where Aj  convenient unit of measurement for the jth subsystem pj  cost per unit for the jth subsystem This symbolic model may be used in the design development phase and later after components of the major systems have been selected and greater accuracy of the cost estimate is feasible. Equation (1.9) gives the construction cost as the sum of the costs of all the subsystems, to which should be added contractors overhead and profit. For more information on cost estimating, see Sec. 19. Optimization. The objective of systems design is to select the single best system for a given set of conditions, a process known as optimization. When more than one property of the system is to be optimized or when there is a single characteristic to be optimized but it is nonquantifiable, an optimum solution may or may not exist. If it does exist, it may have to be found by trial and error with a model or by methods such as those described in Art. 1.8. When one characteristic, such as construction cost, of a system is to be optimized, the criterion may be expressed as Optimize z  (x , x , x , . . . y , y , y , . . .) (1.10) r r 1 2 3 1 2 3 where zr  dependent variable to be maximized or minimized x  controllable variable, identified by a subscript y  uncontrollable variable, identified by a subscript r  objective function Generally, however, there are restrictions on values of the independent variables. These restrictions may be expressed as (x , x , x , . . . y , y , y , . . .)  0 1 1 2 3 1 2 3 (x , x , x , . . . y , y , y , . . .)  0 (1.11) 2 1 2 3 1 2 3 (x , x , x , . . . y , y , y , . . .)  0 n 1 2 3 1 2 3 Simultaneous solution of Eqs. (1.10) and (1.11) yields the optimum values of the variables. The solution may be obtained by use of such techniques as calculus, linear programming, or dynamic programming depending on the nature of the variables and the characteristics of the equations. Direct application of Eqs. (1.10) and (1.11) to a whole building, its systems, and its larger subsystems usually is impractical, because of the large number of variables and the complexity of their relationships. Hence optimization generally has to be attained in a different way, generally by such methods as suboptimization

4.16.3 High-Bond Mortars When polymeric materials, such as styrene-butadiene and polyvinylidene chloride, are added to mortar, greatly increased bonding, compressive, and shear strengths result. To obtain high strength, the other materials, including sand, water, Type I or III portland cement, and a workability additive, such as pulverized ground limestone or marble dust, must be of quality equal to that of the ingredients of standard mortar. The high strength of the mortar enables masonry to withstand appreciable bending and tensile stresses. This makes possible thinner walls and prelaying of single-wythe panels that can be hoisted into place. Portland-cement concrete is a mixture of portland cement, water, coarse and fine aggregates, and admixtures proportioned to form a plastic mass capable of being cast, placed, or molded into forms that will harden to a solid mass. The desirable properties of plastic concrete are that it be workable, placeable and nonsegregating, and that it set in the desired time. The hardened concrete should provide the desired service properties: 1. Strength (compressive and flexural) 2. Durability (lack of cracks, resistance to freezing and thawing and to chemical attacks, abrasion resistance, and air content) 3. Appearance (color, lack of surface imperfections) Each of these properties affects the final cost of the mix design and the cost of the in-place concrete. These properties are available from normal-weight, lightweight, and heavyweight concretes. 4.17.1 Normal-Weight Concrete The nominal weight of normal concrete is 144 lb / ft3 for non-air-entrained concrete, but is less the air-entrained concrete. (The weight of concrete plus steel reinforcement is often assumed as 150 lb / ft3.) Strength for normal-weight concrete ranges from 2000 to 20,000 psi. It is generally measured using a standard test cylinder 6 in in diameter by 12 in high. The strength of a concrete is defined as the average strength of two cylinders taken from the same load and tested at the same age. Flexural beams 6  6  20 in may be used for concrete paving mixes. The strength gains of air-entrained and non-airentrained concretes are graphically shown in Fig. 9.2. As illustrated in Fig. 9.2, the strength of a given mix is determined by the watercement ratio (W/ C), and whether or not air entraining is used. Other factors are the maximum-size aggregate and the desired fluidity (slump) of the concrete at the point of placement. When no historical record is available for the aggregates and cements to be used, the water-cement ratios in Table 9.2 can provide guidance for the initial designs. Each combination of coarse and fine aggregates has a specific water demand for a given mix fluidity, or slump. Two general guidelines are: 1. For a constant slump, the water demand increases with increase in maximumsize

Ductility is developed in reinforced concrete by: Conservative limits on the net flexural tension-steel ratio   0.025, to ensure underreinforced behavior. At least two continuous bars must be provided at both top and bottom of flexural members. Heavy confining reinforcement extending at joints through the region of maximum moment in both columns and beams, to include points where hinges may form. This confining reinforcement may consist of spirals or heavy, closely spaced, well-anchored, closed ties (hoops) with hooked ends engaging the vertical bars or the tie at the far face. (ACI Detailing Manual, SP-66, American Concrete Institute.) Reinforced- and prestressed-concrete, composite flexural members are constructed from such components as precast members with cast-in-place flanges, box sections, and folded plates. Composite structural-steel-concrete members are usually constructed of cast-inplace slabs and structural-steel beams. Interaction between the steel beam and con- crete slab is obtained by natural bond if the steel beam is fully encased with a minimum of 2 in of concrete on the sides or soffit. If the beam is not encased, the interaction may be accomplished with mechanical anchors (shear connectors). Requirements for composite structural-steel-concrete members are given in the AISC Specification for Structural Steel for BuildingsAllowable Stress Design and Plastic Design, and AISC Load and Resistance Factor Design Specification for Structural Steel Buildings, American Institute of Steel Construction. The design strength of composite flexural members is the same for both shored and unshored construction. Shoring should not be removed, however, until the supported elements have the design properties required to support all loads and limit deflections and cracking. Individual elements should be designed to support all loads prior to the full development of the design strength of the composite member. Premature loading of individual precast elements can cause excessive deflections as the result of creep and shrinkage. According to the ACI 318 Building Code, the factored horizontal shear force for a composite member may be transferred between individual concrete elements by contact stresses or anchored ties, or both. The factored shear force Vu at the section considered must be equal to or less than the nominal horizontal shear strength Vnh multiplied by 
0.85. V   V (9.116) u nh When Vu  80bvd, where bv is the section width and d the distance from the extreme compression surface to the centroid of tension reinforcement, the factored shear force may be transferred by contact stresses without ties, if the contact surfaces are clean, free of laitance and intentionally roughened. Otherwise, if the contact surfaces are clean but not intentionally roughened, fully anchored minimum ties [Eq. (9.81)], spaced not over 24 in or 4 times the least dimension of the supported element are required when Vu  80bvd. When fully anchored minimum ties are provided and the contact surfaces are clean, free of laitance and intentionally roughened to a full amplitude of about 1/4 in, the Code permits transferring a factored shear force equal to (260  0.6 vy)bvd but not more than (500 bvd), where v is the ratio of tie reinforcement area to the area of the contact surface, y is the yield strength of shear reinforcement, and  is defined under Eq. (9.117). When Vu exceeds (500bvd), the factored shear force may be transferred by shear-friction reinforcement placed perpendicular to assumed cracks. Shear force Vu should not exceed 800Ac or Ac, where Ac is the area of the concrete section 0.2
c resisting shear transfer, and is the specified concrete compressive strength. Re-
c quired reinforcement area is VA
u (9.117) v   y where y
yield strength of shear reinforcement 
coefficient of friction
1.4 for monolithic concrete
1.0 for concrete cast against hardened concrete with surface intentionally roughened to a full amplitude of about 0.25 in
0.7 for concrete anchored by headed studs or rebars to as-rolled structural steel (clean and without paint)
0.6 for concrete cast against hardened concrete not intentionally