Chapter IV: Introduction (1)

The mechanical properties of wood are its fitness and ability to resist applied or external forces. By external force is meant any force outside of a given piece of material which tends to deform it in any manner. It is largely such properties that determine the use of wood for structural and building purposes and innumerable other uses of which furniture, vehicles, implements, and tool handles are a few common examples.

Knowledge of these properties is obtained through experimentation either in the employment of the wood in practice or by means of special testing apparatus in the laboratory. Owing to the wide range of variation in wood it is necessary that a great number of tests be made and that so far as possible all disturbing factors be eliminated. For comparison of different kinds or sizes a standard method of testing is necessary and the values must be expressed in some defined units. For these reasons laboratory experiments if properly conducted have many advantages over any other method.

One object of such investigation is to find unit values for strength and stiffness, etc. These, because of the complex structure of wood, cannot have a constant value which will be exactly repeated in each test, even though no error be made. The most that can be accomplished is to find average values, the amount of variation above and below, and the laws which govern the variation. On account of the great variability in strength of different specimens of wood even from the same stick and appearing to be alike, it is important to eliminate as far as possible all extraneous factors liable to influence the results of the tests.

The mechanical properties of wood considered in this book are: (1) stiffness and elasticity, (2) tensile strength, (3) compressive or crushing strength, (4) shearing strength, (5) transverse or bending strength, (6) toughness, (7) hardness, (8) cleavability, (9) resilience. In connection with these, associated properties of importance are briefly treated.

In making use of figures indicating the strength or other mechanical properties of wood for the purpose of comparing the relative merits of different species, the fact should be borne in mind that there is a considerable range in variability of each individual material and that small differences, such as a few hundred pounds in values of 10,000 pounds, cannot be considered as a criterion of the quality of the timber. In testing material of the same kind and grade, differences of 25 per cent between individual specimens may be expected in conifers and 50 per cent or even more in hardwoods. The figures given in the tables should be taken as indications rather than fixed values, and as applicable to a large number collectively and not to individual pieces.

FUNDAMENTAL CONSIDERATIONS AND DEFINITIONS

Study of the mechanical properties of a material is concerned mostly with its behavior in relation to stresses and strains, and the factors affecting this behavior. A ~stress~ is a distributed force and may be defined as the mutual action (1) of one body upon another, or (2) of one part of a body upon another part. In the first case the stress is _external_; in the other _internal_. The same stress may be internal from one point of view and external from another. An external force is always balanced by the internal stresses when the body is in equilibrium.

If no external forces act upon a body its particles assume certain relative positions, and it has what is called its _natural shape and size_. If sufficient external force is applied the natural shape and size will be changed. This distortion or deformation of the material is known as the ~strain~. Every stress produces a corresponding strain, and within a certain limit (see _elastic limit_, in FUNDAMENTAL CONSIDERATIONS AND DEFINITIONS, above) the strain is directly proportional to the stress producing it.[1] The same intensity of stress, however, does not produce the same strain in different materials or in different qualities of the same material. No strain would be produced in a perfectly rigid body, but such is not known to exist.

[Footnote 1: This is in accordance with the discovery made in 1678 by Robert Hooke, and is known as _Hooke's law_.]

Stress is measured in pounds (or other unit of weight or force). A ~unit stress~ is the stress on a unit of the sectional { P } area. { Unit stress = --- } For instance, if a load (P) of one { A } hundred pounds is uniformly supported by a vertical post with a cross-sectional area (A) of ten square inches, the unit compressive stress is ten pounds per square inch.

Strain is measured in inches (or other linear unit). A ~unit strain~ is the strain per unit of length. Thus if a post 10 inches long before compression is 9.9 inches long under the compressive stress, the total strain is 0.1 inch, and the unit l 0.1 strain is --- = ----- = 0.01 inch per inch of length. L 10

As the stress increases there is a corresponding increase in the strain. This ratio may be graphically shown by means of a diagram or curve plotted with the increments of load or stress as ordinates and the increments of strain as abscissæ. This is known as the ~stress-strain diagram~. Within the limit mentioned above the diagram is a straight line. (See Fig. 1.) If the results of similar experiments on different specimens are plotted to the same scales, the diagrams furnish a ready means for comparison. The greater the resistance a material offers to deformation the steeper or nearer the vertical axis will be the line.

There are three kinds of internal stresses, namely, (1) ~tensile~, (2) ~compressive~, and (3) ~shearing~. When external forces act upon a bar in a direction away from its ends or a direct pull, the stress is a tensile stress; when toward the ends or a direct push, compressive stress. In the first instance the strain is an _elongation_; in the second a _shortening_. Whenever the forces tend to cause one portion of the material to slide upon another adjacent to it the action is called a _shear_. The action is that of an ordinary pair of shears. When riveted plates slide on each other the rivets are sheared off.

These three simple stresses may act together, producing compound stresses, as in flexure. When a bow is bent there is a compression of the fibres on the inner or concave side and an elongation of the fibres on the outer or convex side. There is also a tendency of the various fibres to slide past one another in a longitudinal direction. If the bow were made of two or more separate pieces of equal length it would be noted on bending that slipping occurred along the surfaces of contact, and that the ends would no longer be even. If these pieces were securely glued together they would no longer slip, but the tendency to do so would exist just the same. Moreover, it would be found in the latter case that the bow would be much harder to bend than where the pieces were not glued together--in other words, the _stiffness_ of the bow would be materially increased.

~Stiffness~ is the property by means of which a body acted upon by external forces tends to retain its natural size and shape, or resists deformation. Thus a material that is difficult to bend or otherwise deform is stiff; one that is easily bent or otherwise deformed is _flexible_. Flexibility is not the exact counterpart of stiffness, as it also involves toughness and pliability.

If successively larger loads are applied to a body and then removed it will be found that at first the body completely regains its original form upon release from the stress--in other words, the body is ~elastic~. No substance known is perfectly elastic, though many are practically so under small loads. Eventually a point will be reached where the recovery of the specimen is incomplete. This point is known as the ~elastic limit~, which may be defined as the limit beyond which it is impossible to carry the distortion of a body without producing a permanent alteration in shape. After this limit has been exceeded, the size and shape of the specimen after removal of the load will not be the same as before, and the difference or amount of change is known as the ~permanent set~.

Elastic limit as measured in tests and used in design may be defined as that unit stress at which the deformation begins to increase in a faster ratio than the applied load. In practice the elastic limit of a material under test is determined from the stress-strain diagram. It is that point in the line where the diagram begins perceptibly to curve.[2] (See Fig. 1.)

[Footnote 2: If the straight portion does not pass through the origin, a parallel line should be drawn through the origin, and the load at elastic limit taken from this line. (See Fig. 32.)]

~Resilience~ is the amount of work done upon a body in deforming it. Within the elastic limit it is also a measure of the potential energy stored in the material and represents the amount of work the material would do upon being released from a state of stress. This may be graphically represented by a diagram in which the abscissæ represent the amount of deflection and the ordinates the force acting. The area included between the stress-strain curve and the initial line (which is zero) represents the work done. (See Fig. 1.) If the unit of space is in inches and the unit of force is in pounds the result is inch-pounds. If the elastic limit is taken as the apex of the triangle the area of the triangle will represent the ~elastic resilience~ of the specimen. This amount of work can be applied repeatedly and is perhaps the best measure of the toughness of the wood as a working quality, though it is not synonymous with toughness.

Permanent set is due to the ~plasticity~ of the material. A perfectly plastic substance would have no elasticity and the smallest forces would cause a set. Lead and moist clay are nearly plastic and wood possesses this property to a greater or less extent. The plasticity of wood is increased by wetting, heating, and especially by steaming and boiling. Were it not for this property it would be impossible to dry wood without destroying completely its cohesion, due to the irregularity of shrinkage.

A substance that can undergo little change in shape without breaking or rupturing is ~brittle~. Chalk and glass are common examples of brittle materials. Sometimes the word _brash_ is used to describe this condition in wood. A brittle wood breaks suddenly with a clean instead of a splintery fracture and without warning. Such woods are unfitted to resist shock or sudden application of load.

The measure of the stiffness of wood is termed the ~modulus of elasticity~ (or _coefficient of elasticity_). It is the ratio of stress per unit of area to the deformation per unit of { unit stress } length. { E = ------------- } It is a number indicative of { unit strain } stiffness, not of strength, and only applies to conditions within the elastic limit. It is nearly the same whether derived from compression tests or from tension tests.

A large modulus indicates a stiff material. Thus in green wood tested in static bending it varies from 643,000 pounds per square inch for arborvitæ to 1,662,000 pounds for longleaf pine, and 1,769,000 pounds for pignut hickory. (See Table IX.) The values derived from tests of small beams of dry material are much greater, approaching 3,000,000 for some of our woods. These values are small when compared with steel which has a modulus of elasticity of about 30,000,000 pounds per square inch. (See Table I.)

|------------------------------------------------------------------------------| | TABLE I | |------------------------------------------------------------------------------| | COMPARATIVE STRENGTH OF IRON, STEEL, AND WOOD | |------------------------------------------------------------------------------| | | Sp. | Modulus of | Tensile | Crushing | Modulus | | MATERIAL | gr., | elasticity | strength | strength | of | | | dry | in bending | | | rupture | |-------------------------+----- +------------+----------+----------+----------| | | | Lbs. per | Lbs. per | Lbs. per | Lbs. per | | | | sq. in. | sq. in. | sq. in. | sq. in. | | | | | | | | | Cast iron, cold blast | | | | | | | (Hodgkinson) | 7.1 | 17,270,000 | 16,700 | 106,000 | 38,500 | | Bessenger steel, | | | | | | | high grade (Fairbain) | 7.8 | 29,215,000 | 88,400 | 225,600 | | | Longleaf pine, | | | | | | | 3.5% moisture (U.S.) | .63 | 2,800,000 | | 13,000 | 21,000 | | Redspruce, | | | | | | | 3.5% moisture (U.S.) | .41 | 1,800,000 | | 8,800 | 14,500 | | Pignut hickory, | | | | | | | 3.5% moisture (U.S.) | .86 | 2,370,000 | | 11,130 | 24,000 | |------------------------------------------------------------------------------| | NOTE.--Great variation may be found in different samples of metals as well | | as of wood. The examples given represent reasonable values. | |------------------------------------------------------------------------------|

TENSILE STRENGTH

~Tension~ results when a pulling force is applied to opposite ends of a body. This external pull is communicated to the interior, so that any portion of the material exerts a pull or tensile force upon the remainder, the ability to do so depending upon the property of cohesion. The result is an elongation or stretching of the material in the direction of the applied force. The action is the opposite of compression.

Wood exhibits its greatest strength in tension parallel to the grain, and it is very uncommon in practice for a specimen to be pulled in two lengthwise. This is due to the difficulty of making the end fastenings secure enough for the full tensile strength to be brought into play before the fastenings shear off longitudinally. This is not the case with metals, and as a result they are used in almost all places where tensile strength is particularly needed, even though the remainder of the structure, such as sills, beams, joists, posts, and flooring, may be of wood. Thus in a wooden truss bridge the tension members are steel rods.

The tensile strength of wood parallel to the grain depends upon the strength of the fibres and is affected not only by the nature and dimensions of the wood elements but also by their arrangement. It is greatest in straight-grained specimens with thick-walled fibres. Cross grain of any kind materially reduces the tensile strength of wood, since the tensile strength at right angles to the grain is only a small fraction of that parallel to the grain.

|--------------------------------------------------------------| | TABLE II | |--------------------------------------------------------------| | RATIO OF STRENGTH OF WOOD IN TENSION AND IN COMPRESSION | | (Bul. 10, U. S. Div. of Forestry, p. 44) | |--------------------------------------------------------------| | | Ratio: | A stick 1 square inch in | | | | cross section. | | | Tensile | | | KIND OF WOOD | strength | Weight required to-- | | | R = ----------- +----------------------------| | | compressive | Pull apart | Crush endwise | | | strength | | | |---------------+-----------------+------------+---------------| | Hickory | 3.7 | 32,000 | 8,500 | | Elm | 3.8 | 29,000 | 7,500 | | Larch | 2.3 | 19,400 | 8,600 | | Longleaf Pine | 2.2 | 17,300 | 7,400 | |--------------------------------------------------------------| | NOTE.--Moisture condition not given. | |--------------------------------------------------------------|

Failure of wood in tension parallel to the grain occurs sometimes in flexure, especially with dry material. The tension portion of the fracture is nearly the same as though the piece were pulled in two lengthwise. The fibre walls are torn across obliquely and usually in a spiral direction. There is practically no pulling apart of the fibres, that is, no separation of the fibres along their walls, regardless of their thickness. The nature of tension failure is apparently not affected by the moisture condition of the specimen, at least not so much so as the other strength values.[3]

[Footnote 3: See Brush, Warren D.: A microscopic study of the mechanical failure of wood. Vol. II, Rev. F.S. Investigations, Washington, D.C., 1912, p. 35.]

Tension at right angles to the grain is closely related to cleavability. When wood fails in this manner the thin fibre walls are torn in two lengthwise while the thick-walled fibres are usually pulled apart along the primary wall.

|--------------------------------------------| | TABLE III | |--------------------------------------------| | TENSILE STRENGTH AT RIGHT ANGLES TO THE | | GRAIN OF SMALL CLEAR PIECES OF 25 WOODS IN | | GREEN CONDITION | | (Forest Service Cir. 213) | |--------------------------------------------| | | When | When | | COMMON NAME | surface of | surface of | | OF SPECIES | failure is | failure is | | | radial | tangential | |------------------+------------+------------| | | Lbs. per | Lbs. per | | | sq. inch | sq. inch | | | | | | Hardwoods | | | | | | | | Ash, white | 645 | 671 | | Basswood | 226 | 303 | | Beech | 633 | 969 | | Birch, yellow | 446 | 526 | | Elm, slippery | 765 | 832 | | Hackberry | 661 | 786 | | Locust, honey | 1,133 | 1,445 | | Maple, sugar | 610 | 864 | | Oak, post | 714 | 924 | | red | 639 | 874 | | swamp white | 757 | 909 | | white | 622 | 749 | | yellow | 728 | 929 | | Sycamore | 540 | 781 | | Tupelo | 472 | 796 | | | | | | Conifers | | | | | | | | Arborvitæ | 241 | 235 | | Cypress, bald | 242 | 251 | | Fir, white | 213 | 304 | | Hemlock | 271 | 323 | | Pine, longleaf | 240 | 298 | | red | 179 | 205 | | sugar | 239 | 304 | | western yellow | 230 | 252 | | white | 225 | 285 | | Tamarack | 236 | 274 | |--------------------------------------------|

COMPRESSIVE OR CRUSHING STRENGTH

~Compression across the grain~ is very closely related to hardness and transverse shear. There are two ways in which wood is subjected to stress of this kind, namely, (1) with the load acting over the entire area of the specimen, and (2) with a load concentrated over a portion of the area. (See Fig. 2.) The latter is the condition more commonly met with in practice, as, for example, where a post rests on a horizontal sill, or a rail rests on a cross-tie. The former condition, however, gives the true resistance of the grain to simple crushing.

The first effect of compression across the grain is to compact the fibres, the load gradually but irregularly increasing as the density of the material is increased. If the specimen lies on a flat surface and the load is applied to only a portion of the upper area, the bearing plate indents the wood, crushing the upper fibres without affecting the lower part. (See Fig. 3.) As the load increases the projecting ends sometimes split horizontally. (See Fig. 4.) The irregularities in the load are due to the fact that the fibres collapse a few at a time, beginning with those with the thinnest walls. The projection of the ends increases the strength of the material directly beneath the compressing weight by introducing a beam action which helps support the load. This influence is exerted for a short distance only.

When wood is used for columns, props, posts, and spokes, the weight of the load tends to shorten the material endwise. This is ~endwise compression~, or compression parallel to the grain. In the case of long columns, that is, pieces in which the length is very great compared with their diameter, the failure is by sidewise bending or flexure, instead of by crushing or splitting. (See Fig. 5.) A familiar instance of this action is afforded by a flexible walking-stick. If downward pressure is exerted with the hand on the upper end of the stick placed vertically on the floor, it will be noted that a definite amount of force must be applied in each instance before decided flexure takes place. After this point is reached a very slight increase of pressure very largely increases the deflection, thus obtaining so great a leverage about the middle section as to cause rupture.

The lateral bending of a column produces a combination of bending with compressive stress over the section, the compressive stress being maximum at the section of greatest deflection on the concave side. The convex surface is under tension, as in an ordinary beam test. (See Fig. 6.) If the same stick is braced in such a way that flexure is prevented, its supporting strength is increased enormously, since the compressive stress acts uniformly over the section, and failure is by crushing or splitting, as in small blocks. In all columns free to bend in any direction the deflection will be seen in the direction in which the column is least stiff. This sidewise bending can be overcome by making pillars and columns thicker in the middle than at the ends, and by bracing studding, props, and compression members of trusses. The strength of a column also depends to a considerable extent upon whether the ends are free to turn or are fixed.

|-------------------------------------------------------| | TABLE IV | |-------------------------------------------------------| | RESULTS OF COMPRESSION TESTS ACROSS THE GRAIN ON | | 51 WOODS IN GREEN CONDITION, AND COMPARISON WITH | | WHITE OAK | | (U. S. Forest Service) | |-------------------------------------------------------| | | Fibre stress | Fiber stress | | COMMON NAME | at elastic | in per cent | | OF SPECIES | limit | of white oak, | | | perpendicular | or 853 pounds | | | to grain | per sq. in. | |-----------------------+---------------+---------------| | | Lbs. per | | | | sq. inch | Per cent | | | | | | Osage orange | 2,260 | 265.0 | | Honey locust | 1,684 | 197.5 | | Black locust | 1,426 | 167.2 | | Post oak | 1,148 | 134.6 | | Pignut hickory | 1,142 | 133.9 | | Water hickory | 1,088 | 127.5 | | Shagbark hickory | 1,070 | 125.5 | | Mockernut hickory | 1,012 | 118.6 | | Big shellbark hickory | 997 | 116.9 | | Bitternut hickory | 986 | 115.7 | | Nutmeg hickory | 938 | 110.0 | | Yellow oak | 857 | 100.5 | | White oak | 853 | 100.0 | | Bur oak | 836 | 98.0 | | White ash | 828 | 97.1 | | Red oak | 778 | 91.2 | | Sugar maple | 742 | 87.0 | | Rock elm | 696 | 81.6 | | Beech | 607 | 71.2 | | Slippery elm | 599 | 70.2 | | Redwood | 578 | 67.8 | | Bald cypress | 548 | 64.3 | | Red maple | 531 | 62.3 | | Hackberry | 525 | 61.6 | | Incense cedar | 518 | 60.8 | | Hemlock | 497 | 58.3 | | Longleaf pine | 491 | 57.6 | | Tamarack | 480 | 56.3 | | Silver maple | 456 | 53.5 | | Yellow birch | 454 | 53.2 | | Tupelo | 451 | 52.9 | | Black cherry | 444 | 52.1 | | Sycamore | 433 | 50.8 | | Douglas fir | 427 | 50.1 | | Cucumber tree | 408 | 47.8 | | Shortleaf pine | 400 | 46.9 | | Red pine | 358 | 42.0 | | Sugar pine | 353 | 41.1 | | White elm | 351 | 41.2 | | Western yellow pine | 348 | 40.8 | | Lodgepole pine | 348 | 40.8 | | Red spruce | 345 | 40.5 | | White pine | 314 | 36.8 | | Engelman spruce | 290 | 34.0 | | Arborvitæ | 288 | 33.8 | | Largetooth aspen | 269 | 31.5 | | White spruce | 262 | 30.7 | | Butternut | 258 | 30.3 | | Buckeye (yellow) | 210 | 24.6 | | Basswood | 209 | 24.5 | | Black willow | 193 | 22.6 | |-------------------------------------------------------|

The complexity of the computations depends upon the way in which the stress is applied and the manner in which the stick bends. Ordinarily where the length of the test specimen is not greater than four diameters and the ends are squarely faced (see Fig. 7), the force acts uniformly over each square inch of area and the crushing strength is equal to the maximum load (P) divided { P } by the area of the cross-section (A). { C = --- } { A }

It has been demonstrated[4] that the ultimate strength in compression parallel to the grain is very nearly the same as the extreme fibre stress at the elastic limit in bending. (See Table V.) In other words, the transverse strength of beams at elastic limit is practically equal to the compressive strength of the same material in short columns. It is accordingly possible to calculate the approximate breaking strength of beams from the compressive strength of short columns except when the wood is brittle. Since tests on endwise compression are simpler, easier to make, and less expensive than transverse bending tests, the importance of this relation is obvious, though it does not do away with the necessity of making beam tests.

[Footnote 4: See Circular No. 18, U.S. Division of Forestry: Progress in timber physics, pp. 13-18; also Bulletin 70, U.S. Forest Service: Effect of moisture on the strength and stiffness of wood, pp. 42, 89-90.]

|-------------------------------------------------------------------------------| | TABLE V | |-------------------------------------------------------------------------------| | RELATION OF FIBRE STRESS AT ELASTIC LIMIT (r) IN BENDING TO THE CRUSHING | | STRENGTH (C) OF BLOCKS CUT THEREFROM, IN POUNDS PER SQUARE INCH | | (Forest Service Bul. 70, p. 90) | |-------------------------------------------------------------------------------| | LONGLEAF PINE | |-------------------------------------------------------------------------------| | | Soaked | Green | 14 | 11.5 | 9.5 | Kiln-dry | | MOISTURE CONDITION | 50 per | 23 per | per | per | per | 6.2 per | | | cent | cent | cent | cent | cent | cent | | -------------------------+--------+--------+-------+-------+-------+----------| | Number of tests averaged | 5 | 5 | 5 | 5 | 4 | 5 | | _r_ in bending | 4,920 | 5,944 | 6,924 | 7,852 | 9,280 | 11,550 | | _C_ in compression | 4,668 | 5,100 | 6,466 | 7,466 | 8,985 | 10,910 | | Per cent _r_ is in | | | | | | | | excess of _C_ | 5.5 | 16.5 | 7.1 | 5.2 | 3.3 | 5.9 | |-------------------------------------------------------------------------------| | SPRUCE | |-------------------------------------------------------------------------------| | | Soaked | Green | 10 | 8.1 | Kiln-dry | | MOISTURE CONDITION | 30 per | 30 per | per | per | 3.9 per | | | cent | cent | cent | cent | cent | |----------------------------------+--------+--------+-------+-------+----------| | Number of tests averaged | 5 | 4 | 5 | 3 | 4 | | _r_ in bending | 3,002 | 3,362 | 6,458 | 8,400 | 10,170 | | _C_ in compression | 2,680 | 3,025 | 6,120 | 7,610 | 9,335 | | Per cent _r_ | | | | | | | is in excess of _C_ | 12.0 | 11.1 | 5.5 | 10.4 | 9.0 | |-------------------------------------------------------------------------------|

When a short column is compressed until it breaks, the manner of failure depends partly upon the anatomical structure and partly upon the degree of humidity of the wood. The fibres (tracheids in conifers) act as hollow tubes bound closely together, and in giving way they either (1) buckle, or (2) bend.[5]

[Footnote 5: See Bulletin 70, _op. cit._, p. 129.]

The first is typical of any dry thin-walled cells, as is usually the case in seasoned white pine and spruce, and in the early wood of hard pines, hemlock, and other species with decided contrast between the two portions of the growth ring. As a rule buckling of a tracheid begins at the bordered pits which form places of least resistance in the walls. In hardwoods such as oak, chestnut, ash, etc., buckling occurs only in the thinnest-walled elements, such as the vessels, and not in the true fibres.

According to Jaccard[6] the folding of the cells is accompanied by characteristic alterations of their walls which seem to split them into extremely thin layers. When greatly magnified, these layers appear in longitudinal sections as delicate threads without any definite arrangements, while on cross section they appear as numerous concentric strata. This may be explained on the ground that the growth of a fibre is by successive layers which, under the influence of compression, are sheared apart. This is particularly the case with thick-walled cells such as are found in late wood.

[Footnote 6: Jaccard, P.: Étude anatomique des bois comprimés. Mit. d. Schw. Centralanstalt f.d. forst. Versuchswesen. X. Band, 1. Heft. Zurich, 1910, p. 66.]

|-------------------------------------------------------| | TABLE VI | |-------------------------------------------------------| | RESULTS OF ENDWISE COMPRESSION TESTS ON SMALL CLEAR | | PIECES OF 40 WOODS IN GREEN CONDITION | | (Forest Service Cir. 213) | |-------------------------------------------------------| | | Fibre | | Modulus | | COMMON NAME | stress at | Crushing | of | | OF SPECIES | elastic | strength | elasticity | | | limit | | | |-------------------+-----------+----------+------------| | | Lbs. per | Lbs. per | Lbs. per | | | sq. inch | sq. inch | sq. inch | | | | | | | Hardwoods | | | | | | | | | | Ash, white | 3,510 | 4,220 | 1,531,000 | | Basswood | 780 | 1,820 | 1,016,000 | | Beech | 2,770 | 3,480 | 1,412,000 | | Birch, yellow | 2,570 | 3,400 | 1,915,000 | | Elm, slippery | 3,410 | 3,990 | 1,453,000 | | Hackberry | 2,730 | 3,310 | 1,068,000 | | Hickory, | | | | | big shellbark | 3,570 | 4,520 | 1,658,000 | | bitternut | 4,330 | 4,570 | 1,616,000 | | mockernut | 3,990 | 4,320 | 1,359,000 | | nutmeg | 3,620 | 3,980 | 1,411,000 | | pignut | 3,520 | 4,820 | 1,980,000 | | shagbark | 3,730 | 4,600 | 1,943,000 | | water | 3,240 | 4,660 | 1,926,000 | | Locust, honey | 4,300 | 4,970 | 1,536,000 | | Maple, sugar | 3,040 | 3,670 | 1,463,000 | | Oak, post | 2,780 | 3,330 | 1,062,000 | | red | 2,290 | 3,210 | 1,295,000 | | swamp white | 3,470 | 4,360 | 1,489,000 | | white | 2,400 | 3,520 | 946,000 | | yellow | 2,870 | 3,700 | 1,465,000 | | Osage orange | 3,980 | 5,810 | 1,331,000 | | Sycamore | 2,320 | 2,790 | 1,073,000 | | Tupelo | 2,280 | 3,550 | 1,280,000 | | | | | | | Conifers | | | | | | | | | | Arborvitæ | 1,420 | 1,990 | 754,000 | | Cedar, incense | 2,710 | 3,030 | 868,000 | | Cypress, bald | 3,560 | 3,960 | 1,738,000 | | Fir, alpine | 1,660 | 2,060 | 882,000 | | amabilis | 2,763 | 3,040 | 1,579,000 | | Douglas | 2,390 | 2,920 | 1,440,000 | | white | 2,610 | 2,800 | 1,332,000 | | Hemlock | 2,110 | 2,750 | 1,054,000 | | Pine, lodgepole | 2,290 | 2,530 | 1,219,000 | | longleaf | 3,420 | 4,280 | 1,890,000 | | red | 2,470 | 3,080 | 1,646,000 | | sugar | 2,340 | 2,600 | 1,029,000 | | western yellow | 2,100 | 2,420 | 1,271,000 | | white | 2,370 | 2,720 | 1,318,000 | | Redwood | 3,420 | 3,820 | 1,175,000 | | Spruce, Engelmann | 1,880 | 2,170 | 1,021,000 | | Tamarack | 3,010 | 3,480 | 1,596,000 | |-------------------------------------------------------|

The second case, where the fibres bend with more or less regular curves instead of buckling, is characteristic of any green or wet wood, and in dry woods where the fibres are thick-walled. In woods in which the fibre walls show all gradations of thickness--in other words, where the transition from the thin-walled cells of the early wood to the thick-walled cells of the late wood is gradual--the two kinds of failure, namely, buckling and bending, grade into each other. In woods with very decided contrast between early and late wood the two forms are usually distinct. Except in the case of complete failure the cavity of the deformed cells remains open, and in hardwoods this is true not only of the wood fibres but also of the tube-like vessels. In many cases longitudinal splits occur which isolate bundles of elements by greater or less intervals. The splitting occurs by a tearing of the fibres or rays and not by the separation of the rays from the adjacent elements.

Moisture in wood decreases the stiffness of the fibre walls and enlarges the region of failure. The curve which the fibre walls make in the region of failure is more gradual and also more irregular than in dry wood, and the fibres are more likely to be separated.

In examining the lines of rupture in compression parallel to the grain it appears that there does not exist any specific type, that is, one that is characteristic of all woods. Test blocks taken from different parts of the same log may show very decided differences in the manner of failure, while blocks that are much alike in the size, number, and distribution of the elements of unequal resistance may behave very similarly. The direction of rupture is, according to Jaccard, not influenced by the distribution of the medullary rays.[7] These are curved with the bundles of fibres to which they are attached. In any case the failure starts at the weakest points and follows the lines of least resistance. The plane of failure, as visible on radial surfaces, is horizontal, and on the tangential surface it is diagonal.

[Footnote 7: This does not correspond exactly with the conclusions of A. Thil, who says ("Constitution anatomique du bois," pp. 140-141): "The sides of the medullary rays sometimes produce planes of least resistance varying in size with the height of the rays. The medullary rays assume a direction more or less parallel to the lumen of the cells on which they border; the latter curve to the right or left to make room for the ray and then close again beyond it. If the force acts parallel to the axis of growth, the tracheids are more likely to be displaced if the marginal cells of the medullary rays are provided with weak walls that are readily compressed. This explains why on the radial surface of the test blocks the plane of rupture passes in a direction nearly following a medullary ray, whereas on the tangential surface the direction of the plane of rupture is oblique--but with an obliquity varying with the species and determined by the pitch of the spirals along which the medullary rays are distributed in the stem." See Jaccard, _op. cit._, pp. 57 _et seq._]

SHEARING STRENGTH

Whenever forces act upon a body in such a way that one portion tends to slide upon another adjacent to it the action is called a ~shear~.[8] In wood this shearing action may be (1) ~along the grain~, or (2) ~across the grain~. A tenon breaking out its mortise is a familiar example of shear along the grain, while the shoving off of the tenon itself would be shear across the grain. The use of wood for pins or tree-nails involves resistance to shear across the grain. Another common instance of the latter is where the steel edge of the eye of an axe or hammer tends to cut off the handle. In Fig. 10 the action of the wooden strut tends to shear off along the grain the portion _AB_ of the wooden tie rod, and it is essential that the length of this portion be great enough to guard against it. Fig. 11 shows characteristic failures in shear along the grain.

[Footnote 8: Shear should not be confused with ordinary cutting or incision.]

|---------------------------------------------| | TABLE VII | |---------------------------------------------| | SHEARING STRENGTH ALONG THE GRAIN OF SMALL | | CLEAR PIECES OF 41 WOODS IN GREEN CONDITION | | (Forest Service Cir. 213) | |---------------------------------------------| | | When | When | | COMMON NAME | surface of | surface of | | OF SPECIES | failure is | failure is | | | radial | tangential | |-------------------+------------+------------| | | Lbs. per | Lbs. per | | | sq. inch | sq. inch | | | | | | Hardwoods | | | | | | | | Ash, black | 876 | 832 | | white | 1,360 | 1,312 | | Basswood | 560 | 617 | | Beech | 1,154 | 1,375 | | Birch, yellow | 1,103 | 1,188 | | Elm, slippery | 1,197 | 1,174 | | white | 778 | 872 | | Hackberry | 1,095 | 1,161 | | Hickory, | | | | big shellbark | 1,134 | 1,191 | | bitternut | 1,134 | 1,348 | | mockernut | 1,251 | 1,313 | | nutmeg | 1,010 | 1,053 | | pignut | 1,334 | 1,457 | | shagbark | 1,230 | 1,297 | | water | 1,390 | 1,490 | | Locust, honey | 1,885 | 2,096 | | Maple, red | 1,130 | 1,330 | | sugar | 1,193 | 1,455 | | Oak, post | 1,196 | 1,402 | | red | 1,132 | 1,195 | | swamp white | 1,198 | 1,394 | | white | 1,096 | 1,292 | | yellow | 1,162 | 1,196 | | Sycamore | 900 | 1,102 | | Tupelo | 978 | 1,084 | | | | | | Conifers | | | | | | | | Arborvitæ | 617 | 614 | | Cedar, incense | 613 | 662 | | Cypress, bald | 836 | 800 | | Fir, alpine | 573 | 654 | | amabilis | 517 | 639 | | Douglas | 853 | 858 | | white | 742 | 723 | | Hemlock | 790 | 813 | | Pine, lodgepole | 672 | 747 | | longleaf | 1,060 | 953 | | red | 812 | 741 | | sugar | 702 | 714 | | western yellow | 686 | 706 | | white | 649 | 639 | | Spruce, Engelmann | 607 | 624 | | Tamarack | 883 | 843 | |---------------------------------------------|

Both shearing stresses may act at the same time. Thus the weight carried by a beam tends to shear it off at right angles to the axis; this stress is equal to the resultant force acting perpendicularly at any point, and in a beam uniformly loaded and supported at either end is maximum at the points of support and zero at the centre. In addition there is a shearing force tending to move the fibres of the beam past each other in a longitudinal direction. (See Fig. 12.) This longitudinal shear is maximum at the neutral plane and decreases toward the upper and lower surfaces.

Shearing across the grain is so closely related to compression at right angles to the grain and to hardness that there is little to be gained by making separate tests upon it. Knowledge of shear parallel to the grain is important, since wood frequently fails in that way. The value of shearing stress parallel to the grain is found by dividing the maximum load in pounds (P) by the area of the cross section in inches (A).

{ P }
{ Shear = --- }
{ A }

Oblique shearing stresses are developed in a bar when it is subjected to direct tension or compression. The maximum shearing stress occurs along a plane when it makes an angle of 45 degrees P with the axis of the specimen. In this case, shear = -----. When 2 A the value of the angle [Greek: theta] is less than 45 degrees, P the shear along the plane = --- sin [Greek: theta] cos [Greek: A theta]. (See Fig. 13.) The effect of oblique shear is often visible in the failures of short columns. (See Fig. 14.)

|---------------------------------------------------------------------------| | TABLE VIII | |---------------------------------------------------------------------------| | SHEARING STRENGTH ACROSS THE GRAIN OF VARIOUS AMERICAN WOODS | | (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.]

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.)

_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.