Metal Plate-Connected Wood Trusses

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Metal plate-connected (MPC) wood trusses were first used in the early 1950s. Today they are used in more than 75% of all new residential roofs. Basically they are dimension lumber engineered and connected with metal plates. Less expensive than alternative roof systems, these trusses can also span longer distances. The “Pitched Truss Parts" illustration shows the parts of a single pitched truss on the next page.

Because MPC trusses are engineered products, they should never be cut, notched, spliced, or drilled without first checking with the designing engineer.

Building codes require that a truss design drawing be delivered to the job site. The drawings must show, among other things, the layout locations and bracing details. Note that these drawings are typically not made with framers in mind, so it might take some study time to figure out where the engineer wants the braces. The bracing details often show the braces as small rectangles running laterally between the trusses. See “Lateral Truss Bracing" illustration later in this chapter.

When flying trusses, you should attach the cables around the panel points. When the trusses are greater than 30′, a spreader bar should be used. The cables should toe inward to prevent the truss from buckling. If the truss is longer than 60′, you will need a strongback temporarily attached to the truss to stabilize it. (See “Flying Trusses" illustration, later in this chapter.)

If you have multiple trusses, you can build a sub­assembly of several trusses on the ground with cross braces and sheathing, then erect them together.

When trusses sit on the ground, on the building, or in place for any length of time, keep them as straight as possible. They are more difficult to set in place and to straighten if they have not been stored properly on site.

Structural Composite Lumber (SCL)

Structural composite lumber (SCL) is an engineered wood product that combines veneer sheets, strands, or small wood elements with exterior structural adhesives. The most common of these products are laminated veneer lumber (LVL), parallel strand lumber (PSL), and laminated strand lumber (LSL). Their names pretty well describe the differences between them.

Like other engineered products, structural composite lumber requires that you follow the engineered specifications that will appear on the plans.

Sometimes the specifications simply indicate the use of a particular piece of SCL in a particular location. For larger jobs, you will find the SCL requirements called out in the shop drawings or the structural plans.

Because these are engineered products, you must consult the design engineer before you can drill or notch. Some manufacturers provide guidelines for drilling and notching, but this is not typical.

SCL has the advantages of dimensional consistency, stability, and availability of various sizes. It is important to note, however, that where dimensional lumber 4 x 10s, 4 x 12s, etc. can shrink significantly, SCLs have minimal shrinkage. The engineer should allow for this in the design so that you will not have to consider this factor when using SCLs as the plans specify.

Note that SCL studs are becoming common in building tall walls. They provide a degree of straightness that dimensional lumber does not. Although they are heavy and, as a result, not so easy to work with, they make nice, straight walls.

Conclusion

Engineered wood products come in a variety of forms. Becoming familiar with these products is important if you plan to work with them. Always be sure to follow manufacturers’ directions, and always consult an engineer if you plan to cut, notch, or drill engineered wood product components.

Lifting devices should be connected to the truss top chord with a closed-loop attachment using materials such as slings, chains, cables, nylon strapping, etc. of sufficient strength to carry the weight of the truss. Each truss should be set in proper position per the building designer’s framing plan and held with the lifting device until the ends of the truss are securely fastened and temporary bracing is installed.

mid-height

Greater than 60

Contents

The Strength of Good Framing 186

Understanding Structural Loads 186

Building Code Load Requirements 187

Regional Considerations 188

Framing Details 188

Hold-Downs 195

Positive Placement Nail Guns 200

Compaction Factor as a Quotient of the Bulk and Maximum Densities

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Calculating the quotient of the bulk and maximum (the so-called theoretical maxi­mum density [TMD] or the Rice density after ASTM 2041) densities is a popular method of determining the compaction factor. The requirement for compaction of an SMA layer is a minimum 94% (of the maximum density) according to the NAPA SMA Guidelines QIS 122; that makes upto 6% (v/v) of air voids allowable after compaction. In this case the reference density does not depend on conditions for preparing laboratory specimens. As in the previous method, densities obtained in a laboratory during mix design are not taken into account; results from testing the den­sity of a mixture taken from an asphalt plant during trial production are recorded.

The essential strength of that method is making the compaction factor free from various conditions for preparing specimens. The possibility of making mistakes in determining the reference density in a laboratory obviously disappears.

Tips for Installing Glu-Lam Beams

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• For glu-lam beams that are installed at a pitch and need to have the bottom cut to be level, make sure that the end of the bottom cut closest to the bearing edge receives full bearing. (See “Cut Edge Full Beaning" illustration.)

• Ends of beams should not be notched unless approved by the engineer. (See “No Notching End of Glu-Lam Beam" illustration.)

• Glu-lam beams will shrink as they dry out.

If the top of the beam is connected in a way that doesn’t allow for shrinkage, the glu-lam beam will split. (See “Glu-Iam Beam Shrinkage" illustration.)

• When a lateral support plate is used to connect two glu-lam beams, the holes should be slotted horizontally to prevent splitting. (See “Lateral Support Plate" illustration).

• Glu-lams are also used for posts. It is important to keep them away from concrete, which contributes to their decay. Placing a steel shim under the beam will keep it from touching the concrete. (See “Decay Prevention Next to Concrete" illustration.)

• Hinge connectors should be installed so that they don’t cause splitting of the glu-lam

beams. This can be done by using a strap that is independent of the hinge connector, or by vertical slotting the holes in a strap that is connected to the hinge connector. (See “Hinge Connector Slotted Holes" illustration.)

• Glu-lam beams rest on metal post caps that often have a weld or radius in the bottom corner. If you don’t ease the bottom corners of the beam, the beam will sit up in the pocket. Often, the glu-lam beam’s bottom corners are already rounded and won’t need attention.

• In some cases, the sides of the metal post caps are bent in so that the beams will not slide in properly. Check all the sides of the metal post caps before they are installed, so you won’t have a forklift or boom truck and crew standing around waiting while someone labors on top of a ladder to widen the sides of the post cap. (See “Forklift setting glu-lam beams" photograph later in this chapter.)

• Glu-lam beams are often attached to metal caps with bolts. The holes can be drilled either before or after setting the glu-lam beams. If the holes are drilled after the beams are set, use a drill with a clutch. It’s easy to break a wrist or get thrown from a ladder when a W drill motor without a clutch gets caught on the metal.

Glu-lam Beam Shrinkage

Splitting can be caused by shrinkage on large splice plates.

Top bolt prevents shrinking and causes splitting. Splitting can also occur from limiting beam end rotation as the beam deflects under load.

Shrinking or beam rotation due to deflection under loading can cause splitting.

If holes are not slotted, splitting may occur due to beam rotation as the beam deflects under load.

Contact with concrete exposes untreated wood to decay.

Water Content Variations in Pavements

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In the last ten years, significant progress has been made in the measurement of in-situ water contents in pavements, using in particular TDR probes (see Chapter 3, Section 3.2.2). These measurements have shown that, often, significant amounts of water infiltrate in to pavements through the pavement surface and from the shoulders.

Low traffic pavements are particularly exposed to water infiltration. Examples of moisture measurements on a typical flexible pavement (6 cm thick bituminous surfacing and granular base) are shown in Figs. 8.3 and 8.4. Figure 8.3 shows that the daily variations of water content in the granular base and in the clayey sub­grade (near the pavement edge) are important and strongly related with the rainfall. Figure 8.4 shows average water contents measured in the granular base, at different locations, near the centreline of the pavement and near the edge. The critical zone is clearly the pavement edge where the water content is about 2 percentage points higher than near the centreline. In this pavement, subjected to a mild oceanic cli­mate, seasonal variations of water content are low, but they can be more important with more continental climates.

Thick bituminous or cement-treated pavements are less permeable, and water infiltrates mainly when cracking develops, thus accelerating the deterioration. In such pavements, protection against water infiltration, by proper maintenance (crack sealing, renewal of the surface course) is one of the main concerns.

Fig. 8.3 Water content variations in the granular base and subgrade of a low traffic pavement (near the pavement edge)

Date

Fig. 8.4 Monthly average water contents in the granular base, at the centre and near the edge of the pavement

Influence of Water Infiltration on Pavement Deterioration and Mechanical Degradation

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Changes in water content, especially excess moisture, in pavement layers com­bined with traffic loads and freezing and thawing can significantly reduce pavement service life. Failures associated with moisture are detected on roads all over the Europe. There is some evidence to suggest that water has less impact on thick and well-construced pavements than it does on thinner ones (Hall & Crovetti, 2007). It appears that in thicker pavements the effect of water may be more indirect than in thinner ones, reducing material stiffness leading to later distress.

To minimize the negative effects of moisture on pavement performance, first we have to identify the sources of infiltration of water. There are many different possible sources of water infiltration in pavement systems. Mainly, the presence of water in pavement is due to infiltration of rainwater through the pavement surfaces through joints, cracks and other defects, especially in older, somewhat deteriorated, pave­ments and shoulders. An important source is also migration of liquid water upwards to the freezing front. Water may also seep upward from a high groundwater table due to capillary suction or vapour movements, or it may flow laterally from the pavement edges and side ditches.

The significance of the routes of infiltration depends on the materials, climate, and topography.

Many pavement failures are the direct result of water entering the pavement courses and/or the subgrade. Water entry in the compacted unsaturated material will increase water pressure or decrease suction, and in turn, reduce the effective stress (see Chapter 9). Hence, the strength and the elastic and plastic stiffnesses of the pavement material and the subgrade will be reduced. The rate of traffic-induced deterioration of the road will increase during this time. The loss of strength and stiffness can lead, in the extreme, to rutting and other forms of surface deformation or, more commonly, to pavement edge failures. The worst situation occurs with poorly-compacted granular material (as a result of shear strength reduction) with frost susceptible soils or with cohesive swelling clays (as a result of damaging vol­ume changes).

Water seeping through the pavement can also transport soil particles and cause erosion and pumping (i. e. transport) of fines as well as leaching of many materials (see Chapter 6). Moisture entry can also affect the performance of the surface course by causing stripping of bitumen from aggregate, layer separation between bound courses, and pothole formation (see Section 5.5). A probable mechanism of pothole formation is illustrated in Fig. 8.1. A somewhat similar failure mechanism has also been observed in slabbed concrete pavements (Roy & Johnson, 1979): [21]

(ii) Stage 2 Water travels along the interface between the asphalt (AC) and the compacted aggregate base

(iii) Stage 3 Asphalt lifts to allow pressure to dissipate, but also allows more water to be pumped into the crack

Fig. 8.1 Layer separation and pothole formation (from Gerke, R. J. (1979), cited by Lay (1986)). Reproduced by permission of M. G. Lay

• the fast-moving water eroded the finer materials;

• at their edges, the concrete slabs progressively lost support from the underlying layers as material was washed away;

• the deflections of the slab became greater and the erosion more rapid eventually leading to cracking of the unsupported concrete at edges and corners of slabs; and

• dirty water was seen to squirt from the joints in the pavement when the slabs were trafficked.

Pavement damage, associated with water, can be divided into moisture-caused and moisture-accelerated distresses. Moisture-caused distresses are those that are primarily induced by moisture, while moisture-accelerated are those that are initi­ated by different factors, but the rate of deterioration is accelerated by presence of water. Most commonly observed damages due to the moisture are:

• surface defects;

• surface deformations; and

• cracking.

Generally, water threatens the stability of soil and it is a particular problem for pavement structures since they are built through areas of changeable moisture quantities.

It has been generally established that the increased subgrade water content during spring results in increased deformability, i. e. a decreased bearing capacity of the pavement. These changes in bearing capacity are, in particular, obvious for silty and clayey materials.

Water contents contained in materials under flexible pavements are influenced by the amount and intensity of rainfall. Periods of long rainfall of low intensity can be more severe than concentrated periods of high intensity, since the amount of moisture absorbed by the soil is greatest under the former conditions. Also, the combined effects of rainfall and freezing temperatures determine, in part, the extent of pavement damage.

Water can penetrate into the pavement structure in several different ways as fol­lows (Fig. 8.2):

• Seepage from the elevated surrounding soil, which depends on the hydraulic gradient and soil permeability coefficient.

• Rise and fall of the phreatic surface, which depends on the climatic circum­stances and soil composition (e. g. following heavy rains there is an increase in the sub-surface water level in permeable strata, while water remaining on the surface of impermeable ground will drain away or evaporates and there is no risk of a rapid rise in sub-surface water level).

• The penetration of water through damaged pavement surfaces causes high lo­cal concentration of water in penetration areas. Under heavy traffic load, such a condition can result in significant damage if the subgrade made of changeable material is in contact with water. Even worse pavement damage can result from freezing of the structure and subgrade when soaked with water.

• The penetration of water through shoulders (if the shoulders are permeable or their surface is deformed in such a way as to allow the retention of water), which depends on the material permeability, compactness of the surface, inclination and the drainage from pavement surface. The effect is similar to the effect of water penetration through a damaged pavement surface.

RAINWATER INFILTRATION

THROUGH SHOULDER

V V

CONDENSATION

CAPILLARY WATER

RISING

і LM lit

OF WATER I ABLE

Fig. 8.2 Possibilities of water movements into the pavement zone (from Moris & Gray, (1976) cited by Lay (1986)). Reproduced by permission of M. G. Lay

• Capillary rise from the foundation soil. The water rises from the foundation soil through the fine-grained soil up to the pavement structure.

• Evaporation of water from the foundation soil and its condensation under the pavement structure if the pavement structure is colder than the soil.

Moisture behaviour in a pavement can be considered as occurring in three phases:

• an entry phase – which occurs quite rapidly;

• a redistribution phase when water moves within the material in response to suc­tion and gravity; and

• an evaporative phase when water, as vapour, leaves a material or moves to other layers. Water vapour movements can occur under temperature gradients with the water vapour travelling from a warm to cool area when it then condenses.

Water content under road pavements will vary seasonally, annually and over longer periods. Seasonal variations in water content are commonly located in the upper 1-2 m, and in the metre or so of pavement width at the edge of the pave­ment surface (the outer wheel path is clearly the critical zone). Significant moisture changes will only occur immediately after rainfall if very permeable layers exist.

The behaviour of the pavement related to moisture should be considered in ref­erence to the climate, type of soil, groundwater depth and moisture concentration in the soil.

Development of the water mill in the Roman Empire

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About fifty years after Vitruvius, under the reign of Vespasian and Titus, Pliny the Elder[255] wrote in his book, The Natural History:

“All the grains are not easily broken. [….] Throughout the greater part of Italy, however, they employ a pestle that is only rough at the end, and wheels turned by water, by means of which the corn is gradually ground.”[256]

The Roman development of the water mill has been consistently underestimated; but new archaeological findings are becoming more and more numerous.[257] One can find remains of simple mills on small rivers near Hadrian’s Wall at the Scottish border.[258] In eastern Tunisia, at Chemtou (the Roman Simitthus) the remains of an installation com­prising three horizontal wheels set side by side have been found near a dam constructed by Trajan on the river Medjerda.[259] And in North Africa, there are still other horizontal – wheel mills. Vertical-wheel mills are found in Gaul from the 1st or 2nd centuries; in the villas ofVar in the 2nd century; on the Janiculum Hill at Rome from the 3rd century; and at the agora of Athens in the 5th century. There is a depiction of a noria on a mosaic dat­ing from 469 AD at Apamea-on-Orontes; this date comes after the fall of the Roman Empire in the west, but the representation suggests that the use of the noria had largely spread to the east during the Roman period. And then, there is the flour mill of Barbegal.

Let’s now describe this installation,[260] the one we used to begin our discussion of

Development of the water mill in the Roman Empire

Figure 6.22 Roman flour mill at Barbegal: view from the summit of the rock outcrop where the aqueduct arrives; to the left and the right, in the form of a V, one can see the two walls that carried the two divergent lateral canals. These canals delivered water to the two “mill” canals that, in turn, cascaded down the slope and supplied the two banks of water wheels (photo by the author).
Подпись: Figure 6.23 The remains of the Roman hydraulic flour mill, comprising two series of eight paddle wheels. View from below, remains of the right-hand canal. The arrows indicate the flow path (photo by the author).
Development of the water mill in the Roman Empire

Roman mills. We have seen that the aqueduct that is dedicated to the supply of the flour mill lies parallel, in its final stretch, to the aqueduct that supplies the city of Arles. Both aqueducts come out of a junction basin which is supplied, in its turn, by two canals deliv­ering water from different remote sources.

At the mill, another distribution basin[261] conveys water into two parallel headrace canals that descend from the hill with a discharge that modern studies have estimated at about 0.15 m3/sec. Along each of these canals, eight vertical wheels, each about 2 m in diameter, are aligned from the top to the bottom of the slope. Each wheel has a sill or weir immediately downstream of it, controlling flow into a small drop that feeds water onto the wheel below it. Adjacent to each wheel, near its center, is a chamber enclosing the reduction gears, with grinding wheels likely set on a platform above each of these chambers.

One of the astonishingly modern aspects of this installation is its engineered, non-nat­ural water supply, using one of the two branches that earlier had come together to supply Arles. The mill was built at a very convenient location, benefiting from a steep slope and yet providing ready access from the plain below, obviating the need for the installation to accommodate the vagaries of a natural river. The discharge is regular, with no risk of erosion. This installation probably dates from the beginning of the 2nd century AD.[262]

Development of the water mill in the Roman Empire

Figure 6.24 Reconstitution of the plan of the flour mill of Barbegal (after Sellin, 1979). The light lines indicate assumed structures.

Acceptance-rejection methods

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Consider a problem for which random variates are to be generated from a specified probability density function (PDF) fx(x). The basic idea of the

TABLE 6.3 List of Distributions the Cumulative Distribution Function (CDF) Inverses of which Are Analytically Expressible

Distribution

Fx (x) =

x = Fx fiu)

Exponential

1 — exp(-вx), x > 0

—в ln(1 — F)

Uniform

(x — a)/(b — a)

a + ( b – a) F

Gumbel

exp{ exp[ (x — %)/в)]}

% — в ln[ ln(F)]

Weibull

1 — exp{—[(x — % )/в ]a}

% + в[— ln( 1 — F )]1/a

Pareto

1 — x—a

(1 — F)—(1/a)

Wakeby

Not explicitly defined

% + (a/в)[1 — (1 — F )в ] — (y/m — (1 — F )—s ]

Kappa

{1 — h[1 — a(x — % )/в]1/а }1/ h

% + (e/a){1 — [(1 — Fh)/h]a}

Burr

1 — (1 + xa )—в

[(1 — F)—1/e — 1]1/a

Cauchy

0.5 + tan— H x)/n

tan[n (F — 0.5)]

Rayleigh

1 — exp[—(x — % )2/2в2]

% + {—2в2 ln( 1 — F)}1/2

Generalized lambda

Not explicitly defined

% + a F в — y (1 — F )s

Generalized extreme

exp[ exp( y)]

% + в{1 — [— ln(F )]a}/a, a = 0

value

where y = — a—1 ln{1 — a(x — % )/в}, a = (x — % )/в, a = 0

= 0

% — в ln[— ln(F)], a = 0

Generalized

1/[1 + exp(—y)]

% + в{1 — [(1 — F )/F]a}/a, a = 0

logistic

where y = —a—1 ln{1 — a(x — % )/в}, a = (x — % )/в, a = 0

= 0

% — в ln[(1 — F )/F], a = 0

Generalized

1 — exp(— y)

% + в[1 — (1 — F)a]/a, a = 0

Pareto

where y = — a—1 ln{1 — a(x — % )/в}, к = (x — % )/в, к = 0

= 0

% — в ln( 1 — F), a = 0

acceptance-rejection (AR) method is to replace the original fx(x) by an appro­priate PDF hx(x) from which random variates can be produced easily and ef­ficiently. The generated random variate from hx(x), then, is subject to testing before it is accepted as one from the original fx (x). This approach for generating random numbers has become widely used.

In AR methods, the PDF fx (x) from which a random variate x to be generated is represented, in terms of hx(x), by

fx (x) = ehx (x)g(x) (6.9)

Acceptance-rejection methods

in which є > 1 and 0 < g(x) < 1. Figure 6.2 illustrates the AR method in that the constant є > 1 is chosen such that f (x) = єhx (x) over the sample space of the random variable X. The problem then is to find a function f (x) = єhx(x) such that f (x) > fx(x) and a function hx(x) = f (x)^, from which random variates are generated. The constant є that satisfies f (x) > fx(x) can be obtained from

(see Problem 6.4). Intuitively, the maximum achievable efficiency for an AR method occurs when f (x) = fx(x). In this case, є = 1, g(x) = 1, and the corre­sponding probability of acceptance P {U < g(Y)} = 1. Therefore, consideration must be given to two aspects when selecting hx(x) for AR methods: (1) the ef­ficiency and exactness of generating a random number from hx(x) and (2) the closeness of hx(x) in imitating fx(x).

Acceptance-rejection methods

Example 6.3 Consider that Manning’s roughness coefficient X of a cast iron pipe is uncertain with a density function fx(x), a < x < b. Develop an AR algorithm using f (x) = c and hx(x) = 1/(b — a), for a < x < b.

1. Generate ui from U(0, 1).

2. Generate U2 from U(0, 1) from which y = a + (b — a)u2.

3. Determine if

Подпись: U1 < g(y) =fx [a + (b — a)u2]

c

holds. If yes, accept y; otherwise, reject (U1, y), and return to step 1.

In fact, this is the von Neumann (1951) algorithm for the AR method.

AR methods are important tools for random number generation because they can be very fast in comparison with the CDF-inverse method for distribution models the analytical forms of CDF inverse of which are not available. This approach has been applied to some distributions, such as gamma, resulting in extremely simple and efficient algorithms (Dagpunar, 1988).

About Metal Fasteners

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Several companies manufacture metal fasteners for a variety of wood-to-wood and wood-to-foundation applications. Since 1975, I have done a lot of timber framing without using these fasteners (except for truss plates), and I will share my techniques in these pages. But manufactured metal fasteners can make life easier, the building inspector happier, and improve the strength of the structure, so they are a valuable option. There are hundreds of different connectors available, and, while reading this book is a good introduction, I cannot cover all of the products in this relatively small volume. Therefore, you should also 1) go to your local hardware or building supply store and look at what’s readily available in your area (see Fig. 4.12) and 2) contact the companies by mail or through the Internet and look at their catalogs or web pages. There is an engineered code-compatible connector for practically every imaginable situation.

If there is a downside to these manufactured connectors and fasteners, it is that they are made mostly for lumber of finished dimensions. In the mechanical fastener industry, a four – by-four is almost always 3V2 inches by 3V2 inches and a six-by-six is 5V2 inches by 5У2 inches. However, companies do manufacture full-sized connectors for rough-cut posts, and also some joist hangers for rough-cut material. Simpson Strong-Tie Co, Inc., for example, has joist hangers for all depths (up to 14 inches or 35.6 centimeters) for rough-cut two-bys, four-bys, and six-bys, but not for three – bys and five-bys. One local building supply sells USP joist hangers that will work with full-sized five-by-tens.

Подпись: Fig. 4.12: There is a great little hardware store in Pahoa, Hawaii, with quite a selection of metal connectors. Practically all new construction in Hawaii makes use of these connectors, because of severe expected wind loads. Still, over 90 percent of the fasteners in a catalog are for dressed lumber, so you will have to wade carefully through the catalogs to find what you need. You can also call the companies with specific requests, although specialty items will be expensive. Plan ahead for your connectors, so that you have them when you need them. The cost of standard connectors and fasteners is very reasonable, with many simple strap and plate connectors selling for fifty cents or less.

Another potential downside of galvanized metal fasteners is that they are not particularly attractive. However, they are usually installed where they are not seen, or, at any rate, not seen for very long. Also, Simpson Strong-Tie makes a few heavy (12-gauge) ornamental connectors with textured fiat black paint, including straps, T-straps, right angles, and a variety of heavy joist hangers. These are quite a bit more expensive than the standard fare, but can justify their cost if only a few are needed in exposed locations.

Post supports, for example, can be installed so that the metal parts will be hidden in the thickness of whatever infilling material is chosen. Similarly, right angle connectors, used, for example, where a girt is supported on a post, can be hidden in the infilling.

Подпись: !Подпись:Подпись: Fig. 4.13: George Stuort in Bornordsviile, North Carolina, used a homemade T-strap on each side of this post-and-girder join. The girders also have a simple half-lap scarf joint connecting them behind the T-strap, as in Fig. 4.14. Fig. 4.14a: A simple horizontal half-lap scarf joint. Here, two ten- by-tens join over a ten-by-ten post. Stewart Elliott (1977) says that the post must be two inches (51 millimeters) wider than the length of the lap. This is an easy joint to make, even by inexperienced owner-builders. Fig. 4.14b: This triangular metal plate with lag screws is a creative alternative to the T-straps of Fig. 4.13. To work properly, there must be an identical plate on each side of the joint. image49image50image51First floor joists are almost always hidden, but exposed ceiling joists or roof rafters are not.

An option to commercially avail­able fasteners is homemade ones, a favorite of many owner-builders. Several examples are shown in this book. Often, home-made connectors for heavy-timbers are made of one- eighth-inch (3.2 millimeter), three – sixteenth-inch (4.8 millimeter), or one-quarter-inch (6.4 millimeter) flat steel stock, which are all available in regular widths, such as 2-inch, 3-inch, 4-inch, 6-inch, etc. When these steel pieces are painted black, they become an attractive part of the structure. See Figs. 4.13 and 4.14a & b. See also Fig. 5.45 on page 138.

The half-lap joint can cut the girders shear strength in half, but the frame in Fig. 4.13 is overbuilt in the first place, and the heavy metal plates would return much of the shear strength to the member in any case.

Larry Schuth of Hilton, New York built a cordwood home within a post-and-beam frame, and told of his adventure in Chapter 17 of my previous book, Cordivood Building: The State of the Art (see Bibliography), which also has a color picture of the finished home. Larry’s foundation consists of two eight-inch block walls laid side by side, in order to provide 16 inches (40.6 centimeters) of bearing for his cordwood. But rather than go “double-wide” with his post system, as Joe Zinni did in Tenino, Washington (see the photo essay Joe’s Rocket Research Landing Pad at the end of this chapter), Larry built a strong frame using just single eight-by-eights as seen in Fig. 4.15.

Incidentally, sometimes plywood makes an effective fastener, particularly as gussets (plates that cover an area where two or more timbers come together), and where they can be used in a hidden application.

Classifications of Random Variates Generation Algorithms

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6.1.1 CDF-inverse method

Let a random variable X have the cumulative distribution function (CDF) Fx(x). From Sec. 2.3.1, Fx(x) is a nondecreasing function with respect to the value of x, and 0 < Fx(x) < 1. Therefore, F-1(u) may be defined for any value of u between 0 and 1 as F-1(u) is the smallest x satisfying Fx(x) > u.

For the majority of continuous probability distributions applied in hydrosys­tems engineering and analysis, Fx(x) is a strictly increasing function of x. Hence a unique relationship exists between Fx (x) and u; that is, u = Fx (x), as shown in Fig. 6.1. Furthermore, it can be shown that if U is a standard uniform ran­dom variable defined over the unit interval [0, 1], denoted by U ~ U(0,1), the following relationship holds:

X = Fx-1(U) (6.6)

Note that X is a random variable because it is a function of the random variable U. From Eq. (6.6), the one-to-one correspondence between X and U, through the CDF, enables the generation of random numbers X ~ Fx(x) from the standard uniform random numbers. The algorithm using the CDF-inverse method for generating continuous random numbers from a CDF Fx(x) can be stated as follows:

1. Generate n uniform random numbers u1, u2,…, un from U(0,1).

2. Solve for xi = F-1(ui), for і = 1, 2,…, n.

Fx(x) = u

Classifications of Random Variates Generation Algorithms

Figure 6.1 Schematic diagram ofthe inverse-CDF method for generating random variates.

Example 6.1 Consider that the Manning roughness coefficient X of a cast iron pipe is uncertain, having a uniform distribution fx(x) = 1/(6 — a), a < x < b. Develop an algorithm using the CDF-inverse method to generate a random Manning roughness coefficient.

Classifications of Random Variates Generation Algorithms
Подпись: A simple algorithm for generating uniform random variates from U(a, b) is 1. Generate n standard uniform random variates ui, u2,..., un from U(0,1). 2. Calculate the corresponding uniform random variates xi = a + (b — a)ui, i = 1, 2,..., n.
Подпись: In the case that the random variables under consideration are discrete, the value of xj corresponding to the generated standard uniform random variate u must satisfy j—1 j Fx(xj—1) = ^ fx(xi) < u < Fx(xj) = Y, fx(xi) (6.7) i=1 i=1 The CDF-inverse algorithm for generating discrete random variates can be implemented as follows: 1. Generate the uniform random number u from U(0, 1). 2. Initialize i = 0 and set p = 0. 3. Let i = i + 1, and compute p = p + fx(xi). 4. If p < u, go to step 3; otherwise, stop, and xi is the random variate sought. Example 6.2 Suppose that the number of snow storms X at a location in January has a discrete uniform distribution fx(x) = 1/5 for x = 0,1, 2, 3, 4 Develop an algorithm to generate a sequence of random number of snow storms. Solution The CDF for the number of snow storms can be written as Fx(x) = (x + 1)/5 for x = 0,1, 2, 3, 4 The algorithm for this example can be outlined as follows. 1. Generate the uniform random number u from U(0, 1). 2. Initialize x = 0 and p1 = 0, and compute Fx(0).

Solution Using the CDF-inverse method, the expression of the CDF of the random variable is first sought. The CDF for this example can be derived as

3. Test if pi < u < Fx(x). If yes, x is the solution; otherwise, go to step 4.

4. Let p1 = Fx(x), x = x + 1, and compute Fx(x + 1). Go to step 3.

To apply the CDF-inverse method for generating random numbers efficiently, an explicit expression between X and U is essential so that X can be obtained analytically from the generated U. The distributions the inverse forms of which are analytically expressible include exponential, uniform, Weibull, and Gumbel. Table 6.3 lists some distributions that are used in hydrosystems the CDF in­verses of which are analytically expressible.

When the analytical forms of the CDF inverse are not available, applying the CDF-inverse method would require solving

/

x

fx (t) dt (6.8)

-TO

for x from the known u. For many commonly used distributions such as normal, lognormal, and gamma, solving Eq. (6.8) is inefficient and difficult. More effi­cient algorithms have been developed to generate random variates from those distributions; some of these are described in Sec. 6.4.

INSTALLING CUT-IN BOXES

Posted by on 19/ 11/ 15

Cut-in boxes have special mounting devices that enable you to mount them directly to finish sur­faces. But first you’ve got to cut a hole for them. Hold the new box at the same height as other outlet boxes in the room, lightly pencil trace around the box, and then drill a small exploratory hole to locate studs or wood lath behind. Insert an offset screwdriver or a bent coat hanger in the hole and twirl it. If the tool hits a stud, move the box till it’s clear. If you hit wood lath, keep drilling small holes within the opening till you find the edges of the lath. If you position the box

correctly, you should need to remove only one section of lath. Cut-in box ears (also called plaster ears) mount to the lath top and bottom, so you may need to adjust the box height to make that happen.

Once you settle on a final location, level the box, trace its outline firmly onto the wall, and use a utility knife to score along the outline to mini­mize plaster fractures. (If the surface is drywall, don’t bother scoring; just go ahead and cut out the opening.) Plaster is delicate stuff, but a cord­less jigsaw can cut plaster without pulverizing it if you hold the saw shoe slightly above the sur­face. (Drill several J4-in. holes around the outline so you can insert the sawblade.) Finally, when cutting through the lath strip you’ll remove, alter­nate partial cuts from one side of the opening to the other; if you cut completely through one side of the lath first, that end will vibrate wildly and crack the plaster. After you’ve cut out around the box ears, predrill the lath so the mounting screws don’t split it. (If there’s metal lath, use a fine­toothed metal-cutting blade, and proceed slowly.)

Before inserting cut-in boxes, remove knock­outs, insert cable clamps, strip sheathing off the ends of incoming cable, feed cable into the cable clamps, and tighten the clamps down. Box mounts vary. Boxes with plaster ears that screw
to wood lath are the most common. Spring-clip types have metal wings that expand once inserted in an opening. Grip-Tite boxes have side-mounted ears that stick out as you turn screws. Mounting tabs (also called elephant ears) slip into the space between the box and the edge of the hole; pull them tight against the wall, and bend the tabs into the box, using pliers. Flatten tabs so they cannot touch any terminal screw of the outlet. For good measure, wrap electrical tape around the devices, as shown in the photo on p. 243.

FISHING CABLE

© Once you’ve cut an opening for a cut-in box— but before inserting the box—fish the cable that will run from the existing outlet to the new one.

Подпись: TIPПодпись: Don't use cut-in boxes to house receptacles in high-use areas like bathrooms and kitchens. Plugs repeatedly inserted and removed will cause cut-in boxes—and possibly, electrical connections—to work loose. In such locations, you'll need to expose the framing and mount new-work boxes to it. 1111 image522,image523
Fishing wires is easiest with a helper. As one person feeds metal fish tape down into an outlet cutout, the second person, with another fish tape, tries to catch the first. (Fish tape ends are often bent over to create a hook.) That accomplished, the first tape is pulled through, to have electrical cable attached to it. Bend and then twist the cable wires tightly over the fish tape, as shown in the drawing on p. 261, and wrap them well with electrical tape. Taper the electrical tape to a point

Подпись: PROTIP If you need to run cable between new and old outlets directly over each other, use a lightweight chain or fishing line with a weight rather than fish tape. The weight line or chain will be easy to snag because it will hang freely. 1111 Подпись: Retrofitting a I "Remodel" Lighting Fixture Подпись: Fixture can Подпись: This recessed lighting fixture is a special “remodel" unit, which is light enough to mount to finish surfaces. Attach the fixture leads to the incoming cable wires in the junction box, and then insert the fixture into the opening.so it will feed more easily into holes. If one per­son feeds the cable into the opening as the other pulls, the cable should move smoothly.

О To pull cable into an existing box, turn the power off, and then unscrew and gently pull the device out from the box and disconnect the wires attached to it. If the box is plastic, use a screw­driver to create a cable slot. If the box is metal, use a screwdriver to remove a knockout and to loosen the cable clamp nearest the knockout. If a metal box has no internal cable clamp, drill and tap a threaded hole so you can add a clamp. Feed the fish tape through the knockout opening; then pull in new cable and clamp it down.

Run a generous length of cable between the two boxes, so at least 1 ft. of cable protrudes from each box. Strip cable sheathing, tighten cable clamps, group like wires, and attach devices as described earlier. Chapter 15 covers patching finish surfaces.

MOUNTING CEILING FIXTURES

How you mount ceiling fixtures depends mostly on the weight of the fixture and whether framing is accessible. New “remodel” fixtures such as that shown below are light enough to mount to dry – wall or plaster ceilings, but heavier fixtures must be mounted to an adjustable hanger bar attached to ceiling joists or to a metal outlet box secured to the framing. Note: To prevent heat buildup and fire, fixtures in insulated ceilings must be rated

IC (insulation compatible) or insulation must be blocked back at least 3 in. from non-IC-rated fixtures.

Recessed lighting fixtures can usually be installed from below. О is cable in the ceiling cavity—either from an old fixture that was removed or from new cable recently fished to that location—and that the cable is not energized.

If there was no old fixture in the location, use a cordless drill to drill a 18-in. exploratory hole. Then insert a bent coat hanger into the hole and twirl it to locate nearby joists. Trace the template for the fixture can (housing) onto the ceiling.