Jacking and Shoring

Jacking refers to raising or lowering a building so you can repair or replace defective framing or failed foundations or to level a house that has set­tled excessively. Shoring refers to a temporary system of posts, beams, and other structural ele­ments that support building loads. Temporary is the crucial word: Shoring supports the building between jackings. Once repairs are complete, you need to lower the house and remove the shoring as soon as possible. If repairs are extensive—say, replacing foundation sections—have a structural engineer design the new sections; specify jack size; and specify the posts, beams, and bracing needed to safely jack and shore the building.

Jacking a house is nerve-wracking. It requires a deep understanding of house framing and how structures transfer loads. It also requires superb organizational skills and a lot of specialized equipment. For that reason, foundation contrac­tors routinely subcontract house-raising to house movers who have crews who know what they’re doing and have on hand thousands of cribbing blocks, scores of hydraulic jacks, and cranes to lift steel I-beams for bigger jobs. Structural engi­neers will usually know qualified house movers. (By the way, these specialists are still called house movers even when the house stays on the site.)

INDOOR SYMPTOMS OF FOUNDATION FAILINGS

image418

Подпись: Widespread springiness in floors and joists sag-ging in mid-span are caused by joists too small Подпись: Localized springiness or low spots in flooring are probably caused by an undersize pad or by a deteriorated or absent post beneath a girder. If you find wet rot or insect damage at the base of the post, correct that situation first. Подпись: If you're willing to excavate, you can retrofit a first-rate drainage system such as this.

Most foundations that fail were poorly designed, poorly constructed, or subjected to changes— especially hydrostatic pressure or soil movement —that exceeded their load-bearing capacities. Exact causes are often elusive.

Подпись: PROTIP If you're not sure that a foun-dation crack is active (moving), epoxy a small piece of glass to both sides of the crack. If the glass breaks, the crack is active. A glass microscope slide is per-fect for this test. Or substitute a scrap of window glass. 1111 image419Подпись:for a span or by a failed or absent girder. If an existing girder seems sound, adding posts or new pads may fix the problem. Otherwise, add a girder to reduce the distance joists span.

Flooring that crowns above a girder, sloping downward toward the outside walls; doors and windows that are difficult to open; and cracking at the corners of openings are often caused by a failure of all or part of the perimeter foundation.

Foundation cracks often signal foundation fail­ure. Cracks may range from short surface cracks to through-the-wall cracks that should be exam­ined by a structural engineer. Here are some common symptoms and remedies:

► Narrow vertical or diagonal surface cracks with aligned sides are likely caused by foundation settlement or soil movement but are probably not serious. If water runs from cracks after a storm, fill cracks with an epoxy cement, and then apply a sealant.

► Wide cracks in foundations less than 2 ft. tall indicate little or no steel reinforcement, a common failing of older homes in temperate climates.

► Large, /і-in. or wider, vertical cracks through the foundation that are wider at the top usually mean that one end of the foun­dation is sinking—typically at a corner with poor drainage or a missing downspout.

► Large vertical cracks through the foundation that are wider at the bottom are usually caused by footings that are too small
for the load. You may need to replace or reinforce failed sections.

► Horizontal cracks though a concrete foundation midway up the wall, with the wall bowing in, are most often caused by lateral pressure from water-soaked soil. This condition is common to uphill walls on sloping lots.

► Concrete-block walls with horizontal cracks that bulge inward are particularly at risk because block walls are rarely reinforced with steel. If walls bulge more than / in. from vertical and there’s a chronic water problem, foundation failure may be imminent.

► In cold climates, horizontal cracks through the foundation, just below ground level, are usually caused by adfreezing, in which damp soil freezes to the top of the foundation and lifts it. If these cracks are accompanied by buckled basement floors, the foundation’s footings may not be below the frost line for your region.

Gaps between the chimney and the house are usually caused by an undersize chimney pad. If the mortar joints are eroded, too, tear the chim­ney down and replace it.

A Roof Overhead

THE WALLS FOR OUR HOUSE ARE UP and we now have something to show for our work.

At this stage, we can walk through the structure; admire the view through rough window openings; and imagine how the finished siding, painted drywall, and floor­ing will look. But first, we need to raise the roof.

Before we reach for a hammer, we need to make some decisions about. he roof trusses. We also have to prepare the site for their delivery and do some layout work so that the installation process can go smoothly. Once the trusses are installed, well move on to the fascia hoards, sheathing, and shingling.

Roof Trusses

Marly in my building career, 1 was taught how to lay out rafters with a site-made tem­plate containing the plumb and birds-mouth cuts. You can still cut and frame a roof one rafter at a time (see the 11ustrat ion on p. 114), but today most roofs are con­structed with factory-made trusses. Trusses are designed on a computer, built on an assembly line, and delivered to the job site ready to install. If you’re building a simple

5 Подпись: STEP BY STEPAdd Drywall Backing

6 Install the Barge Rafters and the Fascia Boards

7 Sheathe the Roof

8 Seal the Roof with Felt Paper

9 Ishingle the Roof

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A Roof Overhead

Подпись: w RAFTER AND RIDGE-BOARD FRAMING DETAILS Подпись: The run of a rafter is half its span. The span is measured from one outside edge of the building to the other. A 4-in-12 roof rises 4 in. vertically for every 12 in. horizontally.

sable-roof house like the one shown here, roof

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trusses can save time and keep the construc­tion process simple—something that everyone will appreciate.

kach truss includes a pair of rafters and a bottom chord that functions as a ceiling joist inside the house and as level soffit framing outside the house. Short lengths of wood, called webbing, connect the rafters with the bottom chord; barbed steel gusset plates [gus­sets, for short) are pressed into place over the joints to hold all the parts together. For a basic look at the different truss configurations, see the illustration on the facing page. The basic – roof installation process that we’ll use on this house will be very much the same for other

4

houses, regardless of size.

Porch considerations

On the house w»4v building, the main root extends over a small porch. In this situation, you need to have at least some of the porch framed before installing the roof trusses. Some builders choose to erect temporary posts to support the top beams on which the roof trusses (or rafters) will bear. When the porch is completed later, these temporary posts arc replaced with permanent ones. This strategy allows roof framing to follow wall framing directly, without the interruption of porch construction work. For details on how to build a porch or a deck that’s attached to the house, see chapter 7.

Ordering roof trusses

After vou make a few basic decisions about the trusses for vour house, it’s fairlv easv to order them. 1 like the raised-heel design of the trusses we used on this house (see the illustra­tion on the facing page). This type of truss provides an overhang along the cave walls as well as framing for a level soffit. Because the rafter is elevated above the wall’s top plate, there’s ample room for ceiling insulation and ventilation space above.

11 о woven, as shown in the illustration on the facing page, other truss designs are also possible. Depending on your budget and design preferences, you can use a scissor-type truss and have a cathedral ceiling inside the house. Or perhaps you like the rustic appear­ance of an open soffit and exposed rafter tails along the eaves. A good lumber dealer has different truss designs to show you, and it’s worth taking a look. Once you decide, here are the basic specifications the manu­facturer needs to design and construct vour trusses:

4

A Roof Overhead

TYPE OF TRUSS. The major types of trusses for gable roofs are shown in the illustration above.

SPAN BETWEEN EXTERIOR WALLS. The span is measured from one outside edge of the build­ing to the other.

EAVE DETAILS. The amount of overhang at a building’s eaves is usually shown on the plans. The plans should also tell you whether the soffit will be open or closed. For details on different wavs to finish off an eavc, see the sidebar on p. I 1 7.

NUMBER OF TRUSSES. Roof trusses are typi­cally spaced 24 in. o. c. Therefore, if your building is 40 It. long, you’ll need to order 21 trusses.

SPECIAL REQUIREMENTS. If your house has a more complex roof, special trusses may be required where one roof section joins another.

Подпись: Helping HandПодпись: Order trusses at least two to three weeks early. It's best not to order them over the phone, though. Fax the plans directly to the manufacturer or take them to the lumberyard or the man-ufacturer. This is the best way to avoid expensive mistakes.ROOF PITCH. Trusses for most small affordable buildings have either a 4-in-12 or a 5-in-12 pitch (see the illustration on the facing page).

Подпись: I IПодпись: Helping HandПодпись: Get set for bracing. Before you begin to install roof trusses, have plenty of bracing boards on hand. You'll need some 16-ft. 1x4s to nail across rafters, plus a good supply of 2x4s for sway and other braces.

STEP 1 Prepare for Truss Arrival and Installation

When roof trusses are delivered lo a job site, they can be offloaded onto the ground or onto the framed walls of the building (see the photo on the facing page). The choice depends on the builder’s preference and on the delivery truck’s capabilities. Experienced builders prefer delivery on the walls because it saves time.

Whether vour trusses are offloaded onto

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the walls or onto the ground, thev must lie flat. I have seen trusses that were stored on uneven ground come apart at the seams, mak­ing them useless within a couple of weeks.

Until they are nailed upright in place, braced, and sheathed, trusses are actually quite fragile. Prepare а Паї area close to the house where the trusses can be offloaded. The bottom­most truss should not rest directlv on the

4

ground but on wood slickers that provide a flat, level base.

Set up work platforms

One of the most difficult parts of roof-truss installation is working high off the ground.

It takes skill and practice to be able to stand on a narrow top plate and nail trusses to the wall. If this seems dangerous, or if you’re uncomfortable with it, try working on a

A JIG FOR MARKING PLUMB CUTS IN RAFTERS

2. Then nail on a 1×2 fence.

1. Mark cut lines with a small rafter square.

A JIG FOR MARKING PLUMB CUTS IN RAFTERS

 

1x stock, the same width as rafters

 

A JIG FOR MARKING PLUMB CUTS IN RAFTERS

If you need to make a series of plumb cuts to match and align rafter tails along an eave, make a jig like the one shown here. Use a rafter square to mark the plumb-cut angle of the roof (4-in-12 or 5-in-12) on a short length of 1×4 or 1×6, depending on the rafter size. Cut a parallelogram-shaped template with
identical plumb-cut angles, then nail a short 1×2 fence to the upper edge of the template. Once a chalkline has been snapped across all the rafter tails, use this template to mark the plumb cut on them so that they can be cut to length.

Подпись:Подпись: over first board. Nail through miter joint, into end of truss. boards are less demanding. On this house, we installed gutter boards that were later clad with coil aluminum (see Chapter 7). CUTTING AND NAILING GUIDELINES FOR GUTTER AND FASCIA BOARDS.

Use long, straight stock for gutter or fascia boards. Both types must butt together over the solid backing of a rafter tail. Gutter boards can meet in a simple butt joint; fascia boards should meet with a mitered joint, as shown in the illustration at left.

Nail gutter and 2x fascia boards to truss or rafter ends with 16d, hot-dipped galva­nized nails. Nail fascia boards with a finish hammer, being careful not to miss the nail or you’ll leave hammer tracks in the wood. Have a partner hold the other end of the board. Hold the board down a bit on the rafter tail so that the roof sheathing just skims over it (see the illustration on the facing page). To line

Подпись: їм їм їм їм їм їм и їм їм їм їм їм їм їм и їм їм їм їм їм їм їм їм и їм їм їм їм їм їм їм їм и їм їм їм їм їм їм їм їм и їм їм їм їм їм їм їм їм и їм їм їм їм їм MI INSTALLING GUTTER AND FASCIA BOARDS A JIG FOR MARKING PLUMB CUTS IN RAFTERSПодпись: Position the gutter board on the truss or rafter end so the sheathing overlaps the top edge.

up everything correctly, tack a piece of scrap wood on the top edge of a rafter and let it project downward, the way the roof sheathing will. Drive the topmost nail high enough so it will be covered by the drip edge that will be installed later.

At the bottom corner of the roof, the level eave trim meets the pitched barge rafter. If the exterior trim will be covered with cladding, as on this house, plumb-cut the barge rafter and nail it to the plumb-cut end of the gutter board. Install the gutter board so that it runs long, just like the lookout boards, then mark the correct overhang distance on the gutter board and on the topmost lookout. Be sure to take the barge rafter’s thickness into account. Snap a line between these two marks, then cut the gutter board and lookouts in place. Now you’re ready to cut and install the barge rafter.

The procedure is similar if you’re install­ing fascia instead of gutter boards; however, miter the end of the fascia board to fit a mitered plumb cut on the barge rafter (see the photo on p. 128). It’s easier to make both miter cuts before nailing the boards. Miter and install the fascia first, then snap the cutoff lines on the lookouts as described previously.

Plumb-cut barge rafters

Select straight, clear stock for the barge rafters, and start with a board that’s a little longer than you need. Make the bottom plumb cut on the ground. This is a mitered plumb cut, 4-in-12 on this roof, if you’re working with finished trim. With one or two helpers holding the board above, nail the barge rafter to the lookouts and to the gutter or fascia board. I like to make the top plumb cut with the board in place, judging by eye where the top of the cut should be. Or you can pull a string along the top of several trusses. Pull this string out over the barge rafter to mark where it will cut. When you install the opposite barge rafter, mark the top plumb cut against the rafter you’ve already cut to ensure a tight-fitting joint.

How Radon First Came to the Attention of the US Public

Stanley Watras had worked as an engineer for 11 years at a nuclear power plant in Pennsylvania. At the end of each workday, he and other plant em­ployees were checked by a monitor that measured radiation levels. This procedure ensured that they had not been contaminated by unsafe levels of radioactivity while at work.

In December 1984, Watras suddenly began setting off the buzzers on the radiation monitors as he walked by the machine on his way out of the building. The readings showed high levels of con­tamination over his entire body. For several days this scenario was repeated, with Watras subjected to a lengthy decontamination ordeal. Where was Watras picking up this radioactivity and why was it affecting only him?

The mystery was solved when Watras de­cided one morning to go through the monitors at the exit door as he entered the workplace. When the alarms went off, Watras immediately realized that the radiation was coming from somewhere outside the nuclear power plant. The local elec-

plugs the pores and capillary tracts of con­crete. Xypex Concentrate can be used as a single-coat dampproofing membrane or in a two-coat system with Xypex Modified. Xypex Modified can be used alone where dampproofing is required.

Fluid-Applied Dampproofing

• AFM Safecoat DynoFlex: A topcoat for use over DynoSeal

• AFM Safecoat DynoSeal: A flexible vaporproof barrier

• Rub-R-Wall: Asphalt-free moisture resis­tant membrane products for various foun­dation applications

Bentonite Dampproofing

• Volclay: A self-healing bentonite-based moisture resistant panel

Creating a Capillary Break

Under some conditions, water will move up­ward through the soil by capillary action. This type of moisture invasion can be controlled by creating a capillary break. Half-inch mini­mum gravel, free of smaller fines, placed under a slab will stop capillary action. A dampproof­ing coating or membrane should also be ap­plied between the footing and the stem wall or the stem wall and the framing to stop any moisture from being carried up through the concrete and entering the framing.

IMPORTANT ELEMENTS

In many parts of North America, building codes don’t require steel reinforcement in concrete foundations, but steel is a cost-effective means of avoiding cracks caused by lateral pressure on foundation walls.

Steel reinforcement and fasteners. Steel re­inforcing bar (rebar) basically carries and distrib­utes loads within the foundation, transferring the loads from high-pressure areas to lower-pressure areas. It thereby lessens the likelihood of point failure, either from point loading above or from lateral soil and water pressures. Anchor bolts or threaded rods, tied to rebar, attach the overlying structure to the foundation. Steel dowels are usu­ally short pieces of rebar that pin foundation walls to footings, new sections to existing foun­dations, and so on.

There are also a number of metal connectors— such as Simpson Strong-Ties—that tie joists to girders, keep support posts from drifting, and hold down mudsills, sole plates, and such.

Several are shown in Chapter 4.

Quality. Concrete quality is critically important, both in its composition and in its placement. Water, sand, and aggregate must be clean and well mixed with the cement. Concrete with com­pressive strength of 2,500 psi to 3,500 psi (pounds per square inch) is common in residential founda-

THE FOUNDATION WltnlP

Most perimeter foundations have a companion foundation within, consisting of a system of girders (beams), posts (columns), and pads that pick up the loads of joists and interior walls and thus reduce the total load on the perimeter foundation. By adding posts, beams, and pads, you can often stiffen floors, reduce squeaks, avoid excessive point loading, support new partitions, and even avoid replacing a mar­ginally adequate perimeter foundation.

image417GETTING THE

Learn what you can about local soil conditions before hiring a soils engineer. Start with local builders—especially those who’ve worked on nearby properties. Next consult with building and land-use departments, for many have maps indicating watersheds, slide zones, contaminated soil, and the like. Finally, the U. S. Department of Agriculture (USDA) has extensive soil maps. And the U. S. Geological Survey (USGS) topographic (topo) maps show streams, lakes, flood plains, and other natural features that could have an impact on your site.

tions, yet there are many ways to achieve that strength, including chemical admixtures. Discuss your needs with a concrete supplier who’s famil­iar with soil conditions in your area, and read "Ordering Concrete,” on p. 222.

Drainage. The drainage system is not technically a part of the foundation, but the flow of water alongside and under a foundation is important to its success. In some soils, it’s essential to mediate water flow. At the very least, water seeping through foundations can cause damp basements and encourage mold. Worse, excessive water can rot framing, undermine footings, and cause un­reinforced foundations to crack, bulge inward, or fail altogether. Often moisture problems can be mitigated simply by keeping gutters and downspouts clear, grading the soil away from the foundation, and improving drainage around basement window wells. Beyond that, the "cures” are increasingly expensive, such as excavating along the outside of the foundation to add gravel and perimeter drainpipe and to apply water­proofing treatments.

MAKE THE ATTIC ACCESSIBLE

MAKE THE ATTIC ACCESSIBLEПодпись: CREATING USABLE SPACE IN THE ATTICMAKE THE ATTIC ACCESSIBLEПодпись: Some usable storage space can be created in the attic, even if the house has a truss roof, by framing a floor that is supported by interior walls and sits above the trusses' joist chords.

Code requires an access hole for the attic. This al­lows workers to get into the attic to install insulation and wiring. You may also want to get up there some day to check on a roof leak or just to see how the spi­ders are doing.

The standard attic access hole has a rough 2x frame that is 221/2 in. wide by 31 in. long. This rectangu­lar opening fits nicely between ceiling joists installed on 2-ft. centers. Usually, the hole is located in a closet or along a hallway. Once you select a spot for access to the attic, cut a pair of 2xs to fit between the ceiling joists and nail them 31 in. apart. The bottom edges of the blocking should be flush with the bottom edges of the joists. Take the time to measure the hole diagonally to make sure it’s square. After V2-in. drywall is nailed in place, you will have a 21V2-in. by 30-in. access hole.

If you are using blown-in insulation to insulate the ceiling, build a plywood or OSB curb around the access hole in the attic. Make sure the curb is tall enough (at least 12 in.) to keep the insulation from falling on your head every time you open the lid to the

attic. The lid can be cut from drywall and should be insu­lated with rigid foam board.

Instead of a simple lid for the access hole, you may want to install a pull-down stair kit. If you’re using the attic for storage and need to go up there on a regular basis, a pull-down staircase is the way to go. The stairs are factory-made and designed to fit in the 22V2-in. space between the joists. The rough opening must be longer, however—54 in. is common. Check the manufacturer’s specifications before you frame the opening.

CREATE ATTIC STORAGE. There is some usable space in most truss-roof attics. To make use of it, don’t lay sheathing directly on the joist chords of roof trusses, as trusses are not engineered for this. Instead, frame a storage platform above the joist chords, making sure that there’s room for insulation beneath the platform. You can support the platform’s joists on short (8-in.- to 12-in.-long) cripple studs nailed to the double top plates of nearby walls. Sheathe the platform with 1/2-in. plywood or OSB.

Foundations and Concrete

Подпись: wood underpinning above the soil so it doesn't rot or get eaten by insects. And it should be sturdy enough to keep walls plumb and floors level despite wind, water, soil movement, and earthquakes. FOUNDATION TYPES Foundations should be appropriate to the site. For example, on sandy well-drained soil, unmortared stone foundations can last for centuries. But an unstable clay hillside may dictate an engineered foundation on piers extending down to bedrock. The tee, or spread, foundation is perhaps the most commonly used type, so named because its cross section looks like an inverted T. It's remark- Foundation issues

before starting extensive remedial work, such as replacing failed foundation sections or adding a second story to your house, have a structural engi­neer evaluate your foundation. In addition, bring in a soils engineer if the site slopes steeply or if the foundation shows any of the following dis­tress signs: bowing, widespread cracking, uneven settlement, or chronic wetness. Engineers can also assess potential concerns such as slide zones, soil load-bearing capacity, and seasonal shifting.

An Overview

A foundation is a mediator between the loads of the house and the soil on which the foundation rests. A well-designed foundation keeps a house’s

These concrete forms are half complete, showing oiled inside form boards, a new mudsill nailed up, and the bottom outer form board in place. Rebar, as shown, will be wire – tied to anchor bolts after they’ve been inserted into predrilled holes in the mudsill.

 

image413

Подпись: Foundation Typesimage414Подпись: Pouring VERSUS PLACING CONCRETE Every trade has its jargon. Concrete snobs, for example, insist on using the phrase placing concrete, though concrete coming out of a 4-in. hose looks more like a pour, albeit a sluggish one. Perhaps "placers" want to emphasize that concrete is so weighty that you should place it as close to its final location as possible. Point taken. But to denote the general movement of concrete from truck to forms, pour us a tall one.

ably adaptable. On a flat site in temperate regions, a shallow tee foundation is usually enough to support a house, while creating a crawl space that allows joist access and ventilation.

Where the ground freezes, foundation footings need to be dug below the frost line, stipulated by local codes. Below the frost line, footings aren’t susceptible to the potentially tremendous lifting and sinking forces of freeze-thaw cycles in moist soil. (Thus most houses in cold climates often have full basements.)

When tee foundations fail, they often do so because they’re unreinforced or have too small or too shallow a footprint. Unreinforced tee founda­tions that have failed are best removed and replaced. But reinforced tees that are sound can be underpinned by excavating and pouring larger footings underneath a section at a time.

Slab on grade is a giant pad of reinforced con­crete, poured simultaneously with a slightly thicker perimeter footing that increases its load­bearing capabilities. Beneath the slab, there’s typically a layer of crushed gravel and sheet plastic over that to prevent moisture from wicking up from the soil. Slabs are generally installed on flat lots where the ground doesn’t freeze because, being above frost line, shallow slabs are vulnerable to frost heaves. Although shallow, the large footprint of a slab sometimes makes it the only feasible foundation on soils with weak load-bearing capacity. Because slabs sit on grade, their drainage systems must be meticulously detailed.

Drilled concrete piers in tandem with grade beams are the premier foundation for most situations that don’t require basements. These foundations get their name because pier holes are typically drilled to bearing strata. This foun­dation type is unsurpassed for lateral stability, whether as replacement foundation for old work or for new construction. Also, concrete piers have a greater cross section than driven steel piers and hence greater skin friction against the soil, so they’re much less likely to migrate. The stability of concrete piers can be further enhanced by concrete-grade beams resting on or slightly below grade, which allow soil movement around the piers, without moving the piers.

The primary disadvantages of drilled concrete piers are cost and access. In new construction, a backhoe equipped with an auger on the power takeoff requires 10 ft. or 12 ft. of vertical clear­ance. Alternatively, there are remote-access portable rigs that can drill in tight quarters, even inside existing houses, but they are labor inten­sive to set up and move, increasing the cost.

Подпись: This pier and grade-beam foundation was built on a sloping site with expansive clay soil, so the engineer specified parallel-grade beams and a more massive grade-beam perimeter. (The piers go down 15 ft.) Integral concrete post piers atop grade beams support 4x6 joists spaced 3 ft. on center. Подпись: Before the pour, the rebar spine of this foundation wall is still visible. The green bolt holders will position anchor bolts in the middle of the mudsill. Driven steel pilings are used to anchor founda­tions on steep or unstable soils. Driven to bedrock and capped, steel pilings can support heavy vertical loads. And, as retrofits, they can stabilize a wide range of problem foundations. There are various types of steel pilings, including helical piers, which look like giant auger bits and are screwed in with hydraulic motors, and push piers, which are hollow and can be pushed in, strengthened with reinforcing bar, and filled with concrete or epoxy.

STEP 6 INSTALL THE BARGE RAFTERS AND THE FASCIA BOARDS

As explained earlier in this chapter, barge rafters extend beyond the end of a building, creating an overhang at the gable ends. The rafters are supported by 2×4 lookouts that fit into notches cut in the gable-end rafters. The lookouts run back to the first inboard rafter (or roof truss). At the bottom corner of the roof,

Подпись:a barge rafter meets a gutter or fascia board, which extends along the eave and is fastened to the ends of the rafter tails.

There are a few tricks to making sure that these exterior trim details are done correctly.

If you plan to cover the trim with aluminum cladding, as we did on this house (see Chapter 7), the cutting and installation work is a little eas­ier. On the other hand, if the barge rafter and fascia will be exposed as finished trim, you’ll need to cut tight-fitting miter joints where the boards meet. Both types of installation will go more smoothly if you follow the steps described here.

Let the lookouts run long

I like to install lookouts long, then snap a line from eave to ridge and cut the lookouts in place. If you’re framing with rafters rather than with trusses, the ridge board can also run long at this stage. This process ensures a straight barge rafter. Hoist a supply of 2×4 lookout stock up to the roof, set each lookout in its notch, and secure the end of each one against the face of
the first inboard truss (or rafter) with two 16d nails. Keep the top surface of the lookout flush with the top edge of the truss. Now, before driv­ing a pair of 16d nails into each notch, move the gable-end rafter in or out to make the framing member as straight as possible. Nail all the look­outs in their notches, and leave them like this for now.

Make truss ends straight

If the truss tails have been set straight, begin to cut and install the gutter or fascia boards. If they’re not straight, snap a line across the top or bottom edges, then mark plumb cuts to line up the truss ends in the same plane. This is another exception to the “never cut a truss” rule. As long as you’re only making a small cutoff at the end of the truss (well away from joints and gussets), there’s no chance of structural damage. Check the eave overhang called for in the plans. Measure this distance on the gable-end trusses at opposite ends of the house. Then snap a line across all the trusses, going from one end of the house to the other. Mark the cut lines on the truss ends and make the cuts. This technique also works when building plans call for exposed rafter tails that must be cut in the same plane. Use a jig or pattern, such as the one shown in the top illustration on p. 130, to mark identical plumb cuts on the rafter tails. You can make the cuts from above, while standing on the top plate of the wall, or from below, while standing on a ladder or scaffolding.

Install the gutter or fascia boards

Although both gutter and fascia boards are nailed to the rafter tails, the distinction between the two is that gutter boards are later covered with trim (aluminum or vinyl cladding or 1x finish material). Fascia, on the other hand, is a finished surface. Some builders install 1x fascia boards over 2x gutter boards or subfascia, whereas others use the 2x stock as the finished fascia.

Obviously, smooth, accurate cuts are re-
quired to install fascia boards, whereas gutter

Treatment of correlated normal stochastic variables

When some of the stochastic basic variables involved in the performance func­tion are correlated, transformation of correlated variables to uncorrelated ones is made. Consider that the stochastic basic variables in the performance func­tion are multivariate normal random variables with the mean matrix p, x and the covariance matrix Cx. Without losing generality, the original stochastic basic variables are standardized according to Eq. (4.30) as X’ = D—1|2(X — p, x).

Therefore, the standardized stochastic basic variables X’ have the mean 0 and covariance matrix equal to the correlation matrix Rx. That is, Cx = Rx = [pjjk], with pjk being the correlation coefficient between stochastic basic variables Xj and Xk.

To break the correlative relation among the stochastic basic variables, or­thogonal transformation techniques can be applied (see Appendix 4C). As an example, through eigenvalue-eigenvector (or spectral) decomposition, a new vector of uncorrelated stochastic basic variables U can be obtained as

U = V1X’ (4.64)

in which Vx is the normalized eigenvector matrix of the correlation matrix Rx of the original random variables. The new random variables U have a mean vector 0 and covariance matrix Lx = diag(Xb X2,…, ), which is a diagonal

matrix containing the eigenvalues of Rx. Hence the standard deviation of each uncorrelated standardized stochastic basic variable Uk is the square root of the corresponding eigenvalue, that is, VX*. Further standardization of U leads to

Y = Л – 1/2U (4.65)

in which Y are uncorrelated random variables having a mean vector 0 and covariance matrix I being an identity matrix.

Consider that the original stochastic basic variables are multivariate nor­mal random variables. The orthogonal transformation by Eq. (4.64) is a linear transformation; the resulting transformed random variables U are individu­ally normal but uncorrelated; that is, U ~ N(0, L) and Y = Z’ ~ N(0, I). Then the relationship between the original stochastic basic variables X and the uncorrelated standardized normal variables Z’ can be written as

Z’ = Л-1/2 VxD-1/2(X – »x) (4.66a)

X = vx + D1/2 V x Л1/2 Z’ (4.66b)

in which Лx and Vx are, respectively, the eigenvalue matrix and eigenvector matrix corresponding to the correlation matrix Rx.

In the transformed domain as defined by Z’, the directional derivatives of the performance function in z ‘-space can be computed, according to Eq. (4.37), as

Подпись: (4.67)Vz – W'(z 0 |V* W'(z 0|

in which the vector of sensitivity coefficients in Z’-space sz> = Vz W'(z’) can be obtained from VxW (x) using the chain rule of calculus, according to Eq. (4.66b), as

д X* 9 Zj

 

VxW (x)

 

Sz’ = Vz W (z’)

 

Подпись: (4.68)D1/2VxЛУ2 ) VxW(x) = ЛУ2DУ2VtSx

in which sx is the vector of sensitivity coefficients of the performance function with respect to the original stochastic basic variables X.

Подпись: Vx в Подпись: d Xk Подпись: V* в Подпись: (A-1/2 VX D -1/2)г V* в Подпись: D-i/2VxA-1/2az, (4.69)

After the design point is found, one also is interested in the sensitivity of the reliability index and failure probability with respect to changes in the involved stochastic basic variables. In the uncorrelated standardized normal Z’-space, the sensitivity of в and ps with respect to Z’ can be computed by Eqs. (4.49) and (4.50) with X’ replaced by Z’. The sensitivity of в with respect to X in the original parameter space then can be obtained as

from which the sensitivity for ps can be computed by Eq. (4.50b). A flowchart using the Ang-Tang algorithm for problems involving correlated stochastic basic variables is shown in Fig. 4.12. Step-by-step procedures for the corre­lated normal case by the Hasofer-Lind and Ang-Tang algorithms are given as follows.

The Hasofer-Lind AFOSM algorithm for problems having correlated normal stochastic variables involves the following steps:

Step 1: Select an initial trial solution x(r).

Step 2: Compute W(x(r>) and the corresponding sensitivity coefficient vector

Sx,(r ) *

Подпись:Подпись: (4.70)

Подпись: fx) sx,(r) W(x(r}) sx,(r) CxSx,(r >
Подпись: x (r +1) — fx + Cx SX,(r )

Step 3: Revise solution point xо+i> according to

Step 4: Check if x(r) and x(r+1) are sufficiently close. If yes, compute the reliability index в(г) according to

в AFOSM — [(x* – fix ) Cx1(xif – fix )]1/2 (4.71)

and the corresponding reliability ps — then, go to step 5. Other­

wise, update the solution point by letting x(r) — x(r+1) and return to step 2.

Step 5: Compute the sensitivity of the reliability index and reliability with respect to changes in stochastic basic variables at the design point x* by Eqs. (4.49), (4.50), (4.69), and (4.58).

On the other hand, the Ang-Tang AFOSM algorithm for problems involving correlated, normal stochastic basic variables consists of following steps:

Step 1: Decompose the correlation matrix Rx to find its eigenvector matrix Vx and eigenvalues Vs, using appropriate techniques.

Step 2: Select an initial point x(r) in the original parameter space.

Step 3: At the selected point x(r) compute the mean and variance of the performance function W(X) according to Eqs. (4.56) and (4.43), respectively.

Подпись: Figure 4.12 Flowchart for the Ang-Tang AFOSM reliability analysis involving correlated variables.
Step 4: Compute the corresponding reliability index ) according to Eq. (4.34).

Step 5: Compute sensitivity coefficients sz> in the uncorrelated standard nor­mal space according to Eq. (4.68) and the vector of directional derivatives a’Xr) according to Eq. (4.67).

Step 6: Using в(г) and ) obtained from steps 4 and 5, compute the location

Подпись: zk,(r +1) = ak,(r)P(r) Подпись: for k = 1, 2,..., K Подпись: (4.72)

of expansion point z r +1> in the uncorrelated standard normal space as

Step 7: Convert the obtained expansion point zT +p back to the original pa­rameter space according to Eq. (4.66b).

Step 8: Check if the revised expansion point x(r+p differs significantly from the previous trial expansion point x (r). If yes, use the revised expansion point as the trial point by letting x(r) = x(r+p, and go to step 3 for another iteration. Otherwise, the iteration procedure is considered complete, and the latest reliability index в(г) is used to compute the reliability ps = Ф(во)).

Step 9: Compute the sensitivity of the reliability index and reliability with respect to changes in stochastic basic variables at the design point x+ by Eqs. (4.49), (4.50), (4.69), and (4.68).

Example 4.11 (Correlated, normal) Refer to the data in Example 4.9 for the storm sewer reliability analysis problem. Assume that Manning’s roughness coefficient n and pipe diameter D are dependent normal random variables having a correlation coefficient of -0.75. Furthermore, the pipe slope S also is a normal random variable but is independent of Manning’s roughness coefficient and pipe size. Compute the re­liability that the sewer can convey an inflow discharge of 35 ft3/s by the Hasofer-Lind algorithm.

Solution The initial solution is taken to be the means of the three stochastic basic vari­ables, namely, xq) = p, x = (ц, п, xd, xs/ = (0.015,3.0,0.005/. Since the stochastic basic variables are correlated normal random variables with a correlation matrix as follows:

1.0

Pn, D

Pn, S

1.00

-0.75

0.00

Rx =

pn, D

1.0

PD, S

=

-0.75

1.00

0.00

.Pn, S

PD, S

1.0

0.00

0.00

1.00

by the spectral decomposition, the eigenvalues matrix associated with the correlation matrix Rx is Лх = diag(1.75, 0.25, 1.00), and the corresponding eigenvector matrix Vx is

0.7071 0.7071 0.0000 -0.7071 0.7071 0.0000 0.0000 0.0000 1.0000

At x(i) = (0.015, 3.0, 0.005/, the sensitivity vector for the performance function

W(n, D, S) = (QC – Ql) = 0.463П1 D8/3S1/2 – 35

is sx (1) = (9W/дn, dW/дD, dW/дS) = (-2734, 36.50, 4101/

and the value of the performance function W(x(1)) = 6.010, is not equal to zero. This indicates that the solution point x(1) does not lie on the limit-state surface. Applying Eq. (4.70), the new solution x(2) can be obtained as x(2) = (0.01569, 2.900, 0.004885). The difference between the two consecutive solutions is computed as

9 = |x(1) – x(2)| = [(0.01569 – 0.015)2 + (2.9 – 3.0)2 + (0.004885 – 0.005)2]0 5

= 0.1002

which is considered large, and therefore, the iteration continues. The following table lists the solution point x(r), its corresponding sensitivity vector sx,(r), and the vector of directional derivatives azi,(r), in each iteration. The iteration stops when the Eu­clidean distance between the two consecutive solution points is less than 0.001 and the value of the performance function is less than 0.001.

Iteation

Var.

x (r )

s (r )

a(r)

x(r+1)

r = 1

n

0.1500 x 10—01

—0.2734 x 10+04

—0.9681 x 10+00

0.1599 x 10—01

D

0.3000 x 10+01

0.3650 x 10+02

0.2502 x 10+00

0.2920 x 10+01

S

0.5000 x 10—02

0.4101 x 10+04

0.1203 x 10—01

0.4908 x 10—02

s

= 0.8008 x 10—01

W = 0.6010 x 10+01

в = 0.000 x 10+00

r = 2

n

0.1599 x 10—01

—0.2217 x 10+04

—0.9656 x 10+00

0.1607 x 10—01

D

0.2920 x 10+01

0.3242 x 10+02

0.2583 x 10+00

0.2912 x 10+01

S

0.4908 x 10—02

0.3612 x 10+04

0.2857 x 10—01

0.4897 x 10—02

s =

0.7453 x 10 — 02

W = 0.4565 x 10+00

в = 0.1597 x 10+01

r = 3

n

0.1607 x 10—01

—0.2178 x 10+04

—0.9654 x 10+00

0.1607 x 10—01

D

0.2912 x 10+01

0.3209 x 10+02

0.2591 x 10+00

0.2912 x 10+01

S

0.4897 x 10—02

0.3574 x 10+04

0.2991 x 10—01

0.4896 x 10—02

s

= 0.7101 x 10—04

W = 0.2992 x 10—02

в = 0.1598 x 10+01

After four iterations, the solution converges to the design point x* = (n*, D*, S*)f = (0.01607, 2.912, 0.004896)г. At the design point x*, W = 0.5 7 5 8 x 10—07, and the mean and standard deviation of the performance function W can be estimated, by Eqs. (4.42) and (4.43), respectively, as

jlw* — 5.510 and &w* — 3.448

The reliability index then can be computed as в* = pw* /aw* = 1.598, and the corre­sponding reliability and failure probability can be computed, respectively, as

ps = Ф(в*) = 0.9450 pf = 1 — ps = 0.055

Finally, at the design point x*, the sensitivity of the reliability index and reliabil­ity with respect to each of the three stochastic basic variables can be computed by Eqs. (4.49), (4.50), (4.56), and (4.57). The results are shown in the following table:

Variable

(1)

x

(2)

a*

(3)

дв/д z (4)

д Ps/д z

(5)

дв/д x (6)

д ps/д x (7)

хдв/вдx (8)

xд ps / Psд x (9)

n

0.01607

—0.9654

0.9654

0.1074

690.3

76.81

11.09

1.234

D

2.912

0.2591

—0.2591

—0.02883

—119.6

—13.31

—348.28

—38.76

S

0.004896

0.02991

—0.02991

—0.003328

—1814.

—201.9

—8.881

—0.9885