Mathematicians and inventors of Alexandria and the Hellenistic world

In 335 – 331 BC, Alexander the Great conquered the totality of Greece and the Persian Empire, including Egypt and Mesopotamia. The spirits of analysis and hydraulic know­how now were brought together in the same crucible, fueled by the need to innovate – in order to ensure survival in a world whose boundaries were suddenly pushed enlarged, and to increase agricultural productivity to meet the needs of the new ruling classes. This crucible has a name: Alexandria.

Mathematicians and inventors of Alexandria and the Hellenistic world

Reliability Analysis Considering Load-Resistance Interference

4.1 Basic Concept

The design of a hydrosystem involves analyses of flow processes in hydrology and hydraulics. In a multitude of hydrosystems engineering problems, uncer­tainties in data and in theory, including design and analysis procedures, war­rant a probabilistic treatment of the problems. The risk associated with the potential failure of a hydrosystem is the result of the combined effects of in­herent randomness of external loads and various uncertainties involved in the analysis, design, construction, and operational procedures. Hence, to evaluate the probability that a hydrosystem will function as designed requires uncer­tainty and reliability analyses.

As discussed in Sec. 1.5, failure of an engineering system can be defined as the load L (external forces or demands) on the system exceeding the resistance R (strength, capacity, or supply) of the system. The reliability ps is defined as the probability of safe (or nonfailure) operation, in which the resistance of the structure exceeds or equals to the load, that is,

Ps = P (L < R) (4.1)

in which P(■) denotes the probability. Conversely, failure probability pf can be computed as

Pf = P (L > R) = 1 – ps (4.2)

The definitions of reliability and failure probability, Eqs. (4.1) and (4.2), are equally applicable to component reliability, as well as total system

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reliability. In hydrosystems engineering analyses, the resistance and load frequently are functions of several stochastic basic variables, that is, L = g( Xl) = g( X ь X 2,…, Xm) and R = h( Xr ) = h( Xm+1, Xm+2,…, Xk ), where Xі, X2,…, XK are stochastic basic variables defining the load function g(Xl) and the resistance function h(Xr ). Accordingly, the failure probability and reliability are functions of stochastic basic variables, that is,

ps = P [g(Xl) < h(Xr)] (4.3)

Note that the foregoing presentation of load and resistance in reliability anal­ysis should be interpreted in a very general context. For example, in the design and analysis hydrosystems infrastructures, such as urban drainage systems, the load could be the inflow to the sewer system, whereas the resistance is the sewer conveyance capacity; in water quality assessment, the load may be the concentration or mass of pollutant entering the environmental system, whereas the resistance is the permissible pollutant concentration set by water quality regulations; in the economic analysis of a hydrosystem, the load could be the total cost, whereas the resistance is the total benefit.

Evaluation of reliability or failure probability by Eqs. (4.1) through (4.3) does not consider the time-dependent nature of the load and resistance if statistical properties of the elements in Xl and Xr do not change with time. This procedure generally is applied when the performance of the system subject to a single worst-load event is considered. From the reliability computation viewpoint, this is referred to as static reliability analysis.

In general, a hydrosystem infrastructure is expected to serve its designated function over an expected period of time. Engineers frequently are interested in knowing the reliability of the structure over its intended service life. In such circumstances, elements of service period, randomness of load occurrences, and possible change in resistance characteristics over time must be consid­ered. Reliability models incorporating these elements are called time-dependent reliability models (Kapur and Lamberson, 1977; Tung and Mays, 1980; Wen, 1987). Computations of the time-dependent reliability of a hydrosystem infras­tructure initially require the evaluation of static reliability. Sections 4.3 through

4.6 describe methods for static reliability analysis, and Sec. 4.7 briefly describes some basic methods for dealing with the time-dependent nature of reliability analysis.

As discussed in the preceding chapters, the natural randomness of hydro­logic and geophysical variables, such as flood and precipitation, is an important part of the uncertainty in the design of hydrosystems infrastructures. However, other uncertainties also may be significant and should not be ignored. Failure to account for the other uncertainties in the reliability analysis in the past (as discussed in Sec. 1.3) hindered progress in evaluation of failure probability as­sociated with hydrosystems infrastructures. As noted by Cornell (1969) with respect to traditional frequency-based analyses of system safety:

It is important in engineering applications that we avoid the tendency to model only those probabilistic aspects that we think we know how to analyze. It is far better to have an approximated model of the whole problem than an exact model of only a portion of it.

Techniques LOCKING WALLS TOGETHER

TO TIE JOINING walls together, the double top plate must overlap the top plate at each wall intersection.

1. If you’re nimble, you can perch on top of the framing to hammer overlapping joints together.

2• Otherwise, do the job on a ladder.

3. It’s good practice to drive a toenail (or two) into the overlapping plate. A toenail pulls the joining walls together before the top is nailed off.

Techniques LOCKING WALLS TOGETHER

STEP7 Plumb and Line the Walls

Theword “plumbing,” when used in the con – text of framing a house, means making sure that the walls are standing straight up and down.“Lining’’ means straightening the top plate along the length of each wall. It’s impor­tant that all the walls are plumbed and lined accurately. Anything else is unacceptable.

Badly plumbed or crooked walls cause sig­nificant problems later—cabinets won’t tit Щ properly, doors won’t close correctly, and fm – ishedsurfaces (both inside and outside) will be wavy.

If the exterior walls were squared and sheathed before being raised, they should

be plumb. Otherwise, plumb them now that they are upright. To lest for plumb, use an accurate level that is at least 4 ft. long or make a plumb stick, as shown in the sidebar on p. 106. Plumbing a wall is best done with two people: one to hold the level and one to move the wall and nail off the bracing.

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Plumb the exterior walls first

Hold a level or plumb stick in a corner to see whether the bubble is centered in the vial.

If not, the wall must be moved laterally. Sometimes a wall can be moved a bit with a bodily shove. If you can pv. sh the wall plumb, install a temporary 2x diagonal brace to keep it that way (sec the bottom left photo on

THE EMPIRES OF THE BUILDERS

For thousands of years, the hydraulic know-how of the East was no more than an oral tradition. But with the conquests of Alexander the Great, this oral tradition comes into contact with the Greek spirit of observation and analysis. The city of Alexandria, at the maritime front door of Egypt, for several centuries serves as the scientific center of the known world. The understanding and know-how of the scientists of the Alexandrian school remain unequalled during the following millenium, both in the East and the West, until the advent of the Renaissance.

Roman engineers, the inheritors of Etruscan and eastern techniques, influenced by the Greeks and endowed with a strong practical sense, leave the influence of their hydraulic achievements around the entire Mediterranean perimeter.

The fall of the Roman Empire ushers in the intellectual decadence of the West, but the East continues to develop up until the Mongol invasions of the 12th century AD. Then the East, in its turn, enters a period of profound reversal.

In China, whose hydraulic development began later, technical developments tran­scend the millennia up through modern times. The scale of these developments reflect the vast expanse of the country itself. From the 1st century BC through the 15th centu­ry AD, China serves as mankind’s principal nursery of technical innovations.

Meanwhile, during the Middle Ages the West even forgets that the earth is round. Nevertheless during the West’s demographic expansion of the 12th and 13th centuries, hydraulic development blossoms as lands are drained and mills are constructed.

Pesticide-Treated Lumber

Although Germany has been a leader in the Bau – Biologie and healthy housing movement, it was only two decades ago that the general public there became aware of multiple chemical sensitiv­ity disorder. This awareness followed the experi­ences of thousands of people who were exposed to lumber treated with both the preservative pentachlorophenol and the pesticide lindane. Hundreds of people developed chronic neurologi­cal complaints, chronic fatigue, and an unusually heightened sensitivity to chemicals that were pre­viously tolerated. Lindane has subsequently been banned in Germany as a wood treatment.

dipped in pesticides that are now banned in the US. In some cases, uncontaminated lum­ber can be picked up directly from a local mill, where the sawyer will be closer to the source of the lumber and will know whether pesti­cides are used where the wood was grown.

Wood Selection and Storage

Kiln-dried framing lumber is drier than air – dried lumber. It is therefore more true to size and less susceptible to shrinkage and mold infestation. Certified, sustainably harvested, kiln-dried framing lumber is now becoming widely available. Framing lumber of this type is currently slightly more expensive than stan­dard lumber, which is often logged using un­sustainable practices.

Wood may occasionally be delivered to the site containing mold. It can also become moldy while stacked onsite if it is unprotected. To avoid these problems, you should include the following instructions in your specifica­tions: [5] [6]

• Fir, spruce, and hemlock are preferred over pine where available at no additional cost to the owner.

• Wood stored onsite shall be protected from moisture damage by elevating it off the ground and covering it with a tarp during precipitation.

• Wood that becomes wet must be quickly dried by cross-stacking to promote aera­tion. It should have less than 16 percent moisture content, as tested by a moisture meter, and must be free of all signs of mold in order to be acceptable. (See Division 13 for moisture meter testing.)

Health Concerns with Wood Frame Construction

Wood has historically been used as a compo­nent of a breathing wall system, whether it be the half-timber, wattle-and-daub construc­tions of medieval Europe or the log cabins of our ancestors in North America. (See Divi­sion 4 for an explanation of the breathing wall concept.) Wood is an advantageous material in a healthy home because it has the property of hygroscopicity. This means it has the ability to absorb and release moisture, thus helping to balance humidity levels and the electrocli­mate. However, for many chemically sensi­tive individuals the natural terpenes found in wood, especially soft or aromatic woods such as pine or cedar, are intolerable. Certain woods may need to be eliminated from, or sealed when used in, a home for a chemically sensi­tive person.

In standard home construction, the air space between the wood studs may be filled with insulation laden with chemicals. The exterior sheathing often contains formalde­hyde-based glue or asphalt backing. The gyp­sum board applied to the inside face of the studs may be finished with harmful joint com­pounds. If the home was built before 2004, the studs sit on a sill plate that is most likely pressure treated with a pesticide to prevent rot and insect infestation. When standard con­struction is the only option, we recommend that the most benign wall construction mate­rials available be used and that a barrier be in­stalled between the wall construction and the living space. Refer to Division 7 for air barrier product and installation information.

In some instances, creating a barrier for the purpose of blocking fumes can cause other problems when moisture from condensation becomes trapped inside the wall. Applying the gypsum board in an airtight manner will help block fumes but will not in itself block the nor­mal movement of water vapor. See Division 9 for more specific details on creating a barrier using gypsum board.

Construction lumber is at risk of con­tamination by pesticides when farmed, when milled, during transportation, and in stor­age. For those who have severe sensitivities to pesticides, it is important to locate a source for uncontaminated wood. Wood that is sustain­ably harvested can be traced from source to sawmill to distributor and its pesticide history can be determined. Certified producers and processors are encouraged to use least-toxic pest management. Some regional certification organizations have mandated a ban of pesti­cides for sustainably harvested wood in their jurisdictions. Certain imported woods maybe

CASE STUDY 6.1

FIREWALLS

Growing pressures on land availability and housing affordability are resulting in an increase in demand for attached homes. Zero-lot-line configurations are becoming more popular for detached homes because of more innovative use of small lots. The principal added code consideration for attached and zero-lot-line homes is the requirement for fire barriers.

Requirements of the major model codes are not always clear and are often subject to local interpretation and/or amendment. It is important to under­stand that all major model codes have an "Alternate Materials and Systems" section and that local code officials have the discretion to approve alternate construction. Appropriate documentation is, of course, usually necessary.

To be acceptable, firewall construc­tions must be rated by a recognized testing laboratory in accordance with ASTM El 19, Standard Methods of Fire Tests of Building Constructions and Materials. Many different one – and two-hour wall constructions have been approved and are listed in literature available from the Gypsum Association, 1603 Orrington Avenue, Evanston, IL 60201; the National Concrete Masonry Association, P. O. Box 781, Herndon, VA 22070; and other industry associations and product manufacturers.

For zero-lot-line homes, each unit must have an independent one-hour fire-resistive rating. The Standard Code (Southern) Uniform Building Code (ICBO), and the One and Two Family Dwelling Code (CABO), all require one-hour ratings for homes built less than 3 feet from the property line. The Basic/National Code (BOCA) requires a one-hour rating on exterior walls less than 6 feet from the property line. None of the codes permit unprotected openings through a firewall. Normal electrical, plumbing, and ductwork are generally allowed.

The most common one-hour firewall is of wood frame construction with 5/8- inch type X gypsum wallboard or gypsum sheathing attached to each side with 6d coated drywall nails 7 inches on center. Joists are required to be staggered at least 24 inches on center on each side.

For attached homes, a two-hour firewall is required, either as two separate one-hour walls or as a common two-hour wall at the property line. Check your local Code. The two-hour common wall typically built is a single wood frame wall with two layers of 5/8-inch type X gypsum wallboard on each side or concrete block. Two-hour walls typically have restrictions on electrical wiring, plumbing, and ductwork within the wall.

In addition to the firewall, some provision is required to block the spread of fire to the roof of an adjoining unit. For zero-lot-line detached homes, codes are somewhat vague because there is no adjoining roof.

In addition to confusing major model code firewall and roof treatments, some local codes require that firewalls be built of masonry construction.

This requirement is prohibitive for factory-built construction.

As mentioned, three of the four model codes require a firewall if within 3 feet of the property line. The other code, BOCA, requires a firewall if within 6 feet of the line. If a home is built 37 inches from the property line (73 inches under BOCA), no firewall is needed.

If an easement for use of that narrow strip of land is assigned permanently to the house next door, a zero-lot – line effect is obtained without cost of a firewall or roof parapet. It will be worthwhile to check local interpreta­tion of firewall/roof treatment requirements prior to construction.

Подпись: EXAMPLES FROM THE DEMONSTRATION PROJECTS
Since the major model codes are difficult to interpret and have not seriously addressed detached zero-lot – line homes in many cases, a complete review and rewrite of all codes should be undertaken.

Three JVAH sites, (Lacey, WA; Everett, WA; and Santa Fe, NM), all built under UBC, ran into the problem of firewall and roof treatment requirements.

Подпись: Santa Fe, New MexicoIn Santa Fe, the normal city require­ment is a masonry firewall between attached garages, including a parapet above the wall. The builder obtained the Fire Resistance Design Manual from the Gypsum Association which shows wood frame firewalls.

In addition, he pointed to the 1,000 square foot per floor exception for roof fire treatment in UBC. These convinced the city that a common two-hour wood-framed firewall with no parapet or roof treatment was adequate.

In Lacey, the city required either a parapet extending 30 inches above the roof or that all framing elements (trusses, wall plates, studs, etc.) within 5 feet of the two-hour separation wall be of one-hour fire resistance construction.

Подпись: Lacey, WashingtonThe builder, John Phillips, pointed out that none of the other major model codes had this requirement and that fire-resistive sheathing, installed at least 4 feet from the wall, provides adequate fire safety according to building code experts. The city accepted his documentation which resulted in substantial cost savings.

Подпись: Everett, WashingtonIn Everett, zero-lot-line homes were built with one-hour fire walls. The city accepted the builder’s documenta­tion that type X fire-rated gypsum board under roof sheathing, within 4 feet of firewall was adequate to comply with the intent of the code.

FIREWALLS

The standard roof truss has become the most common and most cost – effective method of roof framing. Light-weight trusses are the most highly engineered component in new home construction and form the basis of a very efficient roof system. They are easy to install and adapt to many basic designs. Therefore, if cost is the primary consideration, standard roof trusses are recommended.

The "in-line" framing concept dis­cussed in the House and Lot Design section of this manual works very well with roof trusses. That is, the 24- inch on-center roof trusses align with the 24-inch on-center wall studs which in turn align with the 24-inch on- center floor joists. The key to this consistent alignment is to start all layout from the same corner.

Simplification of roof overhang and trim details, consistent with design and function, provides opportumties for cost reduction. For example, the ■ rake overhang is essentially nonfunc­tional on a gable end roof. A simple fascia board at the siding/roof junction serves to cover the rough edge of the siding and conceal inaccuracies of fit. Several of the nation’s largest builders use this detail on all their production homes.

Roof overhangs are desirable for most designs and provide rain protection for the front and rear of the house. They also can provide summer shading for some windows. When an overhang is used, an inexpensive "open" soffit will eliminate much of the cost of the traditional cornice. All trim details on the underside of the overhang may be
eliminated, leaving the truss or rafter tails exposed. Blocking between trusses or rafters and a 1×6 fascia board are the only finish items needed. If soffit venting is needed, screening between trusses or rafters can be used instead of blocking.

Three-eights-inchplywood roof sheathing with metal plyclips is an acceptable alternative to 1/2-inch plywood.

Shoring

Shoring temporarily supports loads carried by bearing walls while you modify them—say, to add a window or a door opening. Typically, shoring is installed after removing finish surfaces and rerouting pipes and wires but before cutting into a bearing wall. If you’re not sure if the wall is bearing or whether it can be safely modified, have a structural engineer inspect the house and review your remodel­ing plans. This is hard-hat work.

For first – and second-floor walls, two types of shoring are common: screw jacks used with top and bottom plates, and temporary stud walls built from 2x4s. In either case, position shoring back 2 ft. to 3 ft. from the wall you’re working on so you’ll have room to move tools and materials.

► If you’re using screw jacks, doubled 2×6 top plates will distribute loads better. Here’s how to laminate the top plates in place: Use two or three

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16d common nails to nail the upper 2×6 directly to the ceiling joists; then face-nail the second 2×6 to it. Ideally, the top plates should extend one joist beyond the new opening on both sides. Don’t over­nail, you’re just holding up the plates till you get jacks underneath. Plumb down to mark the location of the single 2×6 sole plate. Place jacks every 4 ft., and plumb them. Tack-nail the top of each jack so it can’t fall over. Then raise one jack in tiny increments before moving to the next. Raise ceiling joists no more than Vs in.—just enough to take pressure off the bearing wall.

► Building a temporarily stud wall is simi­lar: Tack-nail the top plates; then plumb down to
mark the bottom plate. (To keep the bottom plate in place, tack-nail it to the joists underneath.) Cut studs in. longer than the distance between the plates because, here, the studs do the lifting. Toenail the studs to the top plate on 16-in. or 24-in. centers. Then use a sledge to rap the bottom of each stud till the stud is plumb. Recheck each stud for plumb as you progress, and monitor them periodically.

Once shoring supports the loads above, remove the studs from the bearing wall as needed to enlarge openings, add headers, and the like. If the bearing wall transfers loads from upper stories down into girders or foundation walls, study the lengthy sec­tion on jacking and shoring in Chapter 10.

Подпись: TIPПодпись: Today, rough openings are usually 82VHn. high, which accommodates a standard 6 ft. 8 in. preframed door. But if your house is nonstandard, instead try to line up new or enlarged openings in exterior walls to the tops of existing doors and windows. The underside of a new header will usually be 2 in. to 2V2 in. above the window or door frame, but check your unit's installation instructions. Подпись: llll

Structural Remodeling

Once shoring is in place, you should be safe in removing bearing walls. Check with a structural engineer if you have any doubts. Again, wear safety gear (hard hat, eye protection, work boots with thick soles, and so on) and test the electrical outlets to be sure the power is off. If you need to do any jacking, read Chapter 10.

SUPPORTING FLOOR JOISTS

It’s common in many regions to build houses directly on a concrete slab. When this is the case, wall building can begin once the sill plates are down. Other builders use a system of posts and girders to support joists so that a floor can be nailed to them. Still others use manufac­tured joists (called I-joists) that span a basement from sill plate to sill plate without any interior support, leaving a room large enough for a dance hall. [7]
length of the building. Houses fre­quently have an exterior stemwall foundation with girders supported by piers set on concrete footings (see the drawing above).

Post length

Posts in a basement will be quite long, while posts in a crawl space will be shorter—only about 1 ft. to 2 ft. But to keep wood dry and away from termites, make sure that no wood is within 6 in. of the ground. To determine the exact length of each post, pull a chalkline directly over the tops of the concrete piers that will support them, from foun­dation wall to foundation wall (see the top drawing on p. 91). Then place a scrap piece of girder stock (like a 4×6) on a pier. The distance between the string and the top of the girder stock is the length of the post for that pier. Make a list and note the length of every post before beginning to cut them.

Posts are usually cut from 4×4 stock. In some areas, they may need to be pres­sure treated. They can be cut to length with a circular saw or with a chopsaw. Leave the string in place to help align the posts as you nail them to the top of the piers. Toenail three 16d nails (two on one side, one on the opposite side)—or four 8d nails—down through the post into the wooden block on top of the pier (for more on toenailing, see the sidebar below).

Girders

Once the posts are nailed in place, it’s time to cut and nail on girders. Again, use straight stock that isn’t twisted. Pier posts are often 6 ft. apart, so cut the girders to break in the middle of a post, which will ensure solid bearing for all girders. Secure the girders to the posts with three 16d or four 8d nails. Take some simple steps to strengthen the
girder frame structurally, especially if you live in earthquake country: If the posts are over 3 ft. long, nail a 1x brace diago­nally (45°) from the bottom of the post to the girder with five 8d nails in each end (see the bottom drawing on the facing page). Unite the joints with a metal strap or a plywood gusset. I prefer the gusset because it ties girder to girder and girders to the post.

INSTALLING JOISTS

Joists are placed on edge across the sill plates to provide support and a nailing surface for the subfloor and a platform for the walls (see the drawing on p. 92). The joists need to be strong enough to support your grand piano without having it wind up in the basement.

Joist systems are made from either stan­dard 2x lumber or from manufactured joists. The weight these members can