THE MECHANICAL PROPERTIES OF WOOD INCLUDING A DISCUSSION OF THE FACTORS AFFECTING THE MECHANICAL PROPERTIES, AND METHODS OF TIMBER TESTING TEXT

| (J.C. Trautwine. Jour. Franklin Institute. Vol. 109, 1880, pp. 105-106) | |---------------------------------------------------------------------------| | KIND OF WOOD | Lbs. per | KIND OF WOOD | Lbs. per | | | sq. inch | | sq. inch | |-----------------------+----------+-----------------------------+----------| | Ash | 6,280 | Hickory | 7,285 | | Beech | 5,223 | Locust | 7,176 | | Birch | 5,595 | Maple | 6,355 | | Cedar (white) | 1,372 | Oak | 4,425 | | Cedar (white) | 1,519 | Oak (live) | 8,480 | | Cedar (Central Amer.) | 3,410 | Pine (white) | 2,480 | | Cherry | 2,945 | Pine (northern yellow) | 4,340 | | Chestnut | 1,536 | Pine (southernyellow) | 5,735 | | Dogwood | 6,510 | Pine (very resinous yellow) | 5,053 | | Ebony | 7,750 | Poplar | 4,418 | | Gum | 5,890 | Spruce | 3,255 | | Hemlock | 2,750 | Walnut (black) | 4,728 | | Hickory | 6,045 | Walnut (common) | 2,830 | |---------------------------------------------------------------------------| | NOTE.--Two specimens of each were tested. All were fairly seasoned and | | without defects. The piece sheared off was 5/8 in. The single circular | | area of each pin was 0.322 sq. in. | |---------------------------------------------------------------------------| TRANSVERSE OR BENDING STRENGTH: BEAMS When external forces acting in the same plane are applied at right angles to the axis of a bar so as to cause it to bend, they occasion a shortening of the longitudinal fibres on the concave side and an elongation of those on the convex side. Within the elastic limit the relative stretching and contraction of the fibres is directly[9] proportional to their distances from a plane intermediate between them--the ~neutral plane~. (N_{1} P in Fig. 15.) Thus the fibres half-way between the neutral plane and the outer surface experience only half as much shortening or elongation as the outermost or extreme fibres. Similarly for other distances. The elements along the neutral plane experience no tension or compression in an axial direction. The line of intersection of this plane and the plane of section is known as the ~neutral axis~ (N A in Fig. 15) of the section. [Footnote 9: While in reality this relationship does not exactly hold, the formulæ for beams are based on its assumption.] [Illustration: FIG. 15.--Diagram of a simple beam. N_{1} P = neutral plane, N A = neutral axis of section R S.] If the bar is symmetrical and homogeneous the neutral plane is located half-way between the upper and lower surfaces, so long as the deflection does not exceed the elastic limit of the material. Owing to the fact that the tensile strength of wood is from two to nearly four times the compressive strength, it follows that at rupture the neutral plane is much nearer the convex than the concave side of the bar or beam, since the sum of all the compressive stresses on the concave portion must always equal the sum of the tensile stresses on the convex portion. The neutral plane begins to change from its central position as soon as the elastic limit has been passed. Its location at any time is very uncertain. The external forces acting to bend the bar also tend to rupture it at right angles to the neutral plane by causing one transverse section to slip past another. This stress at any point is equal to the resultant perpendicular to the axis of the forces acting at this point, and is termed the ~transverse shear~ (or in the case of beams, ~vertical shear~). In addition to this there is a shearing stress, tending to move the fibres past one another in an axial direction, which is called ~longitudinal shear~ (or in the case of beams, ~horizontal shear~). This stress must be taken into consideration in the design of timber structures. It is maximum at the neutral plane and decreases to zero at the outer elements of the section. The shorter the span of a beam in proportion to its height, the greater is the liability of failure in horizontal shear before the ultimate strength of the beam is reached. _Beams_ There are three common forms of beams, as follows: (1) ~Simple beam~--a bar resting upon two supports, one near each end. (See Fig. 16, No. 1.) (2) ~Cantilever beam~--a bar resting upon one support or fulcrum, or that portion of any beam projecting out of a wall or beyond a support. (See Fig. 16, No. 2.) (3) ~Continuous beam~--a bar resting upon more than two supports. (See Fig. 16, No. 3.) [Illustration: FIG. 16.--Three common forms of beams. 1. Simple. 2. Cantilever. 3. Continuous.] _Stiffness of Beams_ The two main requirements of a beam are stiffness and strength. The formulæ for the _modulus of elasticity (E)_ or measure of stiffness of a rectangular prismatic simple beam loaded at the centre and resting freely on supports at either end is:[10] [Footnote 10: Only this form of beam is considered since it is the simplest. For cantilever and continuous beams, and beams rigidly fixed at one or both ends, as well as for different methods of loading, different forms of cross section, etc., other formulæ are required. See any book on mechanics.] P' l^{3} E = ------------- 4 D b h^{3} b = breadth or width of beam, inches. h = height or depth of beam, inches. l = span (length between points of supports) of beam, inches. D = deflection produced by load P', inches. P' = load at or below elastic limit, pounds. From this formulæ it is evident that for rectangular beams of the same material, mode of support, and loading, the deflection is affected as follows: (1) It is inversely proportional to the width for beams of the same length and depth. If the width is tripled the deflection is one-third as great. (2) It is inversely proportional to the cube of the depth for beams of the same length and breadth. If the depth is tripled the deflection is one twenty-seventh as great. (3) It is directly proportional to the cube of the span for beams of the same breadth and depth. Tripling the span gives twenty-seven times the deflection. The number of pounds which concentrated at the centre will deflect a rectangular prismatic simple beam one inch may be found from the preceding formulæ by substituting D = 1" and solving for P'. The formulæ then becomes: 4 E b h^{3} Necessary weight (P') = ------------- l^{3} In this case the values for E are read from tables prepared from data obtained by experimentation on the given material. _Strength of Beams_ The measure of the breaking strength of a beam is expressed in terms of unit stress by a _modulus of rupture_, which is a purely hypothetical expression for points beyond the elastic limit. The formulæ used in computing this modulus is as follows: 1.5 P l R = --------- b h{^2} b, h, l = breadth, height, and span, respectively, as in preceding formulæ. R = modulus of rupture, pounds per square inch. P = maximum load, pounds. In calculating the fibre stress at the elastic limit the same formulæ is used except that the load at elastic limit (P_{1}) is substituted for the maximum load (P). From this formulæ it is evident that for rectangular prismatic beams of the same material, mode of support, and loading, the load which a given beam can support varies as follows: (1) It is directly proportional to the breadth for beams of the same length and depth, as is the case with stiffness. (2) It is directly proportional to the square of the height for beams of the same length and breadth, instead of as the cube of this dimension as in stiffness. (3) It is inversely proportional to the span for beams of the same breadth and depth and not to the cube of this dimension as in stiffness. The fact that the strength varies as the _square_ of the height and the stiffness as the _cube_ explains the relationship of bending to thickness. Were the law the same for strength and stiffness a thin piece of material such as a sheet of paper could not be bent any further without breaking than a thick piece, say an inch board. |-------------------------------------------------------------------------------------| | TABLE IX | |-------------------------------------------------------------------------------------| | RESULTS OF STATIC BENDING TESTS ON SMALL CLEAR BEAMS OF 49 WOODS IN GREEN CONDITION | | (Forest Service Cir. 213) | |-------------------------------------------------------------------------------------| | | Fibre | | | Work in Bending | | COMMON NAME | stress at | Modulus | Modulus |-------------------------------| | OF SPECIES | elastic | of | of | To | To | | | | limit | rupture | elasticity | elastic | maximum | Total | | | | | | limit | load | | |-----------------+-----------+----------+------------+----------+----------+---------| | | | | | In.-lbs. | In.-lbs. | In.-lb. | | | Lbs. per | Lbs. per | Lbs. per | per cu. | per cu. | per | | | sq. in. | sq. in. | sq. in. | inch | inch | inch | | | | | | | | | | Hardwoods | | | | | | | | | | | | | | | | Ash, black | 2,580 | 6,000 | 960,000 | 0.41 | 13.1 | 38.9 | | white | 5,180 | 9,920 | 1,416,000 | 1.10 | 20.0 | 43.7 | | Basswood | 2,480 | 4,450 | 842,000 | .45 | 5.8 | 8.9 | | Beech | 4,490 | 8,610 | 1,353,000 | .96 | 14.1 | 31.4 | | Birch, yellow | 4,190 | 8,390 | 1,597,000 | .62 | 14.2 | 31.5 | | Elm, rock | 4,290 | 9,430 | 1,222,000 | .90 | 19.4 | 47.4 | | slippery | 5,560 | 9,510 | 1,314,000 | 1.32 | 11.7 | 44.2 | | white | 2,850 | 6,940 | 1,052,000 | .44 | 11.8 | 27.4 | | Gum, red | 3,460 | 6,450 | 1,138,000 | | | | | Hackberry | 3,320 | 7,800 | 1,170,000 | .56 | 19.6 | 52.9 | | Hickory, | | | | | | | | big shellbark | 6,370 | 11,110 | 1,562,000 | 1.47 | 24.3 | 78.0 | | bitternut | 5,470 | 10,280 | 1,399,000 | 1.22 | 20.0 | 75.5 | | mockernut | 6,550 | 11,110 | 1,508,000 | 1.50 | 31.7 | 84.4 | | nutmeg | 4,860 | 9,060 | 1,289,000 | 1.06 | 22.8 | 58.2 | | pignut | 5,860 | 11,810 | 1,769,000 | 1.12 | 30.6 | 86.7 | | shagbark | 6,120 | 11,000 | 1,752,000 | 1.22 | 18.3 | 72.3 | | water | 5,980 | 10,740 | 1,563,000 | 1.29 | 18.8 | 52.9 | | Locust, honey | 6,020 | 12,360 | 1,732,000 | 1.28 | 17.3 | 64.4 | | Maple, red | 4,450 | 8,310 | 1,445,000 | .78 | 9.8 | 17.1 | | sugar | 4,630 | 8,860 | 1,462,000 | .88 | 12.7 | 32.0 | | Oak, post | 4,720 | 7,380 | 913,000 | 1.39 | 9.1 | 17.4 | | red | 3,490 | 7,780 | 1,268,000 | .60 | 11.4 | 26.0 | | swamp white | 5,380 | 9,860 | 1,593,000 | 1.05 | 14.5 | 37.6 | | tanbark | 6,580 | 10,710 | 1,678,000 | 1.49 | | | | white | 4,320 | 8,090 | 1,137,000 | .95 | 12.1 | 36.7 | | yellow | 5,060 | 8,570 | 1,219,000 | 1.20 | 11.7 | 30.7 | | Osage orange | 7,760 | 13,660 | 1,329,000 | 2.53 | 37.9 | 101.7 | | Sycamore | 2,820 | 6,300 | 961,000 | .51 | 7.1 | 13.6 | | Tupelo | 4,300 | 7,380 | 1,045,000 | 1.00 | 7.8 | 20.9 | | | | | | | | | | Conifers | | | | | | | | | | | | | | | | Arborvitæ | 2,600 | 4,250 | 643,000 | .60 | 5.7 | 9.5 | | Cedar, incense | 3,950 | 6,040 | 754,000 | | | | | Cypress, bald | 4,430 | 7,110 | 1,378,000 | .96 | 5.1 | 15.4 | | Fir, alpine | 2,366 | 4,450 | 861,000 | .66 | 4.4 | 7.4 | | amabilis | 4,060 | 6,570 | 1,323,000 | | | | | Douglas | 3,570 | 6,340 | 1,242,000 | .59 | 6.6 | 13.6 | | white | 3,880 | 5,970 | 1,131,000 | .77 | 5.2 | 14.9 | | Hemlock | 3,410 | 5,770 | 917,000 | .73 | 6.6 | 12.9 | | Pine, lodgepole | 3,080 | 5,130 | 1,015,000 | .54 | 5.1 | 7.4 | | longleaf | 5,090 | 8,630 | 1,662,000 | .88 | 8.1 | 34.8 | | red | 3,740 | 6,430 | 1,384,000 | .59 | 5.8 | 28.0 | | shortleaf | 4,360 | 7,710 | 1,395,000 | | | | | sugar | 3,330 | 5,270 | 966,000 | .66 | 5.0 | 11.6 | | west, yellow | 3,180 | 5,180 | 1,111,000 | .52 | 4.3 | 15.6 | | White | 3,410 | 5,310 | 1,073,000 | .62 | 5.9 | 13.3 | | Redwood | 4,530 | 6,560 | 1,024,000 | | | | | Spruce, | | | | | | | | Engelmann | 2,740 | 4,550 | 866,000 | .50 | 4.8 | 6.1 | | red | 3,440 | 5,820 | 1,143,000 | .62 | 6.0 | | | white | 3,160 | 5,200 | 968,000 | .58 | 6.6 | | | Tamarack | 4,200 | 7,170 | 1,236,000 | .84 | 7.2 | 30.0 | |-------------------------------------------------------------------------------------| _Kinds of Loads_ There are various ways in which beams are loaded, of which the following are the most important: (1) ~Uniform load~ occurs where the load is spread evenly over the beam. (2) ~Concentrated load~ occurs where the load is applied at single point or points. (3) ~Live~ or ~immediate load~ is one of momentary or short duration at any one point, such as occurs in crossing a bridge. (4) ~Dead~ or ~permanent load~ is one of constant and indeterminate duration, as books on a shelf. In the case of a bridge the weight of the structure itself is the dead load. All large beams support a uniform dead load consisting of their own weight. The effect of dead load on a wooden beam may be two or more times that produced by an immediate load of the same weight. Loads greater than the elastic limit are unsafe and will generally result in rupture if continued long enough. A beam may be considered safe under permanent load when the deflections diminish during equal successive periods of time. A continual increase in deflection indicates an unsafe load which is almost certain to rupture the beam eventually. Variations in the humidity of the surrounding air influence the deflection of dry wood under dead load, and increased deflections during damp weather are cumulative and not recovered by subsequent drying. In the case of longleaf pine, dry beams may with safety be loaded permanently to within three-fourths of their elastic limit as determined from ordinary static tests. Increased moisture content, due to greater humidity of the air, lowers the elastic limit of wood so that what was a safe load for the dry material may become unsafe. When a dead load not great enough to rupture a beam has been removed, the beam tends gradually to recover its former shape, but the recovery is not always complete. If specimens from such a beam are tested in the ordinary testing machine it will be found that the application of the dead load did not affect the stiffness, ultimate strength, or elastic limit of the material. In other words, the deflections and recoveries produced by live loads are the same as would have been produced had not the beam previously been subjected to a dead load.[11] [Footnote 11: See Tiemann, Harry D.: Some results of dead load bending tests of timber by means of a recording deflectometer. Proc. Am. Soc. for Testing Materials. Phila. Vol. IX, 1909, pp. 534-548.] ~Maximum load~ is the greatest load a material will support and is usually greater than the load at rupture. ~Safe load~ is the load considered safe for a material to support in actual practice. It is always less than the load at elastic limit and is usually taken as a certain proportion of the ultimate or breaking load. The ratio of the breaking to the safe load is called the factor of safety. (Factor of safety = ultimate strength / safe load) In order to make due allowance for the natural variations and imperfections in wood and in the aggregate structure, as well as for variations in the load, the factor of safety is usually as high as 6 or 10, especially if the safety of human life depends upon the structure. This means that only from one-sixth to one-tenth of the computed strength values is considered safe to use. If the depth of timbers exceeds four times their thickness there is a great tendency for the material to twist when loaded. It is to overcome this tendency that floor joists are braced at frequent intervals. Short deep pieces shear out or split before their strength in bending can fully come into play. _Application of Loads_ There are three[12] general methods in which loads may be applied to beams, namely: [Footnote 12: A fourth might be added, namely, ~vibratory~, or ~harmonic repetition~, which is frequently serious in the case of bridges.] (1) ~Static loading~ or the gradual imposition of load so that the moving parts acquire no appreciable momentum. Loads are so applied in the ordinary testing machine. (2) ~Sudden imposition of load without initial velocity.~ "Thus in the case of placing a load on a beam, if the load be brought into contact with the beam, but its weight sustained by external means, as by a cord, and then this external support be _suddenly_ (instantaneously) removed, as by quickly cutting the cord, then, although the load is already touching the beam (and hence there is no real impact), yet the beam is at first offering no resistance, as it has yet suffered no deformation. Furthermore, as the beam deflects the resistance increases, but does not come to be equal to the load until it has attained its normal deflection. In the meantime there has been an unbalanced force of gravity acting, of a constantly diminishing amount, equal at first to the entire load, at the normal deflection. But at this instant the load and the beam are in motion, the hitherto unbalanced force having produced an accelerated