'" ffiPsi *"" .-" :v^'-V-: : m m SIMPLIFIED FORMULAS AND TABLES FOR FLOORS, JOISTS AND BEAMS; ROOFS, RAFTERS AND PURLINS BY N. CLIFFORD RICKER, B.S., M.ARCH., D.AECH. Professor of Architecture, University of Illinois; President Illinois Board of Examiners of Architects; Chairman Illinois Commission on Building Laws; Honorary and Active Fellow American Institute of Architects; Fellow American Association for Advancement of Science; Member Western Society of Engineers, Society for Promotion of Engineering Education, American Federation of Arts, Etc. FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY ART. 17. Special Formulas for Common Materials 11 ART. 18. Properties of Rectangular Sections 11 ART. 19. Properties of Sections of Lintels 12 ART. 20. Tables of Logarithms 14 ART. 21. Application of Formulas to Roofs 14 ART. 22. Notation and Formulas for Roofs 15 ART. 23. Sheathing 16 ART. 24. Rafters 16 ART. 25. Purlins 17 ART. 26. Application of Formulas to Problems 19 1. Steel Girder, Load Uniform 19 2. Cantilever Beam, Load Uniform 19 3. Supported Beam, Load at Middle 2O 4. Steel Floor Beam 2O 5. Pine Floor Joists 2O 6. Pine Floor Joists 21 7. Mill Deck Roof 21 8. Mill Roof Beams 21 9. Mill Roof Girders. . 22 vi TABLE OF CONTENTS PAGE ART. 27. Cast-iron Lintels 22 10. Inverted T-lintel 23 11. Box Lintel with Two Webs 23 12. Box Lintel with Three Webs 23 13. Sheathing of Roof 24 14. Rafters of Roof 25 15. Purlins of Roof 27 Tables of Simplified Formulas 32 Case 1. Beam Cantilever, Load at Free End . . 32, 33 Case 2. Beam Cantilever, Load Uniform 34, 35 Case 2A. Joist Cantilever, Load Uniform 36, 37 Case 2s. Flooring Cantilever, Load Uniform 38, 39 Case 3. Beam Cantilever, Load Irregular : 40, 41 Case 4. Beam Supported, Load at Middle 42, 43 Case 5. Beam Supported, Load Uniform 44, 45 Case 5 A. Joist Supported, Load Uniform . 46, 47 Case SB. Flooring Supported, Load Uniform 48, 49 Case 6. Beam Supported, Load Irregular 50, 51 Case 7. Beam Fixed anc^ Supported, Load at Middle 52, 53 Case 8. Beam Fixed and Supported, Load Uniform 54, 55 Case SA. Joist Fixed and Supported, Load Uniform 56, 57 Case SB. Flooring Fixed and Supported, Load Uniform 58, 59 Case 9. Beam Fixed, Load at Middle 60, 61 Case 10. Beam Fixed, Load Uniform 62, 63 Case 10A. Joist Fixed, Load Uniform 64, 65 Case 10B. Flooring Fixed, Load Uniform 66, 67 Moment of Inertia for Rectangular Section 68 Section Modulus for Rectangular Section : 69 Properties of Cast-iron Lintels 70, 71, 72 Table of Logarithms, to 1000 74, 75 Table of Logarithms, 1000 to 2000 76, 77 SIMPLIFIED FORMULAS AND TABLES ERRATA Page 1, second line from bottom. Read " inch-pounds " instead of " inch-tons." Page 6, Art. 10. Read " 7 =- X0.248L^." C j Also in fourth and second lines from bottom, read F -F instead of F. iii Page 10. Add to the Table of Safe Values: Hemlock. 0.45 (900). 450 (900,000). Page 19, ninth line from top. Read 30,000X360 " 8X16,000 Ninth line from bottom. Read "/ =QM7WL 2 =0.047 X 15 X30 2 =628.5." Seventh line from bottom. Add: " by the use of simplified formulas." Page 20, tenth line from bottom. Read "/ = 0.047TFL 2 ." Page 23, eleventh line from bottom. Read " and 5^ feet high." M' = maximum bending moment in inch-tons acting on the beam. TABLE OF CONTENTS PAGE ART. 27. Cast-iron Lintels 22 10. Inverted T-lintel 23 11. Box Lintel with Two Webs 23 12. Box Lintel with Three Webs 23 13. Sheathing of Roof 24 14. Rafters of Roof. . SIMPLIFIED FORMULAS AND TABLES 1. Ordinary Formulas for Beams. The formulas for beams supporting transverse loads, commonly given in the text-books, are collected in Table A for comparison and reference. They evidently differ according to the distribution of the load along the beam, and also according to the manner in which its ends are supported or fixed. The cross-section of the beam is here assumed to be constant in dimensions and form through- out its entire length, which is always the case for wooden timbers and steel shapes. 2. Notation Employed in the Ordinary Formulas. Table A. Let P = total load in pounds supported by the beam. I = clear span in inches of the beam. S = maximum safe fibre stress in pounds per square inch acting at a cross-section. E' = modulus of elasticity in pounds per square inch for its material. E' = tensile stress which would theoretically stretch a bar 1 inch square to twice its original length. - = section modulus of cross-section of beam. c I = section moment of inertia of the same. c= maximum distance in inches from horizontal gravity axis of cross-section to its most distant fibre. A = maximum deflection of beam in inches, usually limited to ^7. . oOU M ' = maximum bending moment in inch-tons acting on the beam. \ MFiHIED Fi?JliULAS AND TABLES 3. Table A. Formulas for Beams. / PI 3 PI I PP PP 4. _ T~ c s -fe = 8 c 384E7* P/3 7. /^ u 7P13 16 c' 7QSET 10 12 c ~ 384^7* INCONVENIENT USE OF ORDINARY FORMULAS 4. Inconvenient Use of Ordinary Formulas. As an illustration, take the following practical example. A steel beam is to be composed of two steel I-beams, is 30 ft. long and must safely support a uniformly dis- tributed load of 20,000 Ibs. Its most economical cross- section and actual maximum deflection are to be determined. For rolled steel shapes, S = 16,000 Ibs., E' =29,000,000, and 1= 360 ins. By formulas for Case 5, Table A: PT Q I I PI 20000x360 -g- = S -; transposing; - = ^ - 8xl6000 !- 56.12. 5 P I 3 A = , ; transposing ; 75 P I 2 75 X20000 x 129600 16 #' 16X29000000 Since two I-beams are to be used, for each, - = 28.06 and 7 = 209.69. By "Cambria": two 12 in., 31.5 Ib. I-beams will suffice for both values. For the selected section, 1= 2x215.8 = 431.6 for both beams. 5PZ 3 5X20000X46656000 ~ 384 #'/ " 384X29000000X431.6 ~ Since this maximum deflection should not exceed 1 in., this compound beam may be safely employed. Even with the use of logarithms in solving this problem, it is evident that the use of the ordinary formulas requires considerable time and a large number of figures, with possible errors in the computations, and that they are 4 SIMPLIFIED FORMULAS AND TABLES not adapted for use on the slide rule. Also, that if these formulas can be materially simplified, much time and labor can be saved, and it may become entirely possible to make the necessary calculations with four-place loga- rithms or a good slide rule, obtaining results sufficiently accurate for any practical purpose. 5. Method for Simplifying the Ordinary Formulas. The following changes are made in the ordinary formulas in Table A: a. Change load on beam from pounds to tons. b. Change numerical values of S and E' in pounds to F and E in tons. c. Change length of beam from I in inches to L in feet. d. Change bending moments from inch-pounds to foot- tons. Other values remain as before. 6. Notation Employed in the Simplified Formulas. Let W = total load on beam in tons. . w = total load in pounds per square foot of a floor. L= length of beam in feet, or distance between centres of beams. e = distance in inches between centres of floor joists. t= thickness in inches of the flooring. F= maximum safe fibre stress in tons per square inch. E= modulus of elasticity in tons. M = maximum bending moment in foot-tons. A = maximum deflection of beam in inches; should , L not exceed o?j. METHOD OF SIMPLIFICATION 7. Method of Simplification. In simplifying or transforming a formula, care must always be taken to preserve the numerical value of each side of the equation representing the formula. As an example of the application of the method, take the ordinary formulas given for Case 5 in Table A. Substitute 2000 W for P; 12 L for Z; 2000 P for S: 2000 # for E'] and for A. Then reduce the equation oU to its simplest form and transpose to obtain the forms most convenient for use. PI . a J.2000TTxl2L J p 7 8 c 8 c c 5PZ 3 5X2QQOTFX1728L 3 L 22.5 A ~ 384#'I~ 384X2000^7 " 30 El 8. General Simplified Formulas for Case 5. The formulas just obtained may be put into forms more convenient for use. 15WL-F 1 - c 30 " El I 1.5TFL 675 TFL 2 section mom- - = ^ = section modulus. / = ^ = , . c F E ent of inertia. = safe load. = safe length. SIMPLIFIED FOEMULAS AND TABLES 9. Special Formulas for any Material. For example, take the general formulas just found, adapt them to steel by substituting the numerical values for F and E and reduce to simplest form. 5.333 145QQ / / ~ \ 10. Formula for Directly Computing the Numerical Value of I from that of . Evidently for a beam of a given length, load, and material, the numerical values in the preceding general formulas for W are equal, may be equated and simplified for 7. Then T 7? T?T T ~ X 1 K j = an r. r, from which is found I = - X450 L F. c 1.5L 675 L' c Therefore in Case 5, after obtaining the numerical value of , it may save time to multiply this value by 450 L F instead of using the formula given in Art. 9 for 7. This formula may also be simplified by inserting the value of F and reducing, making it very convenient for the slide rule. FORMULA FOR MAXIMUM SAFE FIBRE STRESS 11. Formula for Maximum Safe Fibre Stress and Deflection. The preceding formulas for safety against rupture and excessive deflection are entirely independent of each other. Therefore, if a beam of any given material and uniform cross-section be assumed, its safe load W be computed by both formulas for successive lengths L, and the values of W be plotted, two curves will be obtained and intersect at a common point at which the numerical values of W and L will be respectively equal, as illustrated in Fig. 11. Hence for the intersection, we may equate the 12r 10 10 15 20 L. IN FEET. FIG. 11. values of W in the two equations, obtaining in Case 5, EC L = 450 F' For F and E may then be substituted the numerical values for any material, thus producing a very simple formula, so that L can be found by it directly. For lengths less than L by this formula, the formula for safety against rupture gives safest results; for those greater than L, the formula against excessive deflection is safest. Hence if this value of L be known for any material, it is only necessary to apply one formula below it and the other above it. SIMPLIFIED FORMULAS AND TABLES 12. Actual Maximum Deflection. This formula gives the actual maximum deflection of the beam in inches. 5 PI 3 _ 5X2000TTX1728L 3 _ 22.5 WL 3 A ~384#'7~ 384X2000 ~ 13. General Formulas for Floor Joists. Case 5 a. Let e = distance in inches between centres of joists. w= total live and dead loads in pounds per square foot of floor. e_ w _ _ wL e T2 X 2000~ "24000* Substituting this value for W in the general formulas for W and simplifying, / 1.5 WL 1.5wL 2 e wL 2 e c F " 24000^ " 16000 F' 675 WL 2 675 w L 3 e wL 3 e E '' 24000 # " 35.56^' Then by transposition: I w L 2 e w L 3 e c 16000 F' 35.56 I 16000 F 35.56 7 16000 F 35.56 E I c X wL 2 ' e= wL 3 ' /7 ~ \ c 16000 F L c we' \ w e GENERAL FORMULAS FOR FLOORING 9 By inserting the values of F and E, these general for- mulas are changed into simpler formulas for any particular material. The formulas for directly computing 7 from - and for L c for maximum safe fibre stress and deflection are unchanged from those found for Case 5. For actual deflection of a joist, substituting value of W and simplifying: _ 22.5 wL*e w L*e A ~ 24000 El" 1067 E F 14. General Formulas for Flooring. Case 5 b. Let t= thickness in inches of the flooring. Take e = 12 ins., assuming a strip of floor 1 ft. wide. Then for the rectangular section of a floor board: ~c = ~S~- 6 = 12 = ^2~ : Substituting 12 for e, 2 t 2 for -, and t 3 for / in equations c for floor joists and simplifying: 12j^ 12 wL* * l ~~ 16000 F' ~ 35.56 #' w L 2 w . .96 # 2667 F t 2 2.96 E t 3 These formulas may be further simplified by inserting the values of F and E for the particular material. 10 SIMPLIFIED FORMULAS AND TABLES The general formula for maximum safe fibre stress and deflection is obtained by equating the values just found for w and simplifying. Et L = 933 The general formula for actual deflection is obtained by substituting t 3 for / in the formula for actual deflection of a joist and reducing. w L 4 e A = 1067 E t 3 ' 15. General and Special Formulas for Cases 1 to 10. These are derived from the ordinary formulas given in Table A by the method just explained and applied to those of Cases 5, 5 a and 5 b. 16. Numerical Safe Values recommended for F and E. Material. F. Lbs. E. Lbs. Cedar 0.45 (900) 450 (900,000) Cypress 0.50 (1,000) 550 (1,100,000) Fir, Washington . . . 0.70 (1,400) 700 (1,400,000) Gum 0.55 (1,100) 650 (1,300,000) Iron, cast. Tension 1.50 (3,000) 8,000 (16,000,000) Iron, wrought 6.00 (12,000) 14,000 (28,000,000) Maple, sugar 0.75 (1,500) 800 (1,600,000) Oak, white 0.65 (1,300) 750 (1,500,000) Pine, longleaf 0.70 (1,400) 850 (1,700,000) Pine, Norway 0.50 (1,000) 600 (1,200,000) Pine, pitch 0.55 (1,100) 600 (1,200,000) Pine, shortleaf 0.55 (1,100) 600 (1,200,000) Pine, white 0.45 (900) 500 (1,000,000) Poplar, yellow 0.45 (900) 500 (1,000,000) Redwood 0.40 (800) 350 (700,000) Spruce 0.55 (1,100) 650 (1,300,000) Steel shapes 8.00 (16,000) 14,500 (29,000,000) FORMULAS FOR THE COMMONLY USED MATERIALS 11 These are safe average values, based on the results of experiments and the average requirements of the building ordinances of the principal cities in the United States. The corresponding safe values for any other materials, or those prescribed by any building ordinance, may easily be inserted in the general formulas for the particular case, then simplified to obtain the working formulas. 17. Special Formulas for the Commonly Used Materials. From the simplified general formulas for Cases 1 to 10, by substituting the proper values of F and E taken from Art. 16 and simplifying, are derived the special formulas here given for steel, cast iron, Washington fir, hemlock, white oak, longleaf, shortleaf, and white pine, and for spruce. These materials have been selected because they are more commonly employed in the Middle and Eastern States. These special formulas are then most rapidly applied by using four-place logarithms or a good slide rule. 18. Tables of Properties of Rectangular Sections. Tables 19 and 20 are to be used in determining the dimensions of timbers corresponding to the values of c and I obtained by the formulas. The upper horizontal line of figures represents the horizontal breadth of the section, and the left-hand vertical line contains the vertical depth. The numerical values of - in Table 19 are com- c puted by the usual formula, I bd 2 c~ 6 * Those of / in Table 20 are obtained by the formula, bd* 2 ~ 12' 12 SIMPLIFIED FORMULAS AND TABLES 19. Tables of Properties of Sections of Cast-iron Lintels. These tables include the stock sections and sectional dimensions of lintels usually furnished by the large foundries. It is not economical to design other sections, excepting when a considerable number are to be cast from the new pattern required. Lin- tels are now generally composed of pairs of steel I-beams. Cast-iron lintels should only be used in Case 4, 5, or 6, since their design becomes too complex in the other cases. Fig. 12 is the section of an in- verted T-section, also applicable to an L-section; Fig. 13 is that of a box lintel; and Fig. 14 is a box lintel with ii r- ! 1 t t-- -M 4 _l__t_ 7 F [Q. 12. r-f i i i i ! f~T~E ^ i 4 -- Jr !--^- o FIG. 13. ']' k FIG. 16. three webs. Flanges and webs have equal thickness of metal, and they are to be connected at proper distances by cross webs to prevent crippling. PROPERTIES OF SECTIONS OF CAST-IRON LINTELS 13 The formulas employed in the computations were obtained as follows: Let t = uniform thickness of metal in inches. h = height of webs from flange in inches. b = breadth of flange in inches. A = total sectional area of lintel in square inches. A' = total sectional area of webs in square inches. A" = total sectional area of flange in square inches. d= vertical distance in inches between horizontal gravity axes of webs and flange. d f = vertical distance in inches between gravity axis of webs and neutral axis of entire section. d,, = vertical distance in inches between gravity axis of flange and neutral axis of entire section. c = distance in inches between bottom of flange and neutral axis of section. 7= moment of inertia of the entire section about its neutral axis. /'= moment of inertia of all webs about their hori- zontal gravity axis. /"= moment of inertia of flange about its horizontal gravity axis. - = section modulus of entire section about its neutral c axis on tension side. Then d =^ = half depth of lintel in inches. Also for location of the neutral axis of the entire section. A'd A : A f : d : d lt \ hence d u = r- . A. By the usual formula for / about any axis parallel to its gravity axis: I=r+A f d J 2 +r f +A"d,, 2 = moment of inertia of entire section. Also c = d n +-^, and -= section modulus. c 14 SIMPLIFIED FORMULAS AND TABLES 20. Tables of Logarithms. In order to make this work as convenient as possible, two tables of four-place logarithms have been added in Tables 24 and 25, one extending from to 999, the other from 1000 to 1999. These will be found sufficient for solving problems relating to beams, joists, and flooring. Or a good slide rule may be employed, saving some time and the labor of writing down the logarithms, but with more liability to error in locating the decimal point. 21. Application of Formulas and Tables to Roofs. These simplified formulas may be applied to roofs as well as to floors, in the following manner. Loads on roofs are composed of four different kinds: 1. Permanent loads in pounds per square foot of inclined surface, acting vertically, and consisting of weight of covering, sheathing, rafters, and purlins. 2. Snow load in pounds per horizontal square foot, acting vertically, its magnitude varying from to 35 Ibs., according to latitude. 3. Wind load or pressure in pounds per square foot of inclined surface, acting at right angles to the latter, its magnitude varying from to 50 Ibs., according to exposure and inclination of the roof. 4. Accidental loads, for example, 25 Ibs. per square foot of a flat roof for weight of snow, firemen, etc. Acts vertically. The weight of the trusses supporting the roof is not included here. FORMULAS EMPLOYED FOR LOADS ON ROOFS 15 22. Notation and Formulas Employed for Loads on Roofs. Let p = permanent load in pounds per square foot of inclined surface, s = snow load in pounds per square foot of horizontal surface. w =wind load in pounds per square foot of inclined surface. i = angle of inclination of surface from horizontal. Then s cos i = snow load in pounds per square foot of inclined surface. For a flat roof, cosi = l, w=Q' } the roof is then treated like a floor. p cos i = normal component of permanent load p. p sin i = parallel component of permanent load p. s cos 2 i = normal component of snow load s cos i. s sin i cos i = parallel component of snow load s cos i. w = normal component of wind load w. = parallel component of wind load w. Since the maximum snow load and wind load can scarcely occur simultaneously on the roof surface, we may have either one of two cases. a. Permanent and snow loads form the maximum load- ing. cos i (p+s cos i) = normal component of p and s loads, sin i(p+s cos i) = parallel component of p and s loads. 6. Permanent and wind loads form the maximum loading. p cos i+w = normal component of p and w loads. p sin {+0 = parallel component of p and w loads. Either pair, a or 6, of formulas are to be employed, which corresponds to the mode of loading, that produces the maximum stresses in the roof. 16 SIMPLIFIED FORMULAS AND TABLES 23. Sheathing. Here p= weight of covering -f weight of sheathing per inclined square foot. For an inclined roof the parallel component of this loading may usually be neglected, since it is safely resisted by the edgewise strength of the sheathing. Take the maximum normal component, substitute this for w in the formulas of Case 5 b to determine L = maximum safe distance in feet between centres of the supporting rafters. 24. Rafters. Here p= weight of covering + weight of shea thing + average weight of rafters per inclined square foot. The maximum normal component acts transversely and its value is substituted for w in the formulas of Case 5 a to determine - and 7; the dimensions of cross-section of c rafters are then found. By applying the formula for A, Case 5 a, the maximum deflection A of the rafter is found. The parallel component of the loading acts lengthwise the rafter producing compression. The magnitude of this , . ,. ^ e L X par. component compression at mid-length of ratter = - 4&QQQ in tons. Let u= uniform compression in tons per square inch at this section of rafter. d = depth of rafter in inches, for rectangular, I or channel section. ((\ A\ 1 +-r) = maximum compression in top fibres in tons per square inch. This is then to be deducted from the value of F employed for the material in the formulas of Case 5 a; PUELINS 17 substitute the remainder for F in the general formula and compute anew the proper values of -, 7 and dimensions c of rafter. In all roofs of ordinary inclination, this parallel component may be neglected. 25. Purlins. Here p= weights of covering + sheathing + average for rafters + average for purlins per inclined square foot. Purlins may be set in either of three ways : a. With middle or major axial plane containing resultant of all loads on purlin. But these loads are liable to varia- tion, and this resultant then varies in magnitude and direction. b. Major axial plane at right angles (normal) to roof surface. Let W= total load in tons on purlin uniformly dis- tributed. W = normal component of loads on purlin. W" = parallel component of loads on purlin. c. Major axial plan vertical and making angle j with resultant of maximum simultaneous loads on purlin. W = W cos j = vertical component of loads on purlin. W" = W sin j = horizontal component of loads on purlin. After obtaining the component W, which acts in the major axial plane of the purlin, and W", that acts at right angles to the former, the formulas of Case 5 are applied to obtain - and of I for each component. A section is then c selected that has the required values of - and I in the two c directions. 18 SIMPLIFIED FORMULAS AND TABLES 1. For a timber purlin, the required sectional dimen- sions may be found by Tables 19 and 20, selecting a section possessing the required values of and I in the respective c directions. 2. For a steel purlin, which may be composed of two I-beams latticed together and spaced apart sufficiently to have the required stiffness sidewise. Or, more commonly, a single I-beam is used with the required values of - and / G for the component W '. This beam is then subdivided in equal spans by one or more suspension rods extending up to the ridge of the roof, so that its stiffness sidewise is sufficient for each short span. But since the neutral axis of the purlin is not usually at right angles to its major axial plane, the angles of this section will not be equidistant from this neutral axis, and those more distant will be more stressed, than if the neutral axis were parallel to the top of the purlin. Therefore, the following formula is then to be applied to determine the maximum fibre stress found in these more distant angles, and whether it exceeds the safe limit for the material used. Let b = parallel breadth of the purlin in inches. d = normal depth of purlin in inches. I v = moment of inertia about parallel minor axis of section. I x = moment of inertia about normal major axis of section. /W'd W"b\ Then 0.75 L{ -j I j ) = maximum fibre stress in tons \ l v !,/ per square inch. If this exceeds the safe value for the material, a larger section must be taken, until a sufficient one is obtained. This formula must be applied to purlins of wood or steel excepting when W coincides with the major axial plane of the cross-section. APPLICATION OF FORMULAS TO PROBLEMS 19 26. Application of Formulas to Problems. Some problems will illustrate the practical use of the formulas and tables. PROBLEM 1. A steel girder is 30 ft. long and must safely support a uniform load of 15 tons. To be composed of two I-beams with separators and bolts. a. By ordinary formulas, Case 5, Table A. p 7 a r For safety against rupture: ~Q- = o C I PI 30000+360 Transposing : - = = = 84 ' 38 ' 5P I 3 For safety against excessive deflection : A = Transposing : _ 5X360PZ 2 _ 5X360X30000X129600 384 E 384x29000000 b. By simplified formulas, Case 5, Table 7. For safety against rupture: - = 0.187 W L = 0.187 X 15 X30 = 84.4. c For safety against deflection: /= 1.192TFL 2 = 1.192 X 15 X30 2 = 628.5 By Cambria, 2, 15 in., 42- Ib. I-beams are required. Comparison shows a decided economy in time and labor in computations. PROBLEM 2. Beam cantilever with uniform load, Case 2, Table 2. Free length 10 ft., and supporting a load of 0.5 ton per foot. Washington fir. - = 8.58 W L = 8.58 X 5 X 10 = 429. C/ / = 9.26 W L 2 = 9.26 X5 X 10 2 = 4630. 20 SIMPLIFIED FORMULAS AND TABLES By Table 19 for-: 8x18, 10x16, 12x16, 14x14 ins. O By Table 20 for I: 8x20, 10x18, 12x18, 14x16 ins. Therefore the beam may be made 10x18 or 14x16, as most convenient. PROBLEM 3. Beam supported at ends with load at middle. Case 4, Table 6. Beam of shortleaf pine 16 ft. clear span, which must safely support a load of 3 tons at middle of span. - = 5.45 W L = 5.45 X3 X16 = 262. c 1= 1.802 WL 2 = 1.802 X3X16 2 = 1384. By Table 19 for -: 4x20, 6x18, 8x14, 10x14, 12x12. c By Table 20 for 7: 4x18, 6x16, 8x14, 10x12, 12x12. Most economical to make the section 8x14 ins. PROBLEM 4. Steel floor beam supporting hollow tile floor, Case 5, Table 7. Beam 16 ft. long and set 4 ft. on centres. Must safely support a total live and dead load of 146 Ibs. per square foot of floor. 146 Here W = X4 X 16 = 4.673 tons. - = 0.187 WL =0.187x4.673x16 =13.98. c I = 0.046 W L 2 = 0.047 X 4.673 X 16 2 = 56.22. By Cambria: 1, 8 in., 18 Ib. I-beam just suffices. PROBLEM 5. Joists supporting floor and ceiling, Case 5 a, Table 8. Shortleaf pine joists 18 ft. long and set 16 ins. on centres must safely support a total live and dead load of 65 Ibs. per square foot. I wL 2 e65xl8 2 Xl6 _ 8800 8800 _ _ wL*e _ 65 X 18 3 Xl6 1 ~ 21337 ~ 21337 ~ ^ ' APPLICATION OF FORMULAS TO PROBLEMS 21 By Table 19 for-: If Xl2, 2x12, 3x10, 4x8. c By Table 20 for I : If X 14, 2 X 12, 3 X 12, 4 X 10. Hence the joists should either be If Xl4 or 2 Xl2, full size. PROBLEM 6. Joists for schoolroom floor, Case 5 a, Table 8. Joists of longleaf pine, 24 ft. long, set 12 ins. on centres, safely supporting total live and dead load of 102 Ibs. per square foot of floor. I _ wL 2 e _ 1Q2X24 2 X12 c~ 11200" 11200 "Tv? T _ _ 102x24 3 Xl2 29606" 29606 ~ By Table 19 for-: 3x12,4x10. C/ By Table 20 for /: 3 X 14, 4x12. Therefore it is best to make the joists 3 Xl4 ins. PROBLEM 7. Mill construction for deck roof, Case 5 6, Table 9. Plank roof of 2f ins. shortleaf pine, which must safely support a total live and dead load of 40 Ibs. per square foot. First find maximum safe distance between centres of supporting beams. 38.3 1 38.3X2.625 _ Q L = 7=-= - -7= = 15.89 ft. on centres. V w v40 T I2.lt 12.1X2.625 L = 3 = - 3 /-^ ~ = 9.29 ft. on centres. V40 Therefore, the supporting beams cannot be safely set over 9.4 ft. on centres. PROBLEM 8. Mill roof beams, Case 5, Table 7. Assum- ing the roof beams to be set 8 ft. on centres and to be of shortleaf pine also, and 16 ft. in clear length. 22 SIMPLIFIED FOEMULAS AND TABLES W =8X16X42= 5376 Ibs. = 2.688 tons, allowing 2 Ibs. per square foot for average weight of roof beams. -=2.730 WL =2.730X2.688X16 =117.4. C/ / = 1.125 W L 2 = 1.125 X2.688 X 16 2 = 774.4. By Table 19 : 4 X 14, 6 X 12, 8 X 10. By Table 20 : 4 X 14, 6 X 12, 8 X 12. Therefore 6 Xl2 beams are preferable. PROBLEM 9. Mill roof girders, Case 4, Table 6. Assum- ing that the posts are 16 ft. on centres, that one inter- mediate beam is supported at middle of girder, for shortleaf pine girders. - = 5.45 W L = 5.45 X2.688 X 16 = 234.4. c 1= 1.802 WL 2 = 1.802 X2.688X16 2 = 1240.3. By Table 19: 6x16,8x14,10x12. By Table 20: 6x14, 8x14, 10x12. Hence it will be best to make these girders 8 X 14 ins. 27. Cast-iron Lintels. Although lintels composed of steel shapes are now generally employed to span openings in masonry walls, cast-iron lintels are still frequently used for this purpose. But only certain stock sections and sizes are usually fur- nished by the larger foundries, since a specially made pattern would usually make the cost of a few lintels prohibitive. Tables 21, 22, and 23 comprise the standard forms and dimensions of lintels usually furnished. For these have been carefully computed their properties, CAST-IRON LINTELS 23 i.e., the numerical values of -, /, and c = distance in c inches from bottom of lintel section to its horizontal gravity axis. Thus, it now becomes possible to apply the formulas previously given to determine the required cross- section of a cast-iron lintel as easily as to obtain the dimensions of a beam of wood or of steel shapes. PROBLEM 10. An inverted T-lintel is 16 ft. long with a section 8x12 ins. and 1J in. metal. Determine its safe uniform load W. By Table 22: - = 58.2; / = 120.8. Case 5, Table 7. c W=- X^ = 58.2 X^ = 3.637 tons. c L 16 ' J i Of) Q W = 11.85 , = 11.85 X- = 5.592 tons. Therefore, the maximum safe uniform load of the lintel = 3.637 tons. PROBLEM 11. Box lintel, with two webs and uniformly loaded. Clear span of 12 ft. and must safely support a brick wall 12 ins. thick and 51 ft. high, weighing 120 Ibs. per cubic foot. Weight of wall = 12 X5J X 120 = 7920 Ibs. =3.96 tons. Then -= l.OOOTFL = 1x3.96x12 = 47.52. c I =0.084TFL 2 -0.084 X 3.96 Xl2 2 =47.91. By Table 21 a box lintel 8xl2xf ins. metal will be ample. PROBLEM 12. Box lintel with three webs supporting brick wall. Span 16 ft. and wall 24 ins. thick and solid. If the lintel be shored up until the mortar sets properly, it is generally assumed that the volume of the brick wall 24 SIMPLIFIED FORMULAS AND TABLES actually supported by the lintel is that included below lines drawn at 60 through each end of the clear span of i A the lintel. In this case the altitude of this triangle = -~- tan 60 = 13.86 ft. 13.86X16.0X2 Volume of brickwork = ~ = 221. 76 cu. ft. Weight = 221.76X120 = 26611 Ibs. = 13.30 tons. The ordinary formulas given for such a mode of loading are PI SI PI* -z- = - ana A = 6 ' c &OET Transforming these into simplified formulas in the manner explained in Art. 4, we obtain the following for- mulas for this form of loading on cast-iron lintel: - = 1.333 W L and / = 0.108 W L 2 . c Then - = 1.333 WL = 1.333x13.306X16 = 283.8. c I = 0.108 WL 2 = 0.108 X13.306X16 2 = 359.5. By Table 23 a lintel 12x24x1^ ins. metal will suffice. PROBLEM 13. Sheathing of roof. Shortleaf pine J-in. thick. Inclination of roof 35. Slated on felt and sheath- ing. p = 10 Ibs. (slates) + 1 Ib. (felt) +3 Ibs. (sheathing) = 14 Ibs. per inclined square foot. s = 15 Ibs. per horizontal square foot. s cos i = 12.3 Ibs. per inclined square foot. w = 31.1 Ibs. per inclined square foot (medium exposure). CAST-IRON LINTELS 25 Then (14 + 12.3) cos 35 = 21.6 Ibs. = normal component p+s per inclined square foot. And 14 cos 35 +31.1 = 42.8 Ibs. = normal component p+w per square foot. 14 sin 35 +0.00 = 8.0 Ibs. = parallel component p+w per square foot. The maximum normal component =42.8 Ibs. is to be taken, and the parallel component 8.0 Ibs. may be neglected, because resisted by edgewise stiffness of the sheathing. By formulas for Case 5 b: 38.3* 38.3X0.875 L = -7=^ = ~ / -- - = 5.06 ft. on centres on rafters. Vtc V42.8 12.1 1 12.1 xO.875 L = 57=^ = -- / -- = 3.02 ft. on centres of rafters. v w V42.8 . Hence the rafters cannnot be placed over 3 ft. on centres. PROBLEM 14. Rafters. Case 5 a. Shortleaf pine. The rafters of the same roof are 12.5 ft. long, and their weight averages 3 Ibs. per square foot of inclined surfaces. Then p = 14+3 = 17 Ibs. per inclined square foot of roof. (17 + 12.3) cos 35 = 24.0 Ibs. = normal component of p +s. 17 cos 35 +31.1 =45.0 Ibs. = normal component of p +w. 17 sin 35 +0.00 = 9.8 Ibs. = parallel component of p+w. 1. Assume that rafters are set 3 ft. on centres. 45X12.5 2 X36 8800 8800 45X12.5 3 X36 21337 = 21337 By Table 19: If X 12, 2x10, 3x8, 4x8. By Table 20: If X 12, 2x10, 3x10, 4x8. 26 SIMPLIFIED FORMULAS AND TABLES It would be most economical to use 2x10 rafters if full size. But these would look heavy, and would have a better appearance if set closer and made smaller. 2. Assume a section If X8 and determine c. By Table 19: - = 17; and by Table 20: I = 79, for this c section. By formulas for e, Case 5 a, Table 8. 7 8800 17x8800 6 == ~c * wL 2 = 45 X 12 5 2 = ms * on cen ^ res 213377 21337x79 e = w L 3 = "45x12 5 3 = ms * on centres Best use If -in. rafters set 18 ins. on centres. 45X12.5 4 X18 rhen A = 6402007 = 640200X79 = ' 391 m ' = max " imum deflection. 9.8XL5X12.5 Also -- o v 2000 = 0*04" ton = longitudinal compres- sion at mid-length of the rafter due to parallel com- ponent of its load. And pfTTo = 0.028 ton per sq. in. compression there. Then 0.028 ( 1 + ^r) = - 028 ( l + 6X Q ' 391 ) = 0.0239 = \ a I \ / maximum fibre stress due to longitudinal compression. Also 0.55 - 0.0239 = 0.526 = maximum safe fibre stress for supporting transverse load on rafter. Substituting this value in the general formula of Case 5 a for safety against rupture: 7 wL 2 e _ 45X12.5 2 X18 _ c ~ 16000 F " : 16000X0.526 ~~ = CAST-IRON LINTELS 27 Since this is less than the actual value of -=17 for If X8 section, this size will amply resist both normal and parallel components of loading. This example shows that in ordinary cases the parallel component of the loading on rafters may be neglected. PROBLEM 15. Purlins of roof, one to a panel. Length 16 ft., set normal to inclined surface. 1. Assume shortleaf pine timber, average weight 3 Ibs. per inclined square foot. p = 17+3 =20 Ibs. per inclined square foot. (20 + 12.3) cos 35 = 26.4 Ibs. = normal component of p+s. 20 cos 35 +31.1 = 47.5 Ibs. = normal component of p +w. 20 sin 35 +0 =11.5 Ibs. = parallel component of p +w. 16x12.5 = 200 sq.ft. of inclined surface supported by one purlin. W = 200 X47.5 = 9500 Ibs. = 4.75 tons = normal loading on purlin. W"= 200x11.5 = 2300 Ibs. = 1.1 5 tons = parallel loading on purlin. a. For normal loading W. - = 2.730 W L = 2.730 X4.75 Xl6 = 207.5. c 7 = 1.125 WL 2 = 1.125 X4.75X16 2 = 1368. By Table 19: 4x18, 6x16, 8x14, 10x12. By Table 20: 4x16, 6x14, 8x14, 10x12. b. For parallel loading W". ^=2.730 WL= 2.730X1.15X16= 50.2. 7 = 1.125TFL 2 = 1. 125X1. 15X16 2 = 331.2. 28 SIMPLIFIED FORMULAS AND TABLES By Table 19: 18x6, 16x6, 14x6, 12x6, 10x6. By Table 20: 18x8, 16x8, 14x8, 12x8, 10x8. Hence 8x14 might suffice for the dimensions required by both loadings. Since the neutral axis of the cross-section of the purlin cannot coincide with its minor axis in this case, it becomes necessary to determine the actual maximum fibre stresses occurring in the corners most distant from the neutral axis, by the formula of Art. 25. By Table 20, for 8x14 section, I v = 1829; for 14x8 section, I x = 597. 4.75X141.15X8\ =0 ' 622 ton per square inch equals actual maximum fibre stress, which exceeds the maximum safe fibre stress of 0.55 ton per square inch for shortleaf pine. Hence it will be necessary to enlarge the section of the purlin, say, to 10x14. By Table 20, for 10x14, I v = 2287; for 14x10, I x = 1167. Then 0.65Xl6 > g 7 + ' 117 =0.449 ton per sq. in., which is amply safe. 2. Assume purlin composed of two latticed steel channels spaced apart to make purlin equally stiff in both directions. Average weight of steel purlins 4 Ibs. per inclined square foot of roof. Then p = 17 +4 =21 Ibs. per inclined square foot. 21 cos 35 +31.1 = 48.3 Ibs. = normal component of p +w. 21 sin 35 +0.00 = 12.0 Ibs. = parallel component of p +w. W = 200 X48.3 = 9660 Ibs. = 4.83 tons = normal loading. W" = 200 X 12.0 = 2400 Ibs. = 1.20 tons = parallel loading. CAST-IRON LINTELS 29 a. For normal loading. - = 0.187 WL =0.187X4.83X16 = 14.4 c I = 0.047 W L 2 = 0.047 = 4.83 X 16 2 = 58.1. By Cambria, 2, 8 in., 11J Ib. channels will suffice. It is evidently unnecessary here to compute - and / for c the parallel loading, since their values are much smaller and the purlin is made to be equally stiff in both directions. But it will be well to apply the formula to determine the actual maximum fibre stresses occurring in the section. Here I y = L = 64.6, and b = 9.43 ins., = width of two flanges + spacing. tons per square inch, which exceeds the maximum safe fibre stress of 8 tons for steel. Hence, the purlin must be composed of 2, 8 in., 13f Ib. channels, which will be amply strong. 3. Assume that purlin is composed of a single I-beam with supporting rods as required. Since for W, - = 14.4, and I = 58.1, as already found, use V 7 4 04 1, 8 in., 20 J Ib. I-beam, for which sidewise - = ^ = 1.98 and I = 4.04. Then by formulas for Case 5 : / 5.333 _ 1.98X5.333 _ 8.80 ft. between supporting L ~ c X W" '' 1.20 "~ rods. / T /4 04 L = 4.64^^7? = 4.64^:^0 = 10.22 ft. between rods. 30 SIMPLIFIED FORMULAS AND TABLES Therefore, one supporting rod at mid-length of purlin will suffice, and this will be much lighter and more economi- cal than the latticed purlin composed of two channels. Since this beam is supported sidewise at the middle of its length, the maximum fibre stress at that point is only that produced by W. Transposing for F the general formula in Case 5: /_ 1.5 WL c~ F > we find ,, 1.5 WL 1.5X4.83X16 r = j = ^ - = 7.73 tons per square inch, which is entirely safe there. Apply formula for actual maximum fibre stress, the free span being here reduced to 8 ft. instead of 16 ft., and W' and W" are likewise halved. 42x8 . 0.60X4.08 60.2 4.04 5.57 tons per square inch, which is amply safe, so that this I-beam may be used. TABLES 32 SIMPLIFIED FORMULAS AND TABLES CASE 1. BEAM CANTILEVER. LOAD AT FREE END. TABLE 1 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. I 12WL c~ F 1.5WL 8.0TFL 17.2TFL 26.7TFL F = -> 0.069 L 0.26^^ For directly computing 7 from . C -X0.596L -X0.203L -X1.08L -X1.08L s* //*/ c c For maximum safe fibre stress and deflection. 1.68c 4.94c Actual maximum deflection. 216TFL 3 WL* 67.207 JFL^ 37.057 JF7,_ 3 3.247 2.087 TABLES 35 CASE 2. BEAM CANTILEVER. LOAD UNIFORM. TABLE 2 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. Spruce For maximum safe fibre stress F. - = 9.28TFL 7 0.108 = 8.65TFL 2 : 0.116-^; 7=-X0.938L 8.54TFL 10.91TFL 13.33TFL 10.91TFL 7 0.117 I 0.092 7 0.075 7 0.092 c X L c X L c X L c X L I 0.117 7 0.092 7 0.075 I 0.092 c X W c X W c X W c X TT For maximum safe deflection L 30' 7.63 TFL 2 10.80TFL 2 12.96TFL 2 9.98TFL 2 0.131 L 0.093 ^ 2 0.077 ^ 2 o.ioo 0-^W - 3 WJ 0.28VJ 0.32^1 For directly computing 7 from . -X0.891L -X0.991L -X0.973L -X0.915L c c c c = 1.07c WL* For maximum safe fibre stress and deflection 1.12c l.Olc 1.03c Actual maximum deflection. 3.477 JFL3 3.947 _ 2.787 WL? 2.327 1.09c WL 3 3.017 36 SIMPLIFIED FORMULAS AND TABLES CASE 2A. JOIST CANTILEVER. LOAD UNIFORM. TABLES General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 7_ wL*e c~4WOF 32000 6000 wL 2 e 2800 1800 I 4000* 1 7 32000 7 6000 7 2800 7 X 1800 c L z e c 7/ 2 e c L 2 e c X L 2 e c L*e I 4000F " c X wL z 7 32000 c wT/ 2 7 6000 c wL 2 7 2800 I X 1800 c wL 2 c wL z L = \\ X * c we 178.9^ 77 W^ 52>9 \^ 42.4-y/ For maximum safe deflection L 30' wL 3 e wL% it?L 3 e wL 3 e wL'e 3.70E 53650 29580 2590 1665 3.70#7 536507 295807 25907 16657 L 3 e L 3 e 3.70#7 536507 295807 25907 16657 wL 3 wL 3 wL* . ^/ we 37.7^/1 * UD(s 30.9^ e 13.7^/1 "Ws For directly computing 7 from . T 7 . 1080LF 7 X0.596L maximum safe -X0.203L fibre stress and ( -X1.08L c leflection. -X1.UH, 7 c ' # For ' 7?c 1.68c 4.94c 0.93c 0.93c 1080F Actual maximum deflection. A wL 4 e wL*e wL*e wL w;e = 1.07c _ ~832~50 For directly computing 7 from . -X0.890L -X0.990L -X0.973L c c c For maximum safe fibre stress and deflection. 1.12c l.Olc 1.03c Actual maximum deflection. 94350 66600 55500 -X0.9HL c 1.09c wL*e 72150 38 SIMPLIFIED FORMULAS AND TABLES CASE 2s FLOORING CANTILEVER. LOAD UNIFORM. TABLE 4 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. , \wtf_ V667F 21.6 ~17.3~ = 667F 466.9 300.2 L 2 L 2 ~L 2 ~ /667W 2UH 17.31 L ~\ w V w For maximum safe deflection . _ 3/3.24WL 3 L\/w ~\ E 6.00 "5705" Et 3 t 3 /s 216 F3 128.9 6.00^ >^ For maximum safe fibre stress and deflection. Actual maximum deflection. wL 4 wL* wL* 6482l TABLES 39 CASE 2B. FLOORING CANTILEVER. LOAD UNIFORM. TABLE 4 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. Spruce For maximum safe fibre stress F. t LV~w LVw LVw LVw LVw 20.8J 21.6 19.2 17.3 19.2 ?/ 433.6 466.9 366.9 300.2 - 366.9 L 2 L 2 L 2 L 2 L 2 T 20.8* 21.6* 19.2* 17.3* 19.2* '7Ti4 L= - For maximum safe deflection . 6.40 5.70 262.5 j^ 185.2 ~ 6.40* 5.70* 5.37 154.3 - 5.37^ For maximum safe fibre stress and deflection. 0.56* 0.51* 0.52* Actual maximum deflection. 6945* wL* 7871* 3 _ 5556* 3 wL* 4630* 3 5.86 20L5 5.86* 0.55* 6019* 3 40 SIMPLIFIED FOEMULAS AND TABLES CASE 3. BEAM CANTILEVER. LOAD IRREGULAR. TABLE 5 General [Steel Cast Iron Fir, Wash. Spruce For maximum fibre stress F. 7 12M c F 1.50M 8.00M 17.16M 26.70M -H 0.667- 0.125- 0.058- c c c 0.038- c For maximum safe deflection . Load at free end. , 17280ML 1.193ML 2.160ML 24.70ML o 8d.o n d.fi^ n 041 38.45ML 0.026 j- E I/- EI 17280L Li ^ Li- Li o QAO n Aft 4 ? n OAT 0.026^ 17280M Af M M For directly computing 7 from . 7 1440LF -X0.795L -X0.270L -X1.44L c c c 7 c Xl.44L Actual maximum deflection. 576ML 2 ML 2 ML 2 ML* ML 2 EI 25.207 13.897 1.2157 For maximum safe deflection 5^. Load uniform. 0.7827 T 12960ML 0.894ML 1.620ML 18.52ML 1.120 1 0.618 ^ 0.054 ^ 28.80ML 0.035 ^ E M EI 12960L 11 on * OfilR 00^4 / "12960M M M M M For directly computing 7 from . 7 1080LF 7 X0.596L 7 X0.203L -X1.08L c c c -X1.08L i X j-, C I'j Actual maximum deflection. 432ML 2 ML 2 ML 2 ML 2 ML 2 EI 33.607 18.537 1.627 1.047 TABLES 41 CASE 3. BEAM CANTILEVER. LOAD IRREGULAR. TABLE 5 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. Spruce For maximum safe fibre stress F. - = 18.48M 17.16M 21.84M 26.70M 21.84M M = 0.054 j 0.058 - 0.046 - 0.038 - c c c 0.046 - For maximum safe deflection . Load at free end. / = 23.07 ML 20.36ML 28.85ML 34.60ML 26.62ML M= 0.043 Y Li 0.049 - 0.035^ 0.029^ LI LI LI 0.038 y LI L= 0.043-^ 0.049^ 0.035^ 0.029^ 0.038 ^ M. For directly computing 7 from . C 7=^-Xl.25L -X1.19L -X1.32L -X1.30L c c c -X1.22L c Actual maximum deflection. A- ML2 ML 2 ML 2 ML 2 ML 2 -1.327 1.487 1.047 0.877 1.137 For maximum safe deflection . Load uniform. 7 = 17.30ML 15.26ML 21.60ML 25.92ML 19.96ML M= 0.058 Y LI 0.066 Y 0.047^ 0.039^ LI Ju LI 0.050 y Ju L= 0.058-^ 0.066 -^ 0.047 ^ 0.039 ^ M M M 0.050 For directly computing 7 from . 7=^X0.936L -X0.891L -X0.991L -X0.973L -X0.915L Actual maximum deflection. A- ML2 ML 2 ML 2 ML* ML 2 A ~ 1.747 1.927 1.397 1.167 1.517 42 SIMPLIFIED FORMULAS AND TABLES CASE 4. BEAM SUPPORTED AT ENDS. LOAD AT MIDDLE TABLE 6 General 7 = 1080TFL 2 Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. I_3WL I75TFL ,2.670 < ~T 2.670 2.000PFL 7 X ~ZT 7 0.500 4.29TFL 7 0.233 6.67TFL 7 0.150 c F v= T -x- -: c 37/ c L= L X JL L> c L I 0.233 c L 7 0.150 ( W c X W c X W c X W For maximum safe deflection . 0.135JFL 2 1.544WL 2 2.403T7L 2 1080L 1080TF 7 360LF 13.430 f- 2 7.410 f- 2 i> A/ 0-648 For directly computing 7 from . 0.417 i -W^ o-W^ -X0.199L c c -X0.360L -X0.360L c c EC ~~3GOF 36TTL 3 El For maximum safe fibre stress and deflection. 5.04c 14.82c 2.78c 2.78c Actual maximum deflection. WL 3 WL* WL* 4037 2227 19.457 TFL 3 12.507 TABLES 43 CASE 4. BEAM SUPPORTED AT ENDS. LOAD AT MIDDLE. TABLE 6 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. Spruce = 4.61TFL c w=L x ^l L- 7 X- 217 L -^ x ~w~ 4.29TFL 7 v ,0.233 X f c L 7 0.233 c X W 5A5WL 7^0183 7 X ~I7~ I 0.183 c X W 6.67TFL 7 0.150 7 X ~L~ 7 0.150 c X W 5.45TFL I 0.183 7 X L -X c 0.183 For maximum safe deflection r. 1.442TFL 2 = 0.695^ 1.272TFL 2 1.802T^L 2 2.163TFL 2 0.787 ^ 0.555 ~ 0.463 ^ 1.965TFL 2 0.602 ^ .887^ 0.745^ 0.681^ 0.776^ 7=-X0.312L For directly computing 7 from . C -X0.296L -X0.330L -X0.324L -X0.330L c c c c 3.20c For maximum safe fibre stress and deflection. 3.37c 3.03c 3.09c Actual maximum deflection. WL 3 WL* WL* 20.847 23.637 WL* 16.677 13.887 3.29c WL* 18.057 44 SIMPLIFIED FORMULAS AND TABLES CASE 5. BEAM SUPPORTED AT ENDS. LOAD UNIFORM TABLE 7 General 7 = 1.5TFL c~ F W-i-X-?- ~c A 1.5L T * \* | c X 1.5TF 7 = 675TFI/ 2 E El 1 El \675FF Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 0.187T7L l.OOOTFL 2.144TFL 3.336TFL 7 _ 0.300 21.50^ 11.86 ri 1.22 ri 4.64 7 0.300 c X TF 7 5.333 ^ V LOOO ^ V 0^67 c X L c X L c X L 7^ 5.333 7 1.000 7 0.467 c X W c X W c X W For maximum safe deflection . oU 0.047TFL 2 0.084TFL 2 0.965TFL 2 1.500TFL 2 0.67 EC For directly computing 7 from . -X0.248L -X0.085L -X0.450L - c c c c For maximum safe fibre stress and deflection. 4.02c 11.90c 2.22c 2.22c Actual maximum deflection. A = 22.5TFL El WL* 5807 WL* 3567 WV 31.17 WL* 20.07 TABLES 45 CASE 5. BEAM SUPPORTED AT ENDS. LOAD UNIFORM TABLE 7 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. Spruce - = 2.310TFL c L _7 0.433 X 2.144TFL 2.730TFL 3.336TFL W 2.730TFL 7 0.467 ^ V 0^67 _/ 0.300 7 0.367 ^N/- y.^7 ^ T ^7 c L c L c L c L 7 0.467 7 0.367 7 0.300 L V 0<367 ^\ TTT _ /\ TTT ^N TTT /X " W For maximum safe deflection ^:. = 0.900TFL 2 0.795TFL 2 1.125TFL 2 1.350PTL 2 1.038TFL 2 1.26 0.89 0.73 0.96 0.98A ~ 7=-X0.390L c For directly computing 7 from . -X0.370L -X0.412L -X0.405L -X0.381L c c c c L=2.56c For maximum safe fibre stress and deflection. 2.70c 2.42c 2.47c > Actual maximum deflection. 33.37 WL* 37.87 26.77 WIS 22.27 2.63c WL* 28.97 46 SIMPLIFIED FORMULAS AND TABLES CASE SA. JOIST SUPPORTED AT ENDS. LOAD UNIFORM TABLE 8 General Steel Cast Iron Fir, Wash. Spruce For maximum safe fibre stress F. 1 wL 2 e wL 2 e wL 2 e wL*e wL*e c mOOF 7 16000F 128000 7 128000 24000 7 24000 11200 7 X 11200 7200 7 x 7200 7 16000F c L 2 e I 128000 c L 2 e 7 24000 7 11200 c X L 2 e 7 7200 "c L c wL 2 c L c ^ wL 2 c wL* // 16000F 357 V^ For maximu wL 3 e 15fr%/ 7 10fU/ 7 84.9 V \w;ec A/ A/ A ' c . we j wL 3 e lODA/ * 1VCC .m safe deflection 1UOAI L 30' 35.56^7 515620 5156207 284480 2844807 24927 249277 16000 160007 L*e 35.56^7 5156207 wL 3 L 3 e 2844807 L 3 e 249277 L 3 e 160007 wL' wL 3 wL 3 wL 3 r s/35.56^7 80.2-v For directly Vfl 94.87^ 65.8^ 29.2^ * we * iye computing 7 from . 3/7 * we 7 450LF >w;e VO d^OT c X # For C C C maximum safe fibre stress and deflection. 4.02c ll.QOc 2.22c Actual maximum deflection. C 2.22c L 450F ty7/ 4 e 1067EI 154715007 85360007 7469007 4801507 TABLES 47 CASE SA. JOIST SUPPORTED AT ENDS. LOAD UNIFORM TABLE 8 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. Spruce 7 wL 2 e wL 2 e~ wL 2 e wL 2 e 8800 7200 wL 2 e ^"10400 11200 8800 7 10400 7 11200 7 8800 7 7200 7 8800 7 10400 7 11200 S\* T n ' S\ T n c Li*e c L**e I 8800 7 7200 c X L 2 e I 8800' e 'c X wL 2 c X wL 2 c wL 2 c wL 2 93.8A/ 84.9-\/ * wee ~ wee 93.8A/ > wee For maximum safe deflection . wL 3 e wL 3 e wL 3 e wL 3 e w;L 3 e- 26670 29606 21337 17780 23114 266707 296067 213377 177807 231147 266707 296067 L 3 e L 3 e 213377 177807 wLi 3 wL 3 27.7jl 26.1-4/1 ' we V we L 3 e 231147 wL 3 wL 3 L = 29.9-\f 30.9^/ wL 3 28.5^.' For directly computing 7 from . C 7=-X0.390L -X0.370L c c -X0.412L -X0.405L -X0.381L For maximum safe fibre stress and deflection. L=2.56c 2.70c 2.42c 2.47c 2.63c Actual maximum deflection. wL 4 e wL 4 e wL*e wL*e wL*e 8002507 9069507 6402007 5335007 6402007 48 SIMPLIFIED FORMULAS AND TABLES CASE SB. FLOORING SUPPORTED AT ENDS. LOAD UNIFORM TABLE 9 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. I wL z L/Vw LVw = \2667F ~43ir ~34lT 2667F* 2 1867* 2 1200* 2 w = -&- -JT "IT- r _ / 2667/^*2 43.2* 34.6* Ll ~-\ 7=r == W -\/ w -\/ w For maximum safe deflection ^r. t= i2.96E 14.4 TOT 2 96-&* $ 2072* 3 1332* 8 W = -VT- ~V~ ~~IJ~ L= 8/2.96#< 3 14.4* 11. Ot ' iy -\/w "Vw For maximum safe fibre stress and deflection. L= ^0.245 I 0.200 7 N 0.245 c X W c X W c ' < W c X W c ' K Tf For maximum safe deflection . 7=0.630WL a W = 1.588 0.556WL 2 0.788 WL 2 0.946 WL* 0.728 TFL 2 7 1.800 1.270 1.058 5.50c A = WL* 47.67 For directly computing / from . -X0.181L -X0.203L -X0.198L c c c For maximum safe fibre stress and deflection. 5.79c 5.20c 5.30c Actual maximum deflection. WL* WL* TfL 3 53.07 -X0.186I- c 5.63c WL* 37.17 31.87 41.37 54 SIMPLIFIED FORMULAS AND TABLES CASE 8. BEAM FIXED AND SUPPORTED AT ENDS. LOAD UNIFORM. TABLE 12 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 7 1. 5WL 0.187TFL 7 5.33 l.OOOTFL 7 1.00 2.14TFL 7 ^.0.467 3.33IFL 7 N 0.300 c IF F X/ 0.667F c L -7 X L VX 0.667F c X L 7 5.33 c X JF c X 7 1 L 1.00 c X L 7 0.467 c L 7 0.300 Tf c FT c X TF c X W 270TFL 2 7 180LF c X E For maximum safe deflection . oO ^ ir= E/ U.UJ-OOKKl^- 53.807 L 2 7 9^-% / u.uooo kr x>" 29.657 2.597 1.677 270L 2 L 2 54K/ T L 2 L 2 L29 \W L J EI L \270PF /.dd'Y yy 5.44^^ L61 A/TT For directly computing 7 from . -X0.099L -X0.0347, -X0.180L -X0.180L 1SOF For maximum safe fibre stress and deflection. lO.lc 29.7c 5.56c 5.56c Actual maximum deflection. EI _ 16127 WL* 8857 JFL* 77.87 50.07 TABLES 55 CASE 8. BEAM FIXED AND SUPPORTED AT ENDS. LOAD UNIFORM. TABLE 12 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum fibre stress F. Spruce * = 2.31WL 2.14TFL 2.72WL 3.33T7L 2.72TFL w J x- 438 7 ^0.467 I ^0.367 7 0.300 7 W 0.367 C X L c X L c X L c X L c X L I 0.438 ~^ X ~~W I 0.467 c X W 7 0.367 c X TF ^0.300 7 0.367 c X W~ For maximum safe deflection . 30 7 = 0.360TFL 2 0.318TFL 2 0.450TFL 2 0.540WL 2 0.416TFL 2 w 2 ' 7SI 3.157 2.227 1.857 2.417 ' L* L 2 L 2 L 2 L 2 For directly computing 7 from . 7=-X0.156L we -X0.180L Ec For maximum safe fibre stress and deflection. lO.lc 29.7c 5.56c Actual maximum deflection. A = wL 4 e 2667^7 5.56c 3867150C7 213360007 18669007 12001507 TABLES 57 CASE SA. JOIST WITH ENDS FIXED AND SUPPORTED. LOAD UNIFORM. TABLE 13 Oak, Wh. I wL*e Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. For maximum safe deflection 57;. oU Spruce c 10400 7 10400 '~c X L 2 e 11200 7 11200 8800 7 8800 c Z/ 2 e 7200 7 7200 8800 7 8800 c X L 2 e c L 2 e I 10400 7 11200 7 8800 c wL 2 7 7200 /_ 8800 c w;7/ 2 e X T 9 c t&L 2 c X wL* c wL^ |~7~ L = 102V " wee 106 V * -u;ec 93.8V ' i^ec 84.9V 93.8V ^ wee tyL 3 e w;L 3 e 44445 444457 wL 3 e 66668 666687 75557 755577 53334 533347 57779 577797 L 3 e 666687 755577 L 3 e 533347 L 3 e 444457 577797 *-Ws wL 3 42^/1 * 1(^6 wL* 35.4^/1 ' we wL 3 For directly computing 7 from . C 7= X0.156L X0.148L ; X0.162L X0.152L c c 6.42c A = wL 4 e 20002507 For maximum safe fibre stress and deflection. 6.75c 6.07c 6.18c Actual maximum deflection. wL 4 e wL*e wL 4 e 22669507 16002007 13335007 6.58c wL 4 e 17335507 58 SIMPLIFIED FORMULAS AND TABLES CASE SB. FLOORING WITH ENDS FIXED AND SUPPORTED LOAD UNIFORM. TABLE 14 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. I wL* kVw ~\2667^ 43.2 _ 26677^ 18677 2 1200^ : L 2 L 2 L 2 /2667F 43.2 34.6Z /=\ -- ......... ......... -7= r=r \ w V w ^w For maximum safe deflection ^. s/ wL 3 L^/w LL^. 17.3 15.0 3335 3 ~7T 173t 15.0J ^w Vw For maximum safe fibre stress and deflection. ...... '-' 2 - 78( 2 - 78 ' Actual maximum deflection. wL* wL* 9990CH 3 TABLES 59 CASE SB. FLOORING WITH ENDS FIXED AND SUPPORTED LOAD UNIFORM. TABLE 14 Oak, Wh. 17.7 17.7* ** ~ 3/ L = 3.21* Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. For maximum safe deflection ^. 18.5 6299^ 16.4 4446 ^ 15.5 3705 |^ 15.5* For maximum safe fibre stress and deflection. 3.38* 3.04* 3.09* Actual maximum deflection. wL* wL* wL* Spruce LVw 41.6 43.2 38.3 34.6 38.3 1734* 2 1867* 2 1467* 2 1200* 2 1467* 2 L 2 L 2 L 2 L 2 j L 2 41.6* 43.2* 38.3* 34.6* 38.3* 16.9 4817 r* 16.9* 3.04* 166500* 3 133200Z 3 111000* 3 144300* 3 60 SIMPLIFIED FORMULAS AND TABLES CASE 9. BEAM FIXED AT ENDS. LOAD AT MIDDLE TABLE 15 General. I = l.5WL c~ F. ri7_ 7 X/0.667F rr /\ y L = I ..0.667F 7 = W = L = 270TFL 2 270JF 7^ 180LF * ^^ 77T Steel. Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 0.187TFL I 5.325 I 5.325 c X W l.OOTFL 7 1.000 /\ f c L I 1.000 c X W 2.14TFL 7 0.467 I 0.467 c X W 3.33TFL 7 0.300 I 0.300 c X W For maximum safe deflection . E El 53.77 L 2 29.67 L 2 2.597 270L 2 L 2 L677 L 2 ^Vw Ww For directly computing 7 from . c -X0.099L -X0.034L -X0.180L -X0.180L c c c c EC 1SQF A = 9TFL 3 El For maximum safe fibre stress and deflection. lO.lc 29.7c 5.56c Actual maximum deflection. WL 3 WL 3 16117 888.77 77.87 5.56c WL 3 50.07 TABLES 61 CASE 9. BEAM FIXED AT ENDS. LOAD AT MIDDLE TABLE 15 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum fibre stress F. Spruce - = 2.31TFL 2.14TFL c II7 I v/ 0.433 7 ^0.467 2.72WL I 0.367 3.33TFL 7 0.300 2.72TFL 7 0.367 ~c X L c L I 0.433 7 .,0.467 c L I 0.367 c L 7 VX 0.300 7 0.367 c X Tf c X W c W " c X TF 0.360TFL 2 For maximum safe deflection ^. 0.318T7L 2 3.157 L 2 0.450TFL 2 2.227 0.540TFL 2 0.416FL* 1.857 L 2 2.417 L 2 7=-X0.156L For directly computing 7 from . -X0.148L -X0.165L -X0.162L - X0.152L C C CO For maximum safe fibre stress and deflection. = 6.42c 6.75c 6.07c 6.18c 6.58c Actual maximum deflection. TFL 3 WL 3 WL* WL 3 WL 3 = 83^7 94^57 66^77 55.67 72^27 62 SIMPLIFIED FOEMULAS AND TABLES CASE 10. BEAM FIXED AT ENDS. LOAD UNIFORM TABLE 16 General W 135TFL 2 ~^~ EI 135L 2 ^ 135JF Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 7 c WL F 0. 125TFL 0.667TFL 1. 43T7L 2.22WL V I F 7 8.00 7 1.50 7 0.70 I 0.45 \ ~c X L c X L c L c L c L L =I ^ x w 7 c 8.00 A JF 7 1.50 c X TF 7^ c 0.70 X W I 0.45 c X W For maximum safe deflection . 0.0093T7L 2 0.0169TFL 2 0.193TFL 2 0.299TFL 2 107.4 -A 59.3 ^ n 5.18 -f- n 35Vi 7.70^ 3.33 1.83^^ 135F 4.5TFL 3 EI For directly computing I from . C -X0.075L -X0.025L -X0.135L -X0.135L c c c c For maximum safe fibre stress and deflection. 13.43c 39.50c 7.40c Actual maximum deflection. WL 3 WL Z WL 3 32257 WL 3 17787 155.57 7.40c _ 100.07 TABLES 63 CASE 10. BEAM FIXED AT ENDS. LOAD UNIFORM TABLE 16 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. Spruce - = 1.54TFL 1. 43TFL I.S2WL 2.22TFL 1.82TFL V= ^ X ~T~ 7 c 0.70 X L I 0.55 c X L -X - 7 0.55 c X L" L -f x9 r 7 X - 7 0.55 c X TF 7 0.45 c X W 7 0.55 c X W For maximum safe deflection . oU = 0.180TFL 2 0.159TFL 2 0.224 TFL 2 0.269TFL 2 0.208TF / 6.30 L 4 44 L 2 3.70 ^ 2 4.82 1 For directly computing 7 from . ;7 = ^-X0.117L ^XO.lllL . ^X0.124L ^X0.122L -X0.114L For maximum safe fibre stress and deflection. L = 8.55c 8.98c 8.10c 8.24c 8.76c Actual maximum deflection. 186.77 _ 189.07 133.37 _ 111.07 WL 3 144.57 64 SIMPLIFIED FORMULAS AND TABLES CASE 10A. JOIST WITH ENDS FIXED. LOAD UNIFORM TABLE 17 General Steel Cast Iron Fir, Wash. Hemlock For maximum safe fibre stress F. 7 wL z e wL z e 192000 7 192000 wL z e wL z e wL z e c 24000^ 7 24000F 36000 7 36000 16800 7 16800 10800 7 10800 c L z e I 24000F c X L z e 7 192000 c 7> 2 e 7 36000 7 16800 c X L 2 e 7 10800 C' f\ -j- n c wL z c ^ iyL 2 c X wL 2 c X w;L 2 c X w;L 2 wL 3 e II 24000F // 192000 /7 36000 // v 16800 .L/ AJ x * c we , wL 3 e ~177.8E 177.8EI \ C X ^ e For maximum wL 3 e 2578100 25781007 \C X ^6 safe deflection wL 3 e " c t^e L 30' 1422400 14224007 124460 1244607 80010 800107 L 3 e 177.8EI L 3 e 25781007 14224007 1244607 L 3 e 800107 wL 3 wL 3 w;L 3 w;L 3 wL 3 sll77.8EI 137.1A/ ' iye For directly co -X0.0745L c maximum safe fi 13.43c Actual max wL*e me -PI 3/7 3| 7 ~* we J For ec wL*e ' t^e ' iye mputing 7 from . -X0.0253L -X0.135 bre stress and deflection. 39.50c 7.40c imum deflection. wL*e wL 4 e -X0.135L 7.40c 5333^7 773285007 426640007 37331007 23998507 TABLES CASE 10A. JOIST WITH ENDS FIXED. LOAD UNIFORM TABLE 17 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. For maximum safe fibre stress F. wL 3 e For maximum safe deflection . wL 3 e wL 3 e Spruce 7 wUe c 15600 wL?e 16800 wL 2 e 13200 wUe 10800 wL 2 e 13200 *4> 15600 7 16800 7 c 13200 / c 10800 7 c 13200 ^2 e-f. 15600 7 c 16800 X wL* 7 c 13200 X wL* 7 c 10800 X wV 7 c 13200 X wL* 1.12, 3 A wee 130A/ U > wee * wee 1( ^ wee i: * wee 133350 1333507 151130 1511307 106680 1066807 88900 889007 115570 1155707 L 3 e 1155707 1333507 1511307 L 3 e 1066807 889007 wL* L-51.1-J/- * t^e wL 3 53.3^ Vtw wL 3 47.4^- * we wL 3 44.6^- * iye w;L 3 48.7-^/1 7=-X0.117L For directly computing 7 from . C -X0.113L -X0.122L -XO.H4L For maximum safe fibre stress and deflection. L=8.55c 8.98c 8.10c 8.27c Actual maximum deflection. wL 4 e 45330507 31998007 26665007 8.76c wL 4 e 34664507 66 SIMPLIFIED FORMULAS AND TABLES CASE 10s. FLOORING FIXED AT ENDS. LOAD UNIFORM TABLE 18 General >-V wV- w = 400QFf 2 L 2 Hf w;: 14.82# U.82EP 3/14.82ff* 3 ~" \ ? Steel Cast Iron Fir, Wash. Hemlock For maximum fibre stress F 52.8 28001 L 2 52.8* For maximum safe deflection r. 21.8 10374 21>8I For maximum safe fibre stress and deflection. 42.4 1800* 2 L 2 18.9 6669^ 18.9^ 270^ Actual maximum deflection. A = wL* 444.4^^3 3.70 wL* TABLES 67 CASE 10s. FLOORING FIXED AT ENDS. LOAD UNIFORM TABLE 18 Oak, Wh. Pine, L.L. Pine, S.L. Pine, Wh. Spruce For maximum safe fibre stress F. v ~~ n n o For maximum safe deflection ^=. A = 50.9 52.8 46.9 42.4 46.9 2600* 2 2800*2 2200*2 1800* 2 2200* L 2 L 2 L 2 L 2 L 2 50.9* 52.8* 46.9* 42.4* 46.9* 22.3 23.3 20.7 19.5 21.3 12597-j^ 8892-^ 7410 ^ 9633-^ OO Q/ OQ Q/ OO *7-t 1 Q P\/ O1 Q/ ZZ.ot Zo.ot zu./c ly.ot zi.ot For maximum safe fibre stress and deflection. 27* 4.50* 4.04* 4.12* 4.38* Actual maximum deflection. wL* wL* 333600* 3 377740* 3 266640* 3 222200* 3 288860* 3 68 SIMPLIFIED FORMULAS AND TABLES a o g a DQ CD 2 g 00 rH t~- CO O5 CO 00 rH OS CO CO CO rH CO O> OO 'rH rH CO CO T^ O5 S5 CO rJH 00 rH rH I> ^ CO 00 3 8 50 rH g % 00 00 Tt< (N TjH $* t*- rH (M s O l> 00 00 rH CO 00 CO OO s 00 CO CO rH TF 00 CO 00 O (N 88 2 8 S TABLES 69 o a ^Q -^H -HH CO f^ CO rH CM Tfl tO g | i i I I I iO O CM Tt< t^ OO rH O CO CO CO CM O O5 00 t^ IO rH CO 00 CO t^ t>- Oi rH rti l> rH rH rH CM rH R CO CO ^^ CO CO ^^ rH IO CM rH CO GO rH CM CO TJH CO CO ^D CO ^^ LO IO 00 CO rH CO 00 O CO CO r 1 rH rH 1 00 rH CM 00 00 CM O CM rH rH rH CO 05 rH ^ CO C^l OO CO CO O5 O CO 00 O C^ I 3 CO 00 rfri CO CO t^ OO CO CO O2 to O5 >O CO (N Tfl O l> O2 rH rH 1 rH 00 CM - C^ O GO O CD OO O2 CM to rH rH C^ co O CO ^ti rH rH C^ CO . Tf iO CO 00 1 00 U^ rH 00 CO O CO 1 CO rH CO CO O ^ Ol (N O 00 rH CM CO ^t^ "^ CO * rH TjH 'CO t^ CO CO rH CM "^ CO O5 rH rH CO 1> CO CO t^ rH CO (M rH rH CM CM CO I CO - GO O CO r-( rH co to rH rH rH CO CO rH |> T* CO CO T^l 1> rH CO -^t 1 to CO OO 1 fl . . 51 (N -tf CO 00 O (N Tfi CD OO O CM * 0) 70 SIMPLIFIED FORMULAS AND TABLES PROPERTIES OF CAST-IRON LINTELS. TABLE 21 Sect. Dims. 7 c |" Metal. 7 c 1" Metal. 7 c i" Metal. I c 7 c 7 c L 6 x6 6x7 6x8 14.0 15.7 17.3 24.2 25.3 26.3 1.73 1.61 1.52 15.8 17.7 19.6 28.2 29.5 30.8 1.78 1.67 1.57 17.4 19.6 21.7 31.9 33.5 34.9 1.83 1.71 1.61 6 xlO 20.5 27.9 1.36 23.2 32.7 1.41 25.5 37.1 1.46 L 6 x!2 7 x7 7x8 23.5 22.9 25.2 29.2 39.9 41.2 1.24 1.74 1.64 26.5 25.8 28.5 34.2 46.5 48.4 1.29 1.80 1.70 28.9 28.7 31.6 38.8 52.7 54.9 1.34 1.84 1.74 8 x8 35.2 61.6 1.75 40.0 72.3 1.81 44.4 82.5 1.86 1 8x10 8x12 9x8 * .36.6 41.7 36.5 74.6 78.7 97.1 2.04 1.89 2.66 40.6 46.5 40.9 85.2 89.7 110.8 2.10 1.93 2.71 10 x!2 . . . 50.9 130.1 2.56 56.9 147.8 2.60 12 x!2 74.4 246.4 3.28 85.7 284.2 3.32 6 x8 20.8 42.2 2.03 23.5 48.9 2.08 26.0 55.2 2.12 6 xlO 24.4 45.3 1.87 27.7 53.1 1.92 30.6 60.0 1.96 6x12 28.0 48.4 1.73 32.0 56.9 1.78 35.0 63.7 1.82 6 x 14 6 x!6 . . . 35.4 39.3 59.1 61.6 1.67 1.57 39.2 43.3 67.0 69.7 1.71 1.61 6x18 .... 42.5 63.3 1.49 47.1 72.1 1.53 6 x20 46.4 65.4 1.41 50.8 74.1 1.46 8x12 49.5 127.0 2.57 55.4 144.6 2.61 8 x!4 . . . 55.4 133.9 2.42 61.9 152.3 2.46 8 x!6 .... 61.3 139.7 2.28 68.3 159.0 2.33 8 x!8 66.8 144.8 2.17 74.7 165.0 2.21 8 x20 72.6 149.4 2.06 80.8 170.4 2.11 8 x24 92.9 179.2 1.93 8 x28 104.1 186.5 1.79 10x20 100.9 280.4 2.78 114.8 323.6 2.82 10 x24 131.6 342.0 2.60 10 x28 147.6 357.2 2.42 12 x20 ... 130.7 465.3 3.56 148.2 533.7 3.60 12 x24 170.1 566.2 3.33 12 x28 TABLES 71 PROPERTIES OF CAST-IRON LINTELS. TABLE 22 Sect. Dims. I c 1" Metal. 7 c 1J" Metal. / c 1J" Metal. I c / c I c 6 x6 19.0 35.4 1.87 21.4 42.0 1.96 23.4 47.7 2.04 6 x7 21.3 37.2 1.75 23.9 44.2 1.85 26.3 50.3 1.93 6 x8 23.5 38.7 1.65 26.2 45.9 1.75 28.8 52.6 1.83 6 xlO 27.5 41.2 1.50 30.9 49.4 1.60 33.4 56.0 1.68 L 6 x!2 7 x7 7 x8 8x8 31.2 31.1 34.6 48.5 43.1 58.7 61.8 92.2 1.38 1.89 1.79 1.90 34.7 35.4 38.5 43.4 51.3 70.1 72.7 106.6 1.48 1.98 1.89 2.46 37.4 38.9 42.8 45.4 58.7 80.6 84.2 115.6 1.57 2.07 1.97 2.55 8x10 44.3 95.2 2.15 51.0 114.2 2.24 56.7 131.9 2.33 8 x!2 51.0 100.3 1.97 58.2 120.3 2.07 65.4 141.3 2.16 9 x8 45.5 124.3 2.75 52.6 149.6 2.85 59.0 172.7 2.93 M 10x12 63.3 166.92.64 84.0 229.9 2.74 94.4 267.2 2.83 12 x!2 94.6 318.5 3.37 111.6 387.0 3.47 123.4 437.7 3.55 6 X8 28.3 61.5 2.17 32.2 72.7 2.26 35.2 82.4 2.34 6 xlO 33.3 66.6 2.00 37.7 78.9 2.09 41.4 89.8 2.17 6 Xl2 6 x!4 6x16 38.6 42.5 46.9 71.0 74.4 77.4 1.84 1.75 1.65 42.8 47.6 52.1 82.9 87.1 90.6 1.94 1.83 1.74 47.3 52.5 57.5 95.5 100.8 105.2 2.04 1.92 1.83 6x18 51.0 80.0 1.57 56.9 93.8 1.65 62.2 108.9 1.75 6 x20 54.9 82.4 1.50 60.3 95.3 1.58 66.8 112.1 1.68 8 x!2 60.8 161.5 2.66 70.2 192.9 2.75 78.5 221.8 2.83 8x14 68.1 170.2 2.50 77.7 203.8 2.60 87.4 234.2 2.68 8 x!6 75.1 177.9 2.37 85.2 209.4 2.46 96.6 245.1 2.54 8 x!8 82.0 184.5 2.25 94.2 221.3 2.35 104.5 254.7 2.43 8 x20 88.6 190.4 2.15 102.0 228.5 2.24 113.1 263.6 2.33 8 x24 101.3 200.5 1.98 116.4 240.8 2.07 128.7 277.9 2.16 8 x28 113.4 208.7 1.84 129.9 250.7 1.93 144.1 289.6 2.02 10 x20 125.5 360.0 2.87 146.4 428.6 2.93 165.3 503.9 3.05 10 x24 144.1 380.1 2.64 166.3 455.4 2.74 188.5 533.6 2.83 10 x28 153.5 377.6 2.46 188.5 480.6 2.55 211.3 557.8 2.64 12 x20 165.2 600.8 3.64 194.3 728.3 3.75 222.2 849.9 3.83 12 x24 12 x28 189.1 212.8 637.1 667.7 3.37 3.14 223.0 250.5 773.7 811.4 3.47 3.24 254.2 284.7 902.5 948.0 3.55 3.33 72 SIMPLIFIED FORMULAS AND TABLES PROPERTIES OF CAST-IRON LINTELS. TABLE 23 Sect. Dims. / C |" Metal. 7 c \" Metal. 7 c \" Metal. 7 c 7 c 7 c 8 Xl6 59.2 155.5 2.63 68.4 183.9 2.69 75.9 207.7 2.73 8x18 62.2 162.3 2.61 74.6 191.5 2.57 83.1 216.8 2.61 8x20 79.9 196.3 2.46 89.8 224.4 2.50 8 x24 105.0 238.3 2.27 8 X28 10 x20 111.9 366.9 3.28 126.2: 420.4 3.33 10 x24 146.8 447 7 3.05 10 x28 12 x 20 146.6 607.0 4.14 166.0 695.5 4.19 12 x24 185.5 729.4 3.87 12x28 1" Metal. \\" Metal. 1J" Metal. 8x16 84.0 232.4 2.77 97.1 277.4 2.86 107.8 318.0 2.95 8x18 91.5 242.3 2.65 105.2 289.4 2.75 117.5 332.4 2.83 8x20 98.6 251.3 2.55 114.0 300.7 2.64 137.5 375.1 2.73 8x24 112.7 267.1 2.37 130.0 319.8 2.46 144.5 367.1 2.54 8x28 126.1 280.1 2.22 145.4 335.7 2.31 160.7 383.9 2.39 10 X20 139.8 471.3 3.37 163.6 567.4 3.47 184.9 655.9 3.55 10 x24 161.6 508.8 3.15 186.7 604.8 3.24 210.5 700.4 3.33 10 x28 178.4 527.8 2.96 209.2 637.9 3.05 235.7 737.6 3.13 12x20 184.6 782.5 4.24 218.8 947.2 4.33 251.6 1110.8 4.42 12 x24 209.8 834.7 3.98 248.2 1011.0 4.07 283.5 1117.8 4.16 12 x28 234.9 880.0 3.75 278.0 1066.8 3.84 317.4 1246.5 3.93 If" Metal. 12x28 314.0 1072.6 3.42 8 x28 174.5 434.2 2.49 10x28 1 O NX Oft 258.0 OCA Z, 833.0 1 /toQ n 3.23 4 no 14 X Zo OOU.O 14Uo.U .uz TABLE OF LOGAEITHMS 74 SIMPLIFIED FORMULAS AND TABLES TABLE OF LOGARITHMS. TO 499 1 2 3 4 5 6 7 8 9 0000 0000 3010 4771 6021 6990 7782 8451 9031 9542 1 0000 0414 0792 1139 1461 1761 2041 2304 2553 2788 2 3010 3222 3424 3617 3802 3979 4150 4314 4472 4624 3 4771 4914 5051 5185 5315 5441 5563 5682 5798 5911 4 6021 6128 6232 6335 6435 6532 6628 6721 6812 6902 5 6990 7076 7160 7243 7324 7404 7482 7559 7634 7709 6 7782 7853 7924 7993 8062 8129 8195 8261 8325 8388 7 8451 8513 8573 8633 8692 8751 8808 8865 8921 8976 8 9031 9085 9138 9191 9243 9294 9345 9395 9445 9494 9 9542 9590 9638 9685 9731 9777 9823 9868 9912 9956 Diff. 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 41.5 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 37.9 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 34.9 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 32.3 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 30.1 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 28.1 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 26.4 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 25.0 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 23.5 19 2788 2810 2833 2856 2878 2900 2923 2945 2967 2989 22.3 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 21.2 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 20.2 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 19.3 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 18.6 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 17.8 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 17.1 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 16.4 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 15.8 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 15.2 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 14.8 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 14.3 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 13.8 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 13.4 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 13.0 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 12.6 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 12.2 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 11.9 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 11.6 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 11.2 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 11.0 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 10 . 7 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 10.4 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 10.1 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 10.0 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 9.9 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 9.5 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 9.3 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 9.4 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 9.0 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 8.8 TABLES 75 / TABLE OF LOGARITHMS. 500 TO 999 1 1 2 3 4 5 6 7 8 9 Diff. 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 8.6 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 8.5 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 8.3 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 8.1 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 8.0 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 7.8 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 7.7 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 7.6 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 7.4 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 7.2 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 7.1 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 7.1 62 7924 7931 7938 7945 7952 7959 7966 7973 7980 7987 7.0 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 6.9 64 8062 8069 8075 8082' 8089 8096 8102 8109 8116 8122 6.8 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 6.7 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 6.6 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 6.5 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 63 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 6.2 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 6.1 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 6.0 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 6.0 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 5.9 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 5.8 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 5.7 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 5.6 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 5.5 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 5.5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 5.4 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 5.3 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 5.3 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 5.3 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 5.2 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 5.1 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 5.1 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 5.0 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 4.9 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 4.8 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538. 4.8 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 4.8 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 4.8 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 4.7 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 4.7 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 4.6 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 4.6 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 4.5 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 4.5 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 4.5 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 4.5 76 SIMPLIFIED FORMULAS AND TABLES TABLE OF LOGARITHMS. 1000 TO 1499 1 2 3 4 5 6 7 8 9 Diff. 100 0000 0004 0009 0013 0017 0022 0026 0030 0035 0039 4.3 101 0043 0048 0052 0056 0060 0065 0069 0073 0077 0082 4.3 102 0086 0090 0095 0099 0103 0107 0111 0116 0120 0124 4.2 103 0128 0133 0137 0141 0145 0149 0154 0158 0162 0166 4.2 104 0170 0175 0179 0183 0187 0191 0195 0199 0204 0208 4.2 105 0212 0216 0220 0224 0228 0233 0237 0241 0245 0249 4.1 106 0253 0257 0261 0265 0269 0273 0278 0282 0286 0290 4.1 107 0294 0298 0302 0306 0310 0314 0318 0322 0326 0330 4.0 108 0334 0338 0342 0346 0350 0354 0358 0362 0366 0370 4.0 109 0374 0378 0382 0386 0390 0394 0398 0402 0406 0410 4.0 110 0414 0418 0422 0426 0430 0434 0438 0441 0445 0449 3.9 111 0453 0457 0461 0465 0469 0473 0477 0481 0484 0488 3.9 112 0492 0496 0500 0504 0508 0512 0515 0519 0523 0527 3.9 113 0531 0535 0538 0542 0546 0550 0554 0558 0561 0565 3.8 114 0569 0573 0577 0580 0584 0588 0592 0596 0599 0603 3.8 115 0607 0611 0615 0618 0622 0626 0630 0633 0637 0641 3.8 116 0645 0648 0652 0656 0660 0663 0667 0671 0674 0678 3.7 117 0682 0686 0689 0693 0697 0700 0704 0708 0711 0715 3.7 118 0719 0722 0726 0730 0734 0737 0741 0745 0748 0752 3.7 119 0755 0759 0763 0766 0770 0774 0777 0781 0785 0788 3.7 120 0792 0795 0799 0803 0806 0810 0813 0817 0821 0824 3.6 121 0828 0831 0835 0839 0842 0846 0849 0853 0856 0860 3.6 122 0864 0867 0871 0874 0878 0881 0885 0888 0892 0896 3.6 123 0899 0903 0906 0910 0913 0917 0920 0924 0927 0931 3.6 124 0934 0938 0941 0945 0948 0952 0955 0959 0962 0966 3.6 125 0969 0973 0976 0980 0983 0986 0990 0993 0997 1000 3.4 126 1004 1007 1011 1014 1017 1021 1024 1028 1031 1035 3.4 127 1038 1041 1045 1048 1052 1055 1059 1062 1065 1069 3.4 128 1072 1075 1079 1082 1086 1089 1092 1096 1099 1103 3.4 129 1106 1109 1113 1116 1119 1123 1126 1129 1133 1136 3.3 130 1139 1143 1146 1149 1153 1156 1159 1163 1166 1169 3.3 131 1173 1176 1179 1183 1186 1189 1193 1196 1199 1202 3.2 132 1206 1209 1212 1216 1219 1222 1225 1229 1232 1235 3.2 133 1239 1242 1245 1248 1252 1255 1258 1261 1265 1268 3.2 134 1271 1274 1278 1281 1284 1287 1290 1294 1297 1300 3.2 135 1303 1307 1310 1313 1316 1319 1323 1326 1329 1332 3.2 136 1335 1339 1342 1345 1348 1351 1355 1358 1361 1364 3.2 137 1367 1370 1374 1377 1380 1383 1386 1389 1392 1396 3.2 138 1399 1402 1405 1408 1411 1414 1418 1421 1424 1427 3.1 139 1430 1433 1436 1440 1443 1446 1449 1452 1455 1458 3.1 140 1461 1464 1467 1471 1474 1477 1480 1483 1486 1489 3.1 141 1492 1495 1498 1501 1504 1508 1511 1514 1517 1520 3.1 142 1523 1526 1529 1532 1535 1538 1541 1544 1547 1550 3.0 143 1553 1556 1559 1562 1565 1569 1572 1575 1578 1581 3.0 144 1584 1587 1590 1593 1596 1599 1602 1605 1608 1611 3.0 145 1614 1617 1620 1623 1626 1629 1632 1635 1638 1641 3.0 146 1644 1647 1649 1652 1655 1658 1661 1664 1667 1670 2.9 147 1673 1676 1679 1682 1685 1688 1691 1694 1697 1700 2.9 148 1703 1706 1708 1711 1714 1717 1720 1723 1726 1729 2.9 149 1732 1735 | 1738 1741 1744 1746 1749 1752 [ 1755 1758 2.9 TABLES 77 TABLE OF LOGARITHMS. 1500 TO 1999 1 2 3 4 5 6 7 8 9 Diff. 150 1761 1764 1767 1770 1772 1775 1778 1781 1784 1787 2.9 151 1790 1793 1796 1798 1801 1804 1807 1810 1813 1816 2.9 152 1818 1821 1824 1827 1830 1833 1836 1838 1841 1844 2.9 153 1847 1850 1853 1855 1858 1861 1864 1867 1870 1872 2.8 154 1875 1878 1881 1884 1886 1889 1892 1895 1898 1901 2.8 155 1903 1906 1909 1912 1915 1917 1920 1923 1926 1928 2.8 156 1931 1934 1937 1940 1942 1945 1948 1951 1953 1956 2.8 157 1959 1962 1965 1967 1970 1973 1976 1978 1981 1984 2.8 158 1987 1989 1992 1995 1998 2000 2003 2006 2009 2011 2.7 159 2014 2017 2019 2022 2025 2028 2030 2033 2036 2038 2.7 160 2041 2044 2047 2049 2052 2055 2057 2060 2063 2066 2.7 161 2068 2071 2074 2076 2079 2082 2084 2087 2090 2092 2.7 162 2095 2098 2101 2103 2106 2109 2111 2114 2117 2119 2.7 163 2122 2125 2127 2130 2133 2135 2138 2140 2143 2146 2.7 164 2148 2151 2154 2156 2159 2162 2164 2167 2170 2172 2.7 165 2175 2177 2180 2183 2185 2188 2191 2193 2196 2198 2.6 166 2201 2204 2206 2209 2212 2214 2217 2219 2222 2225 2.6 167 2227 2230 2232 2235 2238 2240 2243 2245 2248 2251 2.6 168 2253 2256 2258 2261 2263 2266 2269 2271 2274 2276 2.6 169 2279 2281 2284 2287 2289 2292 2294 2297 2299 2302 2.6 170 2304 2307 2310 2312 2315 2317 2320 2322 2325 2327 2.6 171 2330 2333 2335 2338 2340 2343 2345 2348 2350 2353 2.6 172 2355 2358 2360 2363 2365 2368 2370 2373 2375 2378 2.6 173 2380 2383 2385 2388 2390 2393 2395 2398 2400 2403 2.6 174 2405 2408 2410 2413 2415 2418 2420 2423 2425 2428 2.6 175 2430 2433 2435 2438 2440 2443 2445 2448 2450 2453 2.6 176 2455 2458 2460 2463 2465 2467 2470 2472 2475 2477 2.4 177 2480 2482 2485 2487 2490 2492 2494 2497 2499 2502 2.4 178 2504 2507 2509 2512 2514 2516 2519 2521 2524 2526 2.4 179 2529 2531 2533 2536 2538 2541 2543 2545 2548 2550 2.3 180 2553 2555 2558 2560 2562 2565 2567 2570 2572 2574 2.3 181 2577 2579 2582 2584 2586 2589 2591 2594 2596 2598 2.3 182 2601 2603 2605 2608 2610 2613 2615 2617 2620 2622 2.3 183 2625 2627 2629 2632 2634 2636 2639 2641 2643 2646 2.3 184 2648 2651 2653 2655 2658 2660 2662 2665 2667 2669 2.3 185 2672 2674 2676 2679 2681 2683 2686 2688 2690 2693 2.3 186 2695 2697 2700 2702 2704 2707 2709 2711 2714 2716 2.3 187 2718 2721 2723 2725 2728 2730 2732 2735 2737 2739 2.3 188 2742 2744 2746 2749 2751 2753 2755 2758 2760 2762 2.2 189 2765 2767 2769 2772 2774 2776 2778 2781 2783 2785 2.2 190 2788 2790 2792 2794 2797 2799 2801 2804 2806 2808 2.2 191 2810 2813 2815 2817 2819 2822 2824 2826 2828 2831 2.2 192 2833 2835 2838 2840 2842 2844 2847 2849 2851 2853 2.2 193 2856 2858 2860 2862 2865 2867 2869 2871 2874 2876 2.2 194 2878 2880 2882 2885 2887 2889 2891 2894 2896 2898 2.2 195 2900 2903 2905 2907 2909 2911 2914 2916 2918 2920 2.2 196 2923 2925 2927 2929 2931 2934 2936 2938 2940 2942 2.1 197 2945 2947 2949 2951 2953 2956 2958 2960 2962 2964 2.1 198 2967 2969 2971 2973 2975 2978 2980 2982 2984 2986 2.1 199 2989 2991 2993 2995 2997 2999 3002 3004 3006 3Q08 2.1 /.To i^IL^^^Perwi. ' VUctttoa is mad | g| o j 1 SEP 16 10m-4,'23 308540 UNIVERSITY OF CALIFORNIA LIBRARY