided highways, where each pavement can be superelevated independently, or on one-way roadways such as ramps, this tendency should be recognized to see whether some adjustment in the superelevation rate would be desirable and/or feasible. On grades of 4 percent or greater with a length of 1000 ft (305 m) or more and a superelevation rate of 0.06 or more, the designer may adjust the superelevation rate by assuming a design speed 5 mi/h (8 km/h) less in the upgrade direction and 5 mi/h (8 km/h) greater in the downgrade direction, provided that the assumed design speed is not less than the legal speed. On two-lane, two-way roadways and on other multilane undivided highways, such adjustments are less feasible, and should be disregarded.

Superelevation Methods. There are three basic methods for developing superelevation on a crowned pavement leading into and coming out of a horizontal curve. Figure 2.9 shows each method. In the most commonly used method, case I, the pavement edges are revolved about the centerline. Thus, the inner edge of the pavement is depressed by half of the superelevation and the outer edge raised by the same amount. Case II shows the pavement revolved about the inner or lower edge of pavement, and case III shows the pavement revolved about the outer or higher edge of pavement. Case II can be used where off-road drainage is a problem and lowering the inner pavement edge cannot be accommodated. The superelevation on divided roadways is achieved by revolving the pavements about the median pavement edge. In this way, the outside (high side) roadway uses case II, while the inside (low side) roadway uses case III. This helps control the amount of “distortion” in grading the median area.

Superelevation Transition. The length of highway needed to change from a normal crowned section to a fully superelevated section is referred to as the superelevation transition. This length is shown as X in Fig. 2.9, which also shows the various other elements described below. The superelevation transition is divided into two parts: the tangent runout, and the superelevation runoff.

The tangent runout (T in Fig. 2.9) is the length required to remove the adverse pavement cross slope. As is shown for case I of Fig. 2.9, this is the length required to raise the outside edge of pavement from a normal cross slope to a half-flat section. The superelevation runoff (L in Fig. 2.9) is the length required to raise the outside

TABLE 2.12 Superelevation Rates and Runoff Lengths (ft) for Horizontal Curves on Low-Speed Urban Streets Based on a Maximum Superelevation Rate of 4 Percent

Design speed, mi/h

20 25 30 35 40 45

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with

permission.

TABLE 2.13 Runoff Lengths (ft) for Horizontal Curves with Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 12 Percent Based on One Lane Rotated about the Centerline

e Vd	= 15 mi/h	Vd = 20 mi/h	Vd = 25 mi/h	Vd = 30 mi/h	Vd = 35 mi/h	II О 3	Vd = 45 mi/h
(%)	L (ft)	L (ft)	L (ft)	Lr (ft)	L (ft)	Lr (ft)	L (ft)
1.5	0	0	0	0	0	0	0
2.0	31	32	34	36	39	41	44
2.2	34	36	38	40	43	46	49
2.4	37	39	41	44	46	50	53
2.6	40	42	45	47	50	54	58
2.8	43	45	48	51	54	58	62
3.0	46	49	51	55	58	62	67
3.2	49	52	55	58	62	66	71
3.4	52	55	58	62	66	70	76
3.6	55	58	62	65	70	74	80
3.8	58	62	65	69	74	79	84
4.0	62	65	69	73	77	83	89
4.2	65	68	72	76	81	87	93
4.4	68	71	75	80	85	91	98
4.6	71	75	79	84	89	95	102
4.8	74	78	82	87	93	99	107
5.0	77	81	86	91	97	103	111
5.2	80	84	89	95	101	108	116
5.4	83	88	93	98	105	112	120
5.6	86	91	96	102	108	116	124
5.8	89	94	99	105	112	120	129
6.0	92	97	103	109	116	124	133
6.2	95	101	106	113	120	128	138
6.4	98	104	110	116	124	132	142
6.6	102	107	113	120	128	137	147
6.8	105	110	117	124	132	141	151
7.0	108	114	120	127	135	145	156
7.2	111	117	123	131	139	149	160
7.4	114	120	127	135	143	153	164
7.6	117	123	130	138	147	157	169
7.8	120	126	134	142	151	161	173
8.0	123	130	137	145	155	166	178
8.2	126	133	141	149	159	170	182
8.4	129	136	144	153	163	174	187
8.6	132	139	147	156	166	178	191
8.8	135	143	151	160	170	182	196
9.0	138	146	154	164	174	186	200
9.2	142	149	158	167	178	190	204
9.4	145	152	161	171	182	194	209
9.6	148	156	165	175	186	199	213
9.8	151	159	168	178	190	203	218
10.0	154	162	171	182	194	207	222
10.2	157	165	175	185	197	211	227
10.4	160	169	178	189	201	215	231
10.6	163	172	182	193	205	219	236
10.8	166	175	185	196	209	223	240
11.0	169	178	189	200	213	228	244
11.2	172	182	192	204	217	232	249
11.4	175	185	195	207	221	236	253
11.6	178	188	199	211	225	240	258
11.8	182	191	202	215	228	244	262
12.0	185	195	206	218	232	248	267

(Continued)

TABLE 2.13 Runoff Lengths (ft) for Horizontal Curves with Design Speeds from 15 to 80 mi/h (24 to 129 km/h) and Superelevation Rates to 12 Percent Based on One Lane Rotated about the Centerline (Continued)

e V	= 50 mi/h	Vd = 55 mi/h	Vd = 60 mi/h	Vd = 65 mi/h	Vd = 70 mi/h	Vd = 75 mi/h	Vd = 80 mi/h
(%)	Lr (ft)	Lr (ft)	Lr (ft)	Lr (ft)	Lr (ft)	Lr (ft)	Lr (ft)
1.5	0	0	0	0	0	0	0
2.0	48	51	53	56	60	63	69
2.2	53	56	59	61	66	69	75
2.4	58	61	64	67	72	76	82
2.6	62	66	69	73	78	82	89
2.8	67	71	75	78	84	88	96
3.0	72	77	80	84	90	95	103
3.2	77	82	85	89	96	101	110
3.4	82	87	91	95	102	107	117
3.6	86	92	96	100	108	114	123
3.8	91	97	101	106	114	120	130
4.0	96	102	107	112	120	126	137
4.2	101	107	112	117	126	133	144
4.4	106	112	117	123	132	139	151
4.6	110	117	123	128	138	145	158
4.8	115	123	128	134	144	152	165
5.0	120	128	133	140	150	158	171
5.2	125	133	139	145	156	164	178
5.4	130	138	144	151	162	171	185
5.6	134	143	149	156	168	177	192
5.8	139	148	155	162	174	183	199
6.0	144	153	160	167	180	189	206
6.2	149	158	165	173	186	196	213
6.4	154	163	171	179	192	202	219
6.6	158	169	176	184	198	208	226
6.8	163	174	181	190	204	215	233
7.0	168	179	187	195	210	221	240
7.2	173	184	192	201	216	227	247
7.4	178	189	197	207	222	234	254
7.6	182	194	203	212	228	240	261
7.8	187	199	208	218	234	246	267
8.0	192	204	213	223	240	253	274
8.2	197	209	219	229	246	259	281
8.4	202	214	224	234	252	265	288
8.6	206	220	229	240	258	272	295
8.8	211	225	235	246	264	278	302
9.0	216	230	240	251	270	284	309
9.2	221	235	245	257	276	291	315
9.4	226	240	251	262	282	297	322
9.6	230	245	256	268	288	303	329
9.8	235	250	261	273	294	309	336
10.0	240	255	267	279	300	316	343
10.2	245	260	272	285	306	322	350
10.4	250	266	277	290	312	328	357
10.6	254	271	283	296	318	335	363
10.8	259	276	288	301	324	341	370
11.0	264	281	293	307	330	347	377
11.2	269	286	299	313	336	354	384
11.4	274	291	304	318	342	360	391
11.6	278	296	309	324	348	366	398
11.8	283	301	315	329	354	373	405
12.07	288	306	320	335	360	379	411

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: A Policy on Geometric Design of Highways and Streets, American Association of State Highway and Transportation Officials, Washington, D. C., 2004, with permission.

NOTE: The diagrams below show positioning of the superelevation transition for both simple curves and spiral curves. Only one of these conditions would exist for a given transition.

LEGEND: X = Length of superelevation transition.

L = Length of superelevation runoff.

T = Tangent runout R = Crown removal

G = Equivalent slope rate of Change of outside pavement edge compared to the control line In each case. (See Table 2.13 for values.)

N = Normal cross slope S = Full superelevation rate

FIGURE 2.9 Superelevation transition between tangent and simple or spiral curves for three cases. (From Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission) edge of pavement from a half-flat section to a fully superelevated section. The length of transition required to remove the pavement crown (R in Fig. 2.9) is generally equal to twice the T distance.

The minimum superelevation transition length X should be equal in feet to 3 times the design speed in miles per hour. This includes the tangent runout (T) as previously described. The reason to specify this minimum is to avoid the appearance of a “kink” in
the roadway that a shorter transition would provide. The distance is approximately equal to that traveled by a vehicle in 2 s at design speed. This requirement does not apply to low-speed roadways, temporary roads, superelevation transitions near intersections, or transitions between adjacent horizontal curves (reverse or same direction) where normal transitions would overlap each other. In these cases, the minimum transition length is determined by multiplying the edge of pavement correction by the equivalent slope rate (G) shown in Table 2.14. The rate of change of superelevation should be constant throughout the transition X. Some agencies use a flatter rate of transition through the T or R sections than that recommended in Table 2.14, an acceptable but unnecessary practice.

The values given for Lr in Tables 2.12 and 2.13 are based on one 12-ft (3.66-m) lane revolved about the centerline. Table 2.14 shows methods of calculating L when more lanes are revolved about the centerline. In the equations in Table 2.14, L is substituted for Lr. In addition to the terms described in Fig. 2.9, two additional ones are used. W is the width from the point of revolution to the outside edge of pavement. For example, if three 12-ft (3.66-m) lanes are revolved about the lane edge between lanes 2 and 3, then W = 3 X 12 = 36 ft (11 m); the wider section of pavement is used for the width. B is an adjustment factor for multilane pavements to allow for some reduction in the superelevation transition for roads other than interstates, freeways, expressways, and ramps. Section (a) in Table 2.14 lists the equivalent slope rate values G for the various design speeds. Section (b) provides the multilane adjustments factors B for the speeds. Section (c) calculates the value of the overall transition length X based on the values given in (a) and (b) along with a given W and S for each case in Fig. 2.9. Finally, section (d) tests the values calculated to ensure that the minimum transition length discussed in this section is provided. Values for X, L, and T can be lengthened if necessary to achieve a 2-s transition time.

Superelevation Position. Figure 2.9 shows the recommended positioning of the proposed superelevation transition in relationship to the horizontal curve. For those curves with spirals, the transition from adverse crown removal to full superelevation should occur within the limits of the spiral. In other words, the spiral length should equal the L value, usually rounded to the nearest 25 ft (7.6 m).

For simple curves without spirals, the L transition should be placed so that 50 to 70 percent of the maximum superelevation rate is outside the curve limits (point of curvature PC to point of tangency PT). It is recommended that whenever possible, two-thirds of the full superelevation rate be present at the PC and PT. See the case diagrams in Fig. 2.9 for a graphic presentation of the recommended positioning.

Profiles and Elevations. Breakpoints at the beginning and end of the superelevation transition should be rounded to obtain a smooth profile. One suggestion is to use a “vertical curve” on the edge of the pavement profile with a length in feet equal to the design speed in mi/h (i. e., 45 ft for 45 mi/h). The final construction plans should have the superelevation tables or pavement details showing the proposed elevations at the centerline, pavement edges, and, if applicable, lane lines or other breaks in the cross slopes. Pavement or lane widths should be included where these widths are in transition. Pavement edge profiles should be plotted to an exaggerated vertical profile within the limits of the superelevation transitions to check calculations and to determine the location of drainage basins. Adjustments should be made to obtain smooth profiles. Special care should be taken in determining edge elevations in a transition area when the profile grade is on a vertical curve.

Superelevation between Reverse Horizontal Curves. When two horizontal curves are in close proximity to each other, the superelevation transitions calculated independently

TABLE 2.14 Superelevation Notes for Adjusting Runoff Lengths in Tables 2.12 and 2.13

(a) Maximum relative gradients for profiles between the edge of pavement and the centerline or reference line

Design speed, mi/h	Relative gradient	Equivalent slope rate, G
20	0.74	135:1
25	0.70	143:1
30	0.66	152:1
35	0.62	161:1
40	0.58	172:1
45	0.54	185:1
50	0.50	200:1
55	0.47	213:1
60	0.45	222:1
65	0.43	233:1
70	0.40	250:1

(b) Transition length adjustment factors for wide pavements

Number of lanes	B for interstates, freeways,	B for
from point of rotation	expressways, and ramps	other roadways
1.0	1.00	1.00
1.5	1.00	0.83
2.0	1.00	0.75
2.5	1.00	0.70
3.0	1.00	0.67
3.5	1.00	0.64
(c) Calculate X, L, T
Case I	Cases II and III
X = BW(S + N)G	X = BWSG
L = BWSG	L = BW(S – N/2)G
T = BWNG	T = BW(N/2)G

(d) Check for 2-second minimum transition

(Note: D is the linear ft equivalent of the design speed in mi/h. For example, D = 60 ft for 60 mi/h)

If X > 3D, then the values for X, L, and T from section (c) are valid.

If X < 3D, then recalculate X, L, and T as follows:

Case I	Cases II and III
X = 3D	X = 3D
L = 3D[S/(N + S)]	L = 3D[(2S – N)/2S]
T = 3D[N/(N + S)]	T = 3D(N/2S)

Conversion: 1 mi/h = 1.609 km/h.

General notes:

1. The Lr in Tables 2.12 and 2.13 is the same as L in Table 2.14 and is based on a two-lane 24-ft pavement revolved about the centerline.

2. Adjustments to L for varying two-lane pavement widths can be made by direct proportion. For a 20-ft pavement revolved about the centerline, L’ = L(20/24).

3. Determination of X, L, and T when more than one lane is revolved about the centerline (or other reference line, such as a baseline or edge of pavement) is shown in part (c). Values for G and B in the formulas are given in parts (a) and (b), respectively. The value for W is the pavement width from the point of rotation to the farthest edge.

4. The minimum length of superelevation transition (X) as discussed in the text is the distance traveled in 2 s. This number can be rounded off to a figure in feet equal to 3 times the design speed. In part (d) the calculated X value is compared to the value of 3D, where D is the linear feet equivalent of the design speed in miles per hour. If the value of 3D is larger, X is set equal to this value and L and T are adjusted accordingly.

5. The L value is also the recommended spiral length where spirals are used.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with

permission.

may overlap each other. In these cases, the designer should coordinate the transitions to provide a smooth and uniform change from the full superelevation of the first curve to the full superelevation of the second curve. Figure 2.10 shows two diagrams suggesting ways in which this may be accomplished. In both diagrams each curve has its own L value (L1, L2) depending on the degree of curvature, and the superelevation is revolved about the centerline.

PAV	‘EMENT REVOLVED ABC LI	3UT THE CENTERLINE L2	—
JL.	-50Lito. TOLi,	.50L2to. T0L2	—– .—————– —–
	D.

—	PT«		® PCI	—_______ d) 2 ———————-
S.-5 Ut SIMPLE CURVES

PA	VEMENT REVOLVED ABOUT THE CENTERLINE . L3	—
LI		L2
	D


CS«I	ST[1]I		TS	SC*2
SPIRAL CURVES

LEGEND:

(A) – Centerline Pavement (D – Outside EP. Curve I,

Inside EP. Curve 2 E. P.=Edge of Pavement

L3 = Total Superelevation Transition between Spiral Curves

FIGURE 2.10 Superelevation transition between reverse horizontal curves, simple or spiral. (From Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission)

The top diagram involves two simple curves. In the case of new or relocated alignment, the PT of the first curve and the PC of the second curve should be separated by enough distance to allow a smooth, continuous transition between the curves at a rate not exceeding the G value for the design speed (Table 2.14). This requires that the distance be not less than 50 percent nor greater than 70 percent of L1 + L2. Two-thirds is the recommended portion. When adapting this procedure to existing curves where no alignment revision is proposed, the transition should conform as closely as possible to the above criteria. When the available distance between the curves is less than 50 percent of L1 + L2, the transition rate may be increased and/or the superelevation rate at the PT or PC may be set to less than 50 percent of the full superelevation rate.

The lower diagram involves two spiral curves. Where spiral transitions are used, the spiral-to-tangent (ST) point of the first curve and the tangent-to-spiral (TS) transition of the second curve may be at, or nearly at, the same location, without causing superelevation problems. In these cases, the crown should not be reestablished as shown in Fig. 2.9, but instead, both pavement edges should be in continual transition between the curves, as shown in the lower diagram of Fig. 2.10. The total superelevation transition length is the distance between the curve-to-spiral (CS) point of the first curve and spiral-to-curve (SC) point of the second curve.

Spiral Transitions. When a motor vehicle enters or leaves a circular horizontal curve, it follows a transition path during which the driver makes adjustments in steering to account for the gain or loss in centrifugal force. For most curves, the average driver can negotiate this change in steering within the normal width of the travel lane. However, combinations of higher speeds and sharper curvature may cause the driver to move into an adjacent travel lane while accomplishing the change. To prevent this occurrence, the designer should use spirals to smooth out transitions.

There are several advantages to using spiral transitions for horizontal curves:

• They provide an easy-to-follow path for the driver to negotiate.

• They provide a convenient area in which to place the superelevation transition.

• They provide an area where the pavement width can be transitioned when required for curve widening.

• They provide a smoother appearance to the driver.

The Euler spiral is the one most commonly used in highway design. The degree of curve varies gradually from zero at the tangent end to the degree of the circular arc at the curve end. By definition, the degree of curve at any point along the spiral varies directly with the length measured along the spiral. In the case where a spiral transition connects two simple curves, the degree of curve varies directly from that of the first circular arc to that of the second circular arc. As a general guideline, spirals should be used on roadways where the design speed is 50 mi/h (80 km/h) or greater and the degree of curvature exceeds the values given in Table 2.15 for various design speeds listed.

Horizontal Alignment Considerations. The following items should be considered when establishing new horizontal alignment: •

TABLE 2.15 Maximum Curve without a Spiral

Design speed, mi/h	Design speed, km/h	Max. degree of curve	Min. radius, ft	Min. radius, m
50	80	4°30′	1273	388
55	88	3°45′	1528	466
60	96	3°00′	1910	582
65	105	2°30′	2292	699
70	113	2°15′	2546	776

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission.

• Tangents and/or flat curves should be provided on high, long fills.

• Compound curves should be used only with caution.

• Abrupt alignment reversals should be avoided.

• Two curves in the same direction separated by a short tangent (broken-back or flat – back curves) should be avoided.

Statistical Properties of Random Variables

Posted by admin on 12/ 11/ 15

In statistics, the term population is synonymous with the sample space, which describes the complete assemblage of all the values representative of a particular random process. A sample is any subset of the population. Furthermore, parameters in a statistical model are quantities that are descriptive of the population. In this book, Greek letters are used to denote statistical parameters. Sample statistics, or simply statistics, are quantities calculated on the basis of sample observations.

2.1.3 Statistical moments of random variables

In practical statistical applications, descriptors commonly used to show the statistical properties of a random variable are those indicative of (1) the central tendency, (2) the dispersion, and (3) the asymmetry of a distribution. The frequently used descriptors in these three categories are related to the statistical moments of a random variable. Currently, two types of statistical moments are used in hydrosystems engineering applications: product-moments and L-moments. The former is a conventional one with a long history of practice, whereas the latter has been receiving great attention recently from water resources engineers in analyzing hydrologic data (Stedinger et al., 1993; Rao and Hamed 2000). To be consistent with the current general practice and usage, the terms moments and statistical moments in this book refer to the conventional product-moments unless otherwise specified.

Product-moments. The r th-order product-moment of a random variable X about any reference point X = xo is defined, for the continuous case, as

СО /* CO

(x — xo)r fx(x) dx = / (x — xo)r dFx(x) (2.20a)

-O J — O

whereas for the discrete case,

E [(X — xo)r ] = ]T (xk — xo )rpx (xk) (2.20b)

k = і

where E [■] is a statistical expectation operator. In practice, the first three moments (r = 1,2, 3) are used to describe the central tendency, variability, and asymmetry of the distribution of a random variable. Without losing generality, the following discussions consider continuous random variables. For discrete random variables, the integral sign is replaced by the summation sign. Here it is convenient to point out that when the PDF in Eq. (2.20a) is replaced by a conditional PDF, as described in Sec. 2.3, the moments obtained are called the conditional moments.

Since the expectation operator E [ ] is for determining the average value of the random terms in the brackets, the sample estimator for the product-moments for p’r = E(Xr), based on n available data (x1, x2,…, xn), can be written as

p’r = ^2 wi (n) xr (2.21)

i = 1

where wi(n) is a weighting factor for sample observation xi, which depends on sample size n. Most commonly, wi(n) = 1/n, for all i = 1, 2,…, n. The last column of Table 2.1 lists the formulas applied in practice for computing some commonly used statistical moments.

Two types of product-moments are used commonly: moments about the origin, where xo = 0, and central moments, where xo = px, with px = E[X]. The r th-order central moment is denoted as pr = E [(X — px )r ], whereas the r th – order moment about the origin is denoted as p’r = E(Xr). It can be shown easily, through the binomial expansion, that the central moments pr = E [(X — px)r ] can be obtained from the moments about the origin as

pr = ]T( —1)fCr, i p’x ri—i (2.22)

i = 0

where Cr, i = (r) = i!(/^is a binomial coefficient, with! representing factorial, that is, r! = r x (r — 1) x (r — 2) x-x 2 x 1. Conversely, the moments about the origin can be obtained from the central moments in a similar fashion as

jXr = ‘У ^ Cr, i №x №r —i (2.23)

i = 0

Moment	Measure of	Definition	Continuous variable	Discrete variable	Sample estimator
First	Central	Mean, expected value	Vx = f ° xfx(x) dx	Vx = Eallx’s xkP(xk )	x = E xi/n
	location	E( X) = Vx
Second	Dispersion	Variance, Var(X) = V2 = °X	ax = /Too (x – Vx)2 f x(x) dx	°x = Eallx’s(xk – Vx)2 Px(xk )	s2 = n-1 E(xi – x)2
		Standard deviation, ax	ax = у/Var( X)	ax = у/Var( X)	s=J n-л E( xi – x)2
		Coefficient of variation, &x	£2x = ax/vx	£2x = ax/Vx	Cv = s/x
Third	Asymmetry	Skewness	V3 = f-T (x – Vx )3 fx(x) dx	V3 = ^ ^all x’s (xk – Vx ) px(xk )	m3 = (n-1)(n-2) E (x x)
		Skewness coefficient, yx	Yx = V3/a£	Yx = V3 /a£	g = m3/s’3
Fourth	Peakedness	Kurtosis, кх	V4 = 1° (x – Vx )4 fx(x) dx	V4 = allx’s (xk – V%)4px(xk )	m4 = (n-l)(n-2)(n-3) E(x x)
		Excess coefficient, Ex	Kx = V4/a£	Kx = V4/a£	k = m4/s4
			єx — Kx 3	Єx — Kx 3

Equation (2.22) enables one to compute central moments from moments about the origin, whereas Eq. (2.23) does the opposite. Derivations for the expressions of the first four central moments and the moments about the origin are left as exercises (Problems 2.10 and 2.11).

The main disadvantages of the product-moments are (1) that estimation from sample observations is sensitive to the presence of extraordinary values (called outliers) and (2) that the accuracy of sample product-moments deteriorates rapidly with an increase in the order of the moments. An alternative type of moments, called L-moments, can be used to circumvent these disadvantages.

Example 2.8 (after Tung and Yen, 2005) Referring to Example 2.6, determine the first two moments about the origin for the time to failure of the pump. Then calculate the first two central moments.

Solution From Example 2.6, the random variable T is the time to failure having an exponential PDF as

ft(t) = ^ exp(-1/1250) for t > 0, в> 0

in which t is the elapsed time (in hours) before the pump fails, and в = 1250 h/failure.

The moments about the origin, according to Eq. (2.20a), are

r / e-t/e

E (Tr) = t4 = J t4—J dt

Using integration by parts, the results of this integration are for r = 1, p^ = E(T ) = pt = в = 1250 h

for r = 2, p’2 = E(T 2) = 2в2 = 3,125,000 h2

Based on the moments about the origin, the central moments can be determined, according to Eq. (2.22) or Problem (2.10), as

for r = 1, P1 = E (T — pt) = 0

for r = 2, p2 = E [(T — pt )2] = p’2 — P2 = 2в2 — в2 = в2 = 1, 562, 500 h2

L-moments. The r th-order L-moments are defined as (Hosking, 1986, 1990)

Xr = 1 ^ (—1)j— ^ E (— j r) r = 1,2,… (2.24)

in which Xj :n is the j th-order statistic of a random sample of size n from the

distribution Fx(x), namely, X(1) < X(2) < ■ ■ ■ < X(j) < — < X(n). The “L” in L-moments emphasizes that Xr is a linear function of the expected order statistics. Therefore, sample L-moments can be made a linear combination of the ordered data values. The definition of the L-moments given in Eq. (2.24) may appear to be mathematically perplexing; the computations, however, can be simplified greatly through their relations with the probability-weighted moments,
which are defined as (Greenwood et al., 1979)

Подпись: xr [Fx(x)]p [1 – Fx(x)]q dFx(x)

(2.25)

Compared with Eq. (2.20a), one observes that the conventional product – moments are a special case of the probability-weighted moments with p = q = 0, that is, Mr,0,0 = g’r. The probability-weighted moments are particularly attractive when the closed-form expression for the CDF of the random variable is available.

To work with the random variable linearly, M1,p, q can be used. In particular, two types of probability-weighted moments are used commonly in practice, that is,

ar = M1,0,r = E{X[1 – Fx(X)]r} r = 0,1,2,… (2.26a)

вг = M1,r,0 = E{X[Fx(X)]r} r = 0,1,2,… (2.26b)

In terms of ar or er, the r th-order L-moment Xr can be obtained as (Hosking,

1986)

Подпись: (2.27) Xr + 1 = ( -1)rJ2 Pr, j a j =Yj Pr, j ej r = 0,1 …

j = 0 j = 0

in which

Подпись: p;, j = (-1)r - j rfr + i (-1)r j (r + j )!

JA j J (j!)2(r – j)!

For example, the first four L-moments of random variable X are

X1 — e0 — g-1 — gx	(2.28a)
X2 = 2в1 – в0	(2.28b)
X3 = 6в2 – 6в1 + в0	(2.28c)
X4 = 20вэ – 30в2 + 12в1 – в0	(2.28d)

To estimate sample a – and в-moments, random samples are arranged in ascending or descending order. For example, arranging n random observations in ascending order, that is, X(1) < X(2) < ■ ■ ■ < X(j) < ■ ■ ■ < X(n), the rth-order в-moment er can be estimated as

er = -]T X(i) F (X(i))r (2.29)

i = 1

where F(X(i)) is an estimator for F(X(i>) = P(X < X(i>), for which many plotting-position formulas have been used in practice (Stedinger et al., 1993).

The one that is used often is the Weibull plotting-position formula, that is,

F (X (i)) = i / (n + 1).

L-moments possess several advantages over conventional product-moments. Estimators of L-moments are more robust against outliers and are less biased. They approximate asymptotic normal distributions more rapidly and closely. Although they have not been used widely in reliability applications as compared with the conventional product-moments, L-moments could have a great potential to improve reliability estimation. However, before more evidence becomes available, this book will limit its discussions to the uses of conventional product-moments.

Example 2.9 (after Tung and Yen, 2005) Referring to Example 2.8, determine the first two L-moments, that is, Л1 and Л2, of random time to failure T.

Solution To determine Л1 and Л2, one first calculates 0o and 01, according to Eq. (2.26b), as

0 = E{T [Ft(T)]0} = E(T) = nt = в

$Подпись: рЖ рЖ E {T [Ft (T )^}= / [t Ft (t)] ft (t) dt = [t(1 - e-t/0 )](e—t/0/0) dt = 40 00$ 01 =

00 From Eq. (2.28), the first two L-moments can be computed as

60 0

Л2 = 201 – 00 = —— 0 = —

Framing with Steel

Posted by admin on 12/ 11/ 15

The use of steel framing in residential renovation is increasing, but it’s still rare and generally not advised for novice builders unless working with a builder experienced with it.

Подпись: LIGHT STEEL FRAMING

Light steel framing consists primarily of C – shaped metal studs set into U-shaped top and bottom plates, joined with self-drilling pan-head screws. Fast and relatively cheap to install, light steel framing (20 gauge to 25 gauge) is most often used to create non-load-bearing interior partitions in commercial work.

Its advocates argue that more residential contractors would use it if they were familiar with it. In fact, light steel framing is less expensive than lumber; it can be assembled with common tools, such as aviation snips, screw guns, and locking pliers; and it’s far lighter and easier to lug than dimension lumber. To attach drywall, use type-S drywall screws instead of the type W screws specified for wood.

If you want to hide a masonry wall, light steel framing is ideal. Masonry walls are often irregular, but if you use 158-in. metal framing to create a wall within a wall, you’ll have a flat surface to drywall that’s stable and doesn’t eat up much space.

That said, light steel is quirky. You must align prepunched holes for plumbing and wiring before cutting studs and, for that reason, you must measure and cut metal studs from the same end. If you forget that rule, your studs become scrap. Finally, if you want to shim and attach door jambs and casings properly, you need to reinforce steel-framed door openings with wood.

FLITCH PLATES

Flitch plates are steel plates sandwiched between dimension lumber and are through-bolted to increase span and load-carrying capacity. Flitch plates are most often used in renovation where existing beams or joists are undersize. (You insert plates after jacking sagging beams.)

Ideally, a structural engineer should size the flitch plate assembly, including the size and placement of bolts. Steel plates are typically 58 in. to 58 in. thick; the carriage bolts, 58 in. to 58 in. in diameter. Stagger bolts, top to bottom, 16 in. apart, keeping them back at least 2 in. from beam

STEEL FRAMING

Load-bearing steel framing is heavier (14 gauge to 20 gauge) and costs much more than lumber. Plus it requires specialized tools and techniques. Metal conducts cold, so insulating steel walls can be a challenge. For exterior and loadbearing walls, you’re better off with wood framing.

edges. Put four bolts at each beam end. To ease installation, drill bolt holes 58б in. wider than the bolt diameters.

Flitch plates run the length of the wood members. The wood sandwich keeps the steel plate on edge and prevents lateral buckling. Note: Bolt holes should be predrilled or punched—never cut with an acetylene torch. That is, loads are transferred partly through the friction between the steel and wood faces, thus the raised debris around acetylene torch holes would reduce the

desired steel-wood contact.

STEEL I-BEAMS

Although lately eclipsed by engineered-wood beams, steel I-beams, for the same given depth, are stronger. Consequently, steel I-beams may be the best choice if you need to hide a beam in a relatively shallow floor system—say, among 2 x6s or 2x8s—or if clearance is an issue.

Wide-flange I-beams are the steel beams most commonly used in residences, where they typically range from 4’h in. to 10 in. deep and 4 in. to 10 in. wide. Standard lengths are 20 ft. and 40 ft., although some suppliers stock intermediate sizes. Weight depends on the length of the beam and the thickness of the steel. That is, a 20-ft.,

Подпись: Certifying agency Подпись: Thickness Span rating (rafters/studs) Подпись: Mill number Подпись: Plywood grade stamps. 8×4 I-beam that’s 0.245 in. thick weighs roughly 300 lb.; whereas, a 20-ft., 8x8H I-beam with a web that’s 0.458 in. thick weighs 800 lb. If you order a nonstandard size, expect to pay a premium.

Before selecting steel I-beams, consult with a structural engineer. For installation, use a contractor experienced with these beams. Access to the site greatly affects installation costs, especially if there’s a crane involved.

Darcy’s Law

Posted by admin on 12/ 11/ 15

Water flows though porous media from a point to which a given amount of energy can be associated to another point at which the energy will be lower (Cedergren, 1974, 1977). The energy involved is the kinetic energy plus the potential energy. The kinetic energy depends on the fluid velocity but the potential energy is linked to the datum as well as the fluid pressure. As the water flows between the two points a certain head loss takes place.

From an experimental setup as shown in Fig. 2.6, the total energy of the system between points A and B is given from Bernoulli equation as

Подпись: (2.13) UA V A uB vB

— + TT + ZA = — + ^ + ZB + Ah Pwg 2g Pwg 2g

where u and v are the fluid pressure and velocity respectively, z is elevation above the datum line and h is head loss between point A and B that is generating the flow. As velocities are very small in porous media, velocity heads may be neglected, allowing head loss to be expressed as:

h = ^ + zA -( + zb) (2.14)

Pw g Pw g J

Darcy related flow rate to head loss per unit length through a proportion constant referred to as K, the coefficient of permeability (also known as the coefficient of hydraulic conductivity) as:

P. S

p. g

Datum

Fig. 2.6 Head loss as water flows through a porous media. Where u = pore water pressures, h = heads, z & L = distances

or in more general terms, at an infinitesimal scale:

v = -K = – Ki (2.16)

where dh is the infinitesimal change in head over an infinitesimal distance, dl, and i is the hydraulic gradient of the flow in the flow direction. The above equation is known as Darcy’s law and governs the flow of water through soils (see Eq. 1.2).

It should be pointed out that Darcy’s law applies to laminar, irrotational flow of water in porous media. For saturated flow the coefficient of permeability may be treated as constant provided eddy losses are not significant (see below). Above the groundwater table, in the unsaturated zone, Darcy’s law is still valid but the permeability will be a function of the water content, thus K = K(9w), as described in Section 2.8.

Darcy’s law can easily be extended to two or three dimensions. For three dimensions using a Cartesian coordinate system Darcy’s law is given for a homogeneous isotropic medium as

highway engineering problems the porous media, that is each layer, can be assumed homogenous and isotropic.

For very coarse grained soils or aggregates, some of the voids in the material become quite large and the assumption of a laminar flow of water is no longer valid. Instead of irrotational flow, eddy currents develop in the larger voids and/or the flow may become turbulent involving more energy loss than in a laminar flow. For these circumstances the hydraulic gradient in Darcy’s law can be replaced with Forcheimer’s law:

Подпись: , V V2

– = K + K

where two coefficient of permeability, K1 [LT-1] and K2 [L2T-2], are now required to describe the behaviour.

In highway practice this means that coarse aggregates with large pores – such as those which comprise typical granular base courses – must be tested at low hydraulic gradients to ensure laminar flow is maintained and that appropriate values of K are obtained. This aspect is covered further in Chapter 3, Section 3.3.1 (see Fig. 3.7 in particular).

United States

Posted by admin on 12/ 11/ 15

U. S. requirements for aggregates constitute a compromise between high quality conditions and the necessity of taking into account the economics of manufacturing asphalt mixtures. The number of properties specified are limited, while the requirements themselves are somewhat broad (see Table 5.5), compared with the European standards. The requirements also vary from state to state. Similar requirements are used in the United States for SMA airfield surfacing (ETL 04-8).

Additionally, the possibility of using reclaimed dusts (baghouse fines) from an asphalt mixing plant is a fine example of the pragmatic approach to the selection of aggregates for asphalt mixtures.

Dusts, mineral powders, hydrated lime, and pulverized fly ashes are allowed, while lumps and organic impurities are excluded as fillers in the United States.

Sight Distance

Posted by admin on 12/ 11/ 15

A primary feature in the design of any roadway is the availability of adequate sight distance for the driver to make decisions while driving. In the articles that follow, the text contains conclusions based on information contained in Ref. 1. Derivation of formulas and references to supporting research are contained in that document and will not be repeated here. The reader is encouraged to consult that document for more detailed background information. The following paragraphs discuss various sight distances and the role they play in the design of highways.

Stopping Sight Distance. Stopping sight distance is the distance ahead that a motorist should be able to see so that the vehicle can be brought safely to a stop short of an obstruction or foreign object on the road. This distance will include the driver’s reaction or perception distance and the distance traveled while the brakes are being applied. The total distance traveled varies with the initial speed, the brake reaction time, and the coefficient of friction for wet pavements and average tires. The values in Table 2.2 were developed using a reaction time of 2.5 s and a braking deceleration rate of 11.2 ft/s2 (3.4 m/s2). The height of eye was taken as 3.50 ft (1.07 m) and the height of the object as 2.00 ft (0.61 m).

When considering the effect of stopping sight distance, it is necessary to check both the horizontal and the vertical stopping sight distance. Horizontal sight distance may be restricted on the inside of horizontal curves by objects such as bridge piers, buildings, concrete barriers, guiderail, cut slopes, etc. Figure 2.6 shows a diagram describing how horizontal sight distance is checked along an extended curve. Both formulas and a nomograph are provided to enable a solution. Many times, where the curve is not long enough or there are a series of roadway horizontal curves, a plotted-out “graphic” solution will be required to determine the available horizontal sight distance.

TABLE 2.2 Stopping Sight Distance (SSD) for Design Speeds from 20 to 70 mi/h (32 to 113 km/h)

Design speed, mi/h	Design SSD, ft	Design speed, mi/h	Design SSD, ft
20	115	46	375
21	120	47	385
22	130	48	400
23	140	49	415
24	145	50	425
25	155	51	440
26	165	52	455
27	170	53	465
28	180	54	480
29	190	55	495
30	200	56	510
31	210	57	525
32	220	58	540
33	230	59	555
34	240	60	570
35	250	61	585
36	260	62	600
37	270	63	615
38	280	64	630
39	290	65	645
40	305	66	665
41	315	67	680
42	325	68	695
43	340	69	715
44	350	70	730
45	360

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: Location and Design Manual, Vol. 1, Roadway Design,

Ohio Department of Transportation, with permission.

SIGHT DISTANCE

FIGURE 2.6 Horizontal sight distance along curve. Conversion: 1 ft = 0.305 m. (From Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission)

When a cut slope is the potential restriction, the offset should be measured to a point on the backslope having the same elevation as the average of the roadway where the driver is, and the location of the lane downstream where a potential hazardous object lies. In this way, an allowance of 2.75 ft (0.84 m) of vegetative growth on the backslope can be made, since the driver’s eye is assumed to be 3.5 ft (1.07 m) above the pavement and the top of a 2.0-ft (0.61-m) hazardous object downstream may still be seen.

Vertical sight distance may be restricted by the presence of vertical curves in the roadway profile. The sight distance on a crest vertical curve is based on a driver’s
ability to see a 2.0-ft-high (0.61-m) object in the roadway without being blocked by the pavement surface. The height of eye for the driver used in the calculations is 3.5 ft (1.07 m).

The sight distance on a sag vertical curve is dependent on the driver’s being able to see the pavement surface as illuminated by headlights at night. The height of the headlight is assumed to be 2.0 ft (0.61 m), and the height of the object is 0.0. The upward divergence angle of the headlight beam is assumed to be 1°.

Intersection Sight Distance. A motorist attempting to enter or cross a highway from a stopped condition should be able to observe traffic at a distance that will allow safe movement. In cases where traffic is intermittent or moderate in flow, the motorist will wait to find an acceptable “gap.” The driver approaching the intersection on the through road should have a clear view of the intersection including any vehicles stopped, waiting to cross, or turning. The methods described in the following paragraphs produce distances that provide sufficient sight distance for the stopped driver to make a safe crossing or turning maneuver. If these distances cannot be obtained, the minimum sight distance provided should not be less than the stopping sight distance for the through roadway. This would allow a driver on the through roadway adequate time to bring the vehicle to a stop if the waiting vehicle started to cross the intersection and suddenly stopped or stalled. If this distance cannot be provided, additional safety measures must be provided. These could include, but are not limited to, advance warning signals and flashers and/or reduced speed limit zones in the vicinity of the intersection.

There are three possible maneuvers for a motorist stopped at an intersection to make. The motorist can (1) cross the intersection by clearing oncoming traffic on both the left and right of the crossing vehicle, (2) turn left into the crossing roadway after first clearing the traffic on the left and then making a safe entry into the traffic stream from the right, or (3) turn right into the crossing roadway by making a safe entry into the traffic stream from the left.

In order to evaluate the amount of sight distance available to a stopped vehicle waiting to make a crossing or turning maneuver, the American Association of State Highway and Transportation Officials (AASHTO) adopted the concept of using “sight triangles” (Ref. 1). The vertices of the triangles are (a) the waiting driver’s position, (b) the approaching driver’s position, and (c) the intersection of the paths of the two vehicles, assuming a straight-ahead path for the waiting vehicles. Figure 2.7 shows the concept of sight triangles, emphasizing both the horizontal and vertical elements to be considered. The shaded area in the triangles is to be free of objects that would obstruct the field of vision for either driver. The profile view shows the limiting effect of vertical curvature of the through roadway. Notice that the height of eye of the drivers (3.50 ft or 1.07 m) is used for both the waiting and approaching vehicles. This stresses the importance of both drivers being able to see each other.

Table 2.3A provides intersection sight distance values for through vehicle speeds from 15 to 70 mi/h (24 to 113 km/h). The distances are based on a time gap of 7.5 s for a passenger vehicle turning left and a gap of 6.5 s for a crossing or right-turning vehicle. The height of eye and object were taken as 3.50 ft (1.07 m). The table also provides K values for crest vertical curves that would provide the required sight distance. (See Art. 2.2.4 for a discussion of vertical curvature.) Formulas are provided so that distances can be calculated for trucks requiring a longer time gap and for time adjustments due to upgrades or multiple lane crossings. See the notes in Table 2.3A, which explain how to adjust the timings.

Passing Sight Distance. In Table 2.3B, the “PSD” column lists the distances required for passing an overtaken vehicle at various design speeds. These distances are applicable

Подпись: WITH PAVEMENT AS OBSTRUCTION

Sight Distance PORTION OF ABUTMENT
AS OBSTRUCTION

FIGURE 2.7 Intersection sight triangles. (a) Sight triangles. (b) Vertical components. a1 = the distance, along the minor road, from the decision point to 12 the lane width of the approaching vehicle on the major road. a2 = the distance, along the minor road, from the decision point to 112 the lane width of the approaching vehicle on the major road. b = intersection sight distance (ISD). d = the distance from the edge of the traveled way of the major road to the decision point; the distance should be a minimum of 14.4 ft (4.39 m) and 17.8 ft (5.43 m) preferred. (From Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission) to two-lane roadways only. Among the assumptions that affect the required distance calculations are (1) the passing vehicle averages 10 mi/h (1.61 km/h) faster than the vehicle being passed, (2) the vehicle being passed travels at a constant speed and this speed is the average running speed (which is less than the design speed), and (3) the oncoming vehicle is traveling at the same speed as the passing vehicle. Table 2.3B contains K values for designing crest vertical curves to provide passing sight distance. These values assume that the height of the driver’s eye is 3.5 ft (1.07 m) for both the passing and the oncoming vehicle. The equations at the bottom of the table provide mathematical solutions for sight distance on the crest curves.

On two-lane roadways, it is important to provide adequate passing sight distance for as much of the project length as possible to compensate for missed opportunities due to oncoming traffic in the passing zone. On roadways where the design hourly traffic volume exceeds 400, the designer should investigate the effect of available passing sight distance on highway capacity using procedures outlined in the latest Transportation Research Board “Highway Capacity Manual” (Ref. 10). If the available passing sight distance restricts the capacity from meeting the design level of service requirement, then adjustments should be made to the profile to increase the distance.

Подпись: TABLE 2.3A Intersection Sight Distance (ISD) for Design Speeds from 15 to 70 mi/h (24 to 113 km/h) Design speed, mi/h Passenger cars completing a left turn from a stop (assuming a tg of 7.5 s) Passenger cars completing a right turn from a stop or crossing maneuver (assuming a tg of 6.5 s) ISD, ft K-crest vertical curve ISD, ft K-crest vertical curve 15 170 10 145 8 20 225 18 195 14 25 280 28 240 21 30 335 40 290 30 35 390 54 335 40 40 445 71 385 53 45 500 89 430 66 50 555 110 480 82 55 610 133 530 100 60 665 158 575 118 65 720 185 625 140 70 775 214 670 160

If ISD cannot be provided due to environmental or R/W constraints, then as a minimum, the SSD for vehicles on the major road should be provided.

Подпись: ISD = 1.47 X V. Xt major g Using S = intersection sight distance L = length of crest vertical curve A = algebraic difference in grades (%), absolute value K = rate of vertical curvature

• For a given design speed and an A value, the calculated length L = K X A.

• To determine S with a given L and A, use the following:

For S < L: S = 52.92 VK, where K = L/A For S > L: S = 1400/A + L/2

Note: For design criteria pertaining to collectors and local roads with ADT less than 400, please refer to Ref. 15, Guidelines for Geometric Design of Very Low-Volume Local Roads (ADT < 400).

Time gaps

Time gap(s) at design

Design vehicle speed of major road (tg), s

A. Left turn from a stop	Passenger car	7.5
	Single-unit truck	9.5
	Combination truck	11.5
B. Right turn from a	Passenger car	6.5
stop or crossing	Single-unit truck	8.5
maneuver	Combination truck	10.5

(Continued)

TABLE 2.3A Intersection Sight Distance (ISD) for Design Speeds from 15 to 70 mi/h (24 to 113 km/h) (Continued)

A. Note: The ISD and time gaps shown in the above tables are for a stopped vehicle to turn left onto a two-lane highway with no median and grades of 3 percent or less. For other conditions, the time gap must be adjusted as follows:

• For multilane highways: For left turns onto two-way highways with more than two lanes, add 0.5 s for passenger cars or 0.7 s for trucks for each additional lane, from the left, in excess of one, to be crossed by the turning vehicle.

• For minor road approach grades: If the approach grade is an upgrade that exceeds 3 percent, add 0.2 s for each percent grade for left turns.

B. Note: The ISD and time gaps shown in the above tables are for a stopped vehicle to turn right onto a two-lane highway with no median and grades of 3 percent or less. For other conditions, the time gap must be adjusted as follows:

• For multilane highways: For crossing a major road with more than two lanes, add 0.5 s for passenger cars or 0.7 s for trucks for each additional lane to be crossed and for narrow medians that cannot store the design vehicle.

• For minor road approach grades: If the approach grade is an upgrade that exceeds 3 percent, add 0.1 s for each percent grade.

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio

Department of Transportation, with permission.

If the problem cannot be resolved in this manner, then consideration should be given to providing passing lane sections or constructing a multilane facility.

Decision Sight Distance. Stopping sight distances are usually sufficient to allow reasonably competent drivers to come to a hurried stop under ordinary circumstances. However, these distances may not be sufficient for drivers when information is difficult to perceive, or when unexpected maneuvers are required. In these circumstances, the decision sight distance provides a greater length for drivers to reduce the likelihood of error in receiving information, making decisions, or controlling the vehicle.

The following are examples of locations where it is desirable to provide decision sight distance: (1) exit ramps, (2) diverging roadway terminals, (3) intersection stop bars, (4) changes in cross section, such as toll plazas and lane drops, and (5) areas of concentrated demand where there is apt to be “visual noise” (i. e., where sources of information compete, such as roadway elements, traffic, traffic control devices, and advertising signs).

Table 2.4 shows decision sight distances based on design speed and avoidance maneuvers. The table lists values for five different avoidance maneuvers. Maneuvers A (rural stop) and B (urban stop) are calculated similar to the standard stopping sight distance values, except that perception times are increased to 3.0 s for rural environment and 9.1 s for urban. For maneuvers C (rural area), D (suburban area), and E (urban area), the braking component is replaced by an avoidance maneuver. This can be a change in speed, path, or direction. Values shown are calculated based on distance traveled during the perception-maneuver time. This time varies with speed and ranges from 10.2 to 10.7 s for rural areas, 12.1 to 12.4 s for suburban areas, and 14.0 to 14.1 s for

TABLE 2.3B Minimum Passing Sight Distance (PSD) for Design Speeds from 20 to 70 mi/h (32 to 113 km/h)

PSD

Design speed, mi/h	Minimum PSD, ft	K-crest vertical curve
20	710	180
25	900	289
30	1090	424
35	1280	585
40	1470	772
45	1625	943
50	1835	1203
55	1985	1407
60	2135	1628
65	2285	1865
70	2480	2197

Using S = minimum passing sight distance L = length of crest vertical curve A = algebraic difference in grades (%), absolute value K = rate of vertical curvature

• For a given design speed and an A value, the calculated length L = K X A.

• To determine S with a given L and A, use the following:

For S < L: S = 52.92 VK, where K = L/A. For S > L: S = 1400/A + L/2.

Conversions: 1 mi/h = 1.609 km/h, 1 ft = 0.305 m.

Source: Location and Design Manual, Vol. 1, Roadway Design, Ohio Department of Transportation, with permission.

urban areas. To calculate available distance on a crest vertical curve, the driver’s eye height is 3.5 ft (1.07 m) and the height of the object to be avoided is 2.0 ft (0.61 m).

Where conditions call for the use of a decision sight distance in design that cannot be achieved, every effort should be made to provide the stopping sight distance values from Table 2.2. Consideration should also be given to using suitable traffic control devices to provide advance warning of the unexpected conditions that may be encountered.

Requirements According to European Standard EN 13043

Posted by admin on 12/ 11/ 15

In the Comite Europeen de Normalisation, or European Committee for Standardization (CEN), the member states’ requirements for aggregates designed for asphalt mixtures have been unified in the EN 13043 standard entitled Aggregates for Bituminous Mixtures and Surface Treatments for Roads, Airfields, and Other Trafficked Areas. It provides a set of aggregate properties and a series of requirement levels (categories) for each property.

Each CEN member state adopting this standard has defined its own national requirements, considering such issues as local climatic conditions and experience of engineering, from among the alternative properties and categories provided in the standard.

The requirements for SMA aggregates according to the EN 13043 standard are displayed in Tables 5.1 and 5.2. The following is an explanation of records in the Tables 5.2 through 5.4: [20]

• No requirement category (NR), which means that in a given country’s national specification, the requirement for this property is not used, e. g., MBFNR

• Fractions of aggregates are described as d/D (e. g., 2/5 mm) where:

• D means nominal upper limit of gradation (oversized grains are allowed)

• d means nominal lower limit of gradation (undersized grains are allowed)

Some additional explanations are provided in Table 5.1, but details can be found in EN 13043. An example of a similar type of EN standard is described in Chapter 14.

Tables 5.2 through 5.4 present the requirements from selected European countries. The substantial differences among these countries may be confusing. In the most important properties, the demanded level is more or less similar, e. g., LA index is from 20 to 25% or flat and elongated content is usually as high as 20-25%.

Although it is the same SMA mixture, the requirements for components or aggregates, depend on the following factors:

• Materials available in a specified place—countries specify their requirements based on long-term experience with aggregates, test results, and research. Requirements also depend on accessible sources of aggregate— countries in which sufficient amounts of very good quality materials exist are able to limit the number of required properties. One example of such a situation are the Nordic (Scandinavian) countries. However, several countries have quite a wide range of aggregates with very different qualities. There is a need to balance the technical requirements with a view toward economics.

• Technology and previous experiences with materials also have an impact on the set of requirements. For example, using hydrated lime (in mixed filler) is not very popular in Europe, therefore only a few countries put them into their specifications (filler category Ka).

• Test methods in EN 13043 come from different national practices, as described in Chapter 14, hence many countries do not use some of them in practice. For example, the LA method has not been commonly used in Germany, and the same can be said for Rigden’s method in Poland, Aggregate Abrasion Value (AAV) outside the United Kingdom, and the Nordic abrasion value outside Scandinavian countries. Many countries use declared categories because without past experiences it was hard to establish any reasonable requirement.

• In specified number of countries, production of aggregates for asphalt layers is regulated with precisely prescribed fractions and their individual gradation limits. Such a situation, in which all producers of aggregates make the same fractions and with very similar gradation, is very comfortable for asphalt mix producers. At the same time any additional

TABLE 5.1

Comments on System of Requirements Based on EN 13043

Подпись: Fine aggregates: Gf85 All-in aggregates: GaX Tolerances of typical Coarse aggregates: gradation GX/Y comments

Gradation of aggregate fractions after EN 13043 are labeled as GXY category; the numbers describe allowable amounts of oversized (X) and undersized (Y) material, e. g. Gc90/15 means that only 10% of oversized and 15% of undersized material is allowed, similarly GC85/15 means that only 15% of oversized and 15% of undersized material is allowed; category Gc90/10 is the highest possible choice and GC85/35 is the lowest.

There is only one category for fine aggregate,

GF85, which means the limit for oversized grains (>2 mm) is 15% by mass.

When all-in aggregates are used one of two GaX categories, can be used Ga90 and Ga85, where maximum limits for oversized material are 10% and 15% (by mass), respectively.

The idea of this type control is based on a requirement that the producer will document and declare the typical gradation of any produced aggregate fraction; tolerances of typical gradation, labeled as GX/Y, are used for gradation control within the fraction; depending on the D/d coefficient, the control sieve is chosen as D/2 or D/1.4; the number x means x—overall limits of amount of material passing by control sieve (20 means 20-70% by mass; 25 means 25-80% by mass), and the number y means y – tolerances for typical gradation on control sieve declared by aggregate’s producer (15 means ±15%; 17.5 means ±17.5%); example category G25/15 means that on sieve D/2 or D/1.4 overall limits are 25-80% and the tolerance from the declared value is ±15%

This requirement applies to tolerance of percentage of grains passing by sieves D, D/2, and d compared with the gradation declared by producer;

category Gtc10 mean tolerances on sieves: D ±5%, D/2 ±10%, d ±3%,

category Gtc20 mean tolerances on sieves: D ±5%, D/2 ±20%, d ±3%,

Comments on System of Requirements Based on EN 13043

TABLE 5.1 (CONTINUED)

properties	Label of category	comments
Fines content, according to EN 933-1	fx	Amount of grains passing by 0.063-mm sieve, where x in fx category means maximum allowable content of fines
Fines quality, according to EN 933-9 (methylene blue test); in fine and all-in aggregates	MBFX	Control of harmful fines (e. g., swelling clay); category MBFX means that maximum X methylene blue value (g/kg) is allowed in fines; used only if fines content is between 3% and 10% by mass of material; if fines content >10%, requirement for filler applies
Angularity of fine aggregates according to EN 933-6, p.8	EcsX	Flow coefficient of fine aggegates, labeled with EcsX category, where X means minimum time of flow in seconds
Shape of coarse aggregate (Flakiness Index EN 933-3, Shape Index EN 933-4)	SIX or FIX	FI or SI could be used for determination of grains’ shape; labeled as SIX and FIX, where maximum allowable amount of flat and elongated particles is marked as X (% by mass)
Percentage of crushed and broken surfaces in coarse aggregates, according to EN 933-5	CXY	Percentage of crushed and broken surfaces labeled as Cxy, where X means percentage of completely broken particles (by mass) and Y means percentage of completely rounded particles (by mass); so a requirement to use only crushed coarse aggregates is described in category СЮ0/0. Category C95/1 allows up to 1% of noncrushed particles in aggregates.
Resistance to fragmentation (crushing), according to EN 1097-2 clause 5	LAX	Resistance to crushing (fragmentation) with LA method; labeled as LAX, where maximum allowable LA coefficient is marked as X (%)
Resistance to fragmentation (crushing), according to EN 1097-2 clause 6	SZx	Resistance to crushing (fragmentation) with German Schlagzertrummerungswert method (impact test); labeled as SZX where maximum allowable SZ coefficient is marked as X (%)
Resistance to polishing according to EN 1097-8	PSVx	PSV; labeled as PSVx, where X means required minimum PSV value
Resistance to surface abrasion, according to EN 1097-8 Annex A	AAVx	AAV; labeled as AAV# where X means maximum allowable AAV value
Resistance to wear, according to EN 1097-1	MDEX	Micro-Deval coefficient; labeled Mdex, where X means maximum allowable Micro- Deval value
Resistance to abrasion from studded tires, according to EN 1097-9	ANX	Nordic abrasion value; labeled as ANx, where X means maximum allowable value

TABLE 5.1 (CONTINUED)

Comments on System of Requirements

Properties Label of category

Water absorption, Wcm0.5

according to EN 1097-6 WA24X

Resistance to freezing and Fx

thawing, according to EN 1367-1

Resistance to freezing and MSx

thawing, according to EN 1367-2

Resistance to thermal —

shock, according to EN 1367-5:

Affinity of coarse —

aggregates to bituminous binders, according to EN 12697-11

“Sonnenbrand” of basalt, SBSz or SBla

according to EN 1367-3 and EN 1097-2

Coarse lightweight ^lpcX

contaminators, according to EN 1744-1, p.14.2

based on EN 13043

comments

The method of testing is chosen depending upon the size of the aggregate:

• Using EN 1097-6 clause 7 refers to category WA24X, where X means maximum allowable percentage absorption by mass.

• Using EN 1097-6 Annex B refers to category Wcm^-5, where 0.5 means maximum allowable percentage absorption by mass (there is only one category <0.5%).

Additionally EN 13043 connects water absorption and resistance to freeze-thaw of aggregates; aggregates with small absorption are assumed to be freeze-thaw resistant.

Category Fx is used, where X means maximum allowable percentage loss of mass; test can be conducted in water, salt solution, or urea.

Category MSx is used, where X means maximum allowable percentage loss of mass; test is conducted with magnesium sulfate.

Test of aggregate resistance for high temperature; results are declared

Test of binder adhesion to aggregate; results are declared

This is to check for basalt rock decay, which results in lowering aggregate strength and in most cases in very low freeze-thaw resistance of basalt aggregate; categories SBsz and SBla mean that after test (boiling for 36 hrs) and crushing (in SZ or LA, respectively), the aggregate must meet required values:

• Loss of mass after boiling: max 1.0% (and)

• Increase of impact value: max 5.0% (SBsZ) (or)

• Increase of LA coefficient: max 8.0% (SBla) The content of coarse lightweight organic

contaminants larger than 2 mm should be maximum X% by mass.

Comments on System of Requirements Based on EN 13043

TABLE 5.1 (CONTINUED)

properties	Label of category	comments
Dicalcium silicate	Resistance	Slag aggregate will be free from dicalcium silicate
disintegration of air-cooled blastfurnace slag, according to EN 1744-1, p. 19.1	required	disintegration, the results are declared.
Iron disintegration of	Resistance	Slag aggregate will be free from iron
air-cooled blast furnace slag, according to EN 1744-1, p. 19.2	required	disintegration; the results are declared.
Volume stability of steel slag aggregate, according to EN 1744-1, p. 19.3	Vx	Test applied to basic oxygen furnace slag and electric arc furnace slag; category Vx, where X mean maximum allowed expansion by volume percentage
Water content (added filler), according to EN 1097-5, %		Mass content of water in added filler (commercially produced) is fixed and will be maximum 1%.
Stiffening properties: Voids of dry compacted filler (Rigden), according to EN 1097-4	Vx/Y	The range of Rigden voids in dry compacted filler; categories are labeled as VX/y, where X is a lower limit and Y is an upper limit of voids; note that these are voids according to Rigden’s method not Rigden’s method modified by Anderson
Stiffening properties: Delta ring and ball, according to EN 13179-1:	Ar&bX/Y	The range of increase of softening point (SP) with ring-and-ball method; categories are labeled as Ar&bX/Y, where X is a lower limit and Y is an upper limit of SP increase
Water solubility, according to EN 1744-1	WSx	The water solubility is labeled as WSx, where X is a maximum allowed percentage (by mass).
Water susceptibility, according to EN 1744-4	—	No specified limits; the results are declared
Calcium carbonate content of limestone filler aggregate, according to EN 196-21	CCX	Calcium carbonate content is labeled as CCx, where X is a minimum required percentage (by mass) of CaCO3.
Calcium hydroxide content of mixed filler according to EN 459-2	KaX	Calcium hydroxide (hydrated lime) content is labeled as KaX, where X is a minimum required percentage (by mass) of Ca(OH)2.
Bitumen number of added filler, according to EN 13179-2	BNx/y	The range of bitumen number; categories are labeled as BNx/y where X is a lower limit and Y is an upper limit

Note: AAV = Aggregate abrasion value; FI = flakiness index; LA = Los Angeles; PSV = polished stone value; SZ = Schlagzertrummerungswert; SI = shape index.

TABLE 5.2

Requirements for SMA Coarse Aggregate according to EN 13043 in Selected CEN-Member Countries, Aggregates for SMA at the Highest Traffic Level (Reference Mixture SMA 0/11)

List of Categories by Country
Austria

Germany ONORM В

	TL Gestein StB 04 Anhang F and TL Asphalt StB 07	Slovakia KLK 1/2009	3584:2006 RVS 08.97.05:2007 (Class G1)	Switzerland SN 670130a: 2005	Poland WT-1 Kruszywa 2008
Properties3	1	2	3	4	5
Grading, according to EN 933-1	Gc90/10 for (2/5 mm) Gc90/15 for (2/5, 5/8, 8/11 mm)	Gc90/10	Gc90/15	Gc85/15	Gc90/15
Tolerances of typical	—	Declared	—	f-^20/15	f-^25/15
gradation
Fines content, according to	Fractions 2/5 to 8/11	/1	/1	/1	/2
EN 933-1	mm:/2 (max 2%)	(max 1%)	(max 1%)	(max 1%)	(max 2%)
Fines quality, according to EN 933-9—methylene blue	Declared (value to be reported)	—	—	—	—
test
Shape of coarse aggregate	SI’20 ОГ ^20	SI’20 ОГ ^20	SI15	FI25	SI2o or FI20
(Flakiness Index EN 933-3,	(max 20%)	(max 20%)	(max 15%)	(max 25%)	(max 20%)

Shape Index EN 933-4)

(Continued)

List of Categories by Country

TABLE 5.2 (CONTINUED)

Requirements for SMA Coarse Aggregate according to EN 13043 in Selected CEN-Member Countries, Aggregates for SMA at the Highest Traffic Level (Reference Mixture SMA 0/11)

			Austria
	Germany		ONORM В
	TL Gestein StB 04		3584:2006	Switzerland	Poland
	Anhang F	Slovakia	RVS 08.97.05:2007	SN 670130a:	WT-1 Kruszywa
	and TL Asphalt StB 07	KLK 1/2009	(Class C1)	2005	2008
Properties3	1	2	3	4	5
Percentage of crushed and	Qoo/o,	C100/0	Qoo/o	C95/1	Qoo/o
broken surfaces in coarse	C95/1,
aggregates, according to EN 933-5	Q0/1
Resistance to fragmentation	LA20	LA 25	LA 20	4/8 mm—LA 2 .	LA2o or LA 25
(crushing), according to EN	(max 20%)	(max 25%)	(max 20%)	8/11 mm— LA20	depending on
1097-2 clause 5 (LA				11/16 mm— LA25	petrographic type
method)					of aggregate
Resistance to fragmentation	SZI8	—	—	—	—
(crushing), according to EN 1097-2 clause 6 (German SchlagzertrUmmemngswert)	(max 18%)
Resistance to polishing,	PSV51	psvx	PSV50	PSV50	PSV50
according to EN 1097-8	(min 51)	(min 56)	(min 50)	(min 50)	(min 50)
Resistance to surface	—	—	—	—	—
abrasion, according to EN 1097-8 Annex A

Resistance to wear (Micro – Deval), according to EN 1097-1		Mde20 (max 20%)
Resistance to abrasion from studded tires, according to EN 1097-9
Water absorption according	Wcm0.5	WA241
to EN 1097-6	(max 0.5%)	Wcn.0.5
Resistance to freezing and	F,	F2
thawing, according to EN 1367-1 (in water or salt solution)	(max 1.0%)	(max 2.0%)
Resistance to freezing and thawing, according to EN 1367-2		MSla (max 18%)
Resistance to thermal shock, according to EN 1367-5	Declared	—
Affinity of coarse aggregates to bituminous binders, according to EN 12697-11	Declared
“Sonnenbrand” of basalt, according to EN 1367-3 and EN 1097-2	SBsz (SBLA)
Coarse lightweight	^lpcA 1	mlpcO.1
contaminators according to	(max 0.1%)	(max 0.1%)

EN 1744-1 p.14.2

WA241	Declared WA241 Wcn,0.5
F,	^NaCl^
(max 1.0%)	(max 7% in 1% NaCl solution)

Min 85%	Declared —
method В
SBLA	– SBLA
—	mlpcO.1 mLPC0.1 (max 0.1 %) (max 0.1 %)

(iContinued)

TABLE 5.2 (CONTINUED)

Requirements for SMA Coarse Aggregate according to EN 13043 in Selected CEN-Member Countries, Aggregates for SMA at the Highest Traffic Level (Reference Mixture SMA 0/11)

List of Categories by Country

	Germany TL Gestein StB 04 Anhang F and TL Asphalt StB 07	Slovakia KLK 1/2009	Austria ONORM В 3584:2006 RVS 08.97.05:2007 (Class C1)	Switzerland SN 670130a: 2005	Poland WT-1 Kruszywa 2008
Properties3	1	2	3	4	5
Dicalcium silicate disintegration of air-cooled blast furnace slag, according	Resistance required		Resistance required	According to other regulations	Resistance required
toEN 1744-1, p. 19.1
Iron disintegration of air-cooled blast furnace slag, according to EN	Resistance required		Resistance required	According to other regulations	Resistance required
1744-1, p. 19.2
Volume stability of steel slag	^3.5	^3.5	^3.5	According to	^3.5
aggregate, according to EN 1744-1, p. 19.3	(max 3.5%)	(max 3.5%)	(max 3.5%)	other regulations	(max 3.5%)

Подпись: Stone Matrix Asphalt: Theory and Practice Note: Cells with — mean no requirement (NR) category; FI = flakiness index; LA = Los Angeles; SI = shape index. a Names of properties after EN 13043

Подпись: List of Categories by Country Germany Austria Poland TL Gestein StB 04 ONORM В 3584:2006 Switzerland WT-1 Anhang F Slovakia RVS 08.97.05:2007 SN 670130a: Kruszywa and TL Asphalt StB 07 KLK 1/2009 (Class C1) 2005 2008

TABLE 5.3

Requirements for SMA Fine Aggregate according to EN 13043 in Selected CEN-Member Countries, Aggregates for SMA at the Highest Traffic Level

Properties	і	2	3	4	5
Grading, according to EN 933-1	GF85	GF85	Gf85	Gf85	Gp 85
Tolerances of typical gradation	GrcNR	GTC 20	GTC 20	Grc10	О 7 0
Fines content, according to EN	Declared	fio	/іб	/22	/16
933-1		(max 10%)	(max 16%)	(max 22%)	(max 16%)
Fines quality, according to EN	Declared	MBplO	—	—	MBF10
933-9		(max 10 g/kg)			(max 10 g/kg)
Angularity, according to EN	Ecs 35	—	Ecs 35	Declared	Ecs30
933-6, p.8	(min 35 sec)		(min 35 sec)		(min 30 sec)
Coarse lightweight contaminators,	mLPC0.1	—	—	—	тьрС0Д
according to EN 1744-1, p.14.2	(max 0.1%)				(max 0.1%)

TABLE 5.4

Requirements for SMA Filler according to EN 13043 in Selected CEN-Member Countries, Aggregates for SMA at the Highest Traffic Level

List of Categories by Country

	Germany TL Gestein StB 04	Slovakia	Austria ONORM В	Switzerland	Poland WT-1
	Anhang F	KLK	3584:2006	SN 670130a:	Kruszywa
Properties	and TL Asphalt StB 07	1/2009 RVS 08.97.05:2007	2005	2008
Grading, according to EN 933-10 Harmful fines (fines quality),	Declared	According to Table 24 of standard: Sieve 2.0 mm = 100% passing Sieve 0.125 mm = 85-100% passing Sieve 0.063 mm = 70-100% passing		MBF10
according to EN 933-9 Water content (added filler), according	<%	<i%	<1%	<1%	(max 10 g/kg) <1%
to EN 1097-5, % Stiffening properties: voids of dry	^28/45	_	^28/38	T28/45	T28/45
compacted filler (Rigden), according	(min 28%,		(min 28%,	(min 28%,	(min 28%,
to EN 1097-4	max 45%)		max 38%)	max 45%)	max 45%)
Stiffening properties: “Delta ring and	Ar&r8/25	Ar&b8/16	—	Ar&b8/25	Ar&r8/25
ball,” according to EN 13179-1	(min 8°C,	(min 8°C,		(min 8°C,	(min 8°C,
	max 25°C)	max 16°C)		max 25°C)	max 25°C)

Water solubility, according to EN Н7>10

1744-1 (max 10%)

Water susceptibility, according to EN Declared

1744-4

Calcium carbonate content of CC70

limestone filler aggregate, according (min 70%)

to EN 196-21

Calcium hydroxide content of mixed Declared

filler, according to EN 459-2 Bitumen number of added filler, —

Подпись: WSio (max 10%) according to EN 13179-2

Declared

— Declared

CC80 Declared

(min 80%)

Ka20 To be established

(min 20%) in contract

ПМ _

Dn 28/39

Подпись: WSio (max 10%) CC (min 70%) Declared Declared (min 28 max 39)

requirement for tolerance of gradation is not necessary; this is the situation in Germany, Switzerland, and a few other countries. In Poland, where limits for gradation during aggregate production do not exist, it was necessary to put such a requirement (categories G and GTC) in the national specifications.

• Legal systems and approaches to the requirements’ system also play roles.

In Europe, aggregates for asphalt mixtures are construction products and are produced and placed on the market according to Construction Product Directive[21] regulations. Internal regulations of each country have to be consistent with this directive. The product (aggregates) must fulfill specified requirements for intended use; countries are free to determine how they specify the system of requirements (only a few properties are indispensable). When we see Tables 5.2 through 5.4, it is obvious that some countries built a very broad system and established more detailed specifications than others. The reason is most likely in the existing approach to the requirements for components. In some countries only a few properties are specified because the final mixture (e. g., SMA) features are treated as the most important and a large degree of freedom is left for asphalt mix producers as long as they meet the final desired properties. In other countries everything is specified—both components and final mixture properties as well, which can ultimately lead to overspecification.

ENGINEERED BEAMS

Posted by admin on 12/ 11/ 15

The most daunting part of using engineered beams may be the wide selection. Fortunately, lumberyard staff can usually explain the merits of each type and help you determine correct size.

Glulams, or glue-laminated timbers, are the granddaddy of engineered beams. They’ve been used in Europe since the early 1900s. In North America, they’re fabricated from relatively short pieces of dimension lumber (often Douglas fir or southern pine), which is overlapped or finger – jointed, glued, and pressure clamped. Glulams come in stock widths 3% in. to 6% in., but you can obtain them in almost any size or shape, including curves and arches, as well as pressure treated.

Glulams are expensive, but their stability and strength make them suitable for high loads in clear spans as great as 60 ft. Obviously, you’d need a crane to move such a behemoth.

LVL (Microllam®) is fashioned from thin layers of wood veneer glued together—much like plywood, except the wood grain in all LVL layers runs parallel. It’s stronger than sawn lumber or laminated strand lumber of comparable size, though it’s roughly twice the cost of sawn lumber.

LVL is usually milled as planks 114 in. wide, so it’s typically used as rim joists, cantilever joists, or in-floor headers and beams. It’s a good choice for medium-span beams up to 16 ft., and because individual beams are easy to handle, a small crew can join LVL planks on site to create a built-up girder. LVL is available in other widths, from ЗУ2 in. to 512 in. Depths range up to 20 in.

► Disadvantages: LVL can’t be pressure treated and shouldn’t be used on exteriors. If it gets wet, it will cup. For this reason, keep it covered till you’re ready to use it.

PSL (parallel strand lumber, Parallam®) is created from wood fiber strands 2 ft. to 8 ft. long, running parallel, and glued together under tremendous pressure. PSL is the strongest and most expensive of any structural composite lumber.

Standard PSL sizes are 7 in. to 11 in. wide, up to 20 in. deep, and they can be fabricated to virtually any length—66 ft. is not uncommon. Because they’re stronger than glulams, PSLs are

This engineered beam is a 4-in. by 14-in. Parallam girder secured with a Simpson™ CCQ column cap.

built without camber (a curve built-in to anticipate deflection under load), so they’re easier to align during installation.

PSL beams can be pressure treated and thus can be used outside.

LSL (laminated strand lumber, Timberstrand®) is fabricated from 12-in. wood strands from fastgrowing (but weaker) trees, like aspen and poplar, and then glued together in a random manner. Consequently, LSL carries less load than the beams noted previously, and it costs less.

Still, it is stronger than sawn lumber, though more expensive.

LSL is available in 114-in. to 312-in. widths and in depths up to 18 in., but it’s most often used as short lengths in undemanding locations, such as door or window headers, wall plates, studs, and rim joists.

LSL headers are stable, so they’ll probably reduce nail pops and drywall cracks around doors or windows. But for small openings of 10 ft. or less and average loads, sawn-lumber headers are usually more cost-effective.

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