NRLF AND VALLEY DETAILS, FORMULAE AND GRAPHICS - .-SYLES iOF HIP RAFTER CONNECTIONS HIT HIT IIII Illl HIP AND VALLEY DESIGN DETAILS, FORMULAE AND GRAPHICS ROOFS HOPPERS AND PIPE LINES BY H. L. McKIBBEN and L. E. GRAY Edited by J. E. BANKS, Engineer Bureau of Standards American Bridge Company Published by J. E. BANKS Ambridge. Pa. Price. $2.00 Postpaid Copyright, 1912, by H. L. McKIBBBN and L. E. GRAY Ambridge, Pennsylvania First Edition. Second Thousand, November 1, 1913 PREFACE The difficulty of making working shop drawings for roof connections at Hip and Valley is appreciated by Structural Engineers. This book has been prepared to cover practical working details for such construction and to present the analytic and graphic processes needful for their development. From the presentation of the designs here given, Engineers and Architects can determine the style of connection adapted to their demands readily and can specify the same for the structures they have in charge. To Draftsmen the treatment of the subject will especially appeal, resulting to them in a saving of extra labor and concern. Students will discover the practical training in descriptive geometry and trigonometry as applied to active engineering to be exceptionally valuable. Class room work in the proof of the formulae is recommended to Engineering Schools. H. L. McKlBBEN. L. E. GRAY. Engineers with American Bridge Co. 282177 FOREWORD. On pages 3, 4 and 5 are shown working details for styles A, B, C, D, E and F, or six methods of connection to Hip Rafters from which to select the one that conforms best to the adjoining framing. On pages 6, 7 and 8 are found working details for styles A, B, C, D, E and F, or six methods of similar connections to Valley Rafters from which to choose the one most desirable. * A sketch appears with each style of detail showing the position of the purlin in the main roof section, and small sub-formulae showing solutions for the variables yi to yio, with special attention to yi and yz. After making selection of style desired, the detailer should solve the angles required as shown in details; i. e., L8 and L9 are needed in style C. No other angles need be found ; only those involved in the style chosen. Solution of these angles can be readily made from general formulae on page 10, if the worker be familiar with Trigonometry and Logarithms; if not, results may be obtained from the simple graphics given on pages 11, 12 and 13, making the problem easy for the detailer who is not familiar with formulae. If the case in hand be one that is covered by the tabulated solutions on pages 14, 15 and 16, the worker can take from those tables any or all variables which develop in a roof of pitch 1 /5, V4, 1/3, 30 or 55, if the angle B in plan is 30, 45 or 50. These tabulated solutions give the values of the variables for designs in most common use without the necessity of solving any angles whatever; but the formulae on page 10 and graphics on pages 11, 12 and 13 furnish data for solving angles for any roof pitch and all possible positions of rafter. In styles A, B, C and E the roof line being above the main truss metal line, the worker will need to use formulae on page 9 to locate working point "d." The authors desire to call especial attention to the following: 1st. The known data are in all cases the main roof pitch or Angle A. The position of Hip or Valley Rafter, Angle B, which is the angle formed by rafter and main truss as seen in Plan looking directly perpendicular to lower side of Angle A. No other data than A and B as above described is ever required. Throughout both details and graphics the letter "d" refers always to the same working point; the marks di and da refer also to this same point, viewed from different positions. 2d. All formulae on page 10 are logarithmic, and in terms of tangent functions. 3d. Use of the graphics on pages 11, 12 and 13 expedite the work and give accurate results. 4th. A short method of graphics for solution of Angles L5, L6 and L8 also appears on page 10, which may be used after solving L3 and L4, if desired. 5th. For those desiring to follow out the proofs given on pages 21 to 29, the four major intersecting planes involved are as follows (see page 10) : ROOF PLANE. Seen in Elevation of Truss as line ab. Seen in Plan as inclined surface ai, bi, n. PURLIN WEB PLANE. Seen in Elevation of Truss as line c, d. Seen in Plan as inclined surface di, ci, ei. RAFTER WEB PLANE. Seen in Elevation of Rafter as surface ra, ca, bs. Seen in Plan as line n, bi. RAFTER FLANGE PLANE. Seen in Elevation of Rafter as line rz, ba. Seen in Plan as inclined surface n, bi, ei. 6th. Other formulae which may be used if desired are as follows: Cos L3=Cos R Cos L1 Sec A. Tan L5=Cos A Tan B Cos L1. Tan L5=Tan L2 Cos L1. Tan A7=Sin A Sin B Cos L4. Tan L7=Cos L2 Tan L10. HOPPERS, BINS AND CHUTES (FORMS OF VALLEY CONSTRUCTION). Details for these structures are left to the judgment of the detailer and are usually governed by the main design. The solution of the bend on connecting plate at dihedral intersections is the only difficulty for most draftsmen. Both formulae and graphics are provided on page 17 for ready use. PIPE LINES. Large Pipe Lines often require both horizontal and vertical change of direction at the same point, which condition may give rise to annoying details. Two separate bends are more expensive and produce greater friction on the flow than a single resultant bend. Careful attention to resultant angles "X" and detail angles "Y" will save much trouble in fabrication and improve the efficiency of the finished structure. HIP RAFTER DETAILS Xi=gwidthof rafter flange or more . yi =Xi tanB sinR JPurlln may be cut on atted line if desired < g2 x V \ \\ < , I i- .-.-.. J W 1 L2 2ut flange to clear ,IL / STYLE A. Xi = gwidth of rafter flange or more yi = Xi tan B sin R y2=yi tan 3 STYLE B. HIP RAFTER DETAILS width of rafter flange or more. yi = Xi tan B sin R ya=Xi tan B cos R y4=Xi sec B Developed plate STYLE C. d k 1 ( ( V4, Ii-y4 ? < 8 4 5 . .---- ^ -''r i ( < r \ ^ t d, \ .'- ?' fe J Xi=2 width of rafter flange or more. X2, taken so that bend line will clear connection angle. yi=Xi tan B sin R ys=X2 tan B cos R ye=X2 sec B STYLE D. Developed plate 5 HIP RAFTER DETAILS Xi = 2 width of rafter flange or more, t = thickness of bent plate. yi = Xi tan B sin R ya=Xi tan B cos R sec B STYLE E. width of rafter flange or more, thickness of rafter web. yi = Xi tan B sin R V2=*yi tan L3 y4=,Xi sec B zi = t sec L5 flange cut to clear. STYLE F. VALLEY RAFTER DETAILS -Developed plate STYLE A. 82 L1 plate STYLE B. VALLEY RAFTER DETAILS Xa taken so that Bend Line will clear connection angle. y?=.X3 tanBcosR secB STYLE C. X4 taken so that Bend Line will clear connection angle yg = X4 tan B cos R yio = X4 secB = Xi secB STYLE D. 8 VALLEY RAFTER DETAILS Xi=a width of rafter flange or more. t=thickness of bent plate. yi = Xi tan B sin R V3=Xi tan B cos R X4=Xi sec B STYLE E. Xi= width of rafter flange. ti= thickness of rafter web. t2=thickness of bent plate. y4=Xi sec B Z2=ti sec L5+t2 tan L5 STYLE F. Purlin and bent plate must clear rafter flange as shown. ivelope'd plate bend line LI RELATIONS OF ROOF LINE TO WORKING POINTS USED IN FORMULAE 9 Xi or Xi=5 width of rafter flange or more. P=actual depth of purlin. D=P. sec L10 yi = Xi tan LIO D'=P'sec LW yt=D+yi-D' Xi^y/cot LIO' y2=yi tan L3 y2=yi'tan U? w=D tan L3 w=D'tan L3 r N=P tan LI N = P'tan Lf y=D-D' y2-y tan L3' w =D tan L3 w'=D' tan L3' N =P tan L1 N' = P' tan LT 10 GENERAL FORMULAE Given L3 and L4 \]/L3 to find L5. \fl FIG. 3 Given L3 and to find L6. F1G.1 FIG. 4 Given L3 and L4 to find LS. In figs. 2, 3 and 4, the line xx is the intersection of Rafter Flange and Rafter Web. GIVEN A=Pitch of Roof. B=Angle between Truss and Rafter in plan. FORMULAE Tan R =tanA cosB Tan L /=sin A tan B Tan Z.2=cosA tanB Tan 3=sln A cos A sin B tan B Tan L4=cos a A tan B sec R Tan 5=cos L3 tan L4 Tan /.ff=tan L3 cos L4 Tan /L7=tanB sin R cos L2 Tan L8=cos A tan B Tan L9=tan B sin R Tan/./^=tan B sin R 11 GRAPHIC SOLUTION OF ANGLES bl,CI A = PITCH OF ROOF B = ANCLE BETWEEN TRUSS AND RAFTER IN PLAN R= PITCH OF RAFTER Tan R Tan A Cos B L1 BEVEL ON PURLIN WEB PLANE MADE BY INTERSECTION OF RAFTER WEB PLANE FORMULA Tan i7 = A Tan B GRAPHICS Draw d, c l a, b Draw d, dl | b, bl Revolve d to f , about c Draw f, fl | d, dl Draw dl, fl d, dl Connect fi with ci L2 BEVEL ON ROOF PLANE MADE BY INTERSECTION OF RAFTER WEB PLANE FORMULA Tan B Cos A GRAPHICS Revolve b to g about a Draw g, g\ | b, bi Extend ai, bl to g\ Connect g\ with ri L3 BEVEL ON RAFTER WEB PLANE MADE BY INTERSECTION OF PURLIN WEB PLANE FORMULA Tan Z.3 Sin A Cos A Sin B Tan B GRAPHICS Draw d, c i a, b Draw d, dl | b, bl Draw di, d2 1 bi, b2 Connect d2 with C2 D1.CI b,--' bia 12 GRAPHIC SOLUTION OF ANGLES L4 BEVEL ON RAFTER FLANGE PLANE MADE BY INTER- SECTION OF PURLIN WEB PLANE FORMULA Tan/14 = Cos2 A Tan B Sec R GRAPHICS Draw d, c i a, b Draw d, di || b, bl Drawdi,d2 ||bi,b2 Revolve d2 to h, about r2 Draw h, hi ||bi, b2 Extend b, bl to intersect n, r2 at el Connecfel with hi L5 COMPLEMENT OF ANGLE BE- TWEEN PURLIN WEB PLANE AND RAFTER WEB PLANE FORMULA Tan L5 = Cos L3 Tan L.4 GRAPHICS Draw d, c l a, b Draw d, dl || b, bl Draw dt,d2 || bl, b2 Draw d2, Z2 l b2, 02 Draw Z2, zs 1 d2, 02 Revolve 23 to Z4 about Z2 Draw Z4, zs II bl, b2 Locate ze at intersection of d, dl and ci, C2 Connect zs with ze L6 COMPLEMENT OF ANGLE BE- TWEEN PURLIN WEB PLANE AND RAFTER FLANGE PLANE FORMULA Tan L6 = Tan L3 Cos L.4 GRAPHICS Draw d, c l a, b Draw d, di || b, bi and extend to v Extend b, bi to ei Connect el with di Draw es, V3 || ei, dl Draw ei, 03 and di, V3 l ei, dl Take V3, d3 = d, v Connect es with d3 Through n, draw V4, vs l ei, di Draw V4, p l 63, d3 Revolve p to ve about v4 Draw ve, v? J. ei, di Connect v? with n and vs 13 L7 BEVEL ON PURLIN WEB PLANE MADE BY RAFTER FLANGE PLANE FORMULA Tan Z.7 Tan B Sin B Cos L.2 GRAPHICS Draw d, c i a, b Draw d, di || b, bi Revolve d to f about c Draw f, fi || b, bi Draw di, fi j. d, di Extend b, bi to ei Connect ei with fi L8 ANGLE BETWEEN PURLIN WEB PLANE AND A PLANE PERPENDICULAR TO BOTH RAFTER WEB PLANE AND RAFTER FLANGE PLANE FORMULA Tan L8 = GRAPHICS Tan B, Cos A Draw d, c l a, b Draw d, di || b, bi Draw d, m l b, c Draw m, s l d, c Revolve s to n about m Draw n, p || d, di Draw di p j. d, di Draw di, v l ri, bi to intersect b, bi at v Connect v with p L9 BEVEL ON PLANE PERPENDICULAR TO BOTH RAFTER WEB PLANE AND RAFTER FLANGE PLANE MADE BY INTERSECTION OF PURLIN WEB PLANE FORMULA Tan L9 = Tan B Sin R GRAPHICS Draw d, c l a, b Draw d, di || b, bi Draw di, d2 || bi, b2 Draw d2, k l r2, b2 Draw k, ki JL ri, bi Revolve k to j about ,d2 Drawj, ji || k, ki Draw ki, ji Ik, ki Connect d i withji L10 ANGLE BETWEEN ROOF PLANE AND RAFTER FLANGE PLANE FORMULA Tan L.10 = Tan B Sin R GRAPHICS Take p any point on b2, r2 Draw p, 1 1 b2. r2 Revolve p to u about t Draw t, tl || bl, b2 Draw u, ui || bi, b2 Locate a at intersection of t, ti and a, n Connect ui with s bi ci b c 14 SOLUTIONS, FIVE ORDINARY ROOF PITCHES B=30 A 1/5 PITCH 1/4 PITCH 30 PITCH 1/3 PITCH 65 PITCH >< "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. R L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 L 9 L 10 4%2 2%8 6%2 Uie 2 2%2 2%2 9.53959 9.33127 9.72921 8.99801 9.72159 9.71945 8.94484 9.22157 9.72921 9.27642 9.27642 3% 2 6y 82 6 2% 2% 9.63650 9.41195 9.71298 9.06247 9.70185 9.69897 9.01344 9.30929 9.71298 9.36062 9.36062 6 6 5i%8 5% 6 3%2 3%2 9.69897 9.46041 9.69897 9.09691 9.68496 9.68159 9.05119 9.36350 9.69897 9.41195 9.41195 5% 517/82 3% 58/4 31 %2 9.76144 9.50550 9.68159 9.12462 9.66421 9.66039 9.08268 9.41532 9.68159 9.46041 9.46041 1427/32 38y 3 2 1% 3% 31%2 5% 5% 0.09230 9.67480 9.52003 9.13237 9.48016 9.47620 9.11340 9.62961 9.52003 9.65221 9.65221 "S" = Corresponding Bevels or Slopes to Base of 12 inches. A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 66 PITCH X 1 iH 2K 3 ' t/i 6K 1H &A 3 ' fcK 6M IK 2^ 3 4/4 6^ 1K2K 3 ' iKsM IX i Y 1 Y 2 Y 3 Y 4 T 9 2 A H if! TV TV 2% 3^f 11 A 2i i; MS ^ 1 1 A II if! A iV 2% H A it! Sg"^ fl .> H 2/4 *;! ^16 5 TV ft HI 21 A iA H A IT'S iA 2f\ *f! 1^ 9w] * 5 7 * TV TV if! If A 2H I lA A iii 7A 1 1 A H HI 1/8 ^-3^ A T 3 ? 29 i 3 72 !?2| iji[: 7, 7 ^ For Purlins not exceeding 12" Depth, with 4 l /< values of X 2 , X 3 , X4 , give good Results. Y 5 to Y 1 ' CLEARANCES FOR " Connection Clearance, the following assigned derived therefrom. 9 INCH PURLIN A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 65 PITCH X X2= X 3 = 6 X4= X 2 = 6% X 3 = 6 ^ 7 5 X4= X 2 = X 3 = 5 X4= X 2 = 5 4M Y 5 Y 6 Y 7 Y 8 Y O Y1O 3i7 /82 3'!''i,i 3% 8%2 3% 3%2 725, 1" 2 22^2 2lB/g 2% 5 2 %2 52 %2 2^ ^^o 2% ; 2^4 1% &7J.6 CLEARANCES FOR 12 INCH PURLIN A 1/5 PITCH 1/4 PITCH 30 PITCH 1/3 PITCH 65 PITCH X X 2 = 7 X 3 = X4= X 2 = w X 2 = 8 X 3 = 35 X 2 = Xs 7/2 X4= 1O Xs= ^ Y 6 Y 6 Y 7 Y a Y 9 Y1O 8% 2 3 8 y 82 9* 9is/ 16 36' 82^2 im. 4% 2 3 s y 8 2 3% 82y 82 3% 82y 82 ? 5% 8 82^2 2% : | 14 / 1% 5 3 /io n 15 SOLUTIONS, FIVE ORDINARY ROOF PITCHES A 1/5 PITCH 1/4 PITCH 30 PITCH 1/3 PITCH 55 PITCH x "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. R L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 L 9 L 10 21^6 10% sy 4 3% 9.45154 9.56983 9.96777 9.38709 9.95225 9.93971 9.25914 9.29984 9.96777 9.43483 9.43483 5% 108/4 3% 10^6 10% 4 4 9.54845 9.65051 9.95154 9.45154 9.92867 9.91195 9.33379 9.39524 9.95154 9.52288 9.52288 4% 928/82 9.61092 9.69897 9.93753 9.48599 9.90853 9.88908 9.37641 9.45593 9.93753 9.57745 9.57745 621,82 10 93/16 823/32 10 5% 9.67339 9.74406 9.92015 9.51369 9.88387 9.86190 9.41357 9.51558 9.92015 9.62982 9.62982 6% 4 5% 3% 8i% 2 0.00426 9.91336 9.75859 9.52144 9.66984 9.64710 9.47851 9.78984 9.75859 9.85160 9.85160 " S" = Corresponding Bevels or Slopes to Base of 12 inches. A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 55 PITCH X 1 1 3 A 2^; 3 4K6K 1K2K 2 3 4M6M iH 2H 3 WeM 1^2^ 1 9 -I 1 3 8 * 4 r5 - H H *af||8f| 1 *M 6 221 3" 2" 8ff iKafc J 3 1 i 3 4|f fl o2 7 Y 1 Y 2 Y 3 Y 4 H H iA.iH (L ] 3 ! 9 i i 4 A i- 3 21 ;( \ 1 [ i -j 9 j- i 2 27 |X iH 4 6 19 2% ti i 3;', 11 I/ 23 72 72 T2- A il iHa> 2^31 11 19 ; 23 For Purlins not exceeding 12" Depth, with 4>" Connection Clearance, values of X 2 , X 3 , X 4 give good Results. Y 5 to Yj derived therefrom. CLEARANCES FOR 9 INCH PURLIN the following assigned A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 55 PITCH \7 /\ X 2 = 1 X 3 = 6% 4y 2 fv, ^ X 2 = 8 X 3 = 6% X4= X 2 = X 3 = 6V 2 4 4 y; X 2 = 10 ^ X4= 4% Y 6 Y 6 Y 7 Y 8 Y 9 Y1O 6% 7%, 1019/aa m. 71%o 14% 2 12M.2 5^2 58Ae ,* S 72%2 72%S 6% 4y 4 6% 4%2 6% 6% 3% a 6% I CLEARANCES FOR 12 INCH PURLIN A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 55 PITCH 8 X 3 = X4= X 2 = X 3 = *; X 2 = 9 X 3 = ^ 9M X 3 = S X 2 = X 3 = X4= Y 5 Y 6 Y 7 Y 8 Y 9 Y1O 711/16 8 81 y 3 2 13^6 8% 2 ley* 7%2 101%2 6i %6 62^82 59/82 1019/82 6% 1019/82 6% 3%2 6% 411,62 4y 4 6% 6% 16 SOLUTIONS, FIVE ORDINARY ROOF PITCHES B=50 A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 55 PITCH X "S" Log . Tan. "S" Log . Tan. "S" Log. Tan. "S" Log. Tan. "S" Log. Tan. R L 1 L 2 L 3 L 4 L 5 L 6 L 7 L 8 L 9 L 10 3%2 13% 3 122% 2 lite 21 /32 2% 13%2 3/4e 3Ae 9.41013 9.64602 0.04396 9.49804 0.02563 0.00511 9.33433 9.29881 0.04396 9.47241 9.47241 6!%2 4% 121/32 3%2 3 122%, 4% 9.50704 9.72670 0.02773 9.56250 0.00062 9.97344 9.41167 9.39706 0.02773 9.56188 9.56188 7%2 12% 10% 3^6 12% 9.56951 9.77516 0.01372 9.59694 9.97927 9.94775 9.45655 9.46025 0.01372 9.61767 9.61767 Mi 11 2 %2 5^6 3% 4 11 2 %2 5% 6% 9.63198 9.82024 9.99634 9.62465 9.95309 9.91761 9.49632 9.52286 9.99634 9.67155 9.67155 11 11 2 %2 5%2 6% 5% 8 9.96284 9.98955 9.83478 9.63240 9.72610 9.68942 9.57824 9.82305 9.83478 9.90630 9.90630 " S "=Corresponding Bevels or Slopes to Base of 12 inches. A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 55 PITCH X 1 lM A A iff 2M 2} 3 / fc 3J 4 3 4 M6M lM2M 3 4^ ly 9 ? A *H sM l l A2 l /2 3 4^6>i 1^2^ 3 4 MQM 2 2i^ H IK ifcU 4 3 J4J<6 Y 1 Y 2 Y 3 Y 4 i i! i A 3->S **H M H ^29 7 7 *32 ! '32" A If iA A H M H 9f! I/ -iq I/ 11 -1 1 A 11 H Is is 5 II lAHiij A ^|H; 2. A: 3> 4f 2 6% 9 1| For Purlins not exceeding 12" Depth, with 4%" Connection Clearance, the following assigned values of X 2 , X 3 , X 4 , give good results. Y 5 to Yi derived therefrom. CLEARANCES FOR 9 INCH PURLIN A 1/5 PITCH 1/4 PITCH 3O PITCH 1/3 PITCH 65 PITCH \7 X?= X 5K X 4 = 4/2 X 2 = 8 < X4= X 2 = X 3 = X4= X 2 = 9 X 3 = *A J X 3 = Xt= 5M 4H Y 5 Y 6 Y 7 Y 8 Y 9 Y10 8% m. 13% 2 14 9 7 gl, z a =d'l, z 2 =(d, m) Tan B But d, m=Sin 2 A .'. gl, zs=Sin 2 A Tan B b, g=b, d =(d, c) Tan A =Sin A Tan A ,_Sin 2 A Tan B Sin A Tan A Sin A Tan B Tan A =Cos A Tan B Tangent L4 Tangent L7 fl.kl fl, kl=c,'f =d, c =Sin A a, d=Cos A r2, d2=(a, d) Sec L2 =Cos A Sec L2 T2, k=(r2, d2) Sec R =Cos A Sec L2 Sec R el, kl=(r2, k) Csc B =Cos A Sec L2 Sec R Csc B Tan , 7 Sin A L/ Cos A Sec L2 Sec R Csc B =Tan A Cos L2 Cos R Sin B 1") TanB Cos BCosZ.2 =TanA(^ r Tan A Sin R Tan B Cos B CosZ.2 Tan A Cos B =Sin R Tan B Cos L2 Tangent L8 nl, kl=n,m =m, w =(d, m) Cos A =Sin 2 A Cos A kl, v=(dl, kl)CotB =(d, m) Cot B =Sin 2 A Cot B A Cos A .Tan L8= Sin 2 A Cot B =Cos A Tan B Tangent L9 dl, Z4=d2, j =d2,k jl, zi=kl, z- =(dl, Z B ) Tan B dl, zs=(d2, k) Sin R .'.Jlf Z4=(d2, k) Sin R Tan B 2, k) Sin R Tan B .Tan ^ , =Sin R Tan B Tangent L10 .Tan rl, hl=T2, h =T2, d2 =Cos' 2 A Sec B Sec R rl, el Csc B x_Cos 2 A Sec B Sec R Csc B _Cos 2 A Sin B "Cos B Cos R =Cos 2 A Tan B Sec R 8 is any point on TS, 02 hoose location such that si will fall at a tl, sl=SinB t, t2=tl, rl =(rl, si) Sin B =Tan B Sin B ul, tl=u, t =p, t =(t, 72) Sin R =Tan B Sin B Sin R Tan / ->n_Tan B Sin B Sin R ..T&nUO- SinB =Tan B Sin R 22 ANALYTIC PROOFS Tangent L3 1. 2. 3. 4. 6. 6. 7. 8. 9. Tan L3 = J=Sln 2 A C2=Tan A 81n 2 A CosB . z *-CosR Sin-A ~CosB q=(b2, 02) Sin R 10. 11. 12. 13. 14. 16. 16. 17. 18. =Tan A Sin R Sin A Sin R Cos A r=l-d Sin 2 A Cos B Cos R Sin A Sin R Cos A l=(e+z)SlnR Sin R Sin R Sln 2 A Sin A Sin R Cos A Sin R CosB 23 ANALYTIC PROOFS Sin 2 A Cos A Sin A Sin R Cos B Cos R Cos B~Cos R~Cos A 19 - Sin R CosB Sin 2 A Cos A-Sln A Cos 2 R Tan R Cos B Cos A Cos 2 R Tan R Sin 2 A Cos A Sin A Cos 2 R Tan A Cos 2 B Cos A Cos 2 R Tan A Cos B _ A Q 42. Sin 2 A Cos 2 R Cos 2 B Cos A Sin A ,-<- A 22 - COS A Cos A Cos 2 R Sin A Cos B Cos A 23. 24. Cos 2 A Sin 2 A-Sin 2 A Cos 2 R Cos 2 B Cos A Cos A Cos^R Sin A Cos B Cos A Cos 2 A Sln 2 A-Sln 2 A Cos 2 R Cos 2 B Cos A Cos 2 R Sin A Cos B 25. But, re, 02 = i/ 1 . + Tan 2 A V Cos 2 B 26. y Cos ii B + Cos 2 A _ ./Cos 2 A + Sin 2 A Cos 2 B 27> - " Cos 2 A Cos 2 B no V Sin 3 A Cos 2 B+Cos 2 A 28 - = cosrsrcos^B 29. Cos R= 30. 'Sin 2 A Cos 2 B+Cos 2 A Cos A Cos B Cos A l/Sin 2 A Cos 2 B+Cos 2 A Hence by substitution in No. 24. Sin 2 A Cos 2 A-Sin 2 A 0< ZA 08 . o. ^ Cos 2 B 31. Tan 13=- -y - VSin 2 A OoyB+Cos-A/ - 008 A Sin A COS B Sin 2 A Cos 2 A (Sin 2 A Cos 2 B+Cos 2 A)-Sin 2 A Cos 2 A Gos 2 B 32. = _ Sin 2 A Cos 2 B+Cos' 2 A _ Cos 3 A Sin A Cos B Sin 2 A Cos 2 B +Cos 2 A oo Sin A(Sin 2 A Cos 2 B+Cos 2 A)-Sin A Cos 2 B Cos A Cos B Sin A(Sln 2 A Cos 2 B+Cos 2 A-Cos 2 B) ~ ox Cos A dos~B o= _ Sin A [Cos 2 B (Sin 2 A-l) +Cos g A] Cos A Cos B o ft _ Sin A [Cos 2 B(-Cos 2 A) +Cos 2 A] Cos A Cos B Sin A(Cos 2 A-Cos a A Cos 2 B) Cos A Cos B Sin A Cos 2 A(l-Cos 2 B) a - Cos A C5s~lJ Sin A Cos 2 A Sin 2 B 39 - = Cos^rCos~B~~ 4O. = Sin A Cos A Sin B Tan B 24 ANALYTIC PROOFS Tangent L6 Refer to Page 22. 1. a=Cos A 2. b=Cos 2 A 3. c=Cos A Sin A 4. . 1 d =Sin B Cos 2 A 6. e= CosB 6. f=V / d 2 +e 2 l/Si^B Cos 4 A+Cos 2 B . Cos B Sin B 8. Let M=\/Cos 2 B+Cos 4 A Sin 2 B (for convenience) 9. 1O. Then f- Cos B Sin B 11. ,. /r-~=2 A o<^2 A j.Sin 2 B Cos 4 A+Cos 2 B *\ /OOS A oln A H s \ Cos 2 B Sin 2 B V Cos 2 A Sin 2 A Cos 2 B Sin 2 B+Sin 2 B Cos 4 A+Cos 2 B 12. Cos B Sin B 13. Let P=VCos?A Sin 2 A Cos 2 B Sin 2 B+Sin 2 B Cos 4 A+Cos 2 B 14. Til J-Ll-l 1-1 1 IB Then h- Cos B gin B 15. Sin G4= Cos A Sin A Cos B Sin B 16. P 17. Sin G 2 =f Cos 2 A Cos B M Cos B Sin B 19. Cos 2 A Sin B M! 20. Cos Ga=y- 1 O 1 Sin B 21. M Cos B Sin B Cos B 22. 23. g=d Cos Qa 25 ANALYTIC PROOFS 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. _ ( 1 ^ /Cos B\ Cos B = ErSIn~E m = g Sin G* / Cos B \ /Cos A Sin A Cos B Sin B > \ Cos A Sin A Cos 2 B P ) MP k = d Sin Ga ( 1 \ /Cos 2 A Sin B^ -^SinB/V M t Cos 2 A = M s=V k 2 +m 2 = / (^) % (^^ Sin A Cos 2 B\ 2 ~Tvrp ) = ^p- t /P 2 Cos 2 A+Cos 4 B Sin 2 A Let N=v / P 2 Cos 2 A+Cos 4 B Sin 2 N Cos A Then s= 5jp Sin Gs = ^- Cos A Sin A Cos 2 B A (for convenience) M P N Cos A Sin A Cos 2 B N Cos Ga=g- CQS 2 A - N Cos A P Cos A n = g Tan(B-Ga) Cos B Sin (B Ga) - M Sin B Cos (B-Ga) Cos B (Sin B Cos Ga-Cos B Sin Ga) M Sin B (Cos B Cos Ga+Sin B Sin Ga) 26 ANALYTIC PROOFS -,/SinjBCosB _ 47 \~ M 63 64 65 Cos B Cos 2 A Sin B Cos B (Sin B Cos B Cos B Cos 2 A Sin B) M Sin B(Cos 2 B+Cos 2 A Sin 2 B) Cos B (Cos B Cos B Cos 2 A) : M(Cos 2 B + Cos 2 A Cos 2 B(l-Cos 2 A) 60 - ~M(Cos 2 B+Cos 2 ASin 2 B) Cos 2 B Sin 2 A ~M(Cos 2 B + Cos 2 A Sin 2 B) 52. p=m Tan Ga 53. t=n p 54. =n m Tan Gs 55. y=t Cos Ga 56. =(n m Tan Ga) Cos Gs m 57. v= Cos Q 8 58. w=t Sin Gs 59. =(n m Tan Gs) Sin Gs 60. x=v+w A l ._-, m ^- +(n-m Tan Gs) Sin Gs *** OOS Gs Statement for reduction 62. Tan 0=JL (n-m Tan Gs) (Cos Gs) ^ m +(n-m Tan Gs) Sin Gs Cos Gs (n-m Tan Gs) Cos 2 Gs = m+(n-m Tan Gs) Sin Gs Cos Gs m+(n m Cos Gs/ Sln Gs Cos Gs (n Cos Gs-m Sin Gs) Cos Ga Je - -m+(n Cos Gs-m Sin Ga) Sin Gs n Cos 2 Gs m Sin Gs Cos Ga ~~m+n Cos Gs Sin Gs-m Sin 2 Gs n Cos 2 Gs-m Sin Gs Cos Q 8 ~m (1 Sin 2 Ga)+n Cos Gs Sin Gs _n Cos 2 Gs-m Sin Gs Cos Gs ~m Cos 2 Gs+n Cos Gs Sin Gs 27 ANALYTIC PROOFS __ n Cos Ga m Sin Ga ~~m Cos Gs+n Sin Gs Hence by substitution / Cos 2 B Sin 2 A \ /P Cos A\ /Cos A Sin A Cos 2 B\ /Sin A Cos 2 B\ 71 Tan /ff-\M(Cos 2 B+Cos 2 ASin 2 B)/\ N / \ _ MP / \ N / /Cos A Sin A Cos 2 B\ (P Cos A\ . / Cos 2 B Sin 2 A \/Sin A Cos 2 B\ \ MP A N / \M(Cos 2 B+Cos 2 A Sin a B)/\ N j P Cos 2 B Sin 2 A Cos A Cos A Sln 2 A _ 2 _ MN(Cos 2 B +Cos 2 A Sin 2 B) _ MP N P Cos 2 A Sin A Cos 2 B + _ Cos 4 B Sin 3 A M P N MN(Cos 2 B+Cos 2 A Sin 2 B) = P 2 Cos A Cos 2 B Sin 2 A-Cos 4 B Sln 2 A Cos A (Cos 2 B+Cos 2 A Sin 2 B) P Cos 2 A Cos 2 B Sin A(Cos 2 B+Cos 2 A Sin 2 B) +P Cos*B Sin 3 A _ Sin 2 A Cos 2 B Cos A [P 2 -Cos 2 B(Cos 2 B+Cos 2 A Sin 2 B P Sin A Cos 2 B [Cos 2 A (Cos 2 B+Cos 2 A Sin 2 B) + Cos B Si J3in A Cos A [P^Cos^ (Cos 2 B+Cos 2 A Sin 2 B)] ~ P [Cos 2 A Cos 2 B+Cos 4 A Sin B+Cos 2 B Sin 2 A] J31n A Cos A [F*-CO8*B (Cos 2 B+Cos 2 A Sin 2 B)] P [Cos 2 B (Cos 2 A+Sin 2 A)+Cos 4 A Sin 2 B] __ = Sin A Cos A [P 2 -Cos*B(Cos 2 B+Cos ii A Sin 2 B)] P [Cos 2 B+Cos*A Sin 2 B] 78. Cos L4=% 79. Sin B^Cos B 80. = p 81. .'.PCos /.4=CosB 82. r _ Cos B 83. P 2 =Cos 2 A Sin 2 A Cos 2 B Sin 2 B+Sln 2 B Cos 4 A+Cos 2 B -SinACos ACos/.4[Cos 2 ASin 2 ACos 2 B Sin 2 B+Sin 2 BCos 4 A+Cos 2 B-Cos 4 B-Cos 2 BSin 2 B Cos ? A oft. ian/.o 5 T 5 Cos B [Cos 2 B + Cos 4 A Sin 2 B] _Sin A Cos A Cos L4 [Sin 2 B Cos 2 B Cos 2 A (Sin 2 A-l)+Cos 2 B ( l-Cos 2 B) +Sin 2 B Cos 4 A] 85. Cos B [Cos 2 B+Cos 4 A Sin 2 B] Sin A Cos A Cos L4 [-Cos 4 A Sin 2 B Cos 2 B+Cos 2 B Sin 2 B+Sin 2 B Cos 4 A] T> Cos B [Cos J B+Cos*A Sin-fi] Sin A Cos A Cos L4 [Cos 4 A Sln 2 B (l-Cos 2 B)+Cos 2 B Sin 2 B] 87 ' Cos B (Cos 2 B+Cos 4 A Sin 2 B) Sin A Cos A Cos Z.4(Cos 4 A Sin 4 B+Cos 2 B Sin 2 B) 88. = 89. Cos B (Cos-B +Cos*A Sin J B) Sin A Cos A Cos L4 Sln 2 B (Cos 4 A Sln 2 B+Cos 2 B) Cos B (Cos 2 B+Cos 4 A Sin 2 B) Sin A Cos A Cos L4 Sin 2 B - CosB 91. But, Tan L3=Sin A Cos A Sin B Tan B Sin A Cos A Sln 2 B y ^- CosB 93. .'. Tan Z.ff=Cos L4 Tan L3 28 ANALYTIC PROOFS Tangent L.5 Draw d.2, 22 1 b2, ca Pass a plane thru za 1 d2, 02 This plane seen in plan view appears as surface Z2, ze, ZT Revolve this plane about zs, ZB to TA This plane then seen in plan view appears as surface Z2, ze. 2. 3. 4. 6. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Zl. Z5 =: Z2, Z8 =(z2, d2> Cos Gl =(ml, dl) Cos Gl =(d. m) Sec B Cos Gl =Sin 2 A Sec B Cos Gl zl. za=(d, m) Csc B =Sin 2 A Csc B Tan / c Sln 2 A Sec B Cos Gl Sin 2 A Csc B Sln B Cos Gl Cos B =Tan B Cos Gl But Cos GI= (Seepage 1O) Sln A Cos A f, c Sec L1 ln A Cos A ~Sin A Sec L1 Cos A Cos L 1 Hence Tan A5=Tan B Cos A Cos LI 18. 19. 2O. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. But Cos . C2> C2 f zs=C2. T2 Sin R _Sln R Cos B C2, d2=Sin A Sec L1 Sin R - Cos~BI ---- POR . . Cos Cos Ll Sin R Cos L1 Sin A Cos B And Tan 4=Cos 2 A Tan B Sec R COS L3 _Tan R Cos L1 Cos A Tan B Sin A CoifB _Tan A Cos B Cos L1 Cos' 2 A Tan B Sin A Cos B _Sin A Cos B Cos L1 Cos 2 A Tan B Cos A Sin A Cos B =Cos A Tan B Cos L1 .'.Tan L5=Cos L3 Tan L4 PROOF FOR THE 90 USED JN STY^ES^ AND D 29 EXPLANATION FOR 9O BEND LINE ON STYLES C AND D. Purlin Web Plane seen from Elevation of Main Roof is Line cd. in Plan View is Inclined Surface di t zi, ci. ki. Rafter Lug Plane seen in Rafter Elevation is Line d2, k. " Plan View is Inclined Surface di, za, ki. These Two Planes produced Intersect on Line di, ki. Hence if ki, ci equals in length di, zi, then will angle kt, dii zi be 9O in all cases. STATEMENT OF VALUES. a c= Unity cd=Sin A ad=CosA zl, al=Cos 2 A zl, ml=Sin'-A dl, ml=Sin-'A SecB zl. dl=Sin-'A TanB dl, rl.rs, Z4=Cos 2 A SecB 6.2, r>=Cos 2 A SecB SecR da, zt=d. zs=CosA Sin A z*. k=CosA Sin A TanR da, b2=Sin 2 A SecB SecR k, c2=Sin 2 A SecB Cos A Sin A TanR kl, cl=(k, ca) CscB =[Sin2ASecB-CosASinA TanR] Csc B PROOF. 1 . (Sin* A Sec B-Cos A Sin A Tan R) Csc B=Sin 2 A Tan B 2. CosB -Cos A Sin A Tan R -=Sin 2 A TanB SinB Sin 2 A CosASin ACosBTan R Cos B"Bin B =Sin2A Tan B 4. Sin 2 A-Cos A Sin A CosB Tan R=Sin 2 A TanB CosB Sin B 5. Sin 2 A-Cos A Sin A CosB Tan R=Sin 2 A Sin 2 B 6. Tan R=Tan A Cos B. 7. Sin* A-Cos A Sin A Cos B Tan A Cos B=Sin 2 A Sin2B. 8. Sin s A _CosASlnACos 2 B SinA^.^ sln2B 9. Sln2A-Sin 2 A Cos 2 B=Sin 2 A Sin 2 B 10. I Cos 2 B=Sin 2 B 11. l=Sin2B+Cos 2 B or 1=1 Hence Eq. 1 1 being true, proves Eq. 1 to be true. 30 ANALYTIC . PROOF OF ANGLE X SECOND CASE OF PIPE LINE =C 2 + B 2 C = Sin Al + Sin Aa B =^7(008 Al + Cos Aa Cos B) 2 + (Cos An Sin B) 2 P = v /(Sln Al + Sin Aa) 2 + (Cos Al + Cos A2 Cos B) 2 +(Cds Aa Sin B) 2 ^(Sln^AI +2 Sin Al Sin A2 +Sln 2 Aa) +(Cos 2 AI +2 CosAI CosAa CosB +Cos 2 A^ Cos 2 B) +(Cos 2 A 2 Sln^B) = 1 /Sln2AI+Cos2AI+Cos 2 A 2 (Sln 2 B+Cos 2 B)+Sln2A2+2 Cos Al CosAa Cos B+2 SlnAI Sin A 2 =^ I +Cos2A 2 (l)+Sin 2 A2 +2 Cos Al Cos A2 Cos B+2 Sin Al Sin Az =^/ 2+2 Cos Al Cos Az Cos B+2 Sin Al Sin Aa Y _ __ _ Cos = \ V 2 * 2 Cos Al Cos Aa Cos B + 2 Sin Al Sin Aa = 2(2 * 2 Cos Al Cos A2 Cos B + 2 Sln Al 81n Aa) _ , 4 = COS Al COS Aa Cos B + Sin A! Sin Aa When Al or Aa = O above formula becomes Cos X = Cos A Cos B which Is same as first case The following formula (proof omitted) was developed by Mr. C. W. L. Filkins. _ y SlnAjjCosCa Cos * SinCi Rl RETURN CIRCULATION DEPARTMENT 202 Main Library LOAN PERIOD 1 HOME USE 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS Renewals and Recharges may be made 4 days prior to the due date. Books may be Renewed by calling 642-3405. DUE AS STAMPED BELOW FEB271983 JTO Disc FEB 2 6 1989 FORM NO. DD6 UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 282177 e UNIVERSITY OF CALIFORNIA LIBRARY U.C. BERKELEY LIBRARIES COObDTODSI