Technical Notes 31C - Structural Design of Semicircular Brick Masonry Arches Rev [Feb. 1971] (Reissued August 1986) INTRODUCTION The semicircular arch is among the most popular arch forms used by architects today. Its smooth, continuous curve makes it easily adaptable to many architectural styles and applications. This issue of Technical Notes presents recommended procedures and tables for the structural design of nonreinforced semicircular and segmental arches. Technical Notes 31 and 31A contain further information about general arch forms and their design. DESIGN ASSUMPTIONS Since the brick masonry arch is usually an integral part of a wall and not free-standing, basic design assumptions are made which assist in making the analysis. The spring line is assumed to be located on a horizontal line one fourth the span length above the horizontal axis. The arches are assumed to be fully restrained at the spring line and the portion of semicircular arch above this line is analyzed in a manner similar to that for a parabolic arch, using the formulas in Section 10, Method A from Frames and Arches, by Valerian Leontovich, M.S., McGraw-Hill 1959. Holiday Inn, McLean, Virginia NOMENCLATURE d = arch ring depth, in inches, f = rise of arch, in feet, fm = allowable compressive stress, in psi, H = horizontal thrust, in pounds, HDL = horizontal thrust, in pounds, caused by a uniform dead load, L = span length, in feet, n = number of shear planes (see Technical Notes 31A.) P = allowable concentrated load, in pounds, P' = maximum allowable concentrated load in pounds under combined loading provisions, P* = additional concentrated load capacity caused by the uniform dead load. t = wall thickness, in inches vm = allowable shear stress in brick masonry, in psi, W = allowable uniform load, in pounds per foot, x = length of wall required, in inches, to resist horizontal thrust DEVELOPMENT OF TABLES In the determination of the arch's capacity for uniform loads, the limiting factor was found to be the compressive strength of the brick masonry. Additional stresses due to the circular loading of the masonry above the intrados are also taken into consideration. In determining the capacity for concentrated loads, the limiting factors were found to be bending at the center line of span, shear at the spring line (vm = 40 psi), and maximum compressive stress (fm). Tensile stresses were not permitted to develop at mid span. Since axial forces develop in the arch ring from the concentrated and uniform loads, interaction formulas were developed for each loading condition. These formulas combine the axial stresses with the bending stresses. ALLOWABLE LOADING In all formulas used in this publication, d and t are measured in inches, and L is measured in feet. The following loading conditions were considered for analyzing a semicircular arch. Uniform Load. Tables 1, 2, 3 and 4 give the allowable uniform loads occurring over the entire span length for a 1- in. thick arch ring. Figure 1 illustrates the following requirements and limitations: FIG. 1 1. Uniform load occurring between lines 1 and 3 (0.90 L and 0.70 L) are those provided for in the load tables. 2. Uniform loads occurring above line 1 may be ignored, at the discretion of the designer, provided arching action occurs in the brick masonry above the arch ring. (See discussion in Technical Notes 31 and 31A.) 3. There must be a minimum height of masonry (line 3) equal to 0.70 L above the horizontal axis. No superimposed loads are permitted below this line. 4. The maximum design height of masonry is 0.25 L above the crown for walls higher than line 2. 5. In all cases, the horizontal thrust (H) must be checked as shown in Technical Notes 31A, Fig. 4. For a given arch, the horizontal thrust is directly proportional to the uniform load. 6. The portion of wall that resists the horizontal thrust is assumed to be non-yielding to any lateral movement. Concentrated Load. Table 5 gives the allowable concentrated loads occurring at the center line of span for a 1- in. thick arch ring. Figure 2 illustrates the following requirements and limitations: FIG. 2 1. Concentrated loads occurring between lines 2 and 3 (1.20 L and 0.75 L) are those provided for in the load table. 2. Concentrated loads occurring between lines 1 and 2 may be divided by the span length (L) and considered as equivalent uniform loads. 3. Concentrated loads occurring above line 1 (1.50 L) may be ignored, at the discretion of the designer, provided arching action occurs in the brick masonry above the arch ring. (See discussion in Technical Notes 31 and 31A.) 4. In all cases, condition 4 for uniform loads must be used with the resulting thrusts added to those of the concentrated loads. 5. There must be a minimum height of masonry (line 3) equal to 0.75 L above the horizontal axis. No superimposed loads are permitted below this line. 6. In all cases, the horizontal thrust H must be checked as shown in Technical Notes 31A, Fig. 4. For a given arch, the horizontal thrust is directly proportional to the concentrated load. 7. The portion of wall that resists the horizontal thrust is assumed to be non-yielding to any lateral movement. *Values may be linearly interpolated except where horizontal lines occur. At these lines, the allowable load is ((0.241)(L + 0.083d^3 + 0.134)(L + 0.083d)^2 * d)/(1.34(L + 0.083d) - 0.0778d) or the value above the line whichever is smaller The horizontal: thrust is 0.778P + 0.134(L + 0.083d) or value above the line whichever is smaller. Combined Loading. When the uniform loads are combined with concentrated loads, the concentrated load capacity of the arch ring increases. This additional capacity is due to the compressive stress from the uniform load equalizing the tensile bending stress at mid span due to the concentrated load (M/S = P/A). This additional capacity may be expressed by the following formula: The values of P' and H' in Table 6 are the allowable capacities governed by compression or shear. They should be used only as a check when combined loadings are used. In all cases, the actual load must be less than P', and less than the allowable load, P, plus the additional capacity P*. The total horizontal thrust must be checked and should be less than the maximum allowable for a uniform load. SEGMENTAL ARCHES Any segmental arch with f / L' > 0.29 but < 0.50 can be considered as an equivalent semicircular arch as shown in Fig. 3. Twice the radius is the equivalent L for use with the tables. FIG. 3 ILLUSTRATIVE EXAMPLE Design an arch to meet the requirements as shown in Fig. 4. The arch is semicircular; the horizontal axis is 6 ft above the base; the span, L, is 10 ft; the arch ring depth, d, is 12 in. (11 1/2 in. actual); and the nominal wall thickness, t, is 8 in. (7 1/2 in. actual). A beam reaction of 5000 lb is located at the center line of the span and 17 ft above the base. The uniform load consists of 1000 lb per ft dead load and 500 lb per ft live load occurring 14 ft above the base. Assume fm = 400 psi and the brick masonry weighs 10 psf per 1-in. thickness. Uniform Load FIG. 4 All the following calculations will be with 1 in. of wall thickness; actual t = 7.5 in., fm = 400 psi, d = 11.5 in. and L= 10 ft. Uniform Load Concentrated Load Use Table 2, since 0.7L < 0.8L < 0.9L From Table 2, W = 724 lb per ft and H = 2993 lb Use Table 5 since 0.75L < 1.1 L < 1.2L From Table 5, P = 88 lb and H = 84 lb However, since there is combined loading, advantage can be taken of the increased capacity due to the uniform load. 667 < 688 O.K. From Table 6, P'= 1060 lb 667 < 1060 O.K. Horizontal Thrust H(total) =970 + 636 = 1606 < 2993 O.K. At this point the wall shear caused by the horizontal thrust at the spring line should be checked. Assume Vm = 40 psi and n = 2 The overturning moment of the support due to horizontal thrust should be checked next (see Technical Notes 31A). In this example, the horizontal thrust is 1606 (7.5) (8.5) = 102,000 ft-lb. The resistance to overturning is a function of the overall axial load, wall shape, and reinforcement, if any. This is a separate analysis that should be performed after considering the total loading conditions on the entire structure. CONCLUSION This issue of Technical Notes has presented a simplified but conservative approach to a complex structural design problem. To provide an analysis for all possible assumptions and loading conditions is beyond the scope of this publication. Most loading conditions encountered will be similar to those in Fig. 1 and Fig. 2. To load an arch unsymmetrically defeats its use as a natural load-carrying structure and induces bending stresses that may cause failure. If arches are to be loaded unsymmetrically or do not comply with the assumptions and limitations given in this Technical Notes, consideration should be given to reinforced brick masonry. (See Technical Notes 17A Revised, "Reinforced Brick Masonry - Flexural Design", and 17M, "Reinforced Brick Masonry Girders - Examples".) If conditions exist other than those covered in the tables, special analysis should be made by the designer. The Brick Industry Association cannot assume responsibility for the results obtained when using this Technical Notes Issue. It is beyond the scope of the Institute to anticipate every design situation that may arise. However,