Bending Wood Without Steam

These variables used are:

T = board thickness

s = length of board before being bent

y = distance deflected

z = a convenient parameter approximately equal to s

Ymax = 0.5 P z2/T    

I had to include one more parameter, "P", which is the maximum possible strain on the wood.� My best guess is that the estimated breaking point occurs at at a strain of P = 0.4% =.004.� Of course nobody would want to bend the wood this far. 

Example:   Suppose you wanted to build a 22 ft boat using bottom planks that are 1.5 inches thick.  From the figure, we would be bending about 11 ft of board.  We must convert 1.5 inches to 1.5/12 =0.125 ft.  Taking P = 0.004, we have:

Ymax = 0.002 z2/T��� [Estimated Breaking Point]

Ymax = 0.002 (11 ft)2 �/ (0.125 ft) = 1.94 ft��� [Estimated Breaking Point]

Personally, I would not want to bend wood to within a factor of 4 of this limit.  So I would go to about 6 inches.  Does anybody agree or disagree?

The above formulas work for any consistent set of units.  Hence, if you a bending wood for a small model boat, and the wood is 11 cm long and 0.125 cm thick, the maximum deflection is 1.94 cm.  Sometimes it's nice to use inconsistent units.  I like to measure wood thickness in inches and all the other dimensions in feet.  Hence the following formula:

    Ymax (feet) = 0.024 (Z-ft)2/(T-in)     [Estimated Breaking Point]

Taking the previous example of an 11-ft board that is 1.5 inches thick, we get the same answer: Ymax = .024 x 11 x 11 / 1.5 = 1.94 ft.

 Handy Table:  We can get the same result from the following table, which I made from the formula above (at P = 0.004).  The table does not show a 1.5 inch thick plank that is 11 ft lont.  But the average results for 10 ft and 12 ft is 1.95 ft.

Estimated Ymax (in feet) at rupture in feet for various board lengths and thicknesses.  The maximum strain is taken to be 0.004.  As stated before, with low cost planks, I would divide all values of Ymax shown below by a factor of perhaps 4.

 

 

Thickness of board

 

0.25 in

0.375 in

0.5 in

0.75 in

1in

1.5 in

Length

 

 

 

 

 

 

1 ft

.096

0.064

0.048

0.024

0.02

0.016

2 ft

0.38

0.26

0.19

0.13

0.10

0.06

3 ft

0.86

0.58

0.43

0.29

0.22

0.14

4 ft

1.54

1.02

0.77

0.51

0.38

0.26

5 ft

2.40

1.60

1.20

0.80

0.60

0.40

6 ft

3.46

2.30

1.73

1.15

0.86

0.58

7 ft

4.70

3.14

2.35

1.57

1.18

0.78

8 ft

6.14

4.10

3.07

2.05

1.54

1.02

9 ft

7.78

5.18

3.89

2.59

1.94

1.30

10 ft

9.60

6.40

4.80

3.20

2.40

1.60

12 ft

13.82

9.22

6.91

4.61

3.46

2.30

14 ft

18.82

12.54

9.41

6.27

4.70

3.14

16 ft

24.58

16.38

12.29

8.19

6.14

4.10

18ft

31.10

20.74

15.55

10.37

7.78

5.18

20 ft

38.40

25.60

19.20

12.80

9.60

6.40

 

Mathematics of Bending

Solve for y to get y = .5Pz2/T, where P = dL/L �the maximum strain within the wood..  Actually, the formula for the change in length dL/L = T/R may overestimate the actual strain.  I have assumed no strain on one side of the wood.  This is reasonable because many boatbuilders will not optimize the location of zero strain by subjecting the wood to the right amount of tension or compression.

Shown below are three estimates of P

Estimate #1

P= .001, From personal experience with two sticks (one broke and the other did not).

 

I once bent quarter inch thick piece of clear pine lath to near the breaking point. The deflection was .75 in and z was 6 in.  Hence the maximum strain was (2)(.25)(.75)/36 = .01, or about 1%.  I know I was near the breaking point because the board snapped on a prior attempt.  I also bent it around a single point, so the maximum radius of curvature was less than the  z2/2y

 

Estimate #2

P = 0.004: From Recreational Aviation Australia that has been removed [ http://www.auf.asn.au/scratchbuilder/contents.html ]

The table below shows data for  Douglas fir. As I understand the chart, the only modulus of elacticity (strain/stress) is 1550k psi.  For the ultimate strength parallel to the grain, I take the minimum of 12300 psi and 6500 psi.  This yields a strain of P=6500/1550000 = .004. 

Strength value

Douglas fir

Density at 12% moisture content [pounds per cubic foot]

32.5

Static bending: fibre stress at proportional limit [pounds/sq. inch]

6700

Static bending:   modulus of rupture   [pounds/sq. inch] 

12 300

Static bending:  modulus of elasticity   [thousand pounds/sq. inch] 

1550 k

Compression parallel to grain:  fibre stress at proportional limit  [pounds/sq. inch]

4850

Compression parallel to grain:  maximum crushing strength  [pounds/sq. inch ]

6500

Compression perpendicular to grain  [pounds/sq. inch]

1100

Shearing strength parallel to grain  [pounds/sq] 

1000

Hardness  [load required in pounds] 

640

Tension strength parallel to grain  [= modulus of rupture ]

12300

Tension strength perpendicular to grain  [pounds/sq. inch] 

130

 

Estimate #3

P = 0.004 for plywood from a site by William Watts' that has been removed (http://web.ncf.ca/ag384/Boats.htm) He quotes the following table from "Boatbuilding Manual 3rd edition" by Robt. M. Steward (1987).

 

Plywood thickness

Minimum Radius (across grain)

Minimum Radius
(with grain)

1/4               

24              

60

5/16             

24              

72

3/8               

36              

96

1/2               

72             

144

5/8                

96             

192

3/4              

144             

240

 Using Excel to convert these into strains (T/R), I then graphed the results:

 

Comments from Yahoo Boatbuilder Group on bending wood.

LC wrote: It's important to know how clear the grain is.

NM wrote:  Formulas should work fine for plywood but solid wood is highly variable. Moisture content, ring density, reaction wood, shakes, how it was dried, how it was milled etc can throw off mechanical properties you find in any table. Experimental stress analysis is your best option or use a large factor of safety. Read Bruce Hoadley's books. I do applaud your efforts though.............Now to answer your question (How much can you bend wood?)...Until it breaks.

MC wrote: Formulas should work fine for plywood but solid wood is highly variable. Moisture content, ring density, reaction wood, shakes, how it was dried, how it was milled etc can throw off mechanical properties you find in any table. Experimental stress analysis is your best option or use a large factor of safety. Read Bruce Hoadley's books. I do applaud your efforts though. 

My response:  There is a problem with experimental stress analysis.  Since a planked boat can have over 10 planks, we must multiply the chances of 1 plank breaking by about 10 to get the chances of failure in the boat.  If you want that probability to be less than 1 in 100 (pretty risky sailing!) then you need to test about 1000 planks.  No thanks.

WCS wrote: Anhydrous ammonia can be used to make extreme bends in wood. I've seen photos of square pieces of wood tied in a knot. If the link below won't work, try googling ' bending wood anhydrous ammonia 'for more information.

http://www.allbusiness.com/furniture-related/office-furniture-including/644819-1\.html