Summer 2002 RIGIDITY CALCULATIONS FOR WALLS WITH OPENINGS   The "Spring 2002 Masonry Chronicles" issue addressed the design of piers and inherent in it was rigidity calculation for piers. This issue basically addressed the rigidity of shear walls in general, but not the design of shear walls. Masonry walls often have openings in them for doors, windows or openings to accommodate equipment etc. Calculation of rigidities for these walls is complex and tedious. This article provides a background on the rigidity of walls, the components contributing to it and influence of openings on rigidity. Approximate method for calculating rigidity, which is useful for preliminary design, is also proposed.   INTRODUCTION Walls, when subjected to in-plane lateral loads, undergo deflection. This deflection is a result of the wall behaving in flexural mode and shear mode. The prime behavioral mode is dependent on height to length ratios. Most walls neither behave in pure flexural mode, nor in pure shear mode. Their overall behavior is normally a combination of the two modes. Wall rigidity is the amount of force required to deflect the wall by one unit. Calculation of rigidity primarily serves two purposes : 1. distribution of lateral loads to various lateral load-resisting elements and 2. to calculate overall deflection (drift) of the system. While out-of-plane forces and behavior of the wall subjected to them may be important, it is not important in rigidity calculations for resisting in-plane forces. FIGURE 1 SHEAR WALL DEFLECTION A cantilever shear wall subjected to lateral load will deflect "". Considering flexural and shear deformations, for a unit lateral load can be calculated by:   A = t x L where for uncracked section. t = thickness of wall Ev = shear modulus = 0.4 Em Substituting these values in the above equation: Flexural deformation Shear Deformation Component Component Similarly for "piers" or walls with top and bottom edges fixed against rotation, "" is given by: Whether flexural deformation governs or shear deformation governs, is dependent on the Ratio. For a given wall, rigidity therefore, the lesser the deflection, the more rigid the wall. This can be intuitively seen by imagining trying to deflect a wall in its own plane. A higher ratio will require less force to deflect the wall than wall with a lower ratio by the same amount. FIGURE 2 MODES OF DEFORMATION Schematically, the modes of deformation are shown in Figure 2. As shown in Table 1, even for squat walls, i. e., , flexural deformation is 57% of the total. Whereas with a ratio of 2.5, which is not uncommon in 2-3 story tall buildings, flexural deformation is almost 90% of the total deformation. The purpose of Table 1 is to show the deformation multiplier, not to calculate the actual deformation. To calculate deformation, actual lateral load, Em and t must be used. Table 1 is for cantilever walls. A similar table could be developed for other conditions. TABLE 1 DEFORMATION COMPONENTS For Single story buildings with long walls, where length between control joints could be 1.5 times the height or ratio of 0.67, shear deformation predominates and flexural deformation concerns should be of less importance. Although failure in shear is undesirable as it is considered non-ductile (brittle), the behavior of the wall due to its geometry cannot be changed. A well designed and detailed wall for anticipated shear demand should perform well. Influence of Openings on Rigidity As can be seen from the above discussion, shear deformation is a function of cross-sectional area of the wall for a given height. Therefore in squat walls, openings will reduce the cross-sectional area and the deflection will increase proportionally as opening sizes increase. For flexure dominated walls, since the deformation is a function of moment of inertia, for a given height, the influence of openings on deflection is less if located centrally in the wall as the influence on reduction in moment of inertia is minor. We will consider two cases of walls to demonstrate the effect of openings on the rigidity: 1. where = 0.67 2. where = 1.5   Case 1: Wall with = 0.67 Consider the following 8" CMU solidly grouted wall. Em = 1.5 ksi. Ev = 0.6 ksi. FIGURE 3. WALL ELEVATION (CASE 1) t = 7.625 inches   Substituting all the values For a 1K force, for a solid wall, Please note that 63% of the deflection is due to shear. Rsolid =   Wall with Opening To account for the opening, the wall can be split into three portions , and . Since is a beam with 8' 0" depth, , it can be assumed that its behavior is predominantly in shear and will offer fixity to portions and at top. The deflection of the wall is calculated below. (The entire procedure is given in 1997 Design of Reinforced Masonry Structures, published by CMACN and is not repeated here.) Please note 82% of deflection is due to shear.   wall with openings = wo wo = solid - solid strip + piers 8' 0" strip incorporating opening wo = 0.28 - 0.09 + 0.125 = 0.315 inches Comparing the deflection of solid wall with wall with opening, the deflection is increased by 0.035 inches or 12.5%. Now, if we consider that area of the wall is reduced by 16.7% (4 ft. out of 24 ft.) and influence of shear deformation is approximately The deflection is expected to increase by 0.725 (16.7) = 12.1% » 12.5%. Therefore, without resorting to detailed calculations, for walls with ratio of 0.67, influence of opening can be judged approximately by reduction in % of area for a preliminary estimate. The calculation is closer to the final number as decreases. It is important that in final design, exact calculations should be made. Case 2: Wall with = 1.5. Consider the following 8" solidly grouted wall with Em = 1.5.ksi. and Ev = 0.6 ksi. Consider the same opening size as in Case 1. Since = 1.5   FIGURE 4. WALL ELEVATION (CASE 2)   wo (wall with opening) Following the previous procedure:   wo = T - solid strip + piers solid strip piers The increase in deflection due to opening is 0.14 inches (approximately 9%). The decrease in area at base is 25%. However, This is a flexure governing wall. 75% of deflection is due to flexure. So approximation based upon reduction in area is not correct. However, since flexural deflections are governed by "I," a quick calculation of reduced moment of inertia will yield better approximation for preliminary purposes. For a unit thickness, Reduction in moment of inertia is only 1.5%. However, since 75% is flexural contribution, expected increase in deflection = 0.75 x 0.015 = 0.0112, i.e., 1.12%. 25% deflection is due to shear contribution. Expected increase in deflection based on 25% reduction in area = 25% x 0.25 = 6.25% Total expected increase in deflection = 6.25 + 1.12 = 7.37 % » 9%.   Conclusions This discussion on deflection, and thus the rigidity of wall with openings, provides an insight on influence of flexural and shear deformations. The approximate calculations are good enough for preliminary design considerations.   One should keep in mind ratio before the approximations are used. Generally, the cutoff ratio between shear dominant behavior and flexural dominant behavior is = 1.0.   This issue of "Masonry Chronicles" was written by Dr. Vilas Mujumdar, Executive Director of CMACN.

"Masonry Chronicles" is a publication of the Concrete Masonry Association of California and Nevada. Reproduction is expressly prohibited without written authorization from CMACN. Please contact the Association Executive Director with any comments or suggestions for future issues. The Masonry Chronicles may also be seen on our website. CMACN's web address is http://www.cmacn.org   CMACN ACTIVE MEMBERS Active Members are an individual, partnership, or corporation which is actively engaged in the manufacture and sale of concrete masonry units Air Vol Block, Inc. Angelus Block Company, Inc. Basalite Concrete Products, LLC Blocklite Calstone Company, Inc. Desert Block Co., Inc. McNear Brick & Block Orco Block Company, Inc. RCP Block & Brick, Inc. Valley Block Company, Inc. ANNOUNCING CMACN PUBLICATION SPECIAL   Purchase two or more CMACN publications listed in this ad and receive a complimentary copy of "Rapid Strength Design of CMU Shear Walls" a $20.00 value!  These publications are current to California Code and are a terrific value at 50% discount for engineers, architects and plan checkers within California and No. Nevada. ORDER FORM You may order publications by completing and sending this form with a check to CMACN at: 6060 Sunrise Vista Drive, Suite 1990, Citrus Heights, CA 95610 or by credit card faxed to (916) 722-1819 or by phone to (916) 722-1700. Please ship my order to: Name Firm Street Address City State Zip Code Phone Number Type of credit card Visa Master Card Credit card number Expiration date Name as it appears on the credit card Billing address for credit card City for credit card State for credit card Zip Code for credit card Billing phone if different from ship to These discounted prices are for engineers, architects & plan checkers with offices in California and Northern Nevada only. (Exclude 7 ¾% sales tax for NV.)   Publication   Cost With tax & shipping   Quantity   Subtotal 1997 Design of Reinforced Masonry Structures $32.50 $43.02 Masonry Chronicles CD (no discount) $25.00 $31.44 Seismic Design of Masonry Using the 1997 UBC $13.50 $19.05 Testing of Concrete Masonry $ 8.50 $13.66 Rapid Strength Design of CMU Shear Walls Complimentary with purchase of two books TOTAL $

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