The shear stress at the intersection of the walls should not exceed the permissible shear stress. 5.12.7 Coupled Shear Walls Another method than that described in Art. 5.12.6 for increasing the stiffness of a bearing-wall structure and reducing the possibility of tension developing in masonry shear walls under lateral loads is coupling of coplanar shear walls. Figure 5.89 and 5.90 indicate the effect of coupling on stress distribution in a pair of walls under horizontal forces parallel to the walls. A flexible connection between the walls is assumed in Figs. 5.89a and 5.90a, so that the walls act as independent vertical cantilevers in resisting lateral loads. In Figs. 5.89b and 5.90b, the walls are assumed to be connected with a more rigid member, which is capable of shear and moment transfer. A rigid-frame type action results. This can be accomplished with a steel-reinforced concrete, or reinforced brick masonry coupling. FIGURE 5.89 Stress distribution in end shear walls: (a) with flexible coupling; (b) with rigid-frame-type action; (c) with plate-type action. FIGURE 5.90 Stress distribution in interior shear walls: (a) with flexible coupling; (b) with rigid-frame-type action; (c) with plate-type action. A plate-type action is indicated in Figs. 5.89c and 5.90c. This assumes an extremely rigid connection between walls, such as fully story-height walls or deep rigid spandrels. From the basic principles given in preceding articles, systematic procedures have been developed for determining the behavior of a structure from a knowledge of the behavior under load of its components. In these methods, called finite-element methods, a structural system is considered an assembly of a finite number of finitesize components, or elements. These are assumed to be connected to each other only at discrete points, called nodes. From the characteristics of the elements, such as their stiffness or flexibility, the characteristics of the whole system can be derived. With these known, internal stresses and strains throughout can be computed. Choice of elements to be used depends on the type of structure. For example, for a truss with joints considered hinged, a natural choice of element would be a bar, subjected only to axial forces. For a rigid frame, the elements might be beams subjected to bending and axial forces, or to bending, axial forces, and torsion. For a thin plate or shell, elements might be triangles or rectangles, connected at vertices. For three-dimensional structures, elements might be beams, bars, tetrahedrons,

specified compressive strength, psi, determined in accordance with ASTM C39 from standard 6-  12-in cylinders under standard laboratory curing; unless otherwise specified, is based on tests on cylinders 28 days old
c Ec
modulus of elasticity, psi, determined in accordance with ASTM C469; usually assumed as Ec
w1.5 , or for normal-weight concrete (about 145 (33)

c lb/ft3), Ec
57,000

c w
weight, lb / ft3, determined in accordance with ASTM C138 or C567 t
direct tensile strength, psi ct
average splitting tensile strength, psi, of lightweight-aggregate concretes determined by the split cylinder test (ASTM C496) r
modulus of rupture, psi, the tensile strength at the extreme fiber in bending (commonly used for pavement design) determined in accordance with ASTM Other properties, frequently important for particular conditions are: durability to resist freezing and thawing when wet and with deicers, color, surface hardness, impact hardness, abrasion resistance, shrinkage, behavior at high temperatures (about 500F), insulation value at ordinary ambient temperatures, insulation at the high temperatures of a standard fire test, fatigue resistance, and for arctic construction, behavior at cold temperatures (60 to 75F). For most of the research on these properties, specially devised tests were employed, usually to duplicate or simulate the conditions of service anticipated. (See Index to Proceedings of the American Concrete Institute.) In addition to the formal testing procedures specified by ASTM and the special procedures described in the research references, some practical auxiliary tests, precautions in evaluating tests, and observations that may aid the user in practical applications follow. Compressive Strength, . The standard test (ASTM C39) is used to establish the
c quality of concrete, as delivered, for conformance to specifications. Tests of companion field-cured cylinders measure the effectiveness of the curing (Art. 9.14). Core tests (ASTM C42) of the hardened concrete in place, if they give strengths higher than the specified c
or an agreed-on percentage of
c (often 85%), can be used for acceptance of material, placing, consolidation, and curing. If the cores taken for these tests show unsatisfactory strength but companion cores given accelerated additional curing show strengths above the specified , these tests estab-
c lish acceptance of the material, placing, and consolidation, and indicate the remedy, more curing, for the low in-place strengths. For high-strength concretes, say above 5000 psi, care should be taken that the capping material is also high strength. Better still, the ends of the cylinders should be ground to plane. Indirect testing for compressive strength includes surface-hardness tests (impact hammer). Properly calibrated, these tests can be employed to evaluate field curing. (See also Art. 9.14.) Modulus of Elasticity Ec. This property is used in all design, but it is seldom determined by test, and almost never as a regular routine test. For important projects, it is best to secure this information at least once, during the tests on the trial batches at the various curing ages. An accurate value will be useful in prescribing camber or avoiding unusual deflections. An exact value of Ec is invaluable for longspan, thin-shell construction, where deflections can be large and must be predicted accurately for proper construction and timing removal of forms. Tensile Strength. The standard splitting test is a measure of almost pure uniform tension ct. The beam test (Fig. 9.4a) measures bending tension r on extreme surfaces (Fig. 9.4b), calculated for an assumed perfectly elastic, triangular stress

F 3 yw  h   (7.31) vs  340 where Fyw is the yield stress of the web steel (Table 7.18). This shear also may be reduced in the ratio v /Fv as above. TABLE 7.18 Required Shear Capacity of Intermediate-Stiffener Connections to Girder Web Fyw, ksi vg, kips per lin in Fyw, ksi vg, kips per lin in 36.0 0.034h 60.0 0.074h 42.0 0.043h 65.0 0.084h 45.0 0.048h 90.0 0.136h 50.0 0.056h 100.0 0.160h 55.0 0.065h Combined Stresses in Web. A check should be made for combined shear and bending in the web where the tensile bending stress is approximately equal to the maximum permissible. When v, the shear force at the section divided by the web area, is greater than that permitted by Eq. (7.29a), the tensile bending stress in the web should be limited to no more than 0.6Fyw or Fyw(0.825  0.375v /Fv), where Fv is the allowable web shear given by Eq. (7.29a). For girders with steel flanges and webs with Fy exceeding 65 ksi, when the flange bending stress is more than 75% of the allowable, the allowable shear stress in the web should not exceed that given by Eq. (7.22). Also, the compressive stresses in the web should be checked (see Art. 7.22). 7.21.2 LRFD Procedure for Plate Girders Plate girders are normally proportioned to resist bending on the assumption that the moment of inertia of the gross section is effective. The web must be propor- tioned such that the maximum web depth-thickness ratio h/ t does not exceed h/ t given by (7.32) or (7.33), whichever is applicable.