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The Stiffness Matrix

From Eq. 11.24, it appears that the stiffness matrix is a derivative of the internal forces:

n Fint n

FLhK] = – F – = — *ljBLjdvj (11.25)

1

2

1

Derivative of problem 1 nodal forces with respect to problem 1 nodal unknowns

Derivative of problem 1 nodal forces with respect to problem 2 nodal unknowns

2

Derivative of problem 2 nodal forces with respect to problem 1 nodal unknowns

Derivative of problem 2 nodal forces with respect to problem 2 nodal unknowns

Fig. 11.2 Illustrative layout of stiffness matrix

Two contributions will be obtained (Fig. 11.2). On the one hand, one has to derive the stress state with respect to the strain field, itself depending on the displacement field...

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Visibility: Luminance, Illuminance, and STV

The requirement of adequate visibility is essential for safe traffic operations during both day and night operation. Visibility can be separated into at least three classifications when applied to highway driving: perception, recognition, and decision making [5]. Perception involves the condition of our eyes, the quantity and the direction of the available light, size of the object being viewed, contrast of the object against its background, and the time available for viewing the object. Effective roadway lighting can aid in these tasks by pro­viding the quality of light required by the human eye to increase its visual acuity.

The practice of roadway lighting in the United States is governed by tenets published in the ANSI/IESNA RP-8, American National Standard Practice for Roadway Lighti...

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SIGNING AND ROADWAY LIGHTING

PART 2

ROADWAY LIGHTING

C. Paul Watson, P. E.

Formerly, State Electrical Engineer
Alabama Department of Transportation
Montgomery, Alabama

Nelson Russell, P. E.

Manager, Electrical Department
Volkert & Associates
Mobile, Alabama

Brian L. Bowman, Ph. D., P. E.

Professor of Civil Engineering
Auburn University
Auburn, Alabama

Part 2 of this chapter presents considerations in the selection of lighting for freeways and other types of roadways. Both standard and high mast lighting are addressed. Roadside safety and the application of various types of bases are discussed and illustrated. Information on construction, acceptance testing, and maintenance is presented. An exten­sive list of references, which are noted in the text, concludes the section...

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Control-variate method

The basic idea behind the control-variate method for variance reduction is to take advantage of the available information for the selected variables related to the quantity to be estimated. Referring to Eq. (6.91), the quantity G to be estimated is the expected value of the output of the model g(X). The value of G can be estimated directly by those techniques described in Sec. 6.6. However, a reduction in estimation error can be achieved by indirectly estimating the mean of a surrogate model g(X, Z) as (Ang and Tang, 1984)

g(X, Z) = g(X) – Z(g'(X) – E[g'(X)]} (6.100)

Подпись: Var(g) = Var(g) + Z2Var(g0 - 2ZCov(g, g') The coefficient Z that minimizes Var(g) in Eq. (6.101) is C°v(g, g Q * Var( g 0 and the corresponding variance of g(X, Z) is Var(g) = (1 - pg, g, )Var(g) < Var(g) Control-variate method

in which g'(X) is a control variable with the known expected value E[g'(X)], and Z is a coefficient to be determined in such a way that the variance of g(X, Z) is minimized...

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Sign-Support Straightening

A tool such as that shown in Fig. 7.57 can be constructed out of pipe to straighten twisted U-channel posts [48]. Similar devices with a metal U-shape at the end of a pipe handle can be constructed to realign shaped wood posts and square tubing. A large pipe wrench can also be used to realign U-channel and square-tube supports. Small signs should be mounted at 90° to the road.

U. S. Customary Units, in

A.

1 Pc – 38 I. D. BLACK PIPE – 1270 LONG

1.5 I. D.—48

B.

1 Pc – 16 X 75 X 255 LONG

s/e X 3 X 10

c.

1 Pc-20 X 20 X 15 LONG

3/4 X 3/4 X V2

D.

1 Pc-6 x 20 x 510 LONG

1/4 X 3/4 X 20

E.

1 Pc – 10 X 10 X 75 LONG

3/8 X 3/B X 3

R

1 – CLEVIS SLIP HOOK (REMOVE EYE)

FIGURE 7.57 Shop-fabricated tool to straighten twisted U-channel.

7...

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Three Things to Ask an HVAC Specialist

Upgrading the heating and cooling equipment in your house is a good way to improve interior air quality and con­serve energy. Ask a reputable local HVAC contractor about the following:

► Induced-draft gas furnaces. Roughly two-thirds of North American homes have forced-hot-air (FHA) sys­tems, so replacing an existing FHA furnace is a great way to increase efficiency without disturbing existing ducts and registers. Induced-draft furnaces can achieve annual fuel-use efficiencies of 90 percent to 97 percent because they extract heat from combustion gases. (Older well – maintained furnaces might have efficiencies closer to

70 percent, or less.) Because combustion gases are cooler, they’re less buoyant, so the system uses a fan to expel them—hence the name induced draft.

► Heat-recovery ven...

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Materials SIZES AND TYPES OF DRYWALL

DRYWALL IS MADE by sandwiching a gypsum core between two sheets of paper. The "good" side of the panel is faced with smooth, white paper that takes paint easily. The "bad" side is darker in color, with a rough, porous paper surface. Panels (also called sheets) of drywall are packaged in pairs; to open the package, simply pull off the strips of paper that extend along each end.

The standard width for drywall panels is 48 in. Different lengths are available but, for affordable housing, the most commonly used lengths are 8 ft. and 12 ft. The most common thickness for drywall is Уг in. However, Ye-in.-thick panels are often used on ceilings where the joists are spaced 2 ft. o. c. because they are less prone to sagging. Most codes require Ys-in...

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Solving the Non-Linear Problem — The Newton-Raphson Method

Let us now concentrate on the finite element method. The fundamental equation to be solved is the equilibrium Eq. 11.1 (or the balance Eq. 11.6 for diffusion phenomena). As the numerical methods give an approximate solution, the equilib – rium/balance equation has to be solved with the best compromise. This is obtained by a global weak form of the local equation. Using weighted residuals, for solid mechanics, one obtains:

J [v, j Sel}]dV = J P, Sl, dV + j Р8Ш (11.17)

V V A

And for diffusion phenomena:

j [,SSp – f, d, (5p)]dV = j QSpdV + j qSpdA (11.18)

V V A

where p and q are surface terms of imposed loads/fluxes. The weighting functions are denoted 81 and 8p, and $£ represents a derivative of the weighting function based on the Cauchy’s strain derivate operator...

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Reviewing the Plans & Making Preparations

Plan review will save you time and energy, and make your work more productive. If you are framing a house with a plan you have used before, then you have already done the review. But if you are framing a new house design or, particularly, a multi-unit or commercial building, then it becomes very important to review the plans. Here are some of the most common ways of reviewing plans:

1. Study the plans. Sit down with the plans and figure out how the building is put together. Read the specifications. Most often they

are standard and you can skim through them, but make sure to note anything that is new or different. Know enough about the new material so that you can understand the architect’s explanations...

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Asphalt Pavement Analyzer

The APA is the second generation of Georgia loaded-wheel tester (GLWT) used in the United States for testing resistance to deformation of asphalt mixtures. The test temperature depends on the climatic data for the region where the mixture will be placed, and it is usually close to the highest expected temperature of the pavement. The test conditions include the wheel load and contact pressure, which are individu­ally determined (usually 445 N and 690 kPa, respectively), and the number of load­ing cycles, 8000.

A full report about the APA can be found in the U. S. publication National Cooperative Highway Research Program Report No. 508 (Kandhal and Cooley, 2003).

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