w  (5.231) n  l w where n  integer, 1 for the fundamental mode of vibration, 2 for the second mode, . . . l  span of cable, ft w  load on cable, kips per ft g  acceleration due to gravity  32.2 ft / s2 T  cable tension, kips The spreaders of a cable truss impose the condition that under a given load the change in sag of the cables must be equal. But the changes in tension of the two cables may not be equal. If the ratio of sag to span / l is small (less than about 0.1). Eq. (5.222) indicates that, for a parabolic cable, the change in tension is given approximately by H  (5.232) 3 l 2 where  change in sag A  cross-sectional area of cable E  modulus of elasticity of cable steel Double cables interconnected with struts may be analyzed as discrete or continuous systems. For a discrete system, the spreaders are treated as individual members. For a continuous system, the spreaders are replaced by a continuous diaphragm that ensures that the changes in sag and rise of cables remain equal under changes in load. Similarly, for analysis of a cable network, the cables, when treated as a continuous system, may be replaced by a continuous membrane. (C. H. Mollman, Analysis of Plane Prestressed Cable Structures, Journal of the Structural Division, ASCE, Vol. 96, No. ST10, Proceedings Paper 7598, October 1970, pp. 20592082; D. P. Greenberg, Inelastic Analysis of Suspension Roof Structures, Journal of the Structural Division, ASCE, Vol. 96, No. ST5, Proceedings Paper 7284, May 1970, pp. 905930; H. Tottenham and P. G. Williams, Cable Net: Continuous System Analysis, Journal of the Engineering Mechanics Division, ASCE, Vol. 96, No. EM3, Proceedings Paper 7347, June 1970, pp. 277293; A. Siev, A General Analysis of Prestressed Nets, Publications, International Association for Bridge and Structural Engineering, Vol. 23, pp. 283 292, Zurich, Switzerland, 1963; A. Siev, Stress Analysis of Prestressed Suspended Roofs, Journal of the Structural Division, ASCE, Vol. 90, No. ST4, Proceedings Paper 4008. August 1964, pp. 103121; C. H. Thornton and C. Birnstiel, Threedimensional Suspension Structures, Journal of the Structural Division, ASCE, Vol. 93, No. ST2, Proceedings Paper 5196, April 1967, pp. 247270.) A true membrane is able to withstand tension but is completely unable to resist bending. Although it is highly efficient structurally, like a shell, a membrane must be much thinner than a shell and therefore can be made of a very lightweight material, such as fabric, with considerable reduction in dead load compared with other types of construction. Such a thin material, however, would buckle if subjected to compression. Consequently, a true membrane, when loaded, deflects and assumes a shape that enables it to develop tensile stresses that resist the loads. Membranes may be used for the roof of a building or as a complete exterior enclosure. One way to utilize a membrane for these purposes is to hang it with initial tension between appropriate supports. For example, a tent may be formed by supporting fabric atop one or more tall posts and anchoring the outer edges of the stretched fabric to the ground. As another example, a dish-shaped roof may be constructed by stretching a membrane and anchoring it to the inner surface of a ring girder. In both examples, loads induce only tensile stresses in the membrane. The stresses may be computed from the laws of equilibrium, because a membrane is statically determinate. Another way to utilize a membrane as an enclosure or roof is to pretension the membrane to enable it to carry compressive loads. For the purpose, forces may be applied, and retained as long as needed, around the edges or over the surface of the membrane to induce tensile stresses that are larger than the larger compressive stresses that loads will impose. As a result, compression due to loads will only reduce the prestress and the membrane will always be subjected only to tensile

Manganese in low concentrations promotes hardenability and nondeforming, nonshrinking characteristics for tool steels. In high concentrations, the steel is austenitic under ordinary conditions, is extremely tough, and work-hardens readily. It is therefore used for teeth of power-shovel dippers, railroad frogs, rock crushers, and similar applications. Molybdenum is usually associated with other elements, especially chromium and nickel. It increases corrosion resistance, raises tensile strength and elastic limit without reducing ductility, promotes casehardening, and improves impact resistance. Nickel boosts tensile strength and yield point without reducing ductility; increases low-temperature toughness, whereas ordinary carbon steels become brittle; promotes casehardening; and in high concentrations improves corrosion resistance under severe conditions. It is often used with chromium (see Art. 4.44.2). Invar contains 36% nickel. Silicon strengthens low-alloy steels; improves oxidation resistance; with low carbon yields transformer steel, because of low hysteresis loss and high permeability; in high concentrations provides hard, brittle castings, resistant to corrosive chemicals, useful in plumbing lines for chemical laboratories. Sulfur promotes free machining, especially in mild steels. Titanium prevents intergranular corrosion of stainless steels by preventing grainboundary depletion of chromium during such operations as welding and heat treatment. Tungsten, vanadium, and cobalt are all used in high-speed tool steels, because they promote hardness and abrasion resistance. Tungsten and cobalt also increase high-temperature hardness. The principal effects of alloying elements are summarized in Table 4.19. 4.44.2 Stainless Steels Stainless steels of primary interest in building are the wrought stainless steels of the austenitic type. The austenitic stainless steels contain both chromium and nickel. Total content of alloy metals is not less than 23%, with chromium not less than 16% and nickel not less than 7%. Commonly used stainless steels have a tensile TABLE 4.19 Effects of Alloying Elements in Steel*

Sand should comply with ASTM C35. It should be clean, free of organic material, more than about 5% clay, silt, or other impurities, and should not contain salt or alkali. The proportion of sand in the plaster has an important bearing on the characteristics of the product. Oversanding results in considerable reduction in strength and hardness. A mix as lean as 4:1 by weight should never be used. The Gypsum Association suggests that a 1-ft3 measuring box be used for preparing mixes. Some plasterers use a No.2 shovel, which holds about 16 lb of moist sand, for maintaining proper proportions. Thus, with each 100-lb bag of plaster, a 1:2 mix requires 12 shovels of sand and a 1:3 mix 18 shovels (Manual of Gypsum Lathing and Plastering). Water should be clean and free of substances that might affect the rate of set of the plaster. It is not advisable to use water in which plasterers tools have been washed because it might change the set. Excessive water is undesirable in the mix, because when the water evaporates, it leaves numerous large voids, which decrease the strength of the plaster. Hence, manufacturers recommendations should be observed closely in determining water requirements. Perlite and vermiculite are manufactured lightweight aggregates that are used to produce a lightweight plaster with relatively high fire resistance for a given thickness. Both aggregates should conform with ASTM C35, and the mix should be prepared strictly in accordance with manufacturers recommendations. 11.25.3 Mixing Plaster A mechanical mixer disperses the ingredients of a mix more evenly and therefore is to be preferred over box mixing. Recommended practice is as follows: (1) Place the anticipated water requirements in the mixer; (2) add about half the required sand (or all required perlite or vermiculite); (3) add all the plaster; (4) add the rest of the sand; (5) mix at least 2 mm, but not more than 5 mm, adding water, if necessary, to obtain proper workability; and (6) dump the entire batch at once. The mixer should be thoroughly cleaned when it is not in use. If partly set material is left in it, the set of the plaster might be accelerated. For this reason also, tools should be kept clean. For hand mixing, first sand and plaster should be mixed dry to a uniform color in a mixing box, water added, and the plaster hoed into the water immediately and thoroughly mixed. Undermixed plaster is difficult to apply and will produce soft and hard spots in the plastered surface. Plaster should not be mixed more than 1 hr in advance. Nor should a new mix, or gaging, be mixed in with a previously prepared one. And once plaster has started to set, it should not be remixed or retempered. 11.25.4 Plaster Drying A minimum temperature of 55
F should be maintained in the building where walls are to be plastered when outdoor temperatures are less than 55
F, and held for at least 1 week before plaster is applied and 1 week after the plaster is dry. In hot, dry weather, precautions should be taken to prevent water from evaporating before the plaster has set. Plastered surfaces should not be exposed to drafts, and openings to the outside should be closed off temporarily. After the plaster sets, the excess moisture it contains evaporates. Hence, the room should be adequately ventilated to allow this moisture to escape. (Architect Data BookConstruction Products and Systems, Gold Bond Building Products, a National Gypsum Division, 2001 Rexford Road, Charlotte, NC 28211; Gypsum Products Design Data, Gypsum Association, 810 First Street, NE, #510, Washington, D.C., 20002; Gypsum Construction Handbook, United States Gypsum, 125 South Franklin Street, Chicago, IL 60606.) 11.25.5 Gypsum Bases for Plaster One commonly used base for plaster is gypsum lath. This is a noncombustible sheet generally 16
48 in by 3/8 or 1/2 in thick, or 16
96 in by 3/8 in thick. It is composed principally of calcined gypsum that has been mixed with water, hardened, dried, and then sandwiched between two paper sheets (ASTM C37). Insulating gypsum lath is made by cementing shiny aluminum foil to the back of plain lath. It is used for vapor control, and for insulation against heat loss or gain. Also available is gypsum lath with a core specially formulated with minerals for high resistance to fire. Gypsum lath has the advantage over metal lath that less plaster is required, because the first, or scratch, coat is applied only over the lath surface. Also, when used for suspended ceilings or hollow partitions, plaster on gypsum lath is less susceptible to cracking than on metal lath. Installation of gypsum lath should meet the requirements of ASTM C841, Installation of Interior Lathing and Furring. The lath should be applied to studs with long dimension horizontal, and vertical joints should be staggered (Fig. 11.20). In ceilings, the long sides should span supports. Ends should rest on or be nailed to framing, headers, or nailing blocks. Each lath should be in contact with adjoining FIGURE 11.20 Two coats of plaster applied to perforated gypsum lath attached to wood