BOATBUILDING WITH A DIFFERENCE III
(For Aspiring Amateurs)
by Barend Migchelsen Migchelsen@aol.com
http://ca.geocities.com/bmboats2002/
http://members.aol.com/_ht_a/migchelsen/myhomepage/

Barend Migchelsen, (pronounced Mikkelsen) learned to sail in The Netherlands in 1943. In 1975 he started to build boats and boat models as a hobby.  Today, he organizes and teaches classroom courses in boat building, and has published several books on the subject.  The following is an excerpt from one of these books.  this is the second of four parts.
 Part I Part II Part IV
 STEMS The meeting line fore and aft of the side panels of a Double-ender is not just a curved line; the angle between the panels varies from the sheer diminishing downward to the heel of the stem.  Its Profile view is shown in figure 3-1.  It makes construction of stems difficult for a new amateur who has only beginners luck on his side.

 Fig. 3 - 1   The curved stem has a varying crosscut angle. The degree of variation over the whole stem can be easily calculated, but it is a waste of time and effort unless you insist on a curved stem. Side Panel Modification From station #2 forward, and from station #14 toward aft, the side panels are no longer bent, but are allowed to continue in a straight direction as tangents to the sheer line circle segment. Instead of coming together at station #0, the sheer lines join each other five inches fore of station #0 at the location #0-5", and aft at station #16+5".  This increases the Overall Length of the hull to LOA = 16', 10". The angle that the tangent lines make with the centerline is equal to the bevel angle of cross frame #2 (and #14), which is equal to the angle of the center point angle between the radii R2 and R8 (= the angle between R14 and R8).  That angle is 22.24 degrees.  See figure 3-2.

 Fig. 3 - 2   Side panels tangent line Allowing the side panel to continue as the tangent line to the sheer line arc delivers two important simplifications: 1.         The stem becomes straight. 2.         The bevel angle of the stem, called the crosscut angle becomes constant. What was the most difficult part to construct becomes one of the easiest to cut, especially for the beginning amateur with limited, or no carpenter’s skills at all. SAC (Stem At Chine) In figure 3-3, the Body view, the (maximum) bottom rocker between Beam and the heel of the stems is three inches.  The location where the two chine lines join the heel of the stem I call Stem At Chine.  For easy writing, abbreviated to SAC.  The location of Station SAC on the centerline is between stations #1 and #2.  See figure 3-1.  On the sheer line in the half-Body view of figure 3-3 that location is  hsac = 22 x sin 22.62º = 22 x 0.3846 = 8.46”. The exact location on the centerline of the station line SAC in Profile and half Bread view can be calculated with the formula: hSAC + (R - hBeam) = v(R2 - dsac2), or 8.46 + (190.23 - 26) = v(190.232 - dSAC2)  in which dSAC is the distance between station SAC and  station Beam (station #8). Worked out, dSAC = 79.785", or the location of station SAC = station #1 + 4.215" as is shown in figure 3-1, and more detailed in figure 3-4.  In this illustration the original curved stem and the sheer line the Profile and half-Breadth view are drawn in red. The exact length, and the rake angle of the modified stem in black lines are written in.  Also shown is how one half of the constant crosscut is determined in this to scale drawing.  When the drawing is made on one-inch-grid graph paper a high degree of accuracy is achieved, even if it is done on a one-quarter scale.  It makes the rest of the Profile and half-Breadth views redundant!  All the important values of the measurements are written in. The mathematics is just Pythagoras and the basic trigonometric definitions of Sine, Cosine, and Tangent applied.  It is all junior high school stuff.  If a check of the accuracy of figures of these measurements gives you any difficulty, just send me an email for clarification.

 Fig. 3 - 3   Half- Body view of the Double-Ender In figure 3-4, one-half of the crosscut angle is 31 degrees.  A 2"x3" ripped diagonally gives two right-triangular slats.  The tangent of the angle between the hypotenuse/cut and the 2½"-long-leg side is 1½/2½" = 0.6.  The angle is exactly 31 degrees!  Place the two 2½" sides of the slats back to back.  Cut the rabbet groove.  Miter the stem at 46 degrees.  In The Netherlands, where I was born, they say:  “Even a toddler can do the washing.”  See figure 3-5. With the modified sheer line, the straightened-out stem, the raked tomb stone, or transom board, the increase of the Overall Length, the added guardrails and their capping, varying flare and the quoting of the outside measurements, it becomes more difficult for the untrained eye to recognize the original Double-Ender design from which the hull is developed.  However, it is still there!  But the most important result is that even a person with two left hands can now build the simplified boat.

 Fig. 3 - 4   Profile and Body view of the modified stem

 The flare ratio = 2/3 was a (unconscious?) stroke of genius.  It made the setting up of the frames for the “classic” constant flare extremely easy and accurate.  I have spoken with several professional Dory builders who were not aware of this characteristic.  The older ones had received an elementary school education only, or, forced by bad economic conditions, even less.  Sometimes, they were more real artists than simple boat-builders anyhow. THE SYSTEM In the first article of this series, it is shown that all other hard-chined, constant flared hull forms easily can be developed from the original drawing of the Double-Ender, and the formula:  Tan Flare Angle = Profile height/half-Breadth. Calculating the radius of the sheer line circle arc segment provided the key, and became the basis for determining all the other measurements of a hull.  With the printed tables found here, that information is at your fingertips.  No need to make the calculations yourself. The first table provides the radius R for all the flare ratios from 1/24 up to the maximum Dory flare ratio 16/24.  The second table is the calculation of the locations of the station lines on the hypotenuse/ sheer line in the Body view for the most common flare ratios from 6/24 up to 14/24.  The use of the tables will save you a lot of time and effort. Table of the Calculations of the radius R for the Different Flare Ratios The mathematical equation for the radius R is:  2 x hBm x R = (½ LOA)2 + hBm2.
 Prfl Hght (inches) Flare Ratio Flare Angle (degrees) hBm (inches) hBm2 (½ LOA)2  + hBm2 Radius (inches) 0 0 0 24.00 576 9216 + 576 204.00 1 1/24 2.39 24.02 577 9216 + 577 203.85 2 2/24 4.76 24.08 580 9216 + 580 203.40 3 3/24 7.13 24.19 585 9216 + 585 202.60 4 4/24 9.46 24.33 592 9216 + 592 201.56 5 5/24 11.77 24.52 601 9216 + 601 200.22 6 6/24 14.04 24.74 612 9216 + 612 198.63 7 7/24 16.26 25.00 625 9216 + 625 196.82 8 8/24 18.44 25.30 640 9216 + 640 194.78 9 9/24 20.56 25.63 657 9216 + 657 192.60 10 10/24 22.62 26.00 676 9216 + 676 190.23 11 11/24 24.62 26.40 697 9216 + 697 187.75 12 12/24 26.57 26.83 720 9216 + 720 185.16 13 13/24 28.44 27.30 745 9216 + 745 182.44 14 14/24 30.26 27.78 772 9216 + 772 179.77 15 15/24 32.00 28.30 801 9216 + 801 176.98 16 16/24 33.69 28.84 832 9216 + 832 174.18

Flare ratio table for a hard-chined hull of a Double-Ender:  LOA = 16 ft.

 The underlined figures in the table are the measurements of the Double-Ender described in this chapter.  On the same side panel width, the flare ratio figures above the line make the bottom wider.  The figures below the line will make the bottom of the boat narrower.  This ratio table saves you the trouble of having to make the calculations yourself. Offset Table of Profile Heights and Half-Breadths In general, the designs of most constant-flared, hard-chined hulls have a flare ratio between 6/24 (¼) and 14/24 (7/12), or a flare angle between 14 (14.036) degrees and 30¼ (30.256) degrees. With this in mind, the plotting table for the actual sheer line arc, and the offsets of the Profile heights, and the half Breadths at the different stations is a great time and labour saving tool. The table is based on the sheer line circle segment of the 16-feet double-ender.  The Profile heights at Beam vary from 6 inches to 14 inches on a (constant) half-Breadth width of 24 inches.  It lists the Body view measurements of the hypotenuses hn of the sheer line circle arc at the stations #2 = #14, #4 = #12, #6 = #10, and station #8 (Beam).  ½ LOA = 96".  The distances dn are between each station and station #8 (Beam)  All the measurement figures in the table are given in inches. The mathematical equation is hn = v(R2 - dn2) - (R - hBm).
 Flare Ratio R hBm h6 h10 d = 24 h4 h12 d = 48 h2 h14 d = 72 6/24 198.63 24.74 23.29 18.85 11.23 7/24 196.82 25.00 23.53 19.00 11.36 8/24 194.78 25.30 23.81 19.23 11.50 9/24 192.60 25.63 24.13 19.55 11.67 10/24 190.23 26.00 24.48 19.84 11.85 11/24 187.75 26.40 24.91 20.16 12.04 12/24 185.16 26.83 25.27 20.50 12.26 13/24 182.44 27.30 25.71 20.87 12.49 14/24 179.77 27.78 26.17 21.25 12.73
 The underlined figures are the measurements of the Double-Ender model constructed in these articles. The Flare Ratio Table on the preceding page, and the plotting table for the heights of the sheer line circle arc segment above, eliminate the need to make any calculations.  Plot the dimensions on one-inch-grid graph paper.  Draw the hull lines completely in Body view.  Instead of the 10" Profile height at Beam as found in these articles, change to the Profile height of your choice, which can be any number between 6 inches, and 14 inches. If you want to build a bigger boat, based on an 18', or 24' Double-Ender, just increase the scale of all the measurements by the factor 1½, or 2.  It is that simple with this mathematical system of design. SIDE PANELS In the half-Body view of a Double-Ender, figure3-6A, the chine line is drawn parallel to the sheer line.  The side panels have the same width over the whole length. The rocker from the Beam to the stems is 7.4".  Station BAC (Bow At Chine) has moved forward to station #1 + 0.655”.  The chine lines in Plan and half-Breadth view run parallel to the sheer lines.  This strong rocker is still visible in the McKenzie-River Dories.

 Fig. 3 - 6   Chine lines parallel to the sheer lines Cod’s Head, Mackerel Tail In figure 3 -7A, the original chine lines parallel to the sheer lines are the dotted lines.  In this drawing a new chine line is drawn in red.  Instead of a rocker fore of Beam of 7.4", this rocker is reduced to 3", a difference of 4.4". At the same time the rocker aft of Beam is increased by the same amount of 4.4”. The bottom is no longer parallel to the (horizontal) plane of the two sheer lines, but tilted from fore to aft as shown in the Profile drawing of figure 3-7B.  Although the sheer line itself has not changed, the bow has become substantially higher. (or should I say the bottom fore deeper?)  In Dories this is not so pronounced as in the Punter, but still clearly visible in the photograph of the Dory in this posting.