Filler Content

In the majority of worldwide regulations for SMA, the content of particles passing through the smallest sieve (0.075 or 0.063 mm ) generally ranges between 8% to 13% (m/m). However, adopting extreme quantities may be a risky business—that is, 8% can lead to building too little mastic. On the other hand, a large quantity of filler (e. g., approximately 13%) may generate too high a content of mastic, making it susceptible to overstiffening or increasing the risk of forming fat spots.

It has been discussed in Chapter 3 that the optimum relationship between quanti­ties of filler and binder is best illustrated by the filler-to-binder ratio (by weight or volume). This means that each quantity of filler corresponds to a certain optimum amount of binder. The details behind this assumption are inexact, resulting per­haps from experience with a run-of-the-mill filler in a given country. After reading Chapter 3, it should be clear that there are all sorts of fillers and the differences between them do not lie only in one specific area (e. g., gradation, degree of grind­ing) but also in the different content of voids in the compacted filler (determined using Rigden’s method).

The qanats in Egypt under the Achaemenid Empire

In Chapter 2 we described the invention of the qanat in Urartu and its development in Persia. After the conquest of Egypt by Cambyse, qanats are introduced in Egypt to irrigate the oases as well as the mountainous zones situated along communication routes (the wadi Hammamat, between Thebes and the ports of the Red Sea). Traces of three qanats constructed by order of Darius I (about 500 BC) have been found in the oasis of Kharga, some 300 km to the northwest of Asswan.[127] These qanats are as deep as 75 m, with a gallery several kilometers long and a rather flat slope compared to the usual prac­tice, only about 0.5 per thousand.

Irrigation in the land of the Queen of Sheba Irrigated oases at the threshold of the desert

To the south of the Arabian peninsula, the mountains of Yemen rise to 3,000 m and capture the seasonal monsoon rains. The high valleys are therefore well watered – but the most prosperous regions are not found here. The shrubs from which incense and myrrh can be harvested are located, rather, on the edges of the arid interior desert, at about 1,200 m of altitude, in the lands of Qataban and Hadramawt. These aromatic resins become quite the fashion in all the countries of the East, in Greece, and eventual­ly in Rome, from the 8th century BC. The richness of this land, soon to be called Arabia Felix,[128] is built on incense and myrrh – their harvest, exchange, and associated control of the caravan routes that lead to the north of Arabia along the eastern edge of the great mountains.

However it is only through hard work that the kingdoms of Sheba, Qataban,

The qanats in Egypt under the Achaemenid Empire

Hadramawt, and Ma’in, whose capitals are Maryab (today Ma’arib), Tamna (Hajar Kulhan), Shabwa, Yathill (Baraqish), find prosperity. Nothing is possible without water, without crops. The land between the mountains and the desert is very arid; the only sources of water are the wadis. These are normally dry, but are subject to violent floods when heavy rains fall on the high mountains from which they issue, two or three times a year between March and August. People had learned how to make use of this water resource long before incense came into fashion – very likely from the IIIrd millennium BC along the wadis of Dura, Dhana and Markha; and from the IInd millennium BC in the basin of the wadi Hadramawt (along the wadi Irma upstream of Shabwa, in particular).[129]

The water engineering techniques remained about the same through the era of pros­perity of Arabia Felix. Deflector walls, weirs and small dikes, and occasionally true dams, constructed in the beds of the wadis, direct some of the silt-laden floodwaters toward a system of branching earthen canals. These canals are provided with outtakes – stones with grooves into which beams could be slid to control the water flow. Since the canals have a much gentler slope than the wadi, the currents are slower and therefore only the finest of the sediments are diverted to the crops with the water. The fields are quadrilateral in shape and surrounded by earthen levees, the whole comprising a vast irrigated zone. Gates permit controlled flooding of the fields to a depth of several tens of centimeters. Thus two resources are being used simultaneously: the water itself, making it possible to plant as soon as the field flooding is over; and the sediments, whose

The qanats in Egypt under the Achaemenid Empire

Figure 3.10. Reconstitution of the principal canals and irrigated zones around Shabwa, approximately between the 6th century (founding of the city) and the 2nd century BC – after Gentelle, 1992. The irrigated zones posterior to this period are also indicated. On the right bank, a large stone water gate (V) dates from the 2nd century BC; the deflector weirs (D and B) likely are from even later. On the two smallest ancient irrigated plots, just upstream of Shabwa, the thickness of the silt deposits reaches 15 m. The relatively steep slope of the wadi (5 m per km) makes it possible to cope with this downstream deposit, through modification of the locations of the intakes and canals. The desert is quite close, its first dunes appearing at the northwest extremity of this map.

 

deposition creates fertile layers of soil. Over the centuries, the thickness of topsoil increases, attaining some fifteen meters in certain locations (Figure 3.10), and even thir-

 

ty meters at Maryab. This accumulation of soil requires that the canal system be reworked periodically, the intakes being raised or moved further upstream so that the water can still reach the fields.

The qanats in Egypt under the Achaemenid Empire

Figure 3.11. The irrigated valley of Jawf, and the city of the land of Sheba (Maryab is a little further to the south) – adapted from Robin (1991).

This mastery of complex hydraulic systems could not be achieved without a strong social organization. The kingdoms mentioned earlier are effectively organized and hier­archical. They are dominated by Sheba from the beginning of the 7th to the 1st century BC, under the hand of the great Sheban sovereign Karib’Il Watar, the “Karibilu king of Sheba” mentioned in texts of the Assyrian Sennacherib. The sudarabic writing is origi­nal, using an alphabet related to the Phoenician. Numerous inscriptions still exist, com­memorating for example the redistribution of irrigation water following the destruction by Karib’Il Watar of the city of Nashan, at the convergence of the wadis Madhab and Jawf (Figure 3.11):

“(Karib’Il Watar) took the rights of the king of Nashan and of Nashan to the waters of Madhab (…), took the water of dhu-Qafan at Sumhuyafa and at Nashan and grant­ed it to Yadhmurmalik, king of Haram, took from Sumhuyaga and at Nashan the canals dhat – Malikwaqih and granted them to Nabat’ali, king of Kaminahu, and at Kaminahu the waters of the canals dhat-Malikwaqih (that are) beyond the boundaries of Karib’Il, built the enclo­sure of Nasq and peopled it with Shebans.. ..”45

Another later text attests to the concession of irrigated land to a dignitary of the city of Haram (located several kilometers from Qarnaw):

“Wahab’il son of Ammidhara, of the clan of Thabaran (…), chief of the horse-mounted troops of Haram, honored Matabnatiyan, god of Thabaran, (…), when he dug, bore, and lined with stone his well, and that Yashhurmalik Nabat granted him a flooded public (?) land, of which he took possession, extending from the enclosure of Dhu-Arnab upstream to his property, and the concession that is upstream, between the canal of Dhat Batalan and the road of Ma’in, and he acquired and brought into cultivation a number of 300 (parcels).”[130]

Numerous texts also describe the construction of hydraulic works. The following extract, despite its gaps, describes the construction of a canal and dike, again in the region of Haram:

“(the person in question) bound and dug Yasim (?) with the upper canal of the plain of the city of Haram, next to (?) this canal and the tour, (…) and the land that is subject (?) to the king of Sheba (.) he granted his inscription and his dike.”[131] [132]

There are regulations to protect the waterways against thoughtless incursions. One of the earliest known is from the 5th century BC, after a certain Karib’Il Bayyin had built a runoff channel around the small city of Nasq (in Jawf, populated by Sheban colonists after the destruction of Nashan, as is referenced in one of the extracts cited above). His son is then led to issue a decree:

“Damar Alay Watar, son of Karib’Il, has decided and decreed, for Sheba and its colonists, the clearing of the water collector wall (deflector?) of the city of Nasq, that his father Karib’Il had opened according to the inscription of easement where his father delineated it in writing.

Neither watered nor arid land shall be developed therein, and no irrigated produce or crop

48

shall be harvested therefrom.”

Neither the regularity nor the amplitude of the beneficial floods is guaranteed, as is clear from the following text. Thus, if two clans have neglected the practice of a ritual hunt dedicated to the god Halfan, and there is a water shortage two times in a row, the clans must carve an inscription to do penance for their sin in the hopes that Halfan will again irrigate their lands:

“The clan Anir and the clan Athar have confessed and done penance to Halfan (the principal god of Haram after the 2nd century BC) because they did not render unto him the ceremoni­al (?) hunt in the month of dhu-Mawsab when they took refuge at Yathill during the war of Hadramawt, whereas they made the pilgrimage of dhu-Samawi at Yathill and put off the cer­emonial hunt until the month of dhu-Anthar. Therefore he (the god Halfan) did not grant them water for their irrigation network from the spring to autumn, because of an extremely low quantity of water, and they will be careful not to do the same thing again.”[133]

Thanks to these irrigation systems, artificial oases flourish along the wadis Dhana (we will come back to it later) and Markha, in Hadramawt (see also Shabwa on Figure 3.10). Although these regions have the oldest irrigation infrastructure, other systems are found on the wadi Bayhan in Qataban, where a 45-km continuous ribbon of land is irri­gated and cultivated. Arabia Felix has some thirty cities between the 7th and 1st cen­turies BC, in the grand valley of Jawf and near the wadis Dhana, Ragwan, Juba, Harib, Markha, and Hadramawt. Most of these fortified cities have surface areas of from six to ten hectares; but they have neither sewers nor water supply infrastructure.

In the 1st century BC, merchants of Alexandria discover how to make an annual maritime voyage toward India, using the monsoon. This is a negative development for the economy of Arabia Felix, which thus loses its monopoly on the supply of spices. This century thus marks the beginning of a slow economic recession. Upland tribes (the Himyarites) begin their progressive domination beginning in the 1st century AD.

Content of the Fine Aggregate Fraction

In classic SMA composition and in regulations introduced all over the world, the total content of grains smaller than 2 mm has generally been in the range of 15-30% (m/m). When we add the typical filler content (8-13%), we receive up to 22% from the sand fraction (0.063/2.0 mm). But when designing the content of fine aggregate in SMA, one should remember the increase in the content of fine particles during compaction due to crushing and wearing of the coarse particles.

Is the sand fraction desired in a mix? Looking at the shape of an example SMA gradation, we can imagine a mix designer adding all the permitted quantity of filler (i. e., approximately 13% [m/m]) instead of 0.063/2.0 material. This example is illus­trated in Figure 6.13. As can be seen, the gradation curve stays within the limits up to the 0.85 mm sieve. Then it takes the “low route,” meaning that there are too few fine particles, which are needed for mastic creation. Undoubtedly, composing an SMA without the material 0.063/2.0 is not possible.

The sand fraction is indispensable because building a mastic with only filler grains makes achievement of the expected features of a newly designed SMA impossible. So is it possible to determine the best course of a gradation curve in the area below

Подпись: VO О d Подпись: Sieve, # mmПодпись: a a > .<o Подпись: "ns Сі 100

0

Подпись: FIGURE 6.13Example of a SMA aggregate mix gradation without fine aggregate (0/2 mm).

2 mm? There is not an unequivocal answer to this question because a lot depends on the type and properties of filler and features of the 0.063/2.0 material, too. The intended use of the designed mix is also of great significance. However, the follow­ing is worth bearing in mind:

• Guiding the curve upward enhances the risk of closing the mix and raises the threat of excessive mastic and the appearance of fat spots.

• Guiding the curve downward enhances the risk of an excessive opening of the mix.

• Designing using the maximum quantity of filler and the minimum amount of the sand fraction is disadvantageous and risky.

• Care ought to be taken so that an increase of the sand fraction can be observed on subsequent sieves to supply enough material for making mastic.

• The quantity of filler should fluctuate around the middle of the allowable range (i. e., about 9-10% [m/m]) to make possible the collection of material on sieves smaller than 1.0 mm and to prevent the gradation curve from ris­ing upward.

• Non manufactured (natural) sand may be applied only for SMA layers on roads with low traffic volumes.

A thousand years of traffic on the Necho canal

So what is the actual path of this ancient canal? We know it fairly well, since the remains observed in the twentieth century substantially agree with the descriptions of Greek and Roman travelers. The canal issues from the eastern branch of the Nile, follows the southern edges of the wadi Tumilat valley, passes by Tell er-Retaba and Tell el – Maskhouta (Tjekou), rejoins lake Timsah, then flows directly south toward the bitter lakes and the Gulf of Suez. Let us listen to a continuation of the account of Herodotus: “It takes four days to travel along it, and its width is such that two triremes could be rowed in it side by side. It is fed by the waters of the Nile, and begins a little above Bubastis (Tell Basta) by Pithon (Tell el-Maskhouta), an Arabian town. It ends in the Red Sea. The excava­tion was begun in the part of the Egyptian plain which is nearest to Arabia. The mountains where the stone quarries are and which are close to Memphis, are near this plain. The canal was dug along the foot of these mountains from west to east, passing through a gorge (the wadi Tumilat valley?). It turns to the south out of the hill country toward the Arabian Gulf.”

The outlet of the canal into the sea, at the end of the Gulf of Suez, is obviously an important and critical site. Apparently land transfer of goods, or boat portage (a com­mon practice in Antiquity), is necessary up until the era of Ptolemy II. In 280 BC he built the terminal facilities. Diodore of Sicily describes them thus:

“(Ptolemy II) conceived a barrier adroitly placed at the most favorable location. One opened

it when one wanted to pass and immediately closed it, for it was well 38

designed for this purpose.”

When Strabo visited the delta region, this installation was clearly in use. He certain­ly did not travel as far as the Red Sea, but his account confirms that the flow in the canal is from the Nile toward the lakes:

“There is another canal that flows in to the Red Sea and the Arabian Gulf near the city of Arsinoe (…). It flows through the lakes that are called bitter. Originally, these lakes were, without doubt, bitter, but when the aforementioned canal entered them, their waters, mixed with those of the river, changed nature and are today full of fish and inhabited by aquatic birds.”

Regarding the terminal facility of Ptolemy, Strabo’s account is somewhat vague: “the Ptolemite kings finished the excavation and closed the passage, in such a way that they could, at will, freely exit the canal into the sea outside and reenter into the canal.”[123] [124]

One can get lost among all the conjectures regarding this “barrier”. It was probably a single gate that could be opened when the tide equalized the water levels in the two water bodies, but surely was not a true lock[125]. One can only regret that Diodore and

Strabo did not leave us a more precise description of this “barrier”.

The canal will later be maintained and kept in operation by the Romans, who val­ued a direct pathway to the incense and myrrh of Arabia Felix. They named it Trajan s Canal after the work of this emperor, (who moved the point of entry toward Heliopolis, probably to improve the flow). The canal is also renovated and used by the Arabs, start­ing in 641 AD and up until its closure in 767 or 775 AD, to ship Egyptian wheat to Mecca and Medina. This closure was ordered by the Caliph Abu Jafr al-Mansur.[126] For twelve centuries, Necho’s project assures communication between the Red Sea and Indian Ocean with the Mediterranean Sea.

Few technological achievements can lay claim to such long success.

RIGID PAVEMENT DESIGN PROCEDURE

3.

Подпись: FIGURE 3.10 Load-response curve used to determine resilient modulus. (From M. Thompson, “Factors Affecting the Resilient Modulus of Soils and Granular Materials," Proceedings of Workshop on Resilient Modulus Testing, Oregon State University, Corvallis, 1989, with permission)

Determine the drainage coefficient for the pavement.

4. Select the design serviceability loss.

5. Estimate the total number of 18-kip (80-kN) equivalent single-axle loads for the design period.

6. Select a level of reliability and the overall standard deviation.

7. Determine slab thickness and steel reinforcement.

Подпись: k Подпись: 19.4 Подпись: (3.7)

Determine Effective Subgrade Modulus. The first step in designing the thickness of a rigid pavement is the determination of the effective modulus of subgrade reaction. The effective modulus (or composite modulus) is the modulus of subgrade reaction after cor­rection for use of subbase, seasonal variation in subgrade and subbase strength, rigid foundation within 10 ft (3 m) of the surface, and loss of support. Figure 3.11 is used to determine the composite modulus of subgrade reaction when a subbase will be used under the concrete pavement. If the pavement will be placed directly on the subgrade, the AASHTO Pavement Design Guide recommends a composite modulus of subgrade reaction of:

where k is in lb/in3 and MR is in lb/in2.

When a stiff layer (bedrock, etc.) is located within 10 ft (3 m) of the surface, the stiff layer will provide additional support for the pavement. Figure 3.12 is used to correct the composite modulus of subgrade reaction for this additional support.

Example:

DSB = 6 in (150 mm)

ESB = 20,000 lb/in2 (138 MPa)

MR = 7,000 lb/in2 (48 MPa)

RIGID PAVEMENT DESIGN PROCEDURE RIGID PAVEMENT DESIGN PROCEDURE RIGID PAVEMENT DESIGN PROCEDURE

Solution: k* = 400 lb/in3 (0.109 N/mm3)

2000 3000 5000

7000

10,000

12,000

16,000

Подпись: Roadbed Soil Resilient Modulus, MR (lb/in2)

20,000

FIGURE 3.11 Chart for estimating composite modulus of subgrade reaction kra, assuming a semi­infinite subgrade depth; defined as over 10 ft (3 m) below subgrade surface. (From Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials, Washington, D. C., 1993, with permission)

RIGID PAVEMENT DESIGN PROCEDUREIn regions where large moisture variations, freeze and thaw, etc., will affect the strength of the subgrade soils and subgrade, AASHTO provides a procedure to modify the composite modulus of subgrade reaction. Table 3.22 provides a worksheet, and Table 3.23 shows an example. The seasonal variation in strength is determined using laboratory procedures or nondestructive testing (NDT). The seasonal strength of the subbase and subgrade is entered in columns 2 and 3 of Table 3.22. The composite modulus of subgrade reaction is determined using Fig. 3.11 and entered in column 4. If a rigid foundation is present within 10 ft (3 m) of the surface, the k value is corrected

Example:

MR = 4000 Ib/in2 (27.6 MPa)

□SG = 5 ft (1.5 m)

= 230 Ib/in3 (0.062 N/mm3)

Подпись: 260Подпись:Подпись:RIGID PAVEMENT DESIGN PROCEDURESolution: к = 300 Ib/in3 (0.081 N/mm3)

Подпись: Roadbed Soil Resilient Modulus, MR (Ib/in2)Modulus of Subgrade Reaction, к (ib/in3)

(modified lo account for presence of rigid foundation near surface)

FIGURE 3.12 Chart for modifying modulus of subgrade reaction to consider effect of rigid foundation near the surface. Conversions: 1 lb/in2 = 6.895 X 10-3 MPa, 1 lb/in3 = 271.4 X 10-4 N/mm2, 1 ft = 0.305 m. (From Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials, Washington, D. C., 1993, with permission)

Trial subbase: Type _________________ Depth to rigid foundation, ft

Thickness, in___________ Projected slab thickness, in

Loss of support, LS______

Month

(1)

Roadbed modulus Mr, lb/in2 (2)

Subbase modulus, ESb, lb/in2

(3)

Composite k value, lb/in3 (Fig. 3.11) (4)

k value, lb/in3 on rigid foundation

(Fig. 3.12) (5)

Relative damage u (Fig. 3.13) (6)

January

February

March

April

May

June

July

August

September

October

November

December

Average: ur = ____

n

Summation: =

Effective modulus of subgrade reaction k (lb/in3) =_________

Corrected for loss of support (Fig. 3.14): k (lb/in3) =________

Conversions: 1 lb/in2 = 6.895 X 10_3 MPa, 1 lb/in3 = 271.4 X 10_4N/mm2, 1 in = 25.4 mm, 1 ft = 0.305 m. Source: Guide for Design of Pavement Structures, American Association of State Highway and

Transportation Officials, Washington, D. C., 1993, with permission.

TABLE 3.23 Example Application of Method for Estimating Effective Modulus of Subgrade Reaction

Trial subbase: Type ____ Grammar_____ Depth to rigid foundations, ft__________ 5

Thickness, in ___________ Projected slab thickness, in ___________ 99

Loss of support, LS І-0

Month

(1)

Roadbed modulus Mr, lb/in2 (2)

Subbase modulus, ESB, lb/in2

(3)

Composite k value, lb/in3 (Fig – 3-11) (4)

k value, lb/in3 on rigid foundation

(Fig – 3,12) (5)

Relative damage ur (Fig – 3-13) (6)

January

20,000

50,000

1,100

1,350

0-35

February

20,000

50,000

1,100

1,350

0-35

March

2,500

15,000

160

230

0-86

April

4,000

15,000

230

300

0-78

May

4,000

15,000

230

300

0-78

June

7,000

20,000

410

540

0-60

July

7,000

20,000

410

540

0-60

August

7,000

20,000

410

540

0-60

September

7,000

20,000

410

540

0-60

October

7,000

20,000

410

540

0-60

November

4,000

15,000

230

300

0-78

December

20,000

50,000

1,100

1,350

0-35

Summation: Xu = 7-25

Average: йг = X r =____ °-60________

r n

Effective modulus of subgrade reaction k (lb/in3) = 540 Corrected for loss of support (Fig – 3-14): k (lb/in3) = 170

Conversions: See Table 3-22-

Source: Guide for Design of Pavement Structures, American Association of State Highway and

Transportation Officials, Washington, D-C-, 1993, with permission-

RIGID PAVEMENT DESIGN PROCEDURE

FIGURE 3.13 Chart for estimating relative damage to rigid pavements based on slab thickness and underlying support. Conversions: 1 lb/in3 = 271.4 X 10_4 N/mm2, 1 in = 25.4 m. (From Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials, Washington, D. C., 1993, with permission)

using Fig. 3.12 and entered in column 5. The corrected k value is then used in Fig. 3.13 to determine the seasonal or relative damage factor, which is entered in column 6. The sum of relative damage is divided by the total number of periods to determine the average rel­ative damage factor. This value is entered in Fig. 3.13 to determine the average composite modulus of subgrade reaction for the year. Many concrete pavements fail as a result of pumping or loss of support under the slab. Figure 3.14 is provided to correct the effective modulus of subgrade reaction for loss of support. This figure lowers the k so that the

RIGID PAVEMENT DESIGN PROCEDURE

FIGURE 3.14 Chart for correction of effective modulus of subgrade reaction for potential loss of subbase support. Conversion: 1 lb/in3 = 271.4 X 10~4 N/mm2. (From Guide for Design of Pavement Structures, American Association of State Highway and Transportation Officials, Washington, D. C., 1993, with permission)

stress in the slab will be the same as for a slab with a void. Although the AASHTO procedure includes design for loss of support, it is recommended that a pavement base be designed to prevent or reduce loss of support, especially under pavements supporting a large number of heavy loads in wet areas. The cost of providing a base to resist loss of support may be less than the cost of restoring support in the future.

Select Pavement Material Properties. The next step in the design of a rigid pavement is to select material properties. The reliability level and overall standard deviation consider the variation in material properties. Therefore, average material property values must be used in design. The concrete material values needed for design are the average concrete modulus of elasticity and the average concrete modulus of rupture. These values are not known until after construction of the pavement unless the plans for the pavement contain a performance specification. Material properties from past pave­ment construction may be used for design purposes provided a similar mix will be used. American Society for Testing and Materials’ “Test Method for Static Modulus of Elasticity and Poisson’s Ratio of Concrete in Compression,” ASTM C469, details the laboratory test method for determining the concrete modulus of elasticity. ASTM C78,

TABLE 3.24 Recommended Values of Drainage Coefficient Cd for Rigid Pavement Design

Quality of

drainage

Percent of time pavement structure is exposed to moisture levels approaching saturation

Less than 1%

1-5%

5-25%

Greater than 25%

Excellent

1.25-1.20

1.20-1.15

1.15-1.10

1.10

Good

1.20-1.15

1.15-1.10

1.10-1.00

1.00

Fair

1.15-1.10

1.10-1.00

1.00-0.90

0.90

Poor

1.10-1.00

1.00-0.90

0.90-0.80

0.80

Very poor

1.00-0.90

0.90-0.80

0.80-0.70

0.70

Source: Guide for Design of Pavement Structures, American Association

of State Highway and Transportation Officials, Washington, D. C., 1993, with permission.

“Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading),” details the laboratory test method for determining the concrete flexural strength (modulus of rupture).

Determine Drainage Coefficient. The drainage coefficient is used to modify the design thickness for drainage conditions. Moisture affects the pavement performance by decreasing the strength of the subgrade and subbase material and affects the warping and curling behavior of the concrete slabs. The intent of the drainage coefficient is to allow performance prediction for pavements without a proper drainage system. Increasing the pavement thickness should not be used in lieu of a properly designed drainage system. Recommended values for the drainage coefficient are given in Table 3.24. The Federal Highway Administration’s “Highway Subdrainage Design Manual,” Report No. FHWA-TS-80-224, provides a procedure that may be used to determine drainage times for base material. (See Arts. 5.4.5 and 5.4.6.)

Select Design Serviceability Loss. The design serviceability loss is the amount of serviceability loss the agency will tolerate before rehabilitation. To select a design ser­viceability loss, the designer needs to know the initial serviceability and the terminal serviceability of the pavement. The initial serviceability is the serviceability immedi­ately after construction. Since this value is unknown at the time of construction, the designer will usually use the average initial serviceability of previously constructed pavements. The terminal serviceability is the serviceability of a pavement immediately before rehabilitation. The terminal serviceability is a function of traffic volume and speed. A low-volume road with low speeds may be allowed to deteriorate to a lower serviceability than a high-volume freeway, since the associated user costs will be lower. The terminal serviceability used by an agency is a policy decision. Common terminal serviceabilities are 2.5 for high-volume roads and 2.0 for low-volume roads.

Estimate ESALs. The daily ESAL loadings are determined as outlined in Art. 3.5.2. The total number of ESAL loadings for design is the cumulative number of ESAL loadings expected over the design life of the pavement. This value can be determined by assuming a growth rate or, if the pavement is being built on an existing alignment, by extrapolating past traffic patterns.

Determine Slab Thickness and Reinforcement. The design slab thickness is deter­mined by using the design values as outlined above in the nomograph shown in Fig. 3.15. The design thickness is usually rounded up to the nearest 1/2 or 1 in (13 or 25 mm) depending on the local practice for specifying slab thickness. As mentioned in Art. 3.5.2, if the design thickness varies significantly from the thickness used to determine the equivalency factors, the equivalency factors should be recalculated and the thickness design checked. Determination of steel reinforcement content, if used, is detailed in Art. 3.1.1.

The great accomplishments of Egypt in the first millennium BC: from the last pharaohs to the Persians

Who dug the first “Suez Canal”?

Egypt had a long tradition of maritime commerce with countries on the shores of the Red Sea (in particular with the country of Punt, situated approximately east of Sudan and north of Eritrea). The port of Mersa Gawasis was founded in about 1900 BC under Amenemhat II (Middle Empire), somewhat below the 27th parallel (therefore a bit to the north of Thebes). Merchandise is carried by land to or from Thebes through arid val­leys, typified by the wadi Hammamat.

One can therefore readily appreciate the interest in a direct maritime link between the Nile and the Red Sea. Such a link was in fact realized though the valley of the wadi Tumilat with an east-west orientation. The remains of the canal were still visible in the 19th century, as described by Marice Linant de Bellefonds (1799-1883):

“More than forty years ago one could see, in the northern portions of today’s wadi Toumilat, the remains of an ancient canal that had rather small dimensions; it came from the west and flowed to the east along the desert and cultivated lands…. Near Tel-Retabee, this canal joined another much larger one, at a place called Ras el Wadi. this was the principal ancient canal..

It is there that the other canal from the northern portion joins the latter, which is much larger

30

and looks to be a very old and well built canal. u

All of the classic authors mention the existence of this very ancient and large-scale link. Certain of them (Aristotle, Strabo, Pliny) attribute the paternity of the canal to a pharaoh whom they call “Sesostris.”[115] [116] But archaeology clearly rules out the existence of such a communication link in the middle or new Empire. At best, one may consider the possibility that in the new Empire a small canal was constructed to transport stones from the Nile to the monuments constructed by Ramses II at the site known today as Tell

el Retaba. This is surely the small northern canal of the text cited.[117]

33

Modern studies[118] show that it is once again Herodotus who gives us the most pre­cise information. He situates the real beginning of the construction of the large canal during the reign of the Pharaoh Necho II, of the Saite Dynasty, who reigned in about 600 BC. This pharaoh, like most of his predecessors, pursued a policy of expansion toward the east, taking advantage of the fall of the Assyrian Empire. He builds a fleet of boats and embarks on an African expedition. Necho II founds the city of Tjekou, on the site of today’s Tell el Maskhouta, some fifteen kilometers to the west of the present Ismailia.[119] [120] Most of the canal was therefore built in that period (at least as far as Tjekou).

Two independent sources, that we will cite below, indicate the digging of the canal was not effectively ended until a century later, about 5009 BC, by order of the Persian sovereign Darius I.

Here is what Herodotus says:

“The son of Psammetichus was Necos, and he too became king of Egypt, and he was the first to attempt to dig a canal into the Red Sea; Darius the Persian was the second to dig it.”33

According to this author, some 120,000 workers were employed for the task by Necho. In 1866 Ferdinand de Lesseps, during his preliminary reconnaissance for con­struction of the Suez Canal, recognizes the traces of this ancient canal, well to the east of the vestiges found by Linant de Bellefonds.[121] At Kabret, 130 km from Suez on the isthmus of the same name, de Lesseps discovers a stela of pink granite engraved with the name of Darius, and with the following inscription in several languages:

“the king Darius declares: I am a Persian. From Persia, I took Egypt. I ordered this canal to be dug, from a river of the name of Nile, that flows in Egypt, to the sea that comes from Persia. Therefore, this canal was dug, as I had ordered, and boats go from Egypt, by this canal, toward Persia, as it pleases me.”[122]

The Persian’s strategic interest in this canal is obvious. But it is clear that Darius wrongly attributes the conquest of Egypt to himself, since this conquest was in fact accom­plished by his predecessor Cambyse. Therefore it is not surprising that he also exagger­ates his own role in the digging of the canal. Necho’s channel may have become clogged with silt or sand in the era of the sovereign Darius, who therefore would have re-excavat­ed it. It is also possible that the canal project undertaken by Necho was simply not com­pleted. In any case, three other stelas carrying the name of Darius will be discovered along the route of the canal, the most westerly being at Tell el-Mashkhuta (Figure 3.8).

The great accomplishments of Egypt in the first millennium BC: from the last pharaohs to the Persians

RISER DRAWINGS

Riser drawings are used when figuring out drainage systems, just as they are used with vent systems. I want to give you some sample riser diagrams to look over. The drawings will show you what your drawings might look like. Drains are drawn with solid lines, while vents are indicated by broken lines.

RISER DRAWINGS

FIGURE 4.26 ■ Wet venting top floor single bath group. (Courtesy of Standard Plumbing Code)

RISER DRAWINGS

FIGURE 4.27 ■ Wet venting top floor double bath back to back. (Courtesy of Standard Plumbing Code)

RISER DRAWINGS
RISER DRAWINGS

OR

 

SHOWERS

 

M N

 

RISER DRAWINGS
RISER DRAWINGS
RISER DRAWINGS

RISER DRAWINGS

RISER DRAWINGS

FIGURE 4.28 ■ Wet venting lower floors on multistory buildings. (Courtesy of Standard Plumbing Code)

RISER DRAWINGS

RELIEF VENTS A & В MAY CONNECT TO MAIN—„„

VENT ABOVE FIXTURES ON FLOOR ABOVE

-T—I

 

RELIEF VENTS REQUIRED-
914.1

 

SEE 715 UNLESS SIZED FOR A BUILDING DRAIN

 

RISER DRAWINGS
RISER DRAWINGS

FIGURE 4.29 ■ Riser diagram. (Courtesy of Standard Plumbing Code)

 

Подпись: DRAIN-AND-SEWER CALCULATIONS

RISER DRAWINGSRISER DRAWINGSRISER DRAWINGS

RISER DRAWINGS

4" MIN.

 

STACK VENT

 

i-1 1/4" MIN. I MAN VENT—”j

 

RISER DRAWINGS

1 1/2" MIN;

 

WET VENT 1 II

 

RISER DRAWINGS

REQUIRED VENT

 

‘klT U~i

 

REQUIRED VEfJT

 

WET VENT

 

1 1/4" MIN;

 

VENT NOT REQ

 

RISER DRAWINGS
RISER DRAWINGS
RISER DRAWINGS
RISER DRAWINGS

RISER DRAWINGS

RISER DRAWINGS

FIGURE 4.30 ■ Multistory wet venting. (Courtesy of Standard Plumbing Code)

RISER DRAWINGS

FIGURE 4.31 ■ Riser diagram. (Courtesy of Standard Plumbing Code)

 

Подпись: DRAIN-AND-SEWER CALCULATIONS

Q

SO

 

”I

о

 

Подпись: PLUMBER’S AND PIPE FITTER’S CALCULATIONS MANUAL

YOKE VENT
REQUIRED
SECTIONS
715.4 &
917.3

45’ MAX.
FROM
VERTICAL
TO PREVENT
FOULING

 

-SIZE FOR
NUMBER OF
FIXTURE UNITS
ABOVE OFFSET

FOR

OMISSION ________

SEE SEC. T
715.5

 

BLOWOUT FIXTURES REQUIRE INDIVIDUAL VENTS,7 BATTERY VENTING NOT PERMITTED. /

 

SIZE OF EACH SECTION OF VENT BASED ON FIXTURE UNIT LOAD.

 

RELIEF VENT REQUIRED, SEE 715.4
<St 917.3 {SIZE FOR MAIN VENT)

FOR OMISSION SEE
SEC. 715.5

H—-NO HORIZONTAL BRANCH CONNECTION 715.4

 

OFFSET OF MORE THAN 45* FROM VERTICAL SIZE AS FOR A BUILDING DRAIN. 715.4

 

-NO HORIZONTAL-1 BRANCH CONNECTION 715.4

 

SIZE FOR OFFSET AND TOTAL FIXTURE UNITS ON STACK

 

RISER DRAWINGS

4 /L

 

VENT FIXTURES BACK TO BACK IN BATTERY, 914.3

 

FIGURE 4.32 ■ Riser diagram. (Courtesy of Standard Plumbing Code)

 

RISER DRAWINGSRISER DRAWINGSRISER DRAWINGSRISER DRAWINGS

RISER DRAWINGS

VENT TERMINAL SEC. 904.2 AND 905

 

BATTERY WITH LOOP VENT

 

SIXTH FLOOR

 

RISER DRAWINGS

o4-o o4-o LrTJ

 

FIFTH FLOOR

 

4

 

RISER DRAWINGS

——

 

FOURTH FLOOR

 

RISER DRAWINGS
RISER DRAWINGS

BACK TO BACK IN BATTERY

 

THIRD FLOOR

 

RISER DRAWINGS
RISER DRAWINGS

SECOND FLOOR

 

RISER DRAWINGS

RISER DRAWINGS

RISER DRAWINGS

FIGURE 4.33 ■ Drainage waste and vent reference diagram. (Courtesy of Standard Plumbing Code)

RISER DRAWINGS

FIGURE 4.34 ■ Fixtures back-to-back in battery. (Courtesy of Standard Plumbing Code)

OTHER USEFUL POWER TOOLS

Although not often used by frame car­penters, two other power tools worth mentioning are routers and sanders. These tools are typically used for finish work, like preparing a bookcase for stain or paint, which makes them more suit­able for trim and finish carpenters.

Routers

Routers are classified according to the largest-diameter bit shank that the collet (or chuck) can handle. Common sizes are Уд in., 3/s in., and У2 in. The most powerful routers have З-plus HP motors and can make big cuts through heavy stock in a single pass.

Equipped with the right bit, the router can be adapted to a number of carpen­try jobs. You can cut grooves and notches, round over edges, trim materi­als (like plastic laminate), and make moldings with a router. With the right attachment, you can cut perfect circles and even mortise doors for hinges.

For all its usefulness, though, a router can be a frightening tool. Its motor is directly connected to an unguarded bit, which protrudes from a base plate. The motor can spin the bit up to 25,000 rev­olutions per minute (rpm) or faster and makes more noise than a roomful of rock musicians. Because of the noise and the fact that the bit is exposed, I always wear ear protection and safety glasses

OTHER USEFUL POWER TOOLS

The belt sander (left) can remove a lot of stock quickly. The pad sander (right) is more suited to finish work.

 

Carpenters learn to work without sawhorses, but there are times when a good pair of horses can save your back or help you work overhead. Stock placed on sawhorses allows you to cut, drill, or shape materials without having to bend over. For a tempo­rary workstation, top a pair of horses with a sheet of plywood. Planks placed across horses make a good scaffold.

There are many different styles of sawhorses.

Some are works of art suitable for a living room, and others are simple structures. Here are plans and instructions for a strong sawhorse that can be made in just a few minutes.

Measure and cut the 8-ft. 2×4 into two 48-in.-long pieces. Then measure and cut four 30-in.-long legs from the 12-ft. 1×6.

Begin assembly with the crosspiece of the sawhorse. Turn one 2×4 on edge, center another 2×4 flat on the first so that they form a T, and nail them together with four 16d nails.

Next, position the end of a 1×6 leg flush with the top of the crosspiece (see the drawing on the facing page) and about 2 in. from the end. Nail the legs (or use 1 Уг-іп. drywall screws) with two 8d nails into both pieces of the T. Do the same with the remain­ing three legs, then set the sawhorse upright.

Nail the 48-in.-long 1×4 to the top of the crosspiece with two or three 8d nails. Countersink these nail heads with a nailset so you won’t hit them with a saw blade when you are cutting on the sawhorse.

Now take a plywood square and hold it against the legs on one side of the sawhorse. Make sure the top of the plywood is flush to the bottom of the cross­piece, then trace the outline of the legs onto the ply­wood. Repeat with the other piece of plywood on the other side of the sawhorse. These pieces brace the legs of the sawhorse.

Cut both plywood pieces along the lines you traced. Use several 8d nails or 1[4]/2-in. drywall screws to at­tach the plywood pieces to the legs, one on each end and flush with the bottom of the crosspiece. (You may have to predrill small pilot holes to keep the nails or screws from splitting the wood.)

when I’m working with a router. And I’m always extremely aware of where my hands are.

Retrofit Results

construction Basement: Ceiling sprayed with open-cell spray polyure­thane foam (adds thickness and R-value)

Walls: 2×4 construction filled with cellulose; 4 in. of foil­faced polyisocyanurate foam outside of sheathing for a total of R-39

Roof: 6 in. of polyisocyanurate insulation installed above the existing roof sheathing, topped with a layer of ply­wood; 8 in. of open-cell spray polyurethane foam (Icynene) installed between the existing rafters for a total of R-59

Windows: Double-pane (U-0.33) windows by Pella®

Location:

Arlington, Mass. Size: 3,000 sq. ft. (duplex)

Renovation cost: $47 per sq. ft.; $140,000 total

Mechanicals

Heating: Oil-fired steam boiler in each unit

Water: Main boiler in unit 1; on-demand gas water heater in unit 2

Ventilation: Heat-recovery ventilators (one for each apartment)

Results

Energy reduction: 65% (heat­ing fuel)

Annual savings: $2,300 per year

Payback period: 61 years. If the cost of the roofing and siding are subtracted, pay­back is reduced to a little over 35 years.

Подпись: Air-seal, then insulate. If you can't afford to add insulation to your walls, address simple air-sealing measures such as filling the gaps around windows, electrical boxes, doors, and recessed lights in the ceilings.

Retrofit Results

WALL INSULATION

Deep-energy retrofit A typical 2×4 wall insulated with fiberglass batts has a whole – wall R-value of about 10. Many deep-energy retrofits aim to insulate walls to R-40, which typically requires all of the siding to be re­moved and the addition of 4 in. to 5 in. of polyisocyanurate rigid insulation or spray polyurethane foam.

Practical takeaway Unless your home’s existing siding is in bad shape, it’s hard to justify the cost of installing exterior wall foam. If your existing siding is sound, your best retrofit option is careful air-sealing work from the interior with canned spray foam. Typical leakage areas include the gap be­tween the baseboard and the finished floor; electrical boxes; and cracks behind window and door casing.

Performance comparison Above-grade walls represent most of a typical house’s thermal envelope, and an R-10 wall leaks heat at four times the rate of an R-40 wall. Although air-sealing an R-10 wall will surely increase its performance, it will not rival an R-40 wall.

Cost comparison Installing thick exterior – wall foam and new siding on a typical house costs tens of thousands of dollars. Blower- door-directed air-sealing work might cost $700 to $1,000 per house.

WINDOWS

Deep-energy retrofit Single- or double­glazed windows are usually replaced with new triple-glazed windows with full­thickness (13/8 in.) glazing. This glazing is better than thin 7/8-in. or 1-in. glazing.

Practical takeaway The cost of installing high-quality replacement windows can be staggering; as a less expensive alternative, consider installing low-e storm windows over tuned-up windows in good working order and that have been weatherstripped.

Performance comparison Good triple­glazed windows have a U-factor of 0.17 to

0. 20. A low-e storm window won’t achieve the same performance. Installed over a single-pane wood window, a low-e storm window provides a total U-factor of 0.40, while a low-e storm window installed over a double-pane wood window provides a total U-factor of 0.34. (The lower the U-factor, the better.)

Cost comparison The cost to install a low-e storm window ranges from $120 to $160. The installed cost of a new triple-glazed win­dow is about $800 to $1,200.

HVAC

Deep-energy retrofit Most deep-energy ret­rofits include air-sealing measures. Once in­filtration rates have been reduced, an older house requires a good mechanical ventila­tion system. Options range from low-sone bathroom exhaust fans controlled by timers to heat-recovery ventilation systems with dedicated ductwork.

A new heating unit is also a quintessen­tial upgrade in many deep-energy retrofits. New furnaces or boilers are most often ef­ficient sealed-combustion models. The fuel type is relatively unimportant, because the fuel demands of the newly renovated home will be low.

Practical takeaway If you’ve done any air-sealing work, a mechanical ventilation system is essential. Exhaust-only systems are

much less expensive than a system with a heat-recovery ventilator. If you can’t afford an HVAC overhaul, you should at least have ducts tested for leakage and sealed.

Performance comparison Replacing an 80% AFUE (annual fuel utilization efficien­cy) furnace with a 92% AFUE furnace will cut energy use 13%. Sealing ducts may save an additional 5% to 20% of your energy use.

Cost comparison The installed cost of a new 92% AFUE furnace ranges from $3,000 to $6,000. Duct sealing and repair costs be­tween $250 and $500 per house.

 

Retrofit Results

 

Retrofit Results

Retrofit Results

Martin Holladay is a contributing editor to Fine Homebuilding.

Подпись: Tight ducts save money. Sealing leaky ductwork can be done in several ways, but mastic and fiberglass-mesh tape are among the best options. Подпись: cost: $148K Annual savings: $2,490 Alarmed by the implications of the global climate-change crisis, Cador Price-Jones embarked on a major retrofit of his Massa-chusetts duplex (also pictured on p. 40).

construction

Basement: 3 in. of closed-cell spray foam (R-18) applied be­tween the studs of a 2×4 wall built against an 8-in. block foundation

Walls: Existing 2×4 walls filled with dense-pack cellulose; new 2×2 frame installed on exterior and filled with 4 in. of closed-cell spray foam for a total of R-37

Roof: Attic floor air-sealed and filled with 17 in. of loose-fill cellulose for an R-value of 60; 2 in. of spray foam used to air-seal the eaves

Windows: Main house win­dows are double-glazed, low-e, argon-filled units by Jeld – Wen®; basement windows are double-glazed hopper units by Harvey Industries

Mechanicals

Heating: Modulating condens­ing gas boiler, 22,700- to 75,200-Btu rated output,

95% AFUE

Water: 60-gal. Superstor® indirect hot-water tank

Ventilation: Heat-recovery ventilators (one for each apartment)

Photovoltaic: 5.25kw pack­age system by Nexamp™

Results

Energy reduction: From $5,650 per year to $3,160 per year

Annual savings: $2,490

Payback period: 60 years

Подпись: го
Подпись: INSULATION

Retrofit Results

s

S

aving money on heating-fuel costs is a lot simpler than negotiating with OPEC or your local utility. Here’s how: On a recent upgrade in the attic of a 1950s-era house (one of two projects that is featured here), I air-sealed and spread a 12-in.-deep layer of cellulose throughout 1,500 sq. ft. of space in about a day. As a result of this and other energy-saving improvements that were made to the home, the owner saw his heating and cooling costs reduced by half compared to the previous year, even in the face of higher electricity and heating-fuel costs.

I typically focus my efforts to improve the energy efficiency of an attic in two areas: sealing air leaks in the ceiling and increasing the amount of insulation in the attic itself.

The payback period for tightening a leaky ceiling can be as short as a month. Add­ing insulation might take a few heating or cooling seasons to pay off, but the wait is relatively brief. I estimate the payback for air-sealing and upgrading attic insulation to be realized in three years.

On these projects, I often chose to install a radiant reflective membrane. Besides

Подпись:reducing radiant-heat gain from the roof, the membrane makes the attic more attractive and dust-free for storage use, and it keeps the blown-in insulation I use from blocking the rafter bays. While radiant bar­riers can reduce peak attic temperatures by 10°F to 30°F, they haven’t proved to be cost effective in all geographic regions or in at­tics that are adequately insulated, that are air-sealed, and that have well-insulated, wrapped air-handling equipment and ductwork. In these cases, you may be better off spending the money on more insulation and air-sealing than on a radiant barrier.

Multivariate normal distributions

Multivariate normal distributions

A bivariate normal distribution has a PDF defined as

for k = 1 and 2. As can be seen, the two random variables having a bivariate normal PDF are, individually, normal random variables. It should be pointed out that given two normal marginal PDFs, one can construct a bivariate PDF that is not in the form of a bivariate normal as defined by Eq. (2.108).

Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions

According to Eq. (2.17), the conditional normal PDF of X11 x2 can be ob­tained as

(2.109)
>■;

Подпись: 0.1

Подпись: 0 Подпись: p = 0.0 Подпись: 0 Подпись: 2
Multivariate normal distributions
Multivariate normal distributions
Подпись: 0
Multivariate normal distributions
Подпись: p = - 0.8
Multivariate normal distributions
Подпись: 0
Подпись: 0.2
Подпись: 0
Подпись: 2
Подпись: 2
Подпись: p = - 0.4
Подпись: 0.2
Подпись: 2
Подпись: 2
Подпись: 2

.H.

Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions Multivariate normal distributions

0.1

Multivariate normal distributionsMultivariate normal distributionsMultivariate normal distributionsMultivariate normal distributionsMultivariate normal distributionsMultivariate normal distributionsFigure 2.26 Three-dimensional plots of bivariate standard normal probability density functions. (After Johnson and Kotz, 1976.)

Multivariate normal distributions

O1 > O2 Oj = O2 O1 < O2

Multivariate normal distributions

 

x2

 

Multivariate normal distributions

в = 135° XJ

 

Multivariate normal distributionsMultivariate normal distributionsMultivariate normal distributionsMultivariate normal distributions

Multivariate normal distributions

Figure 2.27 Contour of equal density of bivariate standard normal probability density func­tions. (After Johnson and Kotz, 1976.)

for —to < x1 < to. Based on Eq. (2.109), the conditional expectation and variance of the normal random variable X11 x2 can be obtained as

E(X1 | x2) = P1 + P12(ff1/V2)(x2 – P2) (2.110)

Var(X11 x2) = (1 – P22) (2.111)

Expressions of the conditional PDF, expectation, and variance for X2 | x1 can be obtained immediately by exchanging the subscripts in Eqs. (2.109) through (2.111).

Подпись: for —TO < x < TO (2.112)

Подпись: , , . IC,-1!1'2 f-(x) = ГПт exp
Подпись: (x f-lx) Cx (x f-lx)

For the general case involving K correlated normal random variables, the multivariate normal PDF is

in which і, = (д1, p.2,…, /гК )t, a К x 1 column vector of the mean values of the variables, with the superscript t indicating the transpose of a matrix or vector, and Cx is a К x К covariance matrix:

^11

012

■ ‘ ‘ p1K

021

022

■ ■ ■ 02K

Pk 1

pK 2

■ ■ ■ °KK_

Cov(X) = С, =

This covariance matrix is symmetric, that is, Pjk = okj, for j = k, where ajk = Cov(Xj, Xk). In matrix notation, the covariance matrix for a vector of random variables can be expressed as

Подпись: (2.113)С, = E [(X — і,)(X — і,)4

Подпись: | R —111'2 ( 1 Ф (z) = (2;) К /2 exp (- 2zt R— z Подпись: for —TO < z < TO Подпись: (2.114)

In terms of standard normal random variables, Zk = (Xk — ik)’ak, the stan­dardized multivariate normal PDF, can be expressed as

in which Rx = Cz = E (ZZ *) is a К x К correlation matrix:

1

P12 ■

■ P1K

P21

1 ■

■ P2K

PK1

PK2 ■

■ 1

Corr(X) = Cov(Z) = Rx =

with pjk = Cov(Zj, Zk) being the correlation coefficient between each pair of normal random variables Xj and Xk. For bivariate standard normal variables,

Multivariate normal distributions Подпись: (2m)!(2n)! minm,n) (2p^)2і 2m+n (m - І)!(n - І)! (2j)! (2m + 1)!(2n + 1)! (2p12)2 j 2m+n P12 (m - І )!(n - j )!(2j + 1)! E [Z2mZ2n+1 ] = 0 (2.115)

the following relationships of cross-product moments are useful (Hutchinson and Lai, 1990):

for m and n being positive integer numbers.