(My s /EI ) A  (y2 s /EI )   (cos2  x/AE) A A (S. Timoshenko and D. H. Young, Theory of Structures, McGraw-Hill Book Company, New York; S. F. Borg and J. J. Gennaro, Modern Structural Analysis, Van Nostrand Reinhold Company, Inc., New York.) 5.14.3 Stresses in Arch Ribs When the reactions have been found for an arch (Arts. 5.14.1 to 5.14.2), the principal forces acting on any cross section can be found by applying the equations of equilibrium. For example, consider the portion of an arch in Fig. 5.96, where the forces acting at an interior section X are to be found. The load P, HL (or HR), and VL (or VR) may be resolved into components parallel to the axial thrust N and the shear S at X, as indicated in Fig. 5.96. Then, by equating the sum of the forces in each direction to zero, we get N  V sin   H cos   P sin (  ) (5.172) L x L x x S  V cos   H sin   P cos (  ) (5.173) L x L x x And the bending moment at X is M  V x  H y  Pa cos   Pb sin  (5.174) L 1 The shearing unit stress on the arch cross section at X can be determined from S wit the aid of Eq. (5.59). The normal unit stresses can be calculated from N and M with the aid of Eq. (5.67). In designing an arch, it may be necessary to compute certain secondary stresses, in addition to those caused by live, dead, wind, and snow loads. Among the secondary stresses to be considered are those due to temperature changes, rib shortening due to thrust or shrinkage, deformation of tie rods, and unequal settlement of footings. The procedure is the same as for loads on the arch, with the deformations producing the secondary stresses substituted for or treated the same as the deformations due to loads. A structural membrane or shell is a curved surface structure. Usually, it is capable of transmitting loads in more than two directions to supports. It is highly efficient structurally when it is so shaped, proportioned, and supported that it transmits the loads without bending or twisting. A membrane or a shell is defined by its middle surface, halfway between its extrados, or outer surface and intrados, or inner surface. Thus, depending on the geometry of the middle surface, it might be a type of dome, barrel arch, cone, or hyperbolic paraboloid. Its thickness is the distance, normal to the middle surface, between extrados and intrados. 5.15.1 Thin-Shell Analysis A thin shell is a shell with a thickness relatively small compared with its other dimensions. But it should not be so thin that deformations would be large compared with the thickness. The shell should also satisfy the following conditions: Shearing stresses normal to the middle surface are negligible. Points on a normal to the middle surface before it is deformed lie on a straight line after deformation. And this line is normal to the deformed middle surface. Calculation of the stresses in a thin shell generally is carried out in two major steps, both usually involving the solution of differential equations. In the first, bending and torsion are neglected (membrane theory, Art. 5.15.2). In the second step, corrections are made to the previous solution by superimposing the bending and shear stresses that are necessary to satisfy boundary conditions (bending theory,

Speech and music are fuzzy, indistinct Excessive reverberation Absorption Little sounds are most distracting Background level too low Masking Room too dead Optimum reverberation Theres an annoying echo Echo Proper room shape Flutter Proper room shape Focusing effects Eliminate cause of focusing effects Excessive reverberation Absorption I can hear everything the fellow across the office says Room too dead Optimum reverberation Background level too low Masking Focusing or reflection Eliminate focusing or reflection It doesnt sound natural in here Flutter Alter room shape Add absorption Distortion caused by improper sound system Proper sound system Selective absorption Proper type and amount of absorption Room too dead, low reverberation time It feels oppressive Reverberation time too low Proper amount of absorption Background sound level too low Use of background and masking sound Its not loud enough at the rear of the room Room too large Electronic amplification Improper shape Alter room shape Lack of reflecting surface Add reflecting surfaces Poor distribution Eliminate absorption on surfaces needed for

The protective facing for the membrane should be broken at the expansion joint and the space between the membrane and the line of the facing filled with a joint sealant, as recommended in ACI 504R. Details at pipe sleeves running through the membrane must be carefully prepared. The membrane should be reinforced with additional plies and may be calked at the sleeve. Steam and hot-water lines should be insulated to prevent damage to the membrane. 3.4.10 Above-Grade Walls The rate of moisture penetration through capillaries in above-grade walls is low and usually of minor importance. However, such walls should not permit leakage of wind-driven rain through openings larger than those of capillary dimension. Precast-concrete or metal panels are usually made of dense, highly waterresistant materials. However, walls made of these panels are vulnerable to leakage at the joints. In such construction, edges of the panels may be recessed and the interior of vertical joints filled with grout or other sealant after the panels are