STRUCTURAL DETAILS OF HIP AND VALLEY RAFTERS BY CARLTOX THOMAS BISHOP, C.E. ASSISTANT PROFESSOR OF CIVIL KN' . I \\:i KINi ;. sil II'KIKI.I) SCIENTIFIC SCHOOL OF YAL.K UNIVERSITY M l:l.\ m:\Kis\I\N FOR THE AMERICAN BRUME COMPANY AND CHIEF DRAFTSMAN FOR THE HAY FOUNDRY AND IRON WORKS FIRST EDITION FIRST THOUSAND M.W YORK JOHN WILEY AND SONS LONDON: CHAPMAN AND HALL, LIMITED 1912 COPYRIGHT, 1912, BY CARLTON T. BISHOP Stanhope iprcss F. H.GILSON COMPANY BOSTON, U.S.A. PREFACE Tins hook is written to meet the requirements of structural draftsmen when solving problems in hip an of great assistance to the draftsmen when dealing with such problems. Complete directions arc given for making the shop drawings for the steel work of intersecting roofs and similar structures. The notes for the various cases are arranged for convenient reference, and are illus- trated by general drawings and typical problems. The necessary numeri- cal values may be obtained either algebraically or graphically, and both methods are fully explained. Much of the calculation is simplified by means of tables, which give values for use in the most common cases. During his experience as structural draftsman the author either made or checked many drawings of hip and valley rafters and drawings of a similar nature, using numerous systems of calculation, both published and unpublished. He found, however, that no system fulfilled his needs without alteration or addition, and that while the majority was effective NEW HAVEN, CONN., September, 1912. in showing the derivations of the more difficult angles, few explained the application of these angles, or referred to the additional values required. These values may be easily found by an experienced man, but the author has observed that the average draftsman will waste much time in procuring them, since he lacks a clear understanding of the requirements. It was felt by the author and his fellow draftsmen that there existed a demand for a complete treatise of this nature, and accordingly this book has been developed. It is hoped that it will prove of service to the experienced draftsman, as well as to the novice who attempts to solve a problem of this character. The author desires to express his appreciation of the encouragement and helpful criticisms given by Prof. J. C. Tracy, of Yale University. Grate- ful acknowledgment is also given to Mr. E. R. St. John, of Pittsburgh, for checking the more important drawings and formulas, and to Messrs. L. D. Rights, of New York, and W. H. McKinley, of Pitteburg, for their prepublication criticisms. C. T. B. 259667 iii TABLE OF CONTENTS PREFACE CHAPTER I GENERAL OUTLINE Object 1 Definitions 1 Arrangement 2 Working Lines 2 Column Connections 2 Computation 2 Consistent Accuracy 2 Notation , 3 CHAPTER II FLANGE CONNECTION PAOE PAGE . iii Explanatory Notes and Suggestions 19 Illustrative Problem; Case I (o); Hip Rafter 20 Illustrative Problem; Case I (o); Fig. 9 21 Valley Rafter 22 Fig. 10 23 Calculation 24 I (a); Hip Rafter; Buildings at Right Angle; Unequal Pitches. I (a) I (b); Valley Rafter; Buildings at Right Angle; Unequal Pitches. 1(6) II (a); Hip Rafter; Buildings at Oblique Angle; Unequal Pitches. II (a) Formulas; Case Fig. 2; Case Formulas; Case Fig. 3; Case Formulas; Case Fig. 4; Case Formulas; Case II (b) ; Valley Rafter; Buildings at ObliqueAngle; Unequal Pitches. Fig. 5; Case II (b) Formulas; Case III (o); Hip Rafter; Buildings at Right Angle; Equal Pitches. . Fig. 6; Case III (a) 13 Formulas; Case III (6) ; Valley Rafter; Buildings at Right Angle; Equal Pitches. 14 Fig. 7; Case 111(6) 15 Blank Form for Calculation; Case I 16 Blank Form for Calculation; Case II 17 Blank Form for Calculation; Case III 18 4 5 6 7 8 9 10 11 12 Illustrative Problem; Case I (6); Illustrative Problem; Case I (6); Illustrative Problem; Case I (a); Illustrative Problem; Case I (6); Calculation 25 CHAPTER III WEB CONNECTION Formulas; Case IV (a) ; Hip Rafter; Buildings at Right Angle; Unequal Pitches. 26 Fig. 11; Case IV (a) 27 Formulas; Case IV (6) ; Valley Rafter; Buildings at Right Angle; Unequal Pitches. 28 Fig. 12; Case IV (6) 29 Formulas; Case V (o); Hip Rafter; Buildings at Oblique Angle; Unequal Pitches. 30 Fig. 13; Case V (a) Formulas; Case V (6) ; Valley Rafter; Buildings at Oblique Angle; Unequal Pitches. Fig. 14; Case V (6) Formulas; Case VI (a); Hip Rafter; Buildings at Right Angle; Equal Pitches. 31 32 33 34 Fig. 15; Case VI (a) 35 Formulas; Case VI (6); Valley Rafter; Buildings at Right Angle; Equal Pitches . 36 Fig. 16; Case VI (b) 37 Blank Form for Calculation; Case IV 38 Blank Form for Calculation; Case V 39 Blank Form for Calculation ; Case VI 40 Explanatory Notes and Suggestions 41 Illustrative Problem ; Case IV (a) ; Illustrative Problem; Case IV (a); Illustrative Problem ; Case IV (6) ; Illustrative Problem; Case IV (b); Illustrative Problem; Case IV (o); Illustrative Problem; Case IV (b); Hip Rafter 42 Fig. 17 43 Valley Rafter 44 Fig. 18 45 Calculation 46 Calculation 47 IV TABLE OF CONTENTS ( IIAITF.R IV NOTES ON OTHER CASES PAOI ('Iniinel Purlins wli> tin- Otlirr \\'ny 48 1-l.f.iiii Purlins 48 7,-l.ar Purlins 48 Purlins 48 Purlins 48 Purlin-; Hip Rafter 49 Dallas; Tee Purlins; Valley Rafter 60 P.I; I'..- Purlins Hip Rafter 51 JO; Tee Purlins; Valley Rafter 51 < -II \ITKR V DERIVATION OF FORMULAS Formula (7) 52 Formula (9) " 52 Formula (13) 53 52 52 nula (18) 54 Formula (19) 56 Formula (21) 57 Formula (30) 58 Formula (31) 58 Formula (:>4) 59 Formula (55) 60 Formula i :.7: 61 Formula (59) 61 1 orniula (84) .61 CHAPTF.R VI GRAPHIC METHOD OF DETERMINING ANGLES Fig. 30; Fi K . :tl; Fig. 32; Fig. 33; Fig. 34; Fi K . 3. r , ; Fig. 30; Fig. 37; Fig. 38; Fig. 39; Fig. 40; Fig. 41; Fig. 42; Fig. 43; Flange Connection; Angles W and W" 63 Flange Connection; Angles X' and X" 63 Flange Connection ; Angles Y' and Y" 63 Flange Connection; Combined Layout; Cose I 64 Flange Connection; Combined Layout; Cose II 64 \\i-li('oniieetion; Angles W and W" 65 Wcl) Connection; Angles X' and A'" 65 \\vii Connection; Angles Y' and Y" 65 Web Connection; Angles Z' and Z" 66 \\diConnection; Combined Layout; Cose IV 66 Web Connection; Combined Layout; Case V 66 Tee Purlins; Angles X' and X" 67 Tee Purlins; Combined Layout; Case I 67 Tee Purlins; Combined Layout; Case II j. 67 TAHI.KS VALUES AND LOGARITHMS FOR COMMON CASES Pitches J and 30" 68 Pitches land 1 68 Pitches j and J 68 Pitches 1 and I 60 Pitches 30 and J 69 Pitches 30 and } 69 Pitches 30 and J 70 Pitches J and J 70 Pitches J and J 70 Pitches J and i 71 Equal pitches } 71 Equal pitches 30 71 Equal pitches \ 72 Equal pitches \ 72 Equal pitches } 72 HIP AND VALLEY RAFTERS CHAPTER I GENERAL OUTLINE TIIK structural draftsman will occasionally, if not frequently, encounter a building which contains one or more hip or valley rafters. Work of this character involves a certain amount of calculation peculiar to itself, and unle one comes into contact with a problem of this kind repeatedly, he will probably become so much out of practice that he will be forced to spend considerable time in review before he can ascertain the desired results. Oliji rl. It is the purjwse of this book to present the subject of Hip and Valley construction so completely that anyone with a reasonable knowledge of structural details and of trigonometry can make working drawings which shall give all necessary information to the shop without useless refinements. The work is made definite by being outlined in full, and forms of tabulation are given with the hope that valuable time may be saved. The most common types of hip and valley construction are considered in detail, the notes being arranged for convenient reference. These types involve channel purlins which connect either to the flange or to the web of the hi]> or valley rafter. The rafter web is assumed to be vertical and the flange normal to the web, as, for example, a rolled I-beam, or plate and angles built in the form of an I or a T. Suggestions are also given to aid the draftsman in modifying these types to fulfill his requirements when he meets other conditions. Definitions. In building construction, two roofs often intersect to form either a hip or a valley. If the two roofs are so arranged that the drainage is away from the line of intersection a "Hip " results, but if the drainage is toward the intersection a "Valley" is formed. (See Fig. 1.) The rafter which supports the purlin- of both roofs at their intersection is, accordingly, termed "Hip Rafter" or "Valley Rafter." ELEVATIQS Fia. 1. *: HIP AND VALLEY RAFTERS Arrangement. The notes are classified and arranged to simplify the reference to any desired section. In Chapter II are discussed the connec- tions of channel purlins to the flange of the hip or valley rafter, giving the ordinary, the general, and the special cases of each. A general drawing, showing all values, is given for each case and the corresponding formulas are tabulated on the opposite page. A blank form is given for each case to serve as a guide in calculation and tabulation and to indicate the loga- rithms to be used in later computation, thus reducing the amount of labor involved. A few explanatory notes follow, with suggestions for the use of the values obtained, and, finally, two illustrative problems are given in detail, including both drawings and calculations. In a similar manner, the connections of channel purlins to the web of the hip or valley rafter are considered in Chapter III, followed by notes upon other styles of connection in Chapter IV. The principal formulas of Chapters II, III and IV are derived in Chapter V, and Chapter VI shows the graphic method of determining the principal angles of any of the above cases. At the end of the book, tables are given to assist in the solution of the problems which are most likely to occur in practice. By means of this arrangement, a man may obtain all the desired in- formation from the proper general drawing and the corresponding formulas without the necessity of reading either the explanatory notes or the deriva- tions, although it is recommended that he read them at least once when he uses the system for the first time. Working Lines. A few working lines must be chosen at the outset, and all details must be referred to these lines. In laying out work it is often convenient to work to the intersection of the roof planes, or to the inter- section of the planes through the tops of the mam rafters. It is well, however, to change to the center line of the top flange of the hip or valley rafter before commencing the details. This may be readily accomplished by using the vertical distance v. It is customary to work either to the center lines or to the faces of the columns or main rafters, but this depends upon the conditions of the particular problem and does not affect either the principle or the calculation, after certain parts are chosen at the beginning. In general, the slope of each roof will be given, and, with one side as- sumed (b', b" or e), the remaining values may be determined. But fre- quently the rafter will connect to columns already spaced, which means that both b' and b" are fixed, and that the slope of only one roof is given, in which case the slope of the other roof must be calculated to correspond, by transposing formula (2) or (5). Column Connections. The connections of the hip or valley rafters to the columns or main rafters cannot be treated fully in this volume, partly because they depend upon so many conditions that no standard detail can be given, and partly because any draftsman who undertakes to detail a hip or valley rafter should experience no difficulty in designing bent plate con- nections at the ends of the rafter since they generally involve only the angles which he has already determined. However, several different types are represented in the illustrative problems in Chapter II and it is hoped that these will aid the novice in detailing connections for his particular rafter. Computation. For simplicity, all formulas have been arranged so that they involve the use of only three functions of any angle, namely the sine, the cosine and the tangent. This reduces to a minimum the use of the loga- rithmic tables and the tabulation of logarithms, as well as the possibility of error in referring to either. Furthermore, the average draftsman is doubtless slightly more expeditious in the use of these three functions than in the use of the others. It is assumed that a book of tables will be used which gives the loga- rithms of dimensions which are expressed in feet and inches and fractions of inches. Some books* give, in addition, the logarithms of all the common functions of angles whose tangents are expressed by slopes in inches and fractions per foot. The use of such tables not only simplifies the loga- rithmic work, but often precludes the necessity of finding the tangent of an angle in order to determine the slope. It is considered superfluous to differentiate between logarithms and cologarithms, since the latter may easily be written directly from the former by subtracting each figure from nine, except the right-hand one, which is taken from ten. Consistent Accuracy. Although the degree of precision must be deter- mined to satisfy the individual requirements, yet, in most work of this nature, all linear dimensions should be expressed to the nearest sixteenth of an inch. To accomplish this purpose, five place logarithms should be used, those for the functions of angles being interpolated for seconds. * Smoley's Tables of Logarithms and Squares, McGraw-Hill Book Co., New York, are recommended. i II Al'TER I. CENERAL OUTLINE 3 Notation. All values which deal with the main rafter i>r purlins of the steeper roof bear tliis mark ('). Those dealing with the parts of the other roof bear this murk (") Values which apply to parts of both roofs, or to the hip or valley rafter, are represented by the simple letter without a diMinguishing mark. Angles are represented by capital letters, and their slopes, or the tangents of the angles in inches for a base of one foot, by the corres]H>nding lower case letters. Distances are expressed by lower c:itance measured along the rafter. (Seepage41.) it' = the horizontal distance along the line of bend corresponding to u'- L = the angle between the axes of the two roofs. m = the horizontal distance between the working points of the hip or valley rafter. n = the slope distance between the working points of the hip or valley rafter. = the horizontal distance along the purlin of tfie steeper roof, from the intersection of the purlin plane and the center line of the hip or valley rafter to the working point at the upper end of the hip or valley rafter. t = t' u' = u' = w' = A" = x' = y' = z' = the distance along the steeper main rafter corresponding to v. the distance along the steeper main rafter from the back of the purlin to the upper end of the main rafter, the slope distance along the hip or valley rafter corresponding top'. (Chapter III) the thickness of the web of the hip or valley rafter. (Chapter IV) the thickness of the plate connecting the Tec purlin of the steeper roof to the hip or valley rafter. (Chapter II) the distance normal to the steeper roof corresjxmd- ing to v . (Chapter III) the normal distance from the top of the steeper main rafter to the top of the purlin, the vertical distance which the top of the hip or valley rafter is lowered below the planes through the tops of the main rafters. (See page 19.) the tangent of the angle between the line of bend and the center line of the top flange of the hip or valley rafter. (Steeper roof.) the angle of the bend in the connection plate. (Steeper roof.) the tangent of the angle A"', the tangent of the angle between the line of bend and the flange of the purlin of the steeper roof, the tangent of the angle }>etwecn the purlin of the steeper roof and the hip or valley rafter in the plane of the roof. CHAPTER II FLANGE CONNECTION FORMULAS FOR HIP RAFTER CONNECTIONS CASE I (a). (Ordinary Case.) Channel purlins connecting to flange of hip rafter. Axes of roofs intersecting at right angle. Unequal pitches. Given: a' = the slope of the steeper roof (a' always greater than a"). a" = the slope of the other roof. b' = the horizontal distance between the working points of the steeper roof. (Either b" or e might be given instead.) / = the width of the flange of the hip rafter. g' and g"= the purlin gages, r' and r" = the distances from the working points to the backs of the purlins, measured along the tops of the main rafters. (1) tan A' = a'. (2) e = b' tan A'. (3) rl' b' cos A' (4) tanA" = a". (5) h" e v ~~ tan A"' (6) d" b" cos A"' (7) tanC = tan A" tan A' ' (8) c = tanC. (9) tan# = tan A' sin C. (10) h = tan#. (11) b' sinC m \1*J 11 j-. cos// *(13) v = ^tan4'cosC+ 4 (14) w' = v cos A'. (15) q'= v sin A'. (16) , (r'-q')cosA' tanC (17) ,_ P' cos C cos H t(18) . tan C cos H "W cos 2 A' 1(19) s in X' = smA' cos 2 C cos H . (20) x'= tanZ'. -fa* may be added, if necessary, to eliminate sixteenths from the value of v. (21) (22) (23) (24) (25) t(26) y' = w'sinX'. u" v cos A". q"= vsmA". p" = (r"-q")cosA"ia,iLC. S "= P " sin C cos H cos 2 A "tan C cosH (27) sin X" = sin A " sin 2 C cos H. (28) *" = (29) y" = t If the value of w' or w" is greater than I'D" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side w 7 OT to 7 ' These values are obtained directly from the cologarithms. Care should be taken, however, that the original values of w' and w" are used in formulas (21) and (29). 4 rilAl'TKK II. FIANCE CONNMTION << NOTES 1 =StaTidari rare top flange of hip rafter n 1 JII Make ufnrU-iiUj lantr U> all. .w rlr.-t or bolU to be ptaeed kfter plate U bent . IT A V=All-.w infflrleot ectee dUtanre In porlln when cutiquare at center of hip rafter. VI A VH Determine dUunrr e. la e". bolm by larout Hake aa larce M practical. Wi.llh of plate mar be ma than that of flancr If deatred. Kwp wkltb In even Incbei U p CASE I (a) HIP RAFTER See opposite page. Fio. 2. 6 HIP AND VALLEY RAFTERS FORMULAS FOR VALLEY RAFTER CONNECTIONS CASE I (6). (Ordinary Case.) Channel purlins connecting to flange of valley rafter. Axes of roofs intersecting at right angle. Unequal pitches. Given: a' = the slope of the steeper roof (a' always greater than a"), a" = the slope of the other roof. b' = the horizontal distance between the working points of the steeper roof. (Either b" or e might be given instead.) / = the width of the flange of the valley rafter. g' and g" '= the purlin gages. / and r" = the distances from the working points to the backs of the purlins, measured along the tops of the main rafters. (1) tan A' = a'. (2) e = V tan A (3) d' = b' cos A' (4) tanA" = a". (5) b" = e tan A" (6) d" = b" cos A" n^ n r - tan A" .(8) (9) (10) (11) tan A' c tan C. tan H = tan A' sin C. /i = tan H. b' m = sin C (12) *(13) (14) (15) (16) (17) t(18) (19) (20) n = m cosH f 1' v = *: tan A' cos C + T w'= DCOS A'. g'= i) sin A'. ,_ (/ q'} cos A' tanC s = P' cos C cos H , tan C cos H w = - ^r, cos 2 A sin X' = sin A' cos 2 C cos H. (21) (22) (23) (24) (25) t(26) (27) (28) (29) y' = w' sin X'. u"=vcosA". q"=vsmA". p"= (r"- 3 ")cosA"tan<7. S "= P " sin C cos H cos 2 A" tan C w" = - ^ cosH sin X" = sin A" sin 2 C cos H. x"=tanX". y" = sinX" w" * -fa" may be added, if necessary, to eliminate sixteenths from the value of v. This value of ;; gives sufficient clearance to permit the extension of the purlin beyond the valley rafter. Ordinarily the purlin may be cut at the center line of valley rafter, 1" in which case it is sufficient to assume that value of v which will give the desired clearance at the center line, as, for example, v = ^ . Moreover, if the purlin is cut back far enough to allow ample clearance due to the slope of the valley rafter, then v becomes zero, and formulas (13) to (16) and (22) to (24) are reduced to the following forms: (13a) v = 0. (14a) (15a) g' = (16a) p' = r'cosA' tanC (22a) u" = 0. (23a) q" = 0. (24a) p" = r" cos A" tan C. f If the value of w' or w" is greater than 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side Care should be taken, however, that the original values of w' and w" are used in formulas (21) and (29). -, or 77 . These values are obtained directly from the cologarithms. CHAP'ICl: l[ I I AM1E CONNECTION NOTES I = Standard top Bailee of Taller rafter n 4 III = Make raffle Irntlr lante to allow rtri-la or bolu to be placed after plate l> bent IV * V Allow orlli-ient e sin L when L > 00 tanC P' tan (L C) cos H smA"cos*(L-C)cosH. tanX". smX" b cos L tanC. cos C cos H tan C cos H * If any of these values exceeds 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side the reciprocal of the value found. This reciprocal is obtained directly from the cologarithm. Care should be taken, however, that the original values are used in all further calculation, t iV may be added, if necessary, to eliminate sixteenths from the value of v. CHAPTKH II. FLANGK CONNECTION . NOTES; . ,, 6'sin.L (31) tanC = ,-77 n -- rwhenL > wr. b" - b' cos L * (8) C = tan C " (9) tanH = tan A' sin C. *(18) _ _ cos C cos H tanCcosH - cos 2 A' (35) sinX"= sin A" cos 2 (L - C)cosH. (28) x" = tan X". (29) v" w * If any of these values exceeds 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side the reciprocal of the value ound. This reciprocal is obtained directly from the cologarithm. Care should be taken, however, that the original values are used in all further calculation. t -fg' may be added, if necessary, to eliminate sixteenths from the value of v. This value of v gives sufficient clearance to permit the extension of the purlin beyond the valley rafter. Ordinarily the purlin may be cut at the center line of valley rafter, in which case it is sufficient to assume that value of v which will give the desired clearance at the center line, as, for example, v = j . Moreover, if the purlin is cut back far enough to allow ample clearance due to the slope of the valley rafter, then v becomes zero, and formulas (13) to (16) and (22), (23) and (32) are reduced to the following forms: (13a) t> = 0. (14a) u' = 0. (15a) q' = 0. (16a) (22a) u" = 0. (23a) q" = 0. (32a) - C) ' CHAPTI.l; II. FLANGE CONNECTION 11 NOTES I- Standard nee tap fiance of rallcy niter II 4 III- Make lufflclently Urge to allow rtreu or boltt to be placed after plate U bent IV A V- Allow ufflclent cxlce dtetuce la purUn Sea aim following note VI A VII- Arrance ptatM wlthedce* normal to Uw Uncofbeod. Keep width In ercn Incbe* Determine width from (pacing of bole* JOT rafter connection. Place bole* for 'purlin con neettop approi Imitely oppoalU CASK II (6) V ALLEY RAITKR See opposite page. Fio. 5. 12 HIP AND VALLEY RAFTERS FORMULAS FOR HIP RAFTER CONNECTIONS CASE III (a). (Special Case.) Channel purlins connecting to flange of hip rafter. Axes of roofs intersecting at right angle. Equal pitches. Given: a = the slope of each roof. b = the horizontal distance between working points, (e might be given instead.) / = the width of the flange of the hip rafter. g' and g" = the purlin gages. r' and r" = the distances from the working points to the backs of the purlins, measured along the tops of the main rafters. C = 45. c = 12. 9.84949 = log sin 45 = log cos 45. (36) tan A = a. e = b tan A . b (37) (38) cos A (39) tan H = tan A sin 45. h = tan H. b (10) (40) (12) m = n = sin 45 m cosH *(41) (42) (43) (44) (45) (46) / 1" v = tan A cos 45 + u = q = v sin A . p' = (r' q) cos A. p" = (r" -qjcosA. s = (47) (48) w = cos 45 cos H cos 2 A cosH (49) sin X = sin A cos 2 45 cos H. (50) x = tanX. (51) cos 45 cos H 'j* may be added, if necessary, to eliminate sixteenths from the value of a. y = w CAWS III (a) Hir RAFTER See opposite page. I V * V > Allow lufflclent edge dliuncc In purlin boo cut Kjaan) it center of hip rafter TI * > II- Dvtcnmne dUUncK-.toc. bole* by Ujom tUkru Uicr u practical wi.ith uf pi(e majr be made rreater Uuu iiuitrf'i ilaiiire If dnlrad Keep wiUUi In I-TCU Incnei If poolMe li.. .;. 14 HIP AND VALLEY RAFTERS Given: a = b = f - g' and g" - r' andr" = C -- c 9.84949 (36) tan A (37) e FORMULAS FOR VALLEY RAFTER CONNECTIONS CASE III (b). (Special Case.) Channel purlins connecting to flange of valley rafter. Axes of roofs intersecting at right angle. Equal pitches. the slope of each roof. the horizontal distance between working points, (e might be given instead.) the width of the flange of the valley rafter. the purlin gages. the distance from the working points to the backs of the purlins, measured along the tops of the main rafters. = 45. = 12. = log sin 45 = log cos 45. (38) d = (39) tan H (10) h (40) (12) m n = a. b tan A . b cos A tan A sin 45. tanH. b sin 45' m cosH *(41) (42) (43) (44) (45) (46) / 1" v = -= tan A cos 45 + T u = vcosA. q = v sin A . p' = (r' q) cos A. (47) (48) P' w = cos 45 cos H cos 2 A P s' = (r" - q) cos A. P' cos 45 cos H cosH (49) sin X =sinA cos 2 45 cos H. (50) x = tanX. smX (51) y = w * ^j" may be added, if necessary, to eliminate sixteenths from the value of v. This value of v gives sufficient clearance to permit the extension of the purlin beyond the valley rafter. Ordinarily, the purlin may be cut at the center line of valley 1" rafter, in which case it is sufficient to assume that value of v which will give the desired clearance at the center line, as, for example, v = -r Moreover, if the purlin is cut back far enough to allow ample clearance due to the slope of the valley rafter, then v becomes zero, and formulas (41) to (45) are reduced to the following forms: (41a) v = 0. (42a) 0. (43a) g = (44a) = r'cosA. (45a) r"cosA. ( HAITKK II. I I, \.NG1 . ONN1 < T|M\ 15 T II NOTES = Standard mure top flance of rall.-r ratter = Make ufflrl,-nUr larir,- to allow rlreU or bota to be placed afar plateta bent. TV A V= Allow tnmclrnt edffe distance In purlin. Sec alMi (oUowlDK note. Tl 4 VII = Arraoire plate* wim eOtc* normal to the line of bend. Keep width In eren Inrnea. Determine wldtn frutn apaclnc uf note* for rafter connection. Place bole* for purllnconncrtlon tbennolc*. CASK HI (6) VALLEY RAITKB >. , oppQ .-. ,. ltr , Fio. 7. 16 HIP AND VALLEY RAFTERS CALCULATION FOR HIP AND VALLEY RAFTER CONNECTIONS FLANGE CONNECTION The following outline is given to serve as a guide for the tabulation of the required values, and to indicate the logarithms which will be needed for further computation. All the necessary functions of an angle may thus be determined at the same time. CASE I. (Ordinary Case.) Roofs at right angle; unequal pitches. Given: a', a", V (or b" or e), f, g', g", r', and r". Num b' - Angle. Slope. Logarithm. Logarithm. Sine. Cosine. Tangent. A' o' = A" a" = C c = H h = ^x^ X' x' = ^x^ X" x" = X ber. Logarithm. Number, v' = . e = s' = d' - w' - &"= y' = d"- u"= m . q"= 71 v"= . v ~ s" = u' = . w"- . 9' = CHAITKK II. FLANGE CONNECTION 17 CALCULATION FOR HIP AND VALLEY RAFTER CONNECTIONS FLANGE CONNECTION The following outline i~ mv.-n t<> >erve as a guide for the tabulation of the required values, and to indicate the logarithms which will be needed for further computation. All the necessary functions of an angle may thus be determined at the same time. CASE II. (General Case.) Roofs at oblique angle; unequal pitches. Oioen: a', a", b' (or 6" or e), f, g', g", r f , r", and L. A' L- C- L-C- 11 X" o'- o"- c ft- \nili, r. V = d' = b" = d" = m = n = w' = g' = Logarithm. Ix^arithm. Number. ,r - P" = "- "- "- Logarithm. 18 HIP AND VALLEY RAFTERS CALCULATION FOR HIP AND VALLEY RAFTER CONNECTIONS FLANGE CONNECTION The following outline is given to serve as a guide for the tabulation of the required values, and to indicate the logarithms which will be needed for further computation. All the necessary functions of an angle may thus be determined at the same time. CASE III. (Special Case.) Roofs at right angle; equal pitches. Given: a,b (or e), /, g', g", r', r", C = 45. Nun b - Angle. Slope. Logarithm. Sine. Cosine. Tangent. A a C c = 12 9.84949 9.94949 ^x^ H h = X X x = ^X^ iber. Logarithm. Number. Logarithm, n = ... G p' = d = p"- ffl " s' = n s"= V ' ID ~~ u = . 11 = . ( II A IT I I! 11 FLANGE CONNECTION 19 EXPLANATORY NOTES AND SUGGESTIONS FLANGE CONNECTION It should ! initial that in every case the hip rafter must be lowered a certain distance r to allow the purlin to clear the flange of the rafter. It i~ not necessary to lower the valley rafter provided the purlins do not extend lieyi.inl the renter line. (See note, pp. 6, 10, and 14.) Occasionally it may l>e preferable to prolong the purlin past the center of the rnllry rafter if a better connection can thus \n- obtained. Sometimes purlins are supported by a diagonal rafter when neither hip nor valley is involved, that i-. when no other roof intersects. In such an event, the rafter i- lowered a distance v and the details for valley rafters used in prefer- ence to tho-e for hip rafters, for the sake of appearance and strength. It is important that no purlin be extended past the renter line of the hip rafter, for if prolonged more than a certain distance it will pierce the roof of the other slope, a point overlooked by some authors and many dr;r men. This distance is usually very small and is hardly worth considering, connections to prevent the purlins interfering with each other, and to pro- vide a l>etter connection by riveting the bent plate to both sides of the rafter flange. If necessary, however, purlins may be connected at the same point of a hip rafter by means of independent plates, each of which is fastened by two rivets through one side of the rafter flange, provided the bottom flanges and the bottom part of the webs of the purlins are cut to clear each other. (See Fig. 9, p. 21.) But it is not at all practical to attempt to connect two purlins to a valley rafter at the same point because of extensive interference. Connection plates are kept a uniform width, partly to simplify shearing, but chiefly to avoid the reentrant cut which is expensive, impractical, and entirely unnecessary, although many dctailers still cling to it. It is suggested that plates be cut parallel to the center lines of holes, not only for the sake of appearance, but also to enable the shopmen to cut i\ \ \ \ \ \ I .,.. V in view of the fact that it is difficult to determine. It depends upon the pitches of the roofs, upon the depth of the purlin and also upon the width of the purlin flange. Ordinarily, it is safer to cut the purlin at the center line of the hip rafter and to make the holes in the connection plate corre- spond. This accounts for the apparently distorted form of the plate. It is Ix'tter not to connect purlins from both roofs at or near the same point of the hi]) or valley rafter. They should have entirely separate a complete plate with one stroke of the shears with little or no waste, except at the end plates. (See Fig. 8.) It should be noticed that no development is required to determine the size of the plate for the connection to the hip rafter. The length may be found by adding together the distances from the line of bend to each end of the plate, measured parallel to the sides. Yet there seems to be less liability to mistake in the shop if the development is shown. 20 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE I (a). HIP RAFTEB Flange connection; buildings at right angle; unequal pitches. Given a building whose roof slopes up from the end as well as from the sides, involving two hips as shown in the Key Plan, Fig. 9. One end of each hip rafter will connect to a plate and angle column, and the other to the peak plate of one of the main trusses. The details of these connections determine the working points, and we have : V = 11' 1111" = 12' 0" - T y. b" = 14' 10|" = 16' 0" - (6|" + 7"). If the main roof is j pitch, then a" = 6. Using a 12" I 31 1# rafter and 8"u 11|# purlins we have/ = 5, and g' '= g" = 3. The required values are given below. (tor necessary computation see p. 24.) Number. 9"= H p' = 3' 2V p' = 8' IIH" Logarithm. 1.07862 e = 7' 5V 0.87234 d' = 14' If" b"= 14' 10!" 1.17337 d"= 16' 8" m = 19' H" 1.28164 n = 20' 6 T V v = H 9.09691 0.50162 0.95343 Logarithm. Anglo. Slope. Sine. Cosine. Tangent. A' <*' = 7A 9.72272 9.92900 9.79372 A" a" = 6 9.65051 9.95154 9.69897 C c=9f 9.79698 9.89173 9.90525 H A=4H 9.96930 9.59070 X' z'-3! 9 47548 9.49580 X" z"-2 9 21377 9.21966 Number. s' = 4' 4 T V s 1 = 12' 4 T y - i"= H p"-7'2f" s"= 5'7|" s"= 12'4 T y Logarithm. w' = 0.01655 0.51401 0.85860 9.83903 ( HAITI:!! II. FIANGE CONNECTION 1'. S'u ll<*' **' . . , , ' . "i 1 A **' , . ,,; < T't'' ft ILLCOTRATIVK PROBLEM CAM I (a) HIP RAFTER >.. ..;;...-/. j..,^. Fio. 9. 22 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE I (6). VALLEY RAFTER Flange connection; buildings at right angle; unequal pitches. Two intersecting buildings have lean-to's whose roofs form a valley, as shown in the Key Plan, Fig. 10. The upper end of the valley rafter connects to the face of the main column, and the lower end to the lean-to column outside of the connection of the main lean-to rafter. The pitches of the roofs are \ and \ and the distance from the center line of the lean-to columns supporting the steeper roof to the center line of the main columns is 10' 0". If the columns are 85" and 12j" respectively, back to back of angles, we have b' = 9' If" = 10' 0" - (4|" + |" + 6|"). Also a' = 6, a" = 4||, and g' = g" = 3, r' = r" = 5' 0". The required values are given below. (For necessary computation see p. 25.) Logarithm. Angle. Slope. Sine. Cosine. Tangent. A' a' = 6 9.65051 9.95154 9.69897 A" a"=4H 9.56983 9.96777 9.60206 C c=9f 9.79567 9.89258 9.90309 H h = 3l 9.97979 9.49464 X' z'-3i 9 41546 9 43070 X" z"-lti 9 14096 9 14515 Number. Logarithm. V = 9'1|" 0.95974 e =4'6H" 0.65871 d' = 10' 2 T V' b"= 11' 4J" 1.05665 d" = 12' 3 J" 771 = 14' 7^" 1.16407 n = 15' 3 T V v = u' = Number. P' = 5' 7 T y s r = r e" u"= s"= 6'2f" w"= 8H y"= Logarithm. 0.74742 9.97980 0.56983 9.85884 CHAPTER II. FLAN< , I < T|. >N 23 - LEAN -TO PBOBLJUI I (6) VAI.LET RAFTER Bee opposite page. . 10. 24 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE I (a). Computation of values given 6" = tanA" = e - b' -- tan A' b' cos A' d' 1 . 17337 9.69897 = 0.87234 = 1.07862 = 9.79372 = 1.07862 = 9.92900 = 1 . 14962 b' sin C m cos H 1.07862 9.79698 1.28164 9.96930 n = 1.31234 f/2 tan A' cos C b"-- cos A" = d"= 1.22183 1.17337 9.95154 9.31876 9.79372 : 9. 89173 = 9.00421 tan A" =9. 69897 tan A' = 9.79372 tanC = 9.90525 v cos A' 9.09691 9.92900 u' = 9.02591 tan A' sin C tanH = 9.79372 = 9.79698 = 9.59070 v sin A' 9.09691 9.72272 q' = 8.81963 :s given r' -q' cos A' tanC P' cos C cosH ' s' tan (7 cosH cos 2 A' w' \/w' sin A' cos 2 C cos H sinX' w' y' v cos A" on p = 0. = 9. = 0. .20. 47787 0.92968 92900 9 . 92900 09475 0.09475 v = sinA" = q"= r" - q" = cosA" = tan C 9 9. .09691 65051 1 9 9 00181 .95154 .90525 8. 0. 9 9 74742 65722 ' .95154 .90525 = 0. = 0. = 0. 50162 0.95343 10827 0.10827 03070 0.03070 64059 1.09240 90525 96930 14200 P" = sin C = cosH = s" = cos 2 A" = tanC = 'cosH = w" = oi n A" . = 9. = 9 = o' .51401 .20302 03070 .85860 .20302 .03070 9 9 .74773 90308 90525 .03070 1 .09232 = 0. = 9. = 9. = 9. = 9. 01655 98345 72272 78346 96930 9 9 9 9 83903 ,59396 .96930 = 9. = 0. 47548 01655 sin 2 C = cosH = smX" = w" = y" = 49203 09691 95154 . = 9 = 9! 9 9 ,21377 ,83903 9 .37474 Additional calculation required for the above problem is as follows: At peak, to determine r' and r", etc. T-V =8.19382 cos A' = 9.92900 M"=9.04845 7 =9.76592 tan A" =9. 69897 J =8.26482 / = 3' o|" = 3' : Bent plate at peak: -1" 3| =9.46489 we b = A = 8*19382 tan C = 9.90525 1 = 8.28857 //2 = 9 . 31876 tan C = . 09475 cos H = . 03070 | web = T 3 j? sin C = 8.19382 = 9. 79698 iweb = T \ tanC = 8.19382 = 9.90525 = 8.09907 tanC =9.90525 cos A' = 9.92900 111 =9.97625 7 = 9.76592 cos A" =9. 95154 7 it =9.81438 r "=4'7tV" = 5'3"-71f". Bent plate at column: iweb = -ft =8.19382 cos C = 9.89173 J =8.30209 Cut in purlin flange: tanC =9.90525 cos A" =9. 95154 Cut in rafter flange: //2 = 9. 31876 tan C = 9. 90525 cos H = 0.03070 Say 3}". 2 T 3 F = 9.25471 Say 2". Note. No distinction is made between logarithms and cologarithms, since this is apparent from the formulas. f =9.85679 (HAITI K ii. FLANGE CONNECTION 26 O.O.V.'Tl '.i r,-.i v .7 6' tan A' tan A" b"= 1.05665 cos A" =9. 96777 d" = 1.08888 6' =0" cos A' = 9.95154 d' = 1.00820 tan A" =9. 60206 tan A' = 9.09897 tanC =9.90309 ILLUSTRATIVE PROBLEM CASE I (6). Computation of values given on p. 22. tan A' sin C tan // 6' sin C in 89697 9.79567 9.49464 0.95974 9.79567 1.16407 9.97979 1.18428 r 1 = 0.69897 t*f\a A f O Q " I " 1 \*\JO 4\ ~~ 7 . J / 1 i (~t tanC =0.09691 p' = 0.74742 tot r 0.10742 0.02021 ii Additional calculatbn rccniircd for the above prol)lcm is as follows: Bent plates at columns: iweb = A = 8. 11 It,} tanC = 9.90309 W = 8.21155 tanC . r 9.90309 9.97979 0.09692 sin A' ,-, c COS// sin .V w' tanC P" sin (' DM // 9.65051 9.78516 9.97979 9.41546 9.97980 9.39526 0.69897 9.96777 9.90309 (i .-,;-. iv; 0.20433 0.02021 f" -0.79437 sin C 8.11KH 9.79567 8.31897 coM"- 9.93554 tanC -9.90;i09 cosH -0.02021 w"- 9.85884 sin A" -9. 56983 sin'C -9.591:; I cos// - 9.97979 sin X"- 9. 14096 u>"= 9.85884 I/"- 9.28212 Note. No distinction is made between logarithms and eologarithnw since this is apparent from the formulas. CHAPTER III WEB CONNECTION FORMULAS FOR HIP RAFTER CONNECTIONS CASE IV (a). (Ordinary Case.) Channel purlins connecting to web of hip rafter. Axes of roofs intersecting at right angle. Unequal pitches. Given: a' = the slope of the steeper roof (a' always greater than a"). a" = the slope of the other roof. b' = the horizontal distance between the working points of the steeper roof. (Either b" or e might be given instead.) t = the thickness of the web of the hip rafter. u' and u" = the perpendicular distances from the tops of the main rafters to the tops of the purlins. r' and r"= the distances from the working points to the backs of the purlins measured along the tops of the main rafters. (1) tan A' = a'. m (58) V = u'y'. \\-6) n fj~ ^ (2) e = b'tanA'. * (59) (52) i' = t/2 ~cosA' -=* =*' (60) i- < 61 ) /'-stanC. ~teET>' (ifia) p' = rcosA . v , tanC (24a) p"= r"cosA" tanC. (6) d " = ~c^A 7 '' (17) S ' = ^ (2V S " = p " tanA/ , cos C cos H sin C cos// (7) tanC = t ^ A /' (54) w , ^ cos 2 A' tan// (62) w>" = cos 2 A " tan 2 C tan H. (8) c =tanC. ( 63 ) sin X" = cos A" sin C. (9) tan// = tan A' sin C. (55) cosX' = cos A'cosC. (64) x"=tanX". (10) h =tan//. *(56) x' = tanX'. (65) j/"= sin A"tanC. 6' ,,_ , sin A' (66) k"=u"y". m = iuTC ' y == taHC ' (67) z" = cos A" tan C. * If any of these values exceeds 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side the reciprocal of the value found. This reciprocal is obtained directly from the cologarithm. Care should be taken, however, that the original values are used in all further calculation. 26 < HAITI:!! in. \VI:H < t .\\KTION 27 CAMS IV (a) HIP See opposite Doge I A n -Space rirato u far apart a* pncnemL III * 1 V - PUec rlreta far c nougb from (Uncc to fin rafflclcnt drlTlnc clecnnce. {T AUow raple * " = cos 2 A" tan 2 C tan H. (63) smX" = cos A" sin C. (64) x" = tanX". (65) ?/' = sin A" tan (7. (66) k" = "*" (67) z" = cos A" tan C. * If any of these values exceeds 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side the reciprocal of the value found. This reciprocal is obtained directly from the cologarithm. Care should be taken, however, that the original values are used in all further calculation. '!! \ITl.i: 111. \\ I : CONNECTION 29 Space rtreU w far apart ai practical. * Place rlToU far enoafheromflance to (Ire lufltcJcat drlrlnc dnrmnce. V 4 VI = Allow ample edfte dldance In purlin. KlroU mlKht be placed In a line paral- N 1 1 vim '"' to llne * bcod " Preferred. llowln* (ancient edge dtotence la Plato. Detwmlno elcnulno bjr Uyont. CAM IV (6) VALLEY RAPTOR See opposite page. Fio. 12. 30 HIP AND VALLEY RAFTERS FORMULAS FOR HIP RAFTER CONNECTIONS CASE V (a). (General Case.) Channel purlins connecting to web of hip rafter. Axes of roofs intersecting at oblique angle. Unequal pitches. Given: a' = the slope of the steeper roof (a' always greater than a"), a" = the slope of the other roof. 6' = the horizontal distance between the working points of the steeper roof. (Either b" or e might be given instead.) t = the thickness of the web of the hip rafter. u' and u" = the perpendicular distances from the tops of the main rafters to the tops of the purlins. r' and r" = the distances from the working points to the backs of the purlins measured along the tops of the main rafters. L = the angle between the axes of the two roofs. (1) tan A' = a'. (2) e = b' tan A'. (3) d ' = c^A 7 ' (4) tan A" = a". e n (5) (6) b" d" = tan A" b" cos A" (30) tanC = , (31) tanC = v , *(8) c = tan C. (9) tantf = tan A' sin C. (10) h = tan H. b' < 90. when L > 90. (ID m sinC (12) (52) (53) (16a) (17) *(64) (55) *(56) *(57) (58) *(59) m n / cosH t/2 i sinC t/2 3 pf- Q' tanC r' cos A' tanC P' w' = cosX' = x' = 11' cos C cos H cos 2 A' tan H tan 2 C cos A' cosC. tan X'. sin A' y - k' = z' = tanC u'y'. tanC cos A' (68) (69) (32a) < " Pra ' allowlnif .ufBclcnt odge dtatence In plate. Determine IPBCM by lajout, placlnf hole* In a line normal to axU of hip rafter. It U preferable to cut plate parallel to tbe Unc of bulea. (SeeFlc.8 PaeelB.) II * XII = Place bole* far enoueh from Urn of bend to allow rireU or bulu to be placed. Determine dlmenaloo* bjr layout. Till A XTV = Determine b/ lajout. . u- 3 KEY PLAN CAM V (a) HIP rUrncR . ppMlti IMfft FJG. 13. 32 HIP AND VALLEY RAFTERS FORMULAS FOR VALLEY RAFTER CONNECTIONS CASE V (6). (General Case.) Channel purlins connecting to web of valley rafter. Axes of roofs intersecting at oblique angle. Unequal pitches. Given: a' = the slope of the steeper roof (a' always greater than a"). a" = the slope of the other roof. V = the horizontal distance between the working points of the steeper roof. (Either b" or e might be given instead.) I = the thickness of the web of the valley rafter. u' and "= the perpendicular distances from the tops of the main rafters to the tops of the purlins. r' and r" = the distances from the working points to the backs of the purlins measured along the tops of the main rafters. L the angle between the axes of the two roofs. (1) tan A' = a'. (2) e =b' tan A'. d> = coiA"'' (4) tan A" = a". o b" = (5) (6) d" = (30) tanC = (31) tanC = *(8) c =tanC. (9) tan# = tan A' sin C. (10) h =tantf. b' tan A" V cos A"' b' sin L b" +b'cosL b' sin L when L < 90. when L > 90. (H) m = sinC (12) (52) (53) (16a) (17) *(54) (55) *(56) *(57) (58) *(59) m ;/ cosH t/2 ,v sinC t/2 J _/ tanC r'cos A' p (,' tanC t 111' cos C cos H cos 2 A' tan H cosX' = x' = a- k' = z' = tan 2 C cos A' cos C. tan X'. sin A' tanC u'y'. tan C cos A' (68) (69) (32a) (33) (70) (71) *(64) *(72) (66) *(73) t/2 sin (L - C) ,, ot toalkiw riM. M MM to b. I ... 11. 34 HIP AND VALLEY RAFTERS FORMULAS FOR HIP RAFTER CONNECTIONS CASE VI (a). (Special Case.) Channel purlins connecting to web of hip rafter. Axes of roofs intersecting at right angle. Equal pitches. Given: a = the slope of each roof. b = the horizontal distance between working points, (e might be given instead.) t = the thickness of the web of the hip rafter. u' and u" = the perpendicular distances from the tops of the main rafters to the tops of the purlins. r' and r" the distances from the working points to the backs of the purlins measured along the tops of the main rafters. C = 45. c = 12. 9.84949 = log sin 45 = log cos 45. (36) tan A = a. ^ { ^ 111 (78) w = cos 2 Atanff. (37) e = btanA. sin45 f (79) sin X = cos A sin 45. (38) d = ( 75 > 3 -I' cos A ( 80) z=tanX. (39) tan H = tan A sin 45. ( 76 ) P r = r> cos A - (81) y = sin A. (10) fe = tantf. (77) p"=r"cosA. 5 , (82) fc'-u'y. (40) m = -^W (46) s' = - -ng 5- sin 45 cos 45 cos// (83) k = u y . (12) n--^. (47) ' cos 45 cos H (84) z = cos A. rilAITKU III. WEB CON M.I TH >\ 35 NOTES I Space rirct u tar apart u practical. Ill -Place rlrrU far rnouKh from Haute Co tire iufoclent drlrlnir clearance. V * VI - Allow ami.ii- cdire dUtance In purlla RlrcU micnt be placed In a line pnral- Icl to Uoe of bend If preferred. \ II.IX). Flacc bokiai far apart ai practical, * x ) allowing uftlck-nted(redlUnce In lOate. Determine cpacei by layout, placing bole* In a line normal to ula of hip rafter. It If preferable to cat plate parallel to the lino of bolra/Scc Ft*. 8. Pace It) II 4 XII Placcbulc* far enough from line of bend to allow rlreU or bolu to be placed. Determine dlmcnaloo* br layout XIII Determine br layout. CAM VI (a) HIP RAFTER Fin IS 36 HIP AND VALLEY RAFTERS FORMULAS FOR VALLEY RAFTER CONNECTIONS CASE VI (6). (Special Case.) Channel purlins connecting to web of valley rafter. Axes of roofs intersecting at right angle. Equal pitches. Given: a = the slope of each roof. b = the horizontal distance between working points, (e might be given instead.) t = the thickness of the web of the valley rafter. u' and u" = the perpendicular distances from the tops of the main rafters to the tops of the purlins. r' and r" = the distances from the working points to the backs of the purlins measured along the tops of the main rafters. C = 45. c = 12. 9.84949 = log sin 45 = log cos 45. (36) tan A = a. ,_,. . t/2 (78) w=cos 2 AtanH. v ~- ~ jO* (37) e = btanA. ^ (79) ginX = cos A sin 45 '"*' ' (80) .-teZ. (39) tan H = tan A sin 45. (76) p'=/cosA. ^Oly y bill -il. (10) h = tan tf . (77) P" = (40) = ;io- ( 46 ) (82) k' = tt'y. (g3) fc/ , = wV (47) S " = cos45 Po cosH' (84) 2 =cosA. 37 NOTES I - Space rlreU u tar apart w practical . m Place rlreUlmr fiiouch from nature to a-lTeraOclentdrlrliur clearance. V A VI Allow ample edfo dMancc lu purlin. BJrete might be place* In line pant- VII ITI lc " Uw>o">eUct> hol< " " *" "P" 1 "" Pctacln>bokM In line normal to uM of Taller rafter. It U preferable to cut pUte parallel to the line of CAE VI (6) VALUCY IUrr See oppmitc page. . _ . , XI * XII- Place bolM far enoocn from line o bond to allow rlroti or bolu to be Till- KSrt"" 1 " 11 *' ' WU Via. 10. 38 HIP AND VALLEY RAFTERS CALCULATION FOR HIP AND VALLEY RAFTER CONNECTIONS WEB CONNECTION The following outline is given to serve as a guide for the tabulation of the required values, and to indicate the logarithms which will be needed for further computation. All the necessary functions of an angle may thus be determined at the same time. CASE IV. (Ordinary Case.) Roofs at right angle; unequal pitches. Given: a', a", V (or b" or e), t, u', u", r', and r". Nun V - Angle. Slope Logarithm. Sine. Cosine. Tangent. A' a' = A" a" = c c- H h = ^x^ X' x' = X X" x" = X iber. Logarithm. Number. Logarithm, w' = i> y' - d' - k' = z' = d"- j"- p"- i' s"= V - , y"- s' k"- z"= . CHAPTER III. WEB CONNECTION CALCULATION FOR HIP AND VALLEY RAFTER CONNECTIONS WEB CONNECTION The following outline is given to serve as a guide for the tabulation of the required values, and to indicate the logarithms which will be in < (!(f liciul of the connecting plates cannot lie in the same plain 1 as the working line of the rafter, which is the center line of the top flange. It would IM> impractical to use two working lines in the planes of the well face-, and so the line of bend must be located with reference to that point in the center line of the top flange which is cut by the plane of the purlin web. The horizontal distance from this working point to the ction of the line of bend and the working line of the purlin is rep- <1 by i' or i" when measured along the purlin, and by j' or j" when red in the direction of the rafter. The distinction between i' and ;" is !.e-t shown in Fig. 1 1 . p. 27, or Fig. 12, p. 29. It is generally an advantage to the shop to have the rivets and holes in iimection plates placed in lines normal to the flanges of the beams and channels, and this can usually be accomplished. If this arrangement brings the holes too far from the line of bend, they may be located in a line parallel to the line of lx>nd instead. The flange of the purlin may be sheared diagonally if preferred, but in the more modern practice the flange is blocked out as shown. The slope (z* or z") of the clearance line is given to serve l>oth methods. The amount to be blocked out is best determined by a layout, with allowance for any desired clearance. The connection plate should l>e placed upon the obtuse angle side of the purlin where possible, in order to avoid the sharp bend in the plate and also to facilitate erection. To accomplish this in the case of the valley rafter, the plate must be put on the upper aide of the purlin web. To retain the similarity of details for hip and valley rafters, the purlins are shown with their flanges facing down the slope instead of up. Should it be desired to face them the other way, see Chap. IV, p. 48. In order to simplify erection it is better not to have purlins of both slopes connect to the web of the hip or valley rafter at the same point. It is frequently advisable to frame them opposite, however, and this may be easily accomplished in either hip or valley work, by arranging the holes in I Kith connecting plates to correspond to common holes in the rafter web. 42 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE IV (a). HIP RAFTER Web connection; buildings at right angle; unequal pitches. For comparison, let us consider a hip rafter under the same conditions as the one shown on pp. 20 and 21, except that the purlins connect to the web of the rafter instead of to the flange. The working lines and the column connections remain the same, and it is unnecessary to reproduce the latter in Fig. 17. Given V = 11' lltf", 6" = 14' 101", and a" = 6. Using a 12" I31|# rafter and 8"ulli# purlins, we have t = |, u' = u" = 2,r' = 3' Of" and r" = 10' 1^ ". The required values are given below. (For necessary computation see p. 46.) Number. V = 11' lilt" e = r 5 T y d' = 14' If" fe"= 14' 10|" d"= 16' 8" m = 19' 1|" n = 20' 6 T y i' = & j> = i p' = 3' 2 T |" s' = 4' 54" Logarithm. Sine. Cosine. Tangent. A' o' = 7& 9.72272 9.92900 9.79372 A" a" = 6 9.65051 9.95154 9.69897 C c=9| 9.79698 9.89173 9.90525 H A = 4H 9 96930 9 59070 X' ^7=10 A 9.82073 05414 X" z"=8| 9.74852 9 83039 Logarithm. 1.07862 0.87234 1.17337 1.28164 0.51130 Number. w' = 5 T = I T % z' = ill r- i p"= 7' 31" y"= fc"= I 2"= 81 Logarithm. 9.81747 0.86107 9.55576 HAITI. i: in \\i.r, < ONN1 < i i"\ C L f!-.-.. r iLLDsnuTtvm PROBLBM CAM IV (a) HIP RAITM See opposite page. Fio. 17. 44 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE IV (6). VALLEY RAFTER Web connection; buildings at right angle; unequal pitches. For the sake of comparison, we can so modify the problem shown on pp. 22 and 23 that the purlins connect to the web of the valley rafter instead of to the flange. The working lines and the column connections remain the same and it is unnecessary to show the latter in Fig. 18. Given b' = 9' 1|", a' = 6, and a" = 4||- Using a 10" I 25# rafter and 6"u8^t purlins, we have t = T 5 5 , u' = u" = 2, and r' = r" = 5' 0". The required values are given below. (For necessary computation see p. 47.) logarithm. Sine. Cosine. Tangent. A' o' = 6 9.65051 9.95154 9.69897 A" "-48 9.56983 9.96777 9.60206 C c=9f 9.79567 9.89258 9.90309 H A = 3| 9.97979 9.49464 X' -, =llli 9.84412 0.01060 X" *"-8A 9 76344 9.85249 Number. Logarithm. 6' = 9' If" 0.95974 e = 4' 6H" 0.65871 d' = 10' 2ft" 6"= 11' 4f" 1.05665 d"= 12' 3|" m = 14' 7ft" 1 . 16407 n = 15' 3ft" ?'"* P' = 5 F 7^" 0.74742 s' = 7' 6" Number. Logarithm w' = 4H y\ = &H 9.74742 ?"=ft 4 j"= i a = 3' 8 9 " 0.56983 g' = 6' 2f " W)' -2ft ?/' = 3^ 9.47292 fc' = f z' = 8i| CHATTER III. WEB CONNECTION 45 ; LtM-TO U :i...\ II.I.ISTRATIVE PROBLEM CAMS IV (6) VALLEY IUnr R See opposite page. Fio. 18. 46 HIP AND VALLEY RAFTERS ILLUSTRATIVE PROBLEM CASE IV (a) Computation of values given on p. 42. 5"= 1.17337 tan A" = 9. 69897 e = 0.87234 V = 1.07862 tan A' = 9.79372 V = 1.07862 cos A' = 9.92900 d' = 1.14962 6" = 1.17337 cos A" =9. 95154 d"= 1.22183 tan A" =9. 69897 tan A' = 9.79372 tanC =9.90525 tan A' =9.79372 sinC =9.79698 tan// =9.59070 V = 1.07862 sinC = 9.79698 m = 1.28164 cos# =9.96930 n = 1.31234 t/2 =8.19382 sinC = 9.79698 i'=8.39684 t/2 tanC 8.19382 9.90525 /= 8.28857 r = cos A' = tan C = P r cos C cos H 0.48755 9.92900 0.09475 0.51130 0.10827 0.03070 s'= 0.65027 cos 2 A' - tan// = tan 2 C : 9.85800 9.59070 0.18950 w/=9.63820 cos A' cos C cos X'= 9. 82073 9.92900 9.89173 sin A' = 9. 72272 tan C = 9.90525 y' = 9.81747 u' =9.22185 fc' =9.03932 tan C = 9 . 90525 cos A' = 9.92900 z' =9.97625 t/2 = 8.19382 cosC =9.89173 {"=8.30209 t/2 =8.19382 tan C = 9 . 90525 j"=8.09907 r"= 1.00428 cosA"= 9.95154 tan C = 9.90525 p"= 0.86107 sinC =0.20302 cos// =0.03070 s"= 1.09479 cos 2 A"= 9.90308 tan 2 C = 9.81050 tan H = 9 . 59070 w"= 9.30428 cosA"= 9.95154 sinC =9.79698 sinX"= 9.74852 sin A" =9. 65051 tan C = 9.90525 y"= 9.55576 u"= 9^22185 fc"=8.77761 cosA"= 9.95154 tanC =9.90525 z"=9.85679 Note. No distinction is made between logarithms and cologarithma since this is apparent from the formulas. MIAPTEK III. WEB CONNECTION 47 ILLUSTRATIVE PROBLEM CASE IV (6) Computation of values given on p. 44. b' =0.95974 t:m A' = 9.69897 e =0.65871 tan .1"= J).(Ml-_'iM; 6"= 1.05665 1"= 9. 96777 rf" = 1.08888 b' = 0. <>.V. 17 1 1' =9. 95154 d' = 1.00820 tan .1"= 9. 60206 tan.l' =9.69897 taiiT =9.90309 tan.l' sin (' 9.69897 9.79567 tan// =9.49464 6' = 0.95974 siuC = 9.79567 m = 1 . 16407 DM // = 9.97979 t/2 sinC 1/2 tai.r 8.11464 9.79567 8.31897 8.11464 9.90309 s -JII.V, ^=0.69897 cos A'= 9. 95154 tanC - 0.09691 p'= 0.74742 cosC =0.10742 cos H = 0.02021 s'= 0.87505 oof A' tan// = tan'C '.i 'Mi:;nx 9.49464 0. 19382 w'- 9. 59154 cos A' =9. 95154 cosC = 9.89258 cos X'= 9. 84412 MII .r - 1 tanC =9.90309 y' =9.74742 u' - 9.2-J l v. k' - s.wi'.n. 1 ; tanC =9.90309 owA' -9.95154 z' - 9.95155 8.11464 9.89258 8.22206 1/2 -8.11464 tanC =9. n = 1 . 18428 j"- 8.01773 r"- 0.69897 cos A" =9. 96777 tanC =9.90309 p"- 0.56983 sinC = 0.2(>: cos// -0.02021 "- 6.79437 co8M"-9.9:r.:.l tan'C =9.80618 tan// -9.49464 u>"-9.23636 cos A" =9. 96777 sinC - 9.79567 MM X"- 9.76344 sin A" =9. 56983 tanC -9.90309 y"- 9. 47292 u"- 9.22185 *"- 8.69477 008 A" -9. 96777 tanC -9.90309 z"- 9. 87086 Nole. No distinction is made between logarithms and cologarithms since this is apparent from the formulas. CHAPTER IV NOTES ON OTHER CASES ALTHOUGH it is felt that the majority of the problems in hip and valley work will come within the scope of the notes given in the preceding chap- ters, yet conditions will occasionally arise which demand modifications, and it is hoped that, with the aid of the suggestions which follow, the draftsman will be able to adapt one of the first six cases to his particular requirements. CHANNEL PURLINS WHOSE FLANGES FACE THE OTHER WAY. 1. Flange Connections. The connection plate will generally be placed on the back of the channel web as before, but the angle of bend will be obtuse instead of acute, and the details for the hip rafter connections will resemble more closely those for the valley rafter connections of Cases I, II, and III. In like manner the valley rafter connections will be similar to those for the hip rafters of Cases I, II, and III. The formulas remain the same. If preferred, the bottom flange might be cut away to allow the use of the same details as before, except with the plate placed on the inner face of the web. In this case the working lines must be taken to the back of the bent plate rather than to the back of the channel. The former method is recommended. 2. Web Connections. The connection plate should be placed so as to avoid a sharp bend, in order to facilitate shop work and erection. This may be accomplished in two ways, (a) By cutting the purlin short enough to permit the use of the plate shown in Cases IV, V, and VI, the plate being bent toward the channel instead of away from it. Care should be exercised in spacing the rivets and holes to give sufficient edge distance and driving clearance. (6) By using a plate similar to the above and nar- row enough to clear the flanges, when located on the inner face of the web. The latter arrangement is preferable unless the purlin is too small to allow its adoption. In either event, all formulas are applicable, provided that the working lines are taken to the back of the bent plate rather than to the back of channel. It will not be necessary to cut the flanges as in Cases IV, V, and VI. I-BEAM PURLINS. The details are practically the same as for channel purlins if the work- ing lines are taken to the back of the bent plate and not to the center line of web. Part of the flange on one side must be blocked out to allow for the plate. The top flange may have to be cut to avoid piercing the roof of the opposite slope. Z-BAR PURLINS. The same style of detail may be used to connect Z-bar purlins to the hip or valley rafter as for channel purlins shown in Cases I to VI inclu- sive. The flanges must be cut to clear the rafter and to avoid piercing the roof. ANGLE PURLINS. Angle purlins may be attached to the flange of the rafter by connections similar either to those for channel purlins or to those shown on the follow- ing pages for Tees. The web connections are the same as for channels. TEE PURLINS. The connections of Tee purlins are somewhat peculiar to themselves although similar details might be used under other conditions. The flanges of the Tees are connected to the flanges of the hip or valley rafters by means of bent plates. It is deemed unnecessary to illustrate all cases which might arise, but typical details of both hip and valley connections are shown in Figs. 19 and 20, and the formulas for all cases appear on pp. 49 and 50. The spacing of the rivets and holes is apparent without special notes, standard gages being used as a matter of course. 48 CHAPTER IV. NOTES ON OTHER CASES I'. FORMl'l.VS FOR TEE PURLIN CONNECTIONS HIP RAFTERS The following formulas are arranged to correspond to the three cases of hip rafter connections for channel purlins given in Chapter II. The following values are either given or else obtained from the proper formulas of Cases I, II, or III: A', a', A", a", e, b', b", d', d", C, c, H, h, m, n, r', and r". The additional data required may be found by means of the formulas listed below. CASE I (a). CASE II (a). Mil // tllllC' tnnC (85) x' 2' (86) x"= tan C sin//. (67) 2" = cos A" tan C. t(87) /" = 1(88) v = (89) p'= (90) s' = (91) p" = (92) s" = cos// , _ r' cos A' cos C f sin r _ r \f*JJ . *(59) (93) ; (73) i(R7^ taiif . tanC cos A' _,, sin// tan (L - C) ,, cos A" tan (L - C) x" (89) (90) (94) p (95) YcosA'cosC-f >iuC CASE III (o). (96) x -sin//. (84) 2 - cos A (sec 2" in Fig. 19). ft _ Aft t(97) /"-/' + - - z t (98) * - ' (99) p'- (100) '- cos 45 cos H^ COB U -('tan//. (101) p"-r"cosA- r 00045? r // cosA"co8(L-C)-/ > ' sin (L - C) (io2) "-^^a -t"tan//. COS(L-C)C08// If t ho value of :' or : " U greater than 1 ' 0" the bevel should be reveraed on the drawing go that the longer side beoomea the 12" base and the shorter aide P *V' ' valur-i am obtainnl ilirri-tly fmin I ho rologarithmg. t /' is :is.-iitiii-.l ci|ii.-il ID on,, half of tho flango width phu a distance sufficient to allow for the bend in the plate. /" u found from formula (87) or (07). If/" is too Urge, it may bo rvilurol liy increasing /" l>y moans of filli-rs. t D is the amount which tho hip rafter must be lowered vertically. 50 HIP AND VALLEY RAFTERS FORMULAS FOR TEE PURLIN CONNECTIONS VALLEY RAFTERS The following formulas are arranged to correspond to the three cases of valley rafter connections for channel purlins given in Chapter II. The following values are either given or else obtained from the proper formulas of Cases I, II, or III: A', a', A", a", e, b', b", d', d", C, c, H, h, m, n, r', and r". The additional data required may be found by means of the formulas listed below: CASE I (6). sinff tanC' tanC (85) x' = *(59) z' = - -77- cosA' (86) x"=tanCsin#. *(67) z"= cos A" tan C. x" . f'x' - t' }(104) v = (105) p' = cosH (107) (108) ^- cosC (85) x' = *(59) z' = (93) x" = *(73) z" = t(103) /" = }(104) v = CASE II (b). sinH tanC' tanC cos A' sin ff tan (L - C)' cos A" tan(L-C)' f'x' -t' + 1" x" f'x' -t' cos H (105) (106) p'-f'smC - t' tan H. (110) cos C cos H ,_r"cosA"cos(L-C)+/" sin (L - C) cos (L C) cos CASE III (6). (96) a; = sinH. (84) z = cos A (see z" in Fig. 20). t(9 7) r-r**-;?. }(98) t; = "~ ' " (111) p'=r'cosA sin45' : (112) s' = P' cos 45 cos (113) p"=r"cosA + (114) cos H - ' tan H. cos 45 p" f" sin 45 cos H cos H * If the value of z' or z" is greater than 1' 0" the bevel should be reversed on the drawing so that the longer side becomes the 12" base and the shorter side or These values are obtained directly from the cologarithms. fillers. ~a >u' uvmuceu UJTWUIJF nuiii MM uuiuganuiiiiia. t /' is assumed approximately equal to one half of the flange width. /" is found from formula (103) or (97). If /" is too large, it may be reduced by increasing t' by means of v is the amount which the valley rafter must be raised vertically. CHAPTER IV. NOTES ON OTIII.l; \>l.s 51 HIP RAFTER Fia. 19. VALLEY RAFTER Fia. 20. ' ana nan M Vm ! M M CHAPTER V DERIVATION OF FORMULAS IN this chapter are given the derivations of all formulas which involve anything more than the simple trigonometric relations of parts of a right triangle, or the substitution of other values in formulas already deter- mined. These derivations are by no means essential to the complete solution of a problem, but are inserted rather for reference, for those who have difficulty in understanding certain features, or who wish to investi- gate the subject more thoroughly. Any formulas, other than those given here, which apply to the connec- FORMULA (7). (Fig. 21.) But Substituting: tan A' tan C = ~ and b" = tan A" tan C = tan A" tan A' ' FORMULA (9). (Fig. 21.) tan H = m But Substituting: e = b' tan A', and m = -: b' sin C tan H = tan A' sin C. tions of the purlins in the steeper roof may be obtained from similar expressions by making a few simple changes to comply with the new con- ditions. Any formulas which pertain to the other roof, namely, those which bear this mark ("), are similar to those of the steeper roof and may be derived from them by changing (') to (") and substituting for angle C, either L-C or 90-C., bearing in mind the relation between the functions of an angle and the functions of its complement. FORMULA (16). (Fig. 21.) 1-2 But Substituting: tanC 1-2 = (r'-g')cosA'. ,_ (/ - q') cos A' P tanC FORMULA (17). (Fig. 21.) 1-3 But Substituting: 1-3 = cos H 2-3 cos C P' P' cos C cos C cos H 52 ( H.\ITi:n V. DKRI \.\Tli i\ OF FORMULAS Fio. 21. Fio.22. FORMULA (13). (Fig. 22.) If the center lino of the top flange of the hip rafter were placed in the same plane as the top of the main rafters, the edge of the flange would inter- fere with the purlin. Thus, if the bottom of the purlin cut the center line of the top flange of the hip rafter at point 6, and the top of the main rafter :it jxiiiit 1, the vertical plane through the lx>ttom of the purlin would cut the edge of the hip rafter flange at point 7. Since the web of the hip rafter is vertical, and the llange normal to the web, points 7 and 8 are at the same elevation, as shown at points 2 and 3 respectively. Therefore, the hip rafter should be lowered vertically a distance 1-2 plus any desired clearance, as \" or ft". 1-2 = (2-3) tan A'. But 2-3 = (7-8) cos C, and 7-8 Substituting : 1-2 = { tan A ' cos C, and r = tan 4' cosC + J". ./. 2 Note. In tho derivation of the following formulas, the clearance }" is omittwl for simplicity, since the angle* remain unchanged, 54 HIP AND VALLEY RAFTERS FORMULA (18). (Fig. 23.) From the derivation of Formula (13) it may be seen that if the purlin should rest on the flange of the hip rafter at point 7 in the plan, or 11 in the plane of the flange of the hip rafter, the center line of the top flange of the rafter at point 6 would be a vertical distance 1-2 below the purlin. The plane of the purlin web would cut this center line at point 4, which corresponds to point 9 in the plan and 13 in the plane of the flange of the rafter. The line 11-13 is, therefore, the line of intersection of the plane of the purlin web and the plane of the top flange of the hip rafter, and hence the line of bend of the connection plate. Let w' represent the tangent of the angle which this line makes with the center line of the flange; then , 12-13 w = But 11-12 = 11-12 8-9 3-5 cos H sin C cos H where . 3-5 = (3-4) cos A' = (2-3) cos 2 A', and 2-3 = (7-8) cos C; (7-8) cos C cos 2 A' therefi.r 11-12 = 4 -^ sm C cos H Also 12-13 = I = 7-8. Substituting these values above: 7-8 (7-8) cos 2 A' tan C cos H tan C cos H cos 2 A' < HAITKR v. ni:i:i\.\Ti't\ n- IOUMU.AS B5 l ... _'.; 56 HIP AND VALLEY RAFTERS FORMULA (19). (Fig. 23.) The intersection of the plane of the purlin web and the plane of the top flange of the hip rafter is the line 11-13 or 20-21. The line 7-6, 11-15 or 20-22 is the bottom line of the purlin web. If a plane is passed through the point 13, or 21, perpendicular to the line of intersection it will cut the plane of the purlin web in a line 21-22, or 18-19, and the projection of this line upon the plane of the top flange of the rafter will be 13-14 or 17-19. The angle X', or the complement of the angle between the two planes, is shown in its true size at 17-18-19 and . y , = 17-19 = 13-14 " 18-19 "21-22" But 13-14 = (11-13) tan 13-11-14 = (11-13) tan (W - K), and 21-22 = (20-21) tan 21-20-22 = (11-13) y'. . v , tan (IP' -X) tanTF'-tan-K Therefore sm.X'=- - - = . =7- ^T"'' , tan C cos H 15-16 ,_ 1C / _ But tan W = w' = - 2 ., and tan K = - ^ > where 15-16 5 7-8, COS A. 11 lo (7-8) cos H . tanC and 11-16 = (6-8) cos H = whence tan K = jj tan C cos H tan C . , cos 2 A' cosH tan C (cos 2 H cos 2 A') Substituting these values: sin A = - f ^- cos 2 H - cos 2 A'= cos 2 tf [1 - cos 2 A' (1 + tan 2 #)] = cos 2 H[l - cos 2 A' - cos 2 A' tan 2 A' sin 2 C] See Formula (9). = cos 2 H [sin 2 A' - sin 2 A' sin 2 C] = cos 2 .ff sin 2 4' cos 2 C. cos 2 A' + tan 2 C = [cos 2 C + sin 2 C (1 + tan 2 A')] os2 c + sin2 c + sin2 cos 2 A' ,, Wc [1+tan//] cos" A' cos 2 C cos 2 H From the following derivation (p. 57): , _ sin A ' sin C cos C cos 2 H cos 2 A' Substituting these three values and reducing: sin X' = sin A' cos 2 C cos H. CHAPTER V. 1)1 i;l\ ATION OF FORMULAS 57 MULA (21). (Fig. 23.) If y' represent the tangent of the angle in the plane of the purlin web between the line of bend 7-9 and a horizontal line 7-0, then 2-4 But where Substituting: But Substituting: 7-10 2-4 - (1-2) COB A' = ^ sin A' cos C and 7-10 - (7-6) - (10-6), ,_- 2 9-10 5-2 (2-1) sin A' 7-6 = -i 77 and 10-6 = r -75 ; 7, " ~ sin C tanC tanC tanC . i-in A'cosC / /sin* A 'cos'C' 2 sin C 2 sin C ,_ sin A' sin C cos C y ~ 1 -sin* A' cos* C I - sin* A' cos* C - 1 - sin* A' + sin* A' sin* C = cos* A' + sin* A' sin* C - cos* A' (1 + tan* A' sin* C) = cos* A' (1+ tan*//), cos*A' " cos*//' sin A ' sin C cos C cos*// cos'A' If we multiply Formula (18) by Formula (19) we have: tan C cos H . tc'sinA" Therefore sin A' cos* C cos H y'=u>'sinA". sin A' sin C cos C cos* H cos* A' 58 HIP AND VALLEY RAFTERS FIQ. 24. FIG. 25. tanC FORMULA (30). (Fig. 24.) 1-2 b' (3-4) + (2-3) 6" b' .T; sinL tanL V sin L + b'cosL tanC = FORMULA (31). (Fig. 25.) 1-2 b' (3-4) - (3-2) b" V_ sin L tan L 6'sinL c"-6'cosL CHAPTER V. DERIVATION OF FORMULAS 59 Fia. 26. FORMULA (54). (Fig. 26.) The line 2^-24 is the intersection of the plane of the purlin web and the vertical plane through the center line of the main rafter, and 26-29 the intersection of the plane of the purlin web and the vertical plane through the center line of the hip rafter. Let us consider, for illustration, that part of the purlin web whose vertical projection is measured by the distance ' = 23-25 = 26-27. If the line 26-28 is drawn perpendicular to the top of the hip rafter, the angle 27-26-28 = H. Let the angle 27-26-29, which the line of intersection 26-29 makes with the vertical, be represented by M. Then w' = tan 28-26-29 = the tangent of the complement of the angle between the line of bend and the top flange of the hip rafter. But Substituting: tan Af tc 27-29 ian .fio-zo-zy = ian \ai n ) = r-^ 24-25 tan M tan // tan// -in i ' c'tanA > and tan A' tantf(l -sin'C) 26-27 in' v' v' sin C tann . . sinC C . , ,: ian ti sin 2 C tnn'tf ^sitfC tan// cos* C s^C + tan 1 ^ tan // i-i sin* C + tan* A' sin 1 C sin* C (1 + tan' A')' tanM tan// cos* A' tan H tan*C 60 HIP AND VALLEY RAFTERS FIG. 27. FORMULA (55). (Fig. 27.) The line 26-29 is the intersection of the plane of the purlin web and that of the hip rafter web, and M is the angle which it makes with the vertical. C ( = 33-34-35) is the horizontal projection of the-angle between these two planes, but the true angle is measured at right angles to the line of intersection. If the plane containing this angle is revolved into the horizontal about 33-35, an axis normal to the rafter wet), the point 30 will fall at 31 in the elevation, or 32 in the plan, and the required angle X' between the two planes will appear in its true size at 33-32-35. 33-35 (34-35) tan C _ (27-29) tan C tan X' = 32-35 27-30 v' tan M tan C tan C From the derivation of Formula (54), p. 59, we have: tanM = v' sin M cosM tan A' sinC whence cosM = v' sin M 1 Substituting : But tanX' = tanCVl + 1 I tin 2 A' I y 1 + ^nj = \ tan 2 C + tan 2 A' cos 2 C ' cos X' = - tan- A' sin 2 C cosC Vl+tan 2 X' cosX' = cos A' cos C. s/ l+tan 2 C tan 2 A! Vl+tan 2 A' cos 2 C CIIMTKK V. DERIVATION (H- 61 fc I'M;. 28. FORMULA (57). (Fig. 28.) The horizontal projection of the line of intersection of the web of a purlin of depth u and the rafter web is represented by the line 35-37. The bottom flange of the purlin must be a distance 35-36 shorter than the top flange. The cut is shown in the plane of the purlin web by the angle whose tangent is y'. 36-37 , 35-36 tanC 24-25 tanC u Mil .1 ' u tan C u tan C FORMULA (59). (Fig. 28.) The purlin flanges are cut at an angle C measured horizontally, and this heroines the angle whose tangent is z' when measured in the plane of the purlin flange. 36-37 , = 39-40 = 23-34 cosT' " :$s ;{..i " ::;,-:nd. If a plane is passed perpendicular to this line of hend it will cut the top flange in the line 42-44 and the roof plane in the line 42-45. The angle of bend, or the angle between the planes of the top flange and the roof, is shown in the plan at 44-42-45, and in its true size at 50-49-51. Let x' represent the tangent of this angle; then, tan 50-49-51 - .Mi :,1 I'.' .Ml' But and Substituting, 50-51 - 47-48 - (46-48) sin// = (41-13) sin H, 49-50 = /' = 43-45 = (41^3) tan C. j (41-43) sing sing " (41-43) tan C" tan C* CHAPTER VI GRAPHIC METHOD OF DETERMINING ANGLES THE angles used in the various purlin connections of the preceding chapters may be determined graphically if desired, but the linear dimensions should be calculated in every case, for the ordinary precision of the graphic method is not sufficient to insure the best results. In fact, it is believed that the complete algebraic method is quicker as well as more satisfactory, and the graphic method is recommended merely as a check. Since the angles are the same for either hip or valley work, the drawings of this chapter show only valley rafters. For the sake of clearness, the determination of each angle is illustrated separately, as well as in con- junction with the other angles occurring in the same case. The drawings show only those lines which are necessary in the determination of the different angles, and no attempt has been made to show the complete projections of Descriptive Geometry. The accompanying descriptions, therefore, cannot be given in minute detail, but it seems better not to complicate the drawings in order to simplify the analyses. These descrip- tions refer to the angles used in the connections of the purlins in the steeper slope; they may be made to apply also to the corresponding angles of the other slope by substituting (") for ('), and in some cases taking the complement of the angle thus found. The actual value of the angle is seldom necessary, but the slope, or tangent of the angle for a base of one foot, may be scaled directly from the drawing. There are four planes of projection, one horizontal K'GK", and the other three vertical through the webs of the main and valley rafters, D'B'E', D"B"E" and MGN, respectively. The angles A', A", C and H are plotted from the given data, and the line P'GP", drawn perpendicular to GM, is the H trace (horizontal trace) of the plane of the top flange of the valley rafter. FLANGE CONNECTION ANGLE W (Fig. 30), in the plane of the top flange of the valley rafter, between the line of bend and the center line of the rafter. The plane of the purlin web cuts the center line of the rafter at a' in the plan, or b' in the elevation. It also cuts the H trace of the plane of the top flange of the rafter at P'. Thus we have two points in the line of intersection of the two planes. If the plane of the top flange of the rafter is revolved into H, about its H trace, GP', b' will fall at c' and the line of intersection at c'P'. The angle P'c'G between this line and the center line of the rafter will be the required angle W. ANGLE X' (Fig. 31), the complement of the angle between the plane of the top flange of the valley rafter, and the plane of the purlin web, or the angle of bend. If the line of intersection of the two planes be revolved into H about its H projection, a'P', it will fall at d'P'. A plane normal to the line of intersection at any point g' will cut the H trace of the plane of the top flange of the rafter at /', and the H trace of the plane of the purlin web at j'. If this normal plane is revolved into H about its H trace, f'j', g' will fall at h', and the obtuse angle between the plane of the top flange of the rafter and the plane of the purlin web will show in its true size at f'h'j'. If a line be drawn perpendicular to f'h' at h', the angle between this line and the line h'j' will be the required angle X', or the complement of the acute angle between the two planes. ANGLE Y' (Fig. 32), in the plane of the purlin web, between the line of bend and a horizontal line. If the plane of the purlin web is revolved into H about its H trace, MP', the line of intersection of the plane of the purlin web and the plane of the top flange of the rafter, or the line of bend, will fall at m'P', and the angle m'P'M will be the required angle Y'. 62 CHAITKK VI. ORAl'llH MITHOD 63 Wand Fio. 30. Y' and Y" Fia. 32. FLANGE CONNECTION. For combined layout see Fig. 33 or Fig. 34. 64 HIP AND VALLEY RAFTERS CASE I CASE H FIG. 33. FLANGE CONNECTION. For separate layouts see Figs. 30, 31 and 32. Fio. 34. M (.KM'IIH MITIInD W'andW FlO. 35. \\ i. n CONNECTION. For combined layout sec Fig. 39 or Fig. 40. WEB CONNECTION. AM-.LE II"' (Fig. 35), in the plane of the web of the valley rafter, the complement of the angle between the line of bend and the top flange of the rafter. The phino of the purlin web cuts the center line of the top flange of the rafter at n' in the plan, or 6' in the elevation. It also cuts the // trace of the plane of the rafter \ve!> at M. Ml/ is, therefore, the line of inter- section of the two planes, or the line of bend, and the angle between this line and a line normal to the top flange is the required angle W . AM.I.K A ' Fig. 36), Ix'tween the plum- of the purlin web and the plane of the web of the valley rafter, or the angle of bend. If a plane be passed through a' perpendicular to the line of intersection of the two planes at n', it will cut the // trace of the plane of the rafter web at a', and the H trace of the plane of the purlin web at p'. If this normal plane is revolved into H about its H trace, a'p', n' will fall at o', and the required angle X' between the two planes will be shown in its true size at a'o'p'. ANGLE Y' (Fig. 37), in the plane of the purlin web, the complement of the angle between the line of bend and a horizontal line. If the plane of the purlin web is revolved into // about its H trace MK", the line of intersection of the plane of the purlin web and the plane of the rafter web, or the line of bend, will fall at Mm' and the angle m'MK" will l>e the angle which it makes with a horizontal line. The complement of this angle is shown at Mm'a' and is the required angle 1". 66 HIP AND VALLEY RAFTERS B" Z'andZ" FIG. 38. Case IV FIG. 39. WEB CONNECTION. For separate layouts see Figs. 35, 36, 37 and 38. Case V FIG. 40. WEB CONNECTION. ANGLE Z' (Fig. 38), in the plane of the roof, between the center line of horizontal line MK", the center line -of the rafter will fall at Mq', and the valley rafter and a horizontal line. the angle K"Mq' will be the required angle Z'. If the plane of the roof is revolved into a horizontal position about the ( 1IAITKH VI. GRAPHIC Ml Tl|o|i 1,7 X and X" Fia. 41. CASE I Fio. 42. TEE PtRiJN CONNECTION. For doparate layout for angles Z' and Z" see Fig. 38 (Web Connection). TEE PURLIN CONNECTION. ANCI.K .V (Fig. 41), between the plane of the roof and the plane of the its H trace KV, s' will fall at (', and the line of intersection bet\v< m top flange of the valley rafter, or the angle of bend. the normal plane and the plane of the roof will fall at K'l'. The A plane through any point K' in the H trace of the roof plane, normal angle t'K'r 1 between this line and a horizontal line will IK- the required to the tlai\i;r of the valley rafter would cut the flange in a horizontal line angle X'. through the point s'. If this normal plane is revolved into H about ANGLE Z' (Fig. 38), the same as for web connection, p. 66. 68 HIP AND VALLEY RAFTERS Pitches J and 30- Angle. Slope. Logarithm. Flange connections. 5 -JO* w"=8^ x' =31 x" = 2f 2/'=4 *"-3A Sine. Cosine. Tangent. A' A" a' =8 a" = 6{f 9.74406 9.69897 9.92015 9.93753 9.82391 9.76144 Web connections. w'=4H w"=2ft i= z"=8i 2/' =7H log y' =9.80653 y" = 5t\ log y" = 9. 63650 i=m 2" = 9 C II c = 10| h = 5l 9.81601 9.87848 9.96214 9.93753 9.63992 Tee connections. x' = 5 A x" = 4ft | .... 2" = 9 Pitches 3 and ^ Angle. Slope. Logarithm. Flange connections. i =11 ^ u>" = 7f x' =4A x" = \ti 2/=4 j/"=2| Sine. Cosine. Tangent. A' A" a' =8 a" = 6 9.74406 9.65051 9.92015 9.95154 9.82391 9.69897 Web connections. w' =5tf w" = 2f s ? =1 tt x" = 7| j/' =8|log!/' =9.86900 j," = 4 logy" = 9. 52557 2' =1011 "-8* C H c = 9 A = 4H 9.77815 9.90309 9.96777 9.87506 9.60206 Tee connections. *' =5 ^"=3ft 2 ' =io "-8* P tches IT and t- Logarithm. Flange to' =9H x' =5 2/'=3M Sine. Cosine. Tangent. connections. >" = 6ft *"- 2/" = 2A 4' 4" a' =8 a"=4H 9.74406 9.56983 9.92015 9.96777 9.82391 9.60206 Web connections. >'=7H w"=H x' =HH x" = 6| y' = Hi logy' =9.96591 y"=2ft log j/" = 9. 34798 2' =8| 2" = 6M Q c-7A 9 71138 9.93323 9.77815 x' =6J 2' -8| If A-4i 9.97585 9.53529 Tee connections. r" = 2A z" = 6ti TAIU.KS Pi U.U. 4 and \. Anle. -...),- Ixumrilhm. i ,,, u>'-8A i' -5| *'-3A COMM. Tucwt. w"-5| /"-I V"-1A A' A" a' -8 a"-4 0.74406 9.50000 0.02015 0.07712 0.82301 0.52288 Web connection*. ' -9 >"- z' -101 z"-5| i -10JJ logy' -0.04509 "-U logv"-9. 19807 '-7A "-5H C c-6 0.65051 0.05154 0.60807 z' -61 i' -7A TM 11 A-3A 0.08151 0.47442 z"-Hl *"-5li Pitch* 30 d 1 . Antic. Slop.. LocAnthm. i ; mm .T-IH ir ic"-8I x'-3A x"-2A V'-3A y"-2I BH COKM,. Tu^Bt. .1' A" '-H a"-6 0.69807 0.65051 0.03753 0.95154 0.76144 0.69807 Wb onnitioM ' -4A u>"-2| p-IO| x"-8H y' -6f|log/ -9.70144 y"-4| log y"-9. 58804 *' -12 "-OA C II c-lOj A-4A 0.81601 9.87848 9.07100 0.03753 0.57745 Tea . MM '' x'-4i x"-3H s' -12 *"-OA PitcbM30udi- Anile. i Lacarilbm. 1 ' Ml u>' - 10J tr"-7A *'-4A *"-l| '-3| "-2A Mb COM*. Taocmt. A' .1 a'-6 a"-4 9.60807 9.56983 0.03753 0.96777 9.76144 9.60206 M OOUMCtioU. w' -6A U>"-1| i'-HH z"-7J y' -Stilogy' -9.85835 y"-3A log v"-9. 41045 ' -0| *"-7| C // c-8A *-3}| 9.75540 0.01487 97771 9.84062 9.51693 Tee . 01 :.. MM '-5A i"-2| z' -Of x"-7l 70 HIP AND VALLEY RAFTERS Pitches 30 and 6- Angle. Slope. Logarithm. Flange connections. w' = 81 w"-| z'=4f z"=H y'=3A >r"-i Sine. Cosine. Tangent. A' A" a'=6M a" = 4 9.69897 9.50000 9.93753 9.97712 9.76144 9.52288 Web connections. W =7 w"=lA *' =10 A *" = 6A y' =10f logy' =9.93753 j/" = 2A log j/" = 9. 26144 2 ' =8 *"-8A C H c=m A = 3A 9.69897 9.93753 9.98262 ^9.76144 9.46041 Tee connections. x' =51 *"-m 2' =8 2"=6ft Pitches 1 and i- Angle. Slope. Logarithm. Flange connections. W =11 A w"=m x' =3^ *"-! y> =3 2/" = 2ft Sine. Cosine. Tangent. A' A" o' =6 a" = 4H 9.65051 9.56983 9.95154 9.96777 9.69897 9.60206 Web connections. w' =4H w" = 2& p-lltt z" = 8ft y' =6ttlog2/' =9.74742 y" = 3A log y" = 9. 47292 2' =10J "-8 C H c=9f h = 3l 9.79567 9.89258 9.97979 9.90309 9.49464 Tee connections. x' =4^ i"=2J 2' =10^ 2" = 8M Pitches i and i- Angle. Slope. Logarithm. Flange connections. w' =9| w" = "\ x' =3| *"=H J/' = 2I /'-! Sine. Cosine. Tangent. A' A" a' =6 o" = 4 9.65051 9.50000 9.95154 9.97712 9.69897 9.52288 Web connections. w' =6 w"-lft *' =io| *" = 7A y' =8 A logy' =9.82660 y" = 2i log y" = 9. 32391 2'=8B 2" = 7A C H c = 8 A = 3,^ 9.74406 9.92015 9.98391 9.82391 9.44303 Tee connections. *'=4 x"=2J 2' =8 2" = 7A TABLES 71 PitehMiud 4- Antle. -,;.- Logarithm. 1 ,,.. io' -nj w"-9A x'-2| i"-lj *'-2| "-! CoriM. TUCMI. A' A" o'-4H a"-4 0.50663 9. 60000 9.96777 9.97712 9.60206 9.52288 M BOMMrtlOM. w'-3tt u,"-lH I'-llH x"-9A y' -51 logy' -9.64901 "- 3 A logy" -9. 42082 i' -10] "-9J C 11 c-10 *-3A 9.80631 9.88549 9.98621 9.92082 9.40837 Tm MM .. x'-3A z"-2} t' -101 t"-9J I qul pitch.. i- Logarithm. 1 , ,,. \-. i Slap*. M Caria. TucMt. .-..,,-,,,. io-9A z-3J V-3H A 0-8 9.74406 9.92015 9.82391 W*b tc-3fj i-8l y-6|iloR V -9. 74406 (-10 C-45" c-12 9 84949 9 84949 TM z-51 f-10 // A -5H 9 95642 9 67340 Kqiml pilehw 30- Angle. Slopr. Locu-ithm. i mm COOMCtWO.. v-91 z-2I V-3A .-:,,, Cora*. Tunat. A a = 6 9.69897 9.93753 9.76144 W*b COOBCtkMW. U7-3H z-9A V-6 log y-9. 69897 (-101 C-45 // c-12 A-4I 9.84949 9.84949 9.96653 TM *-4A (-10| 9.61093 72 HIP AND VALLEY RAFTERS B qual pitches I- Logarithm. Flange ' Sine. Cosine. Tangent. connections. w 10f 5 x-2fg j/-3 A = 6 9.65051 9.95154 9.69897 Web connections. u) = 3| z = 9H 2/ = 5|logi/ = 9.65051 z = 10| C-45 c = 12 9.84949 9 84949 Tee' z = 4 z = 10f H h=4l 9.97442 9.54846 Equal pitches 5 Angle. Slope. Logarithm. Flange connections. s = 2^ P-3| Sine. Cosine. Tangent. A o=4H 9.56983 9.96777 9.60206 Web connections. W-SH x = 10A !/ = 4A log y=9. 56983 f-lli C = 45 H c = 12 A = 3f 9.84949 9.84949 9.98328 Tee connections. *3t f-lH 9.45155 Equal pitches ff Angle. Slope. Logarithm. Flange connections. w = ll| |f-2 Sine. Cosine. Tangent. x-\l A = 4 9.50000 9.97712 9.52288 Web connections. w = 2& x = 10J y=Sn log y = 9. 50000 2 = 111 C = 45 H c = 12 A=2if 9.84949 9.84949 9.98826 Tee connections. x = 2J 2 = 111 9.37237 'AT THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL. INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. 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