The Stiffness Matrix

Posted by on 22/ 11/ 15

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. On the other hand, the integral is performed on the volume, and the B ma­trix depends on the geometry. If we are concerned with large strains and if we are using the Cauchy’s stresses, geometry is defined in the current configuration, which is changing from step to step, and even from one iteration to the other. These two contributions, the material one, issued from the constitutive model, and the geometric one, have to be accurately computed in order to guarantee the quadratic convergence rate.

A similar discussion may be given for diffusive problems. However, the geom­etry is not modified for pure diffuse problems, so only the material term is to be considered.

Visibility: Luminance, Illuminance, and STV

Posted by on 22/ 11/ 15

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 Lighting. In all the editions of RP-8 published from its inception in 1928 through 1983, the criteria for roadway lighting design were based on illuminance (horizontal footcandles). Illuminance is a measure of the amount of light that falls upon a roadway surface. In the 1983 version of the document, alternative criteria were used—one in terms of illumi­nance (footcandles or lux) and the other in terms of pavement luminance measured in candelas per square meter (cd/m2). The preferred method was luminance since it more accurately described that which is perceived by the human eye. Further research into visi­bility has led to a new concept and provides alternative design criteria that may be used. This alternative set of criteria is based on the concept of providing an adaptation level on and adjacent to the roadway that aids in recognition of low-contrast objects.

The visibility of a stationary object on the roadway of a fixed size and uniform luminance is a function of the following:

1. The contrast between the luminance of the object and its immediate visual background

2. The general level of adaptation of that portion of the retina of the eye concerned with the object

3. The amount of veiling luminance (disability glare) entering the eye

4. The difference in eye adaptation between successive eye movements (transient adaptation)

5. The size, shape, and color of the object

6. The background complexity and the dynamics of motion

7. Visual capability of the roadway user

Visibility level (VL) is a metric used to combine mathematically the varying effects of the several factors listed above on the visibility of a standard observer. VL for an object at a particular location on the roadway viewed from a specified point and direc­tion is the amount above the visibility threshold as seen by the observer. Visibility level is a ratio and has no units. The VL as commonly used is based on detection of a “small target” that is flat and 7 in (18 cm) on each side. Small-target visibility (STV) is the weighted average VL for an array of targets as calculated by the visibility mode. A full and complete discussion of the STV method is included in Annex F of RP-8-00 published by the Illuminating Engineering Society of North America, New York (www. iesna. org).

SIGNING AND ROADWAY LIGHTING

Posted by on 22/ 11/ 15

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. Portions of this material were derived from studies made under a Federal Highway Administration Project, “Design, Construction and Maintenance of Highway Safety Features and Appurtenances.”

714 BENEFITS AND FUNDAMENTALS OF LIGHTING

Properly designed and installed roadway lighting can result in significant reductions in
nighttime traffic accidents, act as a deterrent to crime, increase commercial activity,

and improve aesthetic value. Roadway lighting increases traffic safety by enhancing the visibility of potential roadway hazards, other vehicles, pedestrians, and roadway geometrics. Pedestrians are among the largest beneficiaries of lighting installed on urban streets. Studies indicate reductions of up to 80 percent in pedestrian accidents and reductions ranging from 20 to 40 percent for all types of night accidents [1]. Another study identified a 40 percent reduction in the ratio of night accidents to day accidents resulting from the installation of roadway lighting on freeways [2]. While these figures are significant, it is anticipated that the safety benefits derived from the installation of roadway lighting will become even more pronounced in the future. This is due to the increasing age of the driving population and the significantly reduced visual abilities of persons over 65 years of age. The savings realized by accident preven­tion alone can often justify the costs of a modern lighting system [3].

Although much progress has been made in improving lighting system efficiency and effectiveness, there are still many streets, particularly in small communities, that are not lighted in accordance with present guidelines. This is primarily due to the scarcity of local funds, which can be mitigated by the use of federal funds on qualifying projects. Roadway lighting has been recognized as a viable countermeasure for increasing traffic safety since 1966, when federal legislation enabled federal aid expenditures for construction and maintenance of roadway lighting [4].

The benefits of providing roadway lighting include enhancing traffic safety, improving pedestrian visibility, deterring crime, improving commercial interests, and promoting com­munity pride. The actual benefits obtained are dependent upon the type of facility and area in which the lighting will be installed. Only the traffic operational and safety benefits obtained from the proper design and installation of roadway lighting are discussed in this chapter. It should be noted that properly designed and installed roadway lighting can result in roadway facilities operating almost as efficiently and safely at night as during the day­time. Lighting cannot, however, be expected to achieve the same safety levels as daytime operation, because of the influence of other factors, such as fatigue, higher speeds, and intoxication, which make a greater contribution to nighttime accident frequency.

Control-variate method

Posted by on 22/ 11/ 15

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. The control variable g'(X) is also a model, which is a function of the same stochastic variables X as in the model g(X). It can be shown that g(X, Z) is an unbiased estimator of the random model output g(X), that is, E[g(X, Z)] = E[g(X)] = G. The variance of g(X, Z), for any given Z, can be obtained as

in which pg, gі is the correlation coefficient between the model output g(X) and the control variable g'(X). Since both model output g(X) and the control vari­able g ‘(X) depend on the same stochastic variables X, correlation to a certain degree exists between g(X) and g'(X). As can be seen from Eq. (6.103), using a control variable g ‘(X) could result in a variance reduction in estimating the expected model output. The degree of variance reduction depends on how large the value of the correlation coefficient is. There exists a tradeoff here. To attain a high variance reduction, a high correlation coefficient is required, which can be achieved by making the control variable g'(X) a good approximation to the model g(X). However, this could result in a complex control variable for which the expected value may not be derived easily. On the other hand, the use of a simple control variable g'(X) that is a poor approximation of g(X) would not result in an effective variance reduction in estimation.

The attainment of variance reduction, however, cannot be achieved from total ignorance. Equation (6.103) indicates that variance reduction for estimating G is possible only through the correlation between g(X) and g'(X). However, the correlation between g(X) and g'(X) is generally not known in real-life situa­tions. Consequently, a sequence of random variates of X must be produced to compute the corresponding values of the model output g(X) and the control variable g'(X) to estimate the optimal value of Z* by Eq. (6.102). The general algorithm of the control-variate method can be stated as follows.

1. Select a control variable g'(X).

2. Generate random variates for X(i) according to their probabilistic laws.

3. Compute the corresponding values of the model g(X(i)) and the control vari­able g’ (X(i)).

4. Repeat steps 2 and 3 n times.

5. Estimate the value Z*, according to Eq. (6.102), by

Подпись: (6.104) (6.105) c En=1(g(° – g)[g/(° – E(gQ]

* n Var(g 0

Z ЕП=1і£(0 – g]fe'(0 – E(g0]

Z* En=1k'(0 – E (g OP

depending on whether the variance of the control variable g ‘(X) is known or not.

6. Estimate the value of G, according to Eq. (6.100), by

Подпись: (6.106)G = -]T (g(i) – Z* g/(i)) + Z* e (g о

i=1

Further improvement in accuracy could be made in step 2 of this above algo­rithm by using the antithetic-variate approach to generate random variates.

This idea of the control-variate method can be extended to consider a set of J control variates g'(X) = [g(X), g2(X), …, gJ(X)]г. Then Eq. (6.100) can be modified as

J

g (X, Z) = g (X) -£ Zj {g j (X) – E[g j (X)]} (6.107)

j=1

The vector of optimal coefficients Z*= (Z*nZ*2,… ,Z* J ) that minimizes the vari­ance of g(X, Z) is

in which c is a J x 1 cross-covariance vector between J control variates g'(X) and the model g(X), that is, c = {Cov[g(X),g[(X)],Cov[g(X),g’2(X)], Cov[g(X), g J (X)]}, and C is the covariance matrix of the J control variates, that is, C = [oij] = [ g i (X), g j (X)], for i, j = 1,2,…, J. The corresponding minimum variance of the estimator g(X, Z) is

Var(g) = Var(g) – ctCc= (1 – pgg,) Var(g) (6.109)

in which pg, gі is the multiple correlation coefficient between g(X) and the vector of control variates g'(X). The squared multiple correlation coefficient is called the coefficient of determination and represents the percentage of variation in the model outputs g(X) explained by the J control variates g'(X).

Sign-Support Straightening

Posted by on 22/ 11/ 15

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.

7.9 REFERENCES ON SIGNING

1. Manual on Uniform Traffic Control Devices, American Association of State Highway Officials and National Conference on Street and Highway Safety, Washington, D. C., 1935.

2. Manual on Uniform Traffic Control Devices for Streets and Highways, Federal Highway Administration, U. S. Department of Transportation, Washington, D. C., 2003.

3. Traffic Control Devices Handbook, Federal Highway Administration, U. S. Department of Transportation, Washington, D. C., 1983.

4. Bowman, Brian L., NCHRP Synthesis of Highway Practice 186: Supplemental Advance Warning Devices, Transportation Research Board, National Research Council, Washington, D. C., 1993.

5. Harwood, Douglas W., NCHRP Synthesis of Highway Practice 191: Use of Rumble Strips to Enhance Safety, Transportation Research Board, National Research Council, Washington, D. C., 1993.

6. Cunard, Richard A., NCHRP Synthesis of Highway Practice 157: Maintenance Management of Street and Highway Signs, Transportation Research Board, National Research Council, Washington, D. C., 1990.

7. “1988 Annual Report on Highway Safety Improvement Programs,” Report of the Secretary of Transportation to the United States Congress, Report No. FHWA-SA-88-0003, Federal Highway Administration, U. S. Department of Transportation, Washington, D. C., 1988.

8. Lewis, Russel M., NCHRP Synthesis of Highway Practice 106: Practical Guidelines for Minimizing Tort Liability, Transportation Research Board, National Research Council, Washington, D. C., 1983.

9. Standard Highway Signs, Including Pavement Markings and Standard Alphabets, 2004 Version, U. S. Department of Transportation. Federal Highway Administration, Washington, D. C., 2004.

10. Roadside Design Guide, American Association of State Highway and Transportation Officials, Washington, D. C., 2002.

11. Delphi V—Forecast and Analysis of the U. S. Automotive Industry through the Year 2000, University of Michigan Transportation Research Institute, Ann Arbor, July 1989.

12. Standard Specifications for Structural Supports for Highway Signs, Luminaires and Traffic Signals, American Association of State Highway and Transportation Officials, Washington, D. C., 2001, and Interim Specifications, 2002, 2003, and 2006.

13. Mickie, J. D., “Recommended Procedures for the Safety Performance Evaluation of Highway Appurtenances,” NCHRP Report 230, Transportation Research Board, Washington, D. C., March 1987.

14. Ross, Hayes, Jr., Dean L. Sicking, Richard A. Zimmer, and Jarvis D. Mickie, “Recommended Procedures for the Safety Performance Evaluation of Highway Features,” NCHRP Report 350, Transportation Research Board, National Research Council, Washington, D. C., 1993.

15. Cunard, Richard A., NCHRP Synthesis of Highway Practice 157; Maintenance Management of Street and Highway Signs, Transportation Research Board, National Research Council, Washington, D. C., 1990.

16. Perkins, David, D., “Manual on Countermeasures for Sign Vandalism,” Report No. FHWA – IP-86-7, Federal Highway Administration, Washington, D. C., September 1986.

17. Texas Transportation Institute, “State of the Practice in Supports for Small Highway Signs,” Technology Sharing Report 80-222, Federal Highway Administration, Washington, D. C., 1980.

18. Heinz, Ronald E., “Request for Technical Assistance: Sign Support Design,” Memorandum, Federal Highway Administration, Highway Design Division, Washington, D. C., July 1985.

19. Hayes, E. Ross, Jr., Jesse L. Buffington, et al., “State of the Practice in Supports for Small Highway Signs,” Federal Highway Administration, Washington, D. C., 1980.

20. Composite Technologies Company, Signs Manufactured from 100% Landfill-Destined Plastic, undated brochure, Composite Technologies Company, Dayton, Ohio.

21. Standard Specifications for Highway Signs, Luminaires and Traffic Signals, American Association of State Highway and Transportation Officials, Technical Committee, Washington, D. C., 1985.

22. Phillips, David K., “Steel Flanged Channel Posts for Small Highway Sign Supports,” techni­cal advisory, Federal Highway Administration, Office of Engineering, Washington, D. C., September 27, 1983.

23. Staron, L. A., “Minute Man Breakaway Device,” letter, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., January 1987.

24. Heinz, Ronald E., “Splicing of U-Channel Steel Posts,” memorandum, Federal Highway Administration, Highway Design Division, Washington, D. C., October 1984.

25. Noel, Leon M., “Timber Sign Supports,” letter, Federal Highway Administration, Highway Design Division, Washington, D. C., August 1982.

26. Hove, R. W., “Ground Mounted Signs: Timber Sign Supports,” memorandum, Federal Highway Administration, U. S. Department of Transportation, Washington, D. C., May 1984.

27. Allied Tube and Conduit, Qwik-Punch and Qwik-Coat Systems, Harvey, Ill., January 1991.

28. Noble, Glen, “Support System Acceptance,” memorandum, Unistrut Corporation, Wayne, Mich., November 1992.

29. Staron, L. A., “Quick-Punch Sign Supports,” letter, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., October 3, 1986.

30. Xcessories Squared, Soil Stabilizer (brochure), Auburn, Ill., February 23, 1995.

31. Staron, Lawrence A., Geometric and Roadway Acceptance Letter No. SS-25, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., June 4,

1991.

32. Staron, L. A., Geometric and Roadside Design Letter No. SS-38, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., November 1993.

33. Dent Bolt, Trinity Industries, Inc., Highway Safety Products, Dallas, Tex.

34. Staron, L. A., ACTION: Breakaway Sign Supports, memorandum, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., September 1993.

35. Franklin Steel Eze-Erect™ Sign Posts, Franklin Steel, Franklin, Pa., May 1989.

36. The Minute Man™ U-Channel Breakaway Signpost System, Marion Steel Co., Marion, Ohio,

1992.

37. Staron, L. A., “Splicing of Steel U-Channel Posts on Small Sign Supports,” memorandum, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., September 1991.

38. Unistrut-Telespar™ Sign Support System, Unistrut Corporation, Wayne, Mich., 1986.

39. Allied Square Tube Signposts, Allied Tube and Conduit, Harvey, Ill., February 1992.

40. 14 Ga. Qwik-Punch System, Allied Tube and Conduit, Harvey, Ill., September 1991.

41. Roadside Improvements for Local Roads and Streets, Office of Highway Safety, Federal Highway Administration, Washington, D. C., October 1986.

42. Staron, L. A., Geometric and Roadside Acceptance Letter No. SS-27, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., May 1992.

43. Staron, L. A., Geometric and Roadside Acceptance Letter No. LS-23, Federal Highway Administration, Federal Aid and Design Division, Washington, D. C., January 1991.

44. Van Ness, Norman, Acceptance Letter to Southwest Pipe Inc., Highway Design Division, Federal Highway Administration, Washington, D. C., July 1986.

45. Hanna, Howard, “Evaluation of the Upper Hinge Mechanism of Multiple Leg Breakaway Sign Supports,” memorandum, Federal Highway Administration, Program Development Division, Washington, D. C., October 1987.

46. “Maintenance of Small Traffic Signs—A Guide for Street and Highway Maintenance Personnel,” Federal Highway Administration, Report No. FHWA-RT-90-002, Washington, D. C., 1991.

47. Nettleson, Tom, Signs Maintenance Guide, Forest Service, U. S. Department of Agriculture, October 1979.

48. McGee, H. W., et al., “Sign Fabrication, Installation and Maintenance,” Federal Highway Administration, Report No. FHWA-SA-91-033, Washington, D. C., May 1992.

49. Horne, Dwight A., “Slip Base Triple Square Supports in Standard Soil,” Letter No. SS68B, U. S. Department of Transportation, Federal Highway Administration, Washington, D. C., 2004.

Three Things to Ask an HVAC Specialist

Posted by on 22/ 11/ 15

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 ventilators. Tightly insulated houses conserve energy, but they may also recycle stale air endlessly. In response to the dilemma of introducing fresh air without expelling conditioned air, heat-recovery ventila­tors (HRVs) were developed. Typically, an HRV has two fans: one to bring in fresh air and one to expel stale air. It also has a heat exchanger that recovers 75 percent to 80 percent of the heat in the outgoing air and preheats incoming air. HRVs can filter pollen and dust from incoming air and, by
equalizing air pressure in tight houses, prevent potentially dangerous situations such as back-drafting (furnace com­bustion air, including carbon monoxide, being sucked back down a vent flue by negative air pressure). Some HRVs also remove excess humidity from incoming air.

► Central air cleaners. Most standard central heating/ air-conditioning filters don’t do a good job. If you’re con­cerned about interior air quality and removing allergens (such as dust mites, mold, and pet dander), you have a variety of air cleaners to choose from, including pleated media, self-charging electrostatic filters, electronic air cleaners, and by-pass HEPA filters. In general, air cleaners vary by fineness of filtering (dust arrestance), ease of installing into existing ductwork, airflow impedance, and cost. To make sense of this and many other HVAC topics, visit www. dulley. com. On air filters, mechanical engineer Jim Dulley notes, "For do-it-yourself installation, self­charging electrostatic models are ideal because they require no sheet-metal ductwork to install."

► Upgrading ducts. Sealing ducts can prevent air leaks and, in many cases, reduce excess moisture. But if ducts are rusty and as tired as the one shown on p. 333, or are uninsulated as they traverse unheated areas, maybe it’s time to replace them. Three popular types are shown above.

Materials SIZES AND TYPES OF DRYWALL

Posted by on 22/ 11/ 15

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. panels between the garage and the house for fire resistance. If you
use Vs-in. drywall on the walls, be sure to order wider doorjambs.

Water-resistant drywall is often used in high – moisture areas, such as bathrooms. Called "green – board" because of its green-paper facing, it is treated to resist moisture damage but is not waterproof. It’s most often used to cover wall areas above tub and shower enclosures. Greenboard can be taped and painted just like regular drywall. It should not be installed on the ceiling, unless the joists are spaced 12 in. o. c. to keep the board from sagging.

The short (48-in.) ends of a drywall panel are cut square, leaving the gypsum core exposed. The long edges of the panel are faced with paper and tapered so that the seams between panels can be leveled with the surrounding drywall during the fin­ishing process.

PREPARING FOR DRYWALL INSTALLATION

Подпись: Electrical switch = $ location centerline holder 24 in. to centerline Подпись: Before installing drywall, mark on the floor with keel the location of wall studs, electrical boxes, and backing.Materials SIZES AND TYPES OF DRYWALL

warm weather. Otherwise, make sure the wood dries out. You can even run a dehumidi­fier inside, if necessary.

Clean and mark the floor

Take time to clean up any scraps of wood or trash on the floor. Once the floor is clean, use a piece of keel (I use red because it shows up well) to mark the stud, trimmer, and cripple locations on the floor and the joist locations on the top plate. Knowing the location of studs and joists makes it easier to nail off dry – wall and, later, baseboard trim.

It’s also a good idea to mark the locations of electrical outlets on the floor (see the top illustration at right). This helps avoid installing drywall panels over outlets, which can easily happen if you’re not paying atten­tion. If it does happen anyway, at least there will be a mark on the floor telling you where the outlet is located. You can also mark the location of the backing placed in the walls to support towel racks, grab bars, toilet-paper holders, and so on.

Check and correct bad studs

Even if all the studs were crowned in one direction during wall framing, it doesn’t ensure a perfectly straight wall. Sight down the length of the walls or lay a straightedge across them to locate bad studs. Replace any badly bowed studs, or fix a bowed stud by making a cut into the bowed area, forcing the stud straight, and bracing it with a lx cleat (see the bottom illustration at right).

Tool up to hang drywall

The tools you need to install drywall are pretty basic. In addition to the chalkline and tape measure you’ve used for the work cov­ered in earlier chapters, you’ll need the follow­ing tools:

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

Posted by on 22/ 11/ 15

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. An equivalent equation could be obtained based on the virtual power principle. The 81 and 8p terms would then be interpreted as virtual arbitrary displacements and pressures. Within the finite ele­ment method, the global equilibrium/balance equation will be verified for a number of fundamental cases equivalent to the degrees of freedom (d. o.f.) of the problem, i. e. the number of nodes times the number of degrees of freedom per node, minus the number of imposed values. The corresponding weighting functions will have simple forms based on the element shape functions.[26]

Giving a field of stress or of flux, using the weighting functions, one will obtain a value for each d. o.f., which is equivalent to a nodal expression of the equilib – rium/balance equation.

More precisely, for solid mechanics problems, one will obtain internal forces equivalent to stresses at each node, L:

Fu = f (JijBLjdV (11.19)

V

where BLj is a member of the matrix, B, of derivatives of the shape functions, N. If equilibrium is maintained from the discretised point of view, these internal forces are equal to external forces (if external forces are distributed, a weighting is necessary):

Fun = FLX (11.20)

Similarly, for diffusion phenomena the nodal internal fluxes are equivalent to the local fluxes:

Fnt = f [SNl – f diNu]dV (11.21)

V

If the balance equation is respected from the discretised point of view, these internal fluxes are equal to external ones:

FLnt = F[xt (11.22)

However, as we are considering non linear-problems, equilibrium/balance can­not be obtained immediately, but requires iteration. This means that the equations (Eqs. 11.20 and 11.22) are not fulfilled until the last iteration of each step.

Non-linear problems have been solved for some decades, and different methods have been used. From the present point of view, the Newton-Raphson method is the
reference method and probably the best one for a large number of problems. Let us describe the method. In Eq. 11.20 the internal forces Fg/ are dependant on the basic unknown of the problem, i. e. the displacement field. Similarly in Eq. 11.22 the internal fluxes are dependant on the pressure (temperature, concentration…) field.

If the external forces/fluxes don’t equilibrate, the question to be treated can be formulated in the following manner. Following the Newton-Raphson method, one develops the internal force as a first order Taylor’s series around the last approxima­tion of the displacement field:

– Fint

Fu – FL (h)) + + O2 = FLx, (11.23)

-/Kj

where the subscript (i) indicates the iteration number and O2 represents second order, infinitely small terms. This is a linearization of the non-linear equilibrium equation. It allows one to obtain a correction of the displacement field:

/А Fint 1

-g FLfQa)) – FUX) – ELikj {F^Qi)) – FUT) (11.24)

Here, the matrix, E, represented here by its member term ELi, Kj, is the so-called stiffness matrix. With the corrected displacement field, one may evaluate new strain rates, new stress rates, and new improved internal forces. Equilibrium should then be improved.

The same meaning may be developed for diffusion problems using Taylor’s development of the internal fluxes with respect to the pressure/temperature/concen – tration nodal unknowns.

The iterative process may be summarised as shown in Fig. 11.1 for a one-d. o.f. solid mechanics problem. Starting from a first approximation of the displacement field /(i) the internal forces Fint (1) (point A(1) in the figure) are computed to be lower then the imposed external forces Fext. Equilibrium is then not achieved and a new approximation of the displacement field is sought. The tangent stiffness matrix is

evaluated and an improved displacement is obtained l(2) (point B(i)) (the target being as in Eq. 11.22. One computes again the internal forces Fint(2) (point A(2)) that are again lower then the external forces Fext. As equilibrium is not yet fulfilled, a new approximation of the displacement field is sought, l(3) (point B(2)). The procedure has to be repeated until the equilibrium/balance equation is fulfilled with a given accuracy (numerical convergence norm). The process has a quadratic convergence, which is generally considered as the optimum numerical solution.

However the Newton-Raphson method has an important drawback: it needs a large amount of work to be performed as well as to be run on a computer. The stiffness matrix, E, is especially time-consuming for analytical development and for numerical inversion. Therefore other methods have been proposed:

• An approximate stiffness matrix, in which some non-linear terms are neglected.

• Successive use of the same stiffness matrix avoiding new computation and inver­sion at each iteration.

It should be noted that each alternative is reducing the numerical convergence rate. For some highly non-linear problems, the convergence may be lost, and then no numerical solution will be obtained.

Some other authors, considering the properties and the efficiency of explicit time schemes for rapid dynamic problems (e. g. for shock modelling) add an artificial mass to the problem in order to solve it as a quick dynamic one. It should be clear that such a technique might degrade the accuracy of the solution, as artificial inertial effects are added and the static equilibrium Eq. 11.1 is not checked.

Reviewing the Plans & Making Preparations

Posted by on 22/ 11/ 15

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. If you can’t figure it out, ask the framing contractor, superintendent, or architect about that particular element. If you are on a large job where the specifications come bound by themselves, you should know that they are probably organized under the Construction Specification Institute’s (CSI) MasterFormat. Under this system, rough carpentry is listed in Division 6 as 06 10 00. This section contains the basic specification information about framing this job.

2. Make a list of questions. While you are studying the plans, have a pad of paper and pencil handy so you can write down any questions. Go over these questions with the superintendent at the pre-start job site review meeting. Often, getting a question answered or a problem solved before the job begins saves an interruption in the framing. Even a little thing like the architect missing a dimension on the plans can cause a delay. If the superintendent okays scaling the missing dimensions, there won’t be a problem; but if you need verification on missing dimensions, it’s best to get them before you begin.

3. Highlight the plans. It’s a big help to highlight easy-to-miss items on your plans. Use the same color highlights on all jobs so that it becomes easy to identify items for you and your crew. An example would be: Orange-Hold-downs Pink—Shear walls Green—Glu-lam beams Blue—Steel Yellow—Special items

Highlighting the plans

4. Establish framing dimensions. Most rough openings are standardized, but because of exceptions and differences in floor covering, it’s important to go over the rough openings before the job begins. The information sheets that follow can be used for reviewing these dimensions with the superintendent.

There is a sheet for 885/в" studs and one for 92s/8" studs. These can be adjusted for different size studs. Go over each item with the superintendent or whoever is in charge of the job. Ask him/her to review the sheet and indicate that you will be using the rough­opening dimensions listed unless you are instructed differently. Note that 885/8" studs are standard because with a 4 x 8 header, they leave a standard 82%" door opening. Note, too, that 92s/8" studs work with a 4 x 12 header.

(See "Standard Framing Dimensions” sheets.)

These sheets apply to residential framing. Commercial framing is not so standardized. Note that the use of hollow metal (H. M.) door and window frames is common in commercial framing. The frames are usually 2" in width. Rough openings (R. O.) for H. M. frames would typically be 2" for the frame plus W" installation space. As an example, a 3′-0" door would have an R. O. width of 3′-4%", which is made up of 3′-0" for the door opening, 4" for the frames on each side, and W for the W" installation space on each side. The R. O. height would be 7′-2W", which would be made up of the 7′-0" for the door opening, 2" for the frame, and W" for installation space.

Tape the plans. Plan deterioration can be a problem, particularly at the end of a job. Use the same set of plans when possible so they include your highlighting and any changes that you have marked. When possible, request water-resistant print paper for the plans. If you’re in a rainy area or season, this will keep the lines from running. Plastic covers are made to cover plans, but they can make it difficult to turn the pages. Clear plastic adhesive covering can be used, but then you can’t write on the plans to note changes. A good system is to use clear plastic wrapping tape to tape the edges of the plans. This treatment usually provides the stability to make it through the job while still allowing for notes written on the plans.

Taping the plans

These dimensions should be checked with the job site superintendent before beginning each job. * Furr = furring under header after header is in place.

** Trimmer heights will increase by IV211 it lightweight concrete is used or %" if gypcrete is used.

*** Cut T. P. —Cut the top plate out and leave the double plate.

R. O. (rough opening) – Any opening framed by the framing members.

These dimensions should be checked with the job site superintendent before beginning each job. * Furr = furring under header after header is in place.

** Trimmer heights will increase by IV211 it lightweight concrete is used or %" if gypcrete is used.

R. O. (rough opening) – Any opening framed by the framing members.

Asphalt Pavement Analyzer

Posted by on 22/ 11/ 15

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).