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. (N1P 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.

Figure 15

Diagram of a simple beam. N1P = 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.)

Figure 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]



P' l3
E=-----------


4 D b h3



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 h3
Necessary weight (P')=----------


l3

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 h2



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 (P1) 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)
COMMON NAME OF SPECIES Fibre stress at elastic limit Modulus of rupture Modulus of elasticity Work in Bending
To elastic limit To maximum load Total
Lbs. per sq. in. Lbs. per sq. in. Lbs. per sq. in. In.-lbs. per cu. inch In.-lbs. per cu. inch In.-lbs. per cu. 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]

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.

(
ultimate strength)
Factor of safety
= -------------------
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:

(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 velocity, and this velocity of the weight and beam gives to them an energy, or vis viva, which must now spend itself in overcoming an excess of resistance over and above the imposed load, and the whole mass will not stop until the deflection (as well as the resistance) has come to be equal to twice that corresponding to the static load imposed. Hence we say the effect of a suddenly imposed load is to produce twice the deflection and stress of the same load statically applied. It must be evident, however, that this case has nothing in common with either the ordinary 'static' tests of structural materials in testing-machines, or with impact tests."[13]

(3) Impact, shock, or blow.[14] There are various common uses of wood where the material is subjected to sudden shocks and jars or impact. Such is the action on the felloes and spokes of a wagon wheel passing over a rough road; on a hammer handle when a blow is struck; on a maul when it strikes a wedge.

Resistance to impact is resistance to energy which is measured by the product of the force into the space through which it moves, or by the product of one-half the moving mass which causes the shock into the square of its velocity. The work done upon the piece at the instant the velocity is entirely removed from the striking body is equal to the total energy of that body. It is impossible, however, to get all of the energy of the striking body stored in the specimen, though the greater the mass and the shorter the space through which it moves, or, in other words, the greater the proportion of weight and the smaller the proportion of velocity making up the energy of the striking body, the more energy the specimen will absorb. The rest is lost in friction, vibrations, heat, and motion of the anvil.

In impact the stresses produced become very complex and difficult to measure, especially if the velocity is high, or the mass of the beam itself is large compared to that of the weight.

The difficulties attending the measurement of the stresses beyond the elastic limit are so great that commonly they are not reckoned. Within the elastic limit the formulæ for calculating the stresses are based on the assumption that the deflection is proportional to the stress in this case as in static tests.

A common method of making tests upon the resistance of wood to shock is to support a small beam at the ends and drop a heavy weight upon it in the middle. ([See Fig. 40].) The height of the weight is increased after each drop and records of the deflection taken until failure. The total work done upon the specimen is equal to the area of the stress-strain diagram plus the effect of local inertia of the molecules at point of contact.

The stresses involved in impact are complicated by the fact that there are various ways in which the energy of the striking body may be spent:

(a) It produces a local deformation of both bodies at the surface of contact, within or beyond the elastic limit. In testing wood the compression of the substance of the steel striking-weight may be neglected, since the steel is very hard in comparison with the wood. In addition to the compression of the fibres at the surface of contact resistance is also offered by the inertia of the particles there, the combined effect of which is a stress at the surface of contact often entirely out of proportion to the compression which would result from the action of a static force of the same magnitude. It frequently exceeds the crushing strength at the extreme surface of contact, as in the case of the swaging action of a hammer on the head of an iron spike, or of a locomotive wheel on the steel rail. This is also the case when a bullet is shot through a board or a pane of glass without breaking it as a whole.

(b) It may move the struck body as a whole with an accelerated velocity, the resistance consisting of the inertia of the body. This effect is seen when a croquet ball is struck with a mallet.

(c) It may deform a fixed body against its external supports and resistances. In making impact tests in the laboratory the test specimen is in reality in the nature of a cushion between two impacting bodies, namely, the striking weight and the base of the machine. It is important that the mass of this base be sufficiently great that its relative velocity to that of the common centre of gravity of itself and the striking weight may be disregarded.

(d) It may deform the struck body as a whole against the resisting stresses developed by its own inertia, as, for example, when a baseball bat is broken by striking the ball.

TABLE X
RESULTS OF IMPACT BENDING TESTS ON SMALL CLEAR BEAMS OF 34 WOODS IN GREEN CONDITION
(Forest Service Cir. 213)
COMMON NAME OF SPECIES Fibre stress at elastic limit Modulus of elasticity Work in bending to elastic limit
Lbs. per sq. in. Lbs. per sq. in. In.-lbs. per cu. inch
Hardwoods


Ash, black 7,840 955,000 3.69
white 11,710 1,564,000 4.93
Basswood 5,480 917,000 1.84
Beech 11,760 1,501,000 5.10
Birch, yellow 11,080 1,812,000 3.79
Elm, rock 12,090 1,367,000 6.52
slippery 11,700 1,569,000 4.86
white 9,910 1,138,000 4.82
Hackberry 10,420 1,398,000 4.48
Locust, honey 13,460 2,114,000 4.76
Maple, red 11,670 1,411,000 5.45
sugar 11,680 1,680,000 4.55
Oak, post 11,260 1,596,000 4.41
red 10,580 1,506,000 4.16
swamp white 13,280 2,048,000 4.79
white 9,860 1,414,000 3.84
yellow 10,840 1,479,000 4.44
Osage orange 15,520 1,498,000 8.92
Sycamore 8,180 1,165,000 3.22
Tupelo 7,650 1,310,000 2.49
Conifers


Arborvitæ 5,290 778,000 2.04
Cypress, bald 8,290 1,431,000 2.71
Fir, alpine 5,280 980,000 1.59
Douglas 8,870 1,579,000 2.79
white 7,230 1,326,000 2.21
Hemlock 6,330 1,025,000 2.19
Pine, lodgepole 6,870 1,142,000 2.31
longleaf 9,680 1,739,000 3.02
red 7,480 1,438,000 2.18
sugar 6,740 1,083,000 2.34
western yellow 7,070 1,115,000 2.51
white 6,490 1,156,000 2.06
Spruce, Engelmann 6,300 1,076,000 2.09
Tamarack 7,750 1,263,000 2.67

Impact testing is difficult to conduct satisfactorily and the data obtained are of chief value in a relative sense, that is, for comparing the shock-resisting ability of woods of which like specimens have been subjected to exactly identical treatment. Yet this test is one of the most important made on wood, as it brings out properties not evident from other tests. Defects and brittleness are revealed by impact better than by any other kind of test. In common practice nearly all external stresses are of the nature of impact. In fact, no two moving bodies can come together without impact stress. Impact is therefore the commonest form of applied stress, although the most difficult to measure.

Failures in Timber Beams

If a beam is loaded too heavily it will break or fail in some characteristic manner. These failures may be classified according to the way in which they develop, as tension, compression, and horizontal shear; and according to the appearance of the broken surface, as brash, and fibrous. A number of forms may develop if the beam is completely ruptured.

Since the tensile strength of wood is on the average about three times as great as the compressive strength, a beam should, therefore, be expected to fail by the formation in the first place of a fold on the compression side due to the crushing action, followed by failure on the tension side. This is usually the case in green or moist wood. In dry material the first visible failure is not infrequently on the lower or tension side, and various attempts have been made to explain why such is the case.[15]

Within the elastic limit the elongations and shortenings are equal, and the neutral plane lies in the middle of the beam. ([See page 23].) Later the top layer of fibres on the upper or compression side fail, and on the load increasing, the next layer of fibres fail, and so on, even though this failure may not be visible. As a result the shortenings on the upper side of the beam become considerably greater than the elongations on the lower side. The neutral plane must be presumed to sink gradually toward the tension side, and when the stresses on the outer fibres at the bottom have become sufficiently great, the fibres are pulled in two, the tension area being much smaller than the compression area. The rupture is often irregular, as in direct tension tests. Failure may occur partially in single bundles of fibres some time before the final failure takes place. One reason why the failure of a dry beam is different from one that is moist, is that drying increases the stiffness of the fibres so that they offer more resistance to crushing, while it has much less effect upon the tensile strength.

There is considerable variation in tension failures depending upon the toughness or the brittleness of the wood, the arrangement of the grain, defects, etc., making further classification desirable. The four most common forms are:

(1) Simple tension, in which there is a direct pulling in two of the wood on the under side of the beam due to a tensile stress parallel to the grain, ([See Fig. 17], No. 1.) This is common in straight-grained beams, particularly when the wood is seasoned.

(2) Cross-grained tension, in which the fracture is caused by a tensile force acting oblique to the grain. ([See Fig. 17], No. 2.) This is a common form of failure where the beam has diagonal, spiral or other form of cross grain on its lower side. Since the tensile strength of wood across the grain is only a small fraction of that with the grain it is easy to see why a cross-grained timber would fail in this manner.

(3) Splintering tension, in which the failure consists of a considerable number of slight tension failures, producing a ragged or splintery break on the under surface of the beam. ([See Fig. 17], No. 3.) This is common in tough woods. In this case the surface of fracture is fibrous.

(4) Brittle tension, in which the beam fails by a clean break extending entirely through it. ([See Fig. 17], No. 4.) It is characteristic of a brittle wood which gives way suddenly without warning, like a piece of chalk. In this case the surface of fracture is described as brash.

Compression failure ([see Fig. 17], No. 5) has few variations except that it appears at various distances from the neutral plane of the beam. It is very common in green timbers. The compressive stress parallel to the fibres causes them to buckle or bend as in an endwise compressive test. This action usually begins on the top side shortly after the elastic limit is reached and extends downward, sometimes almost reaching the neutral plane before complete failure occurs. Frequently two or more failures develop at about the same time.

Figure 17

Characteristic failures of simple beams.

Horizontal shear failure, in which the upper and lower portions of the beam slide along each other for a portion of their length either at one or at both ends ([see Fig. 17], No. 6), is fairly common in air-dry material and in green material when the ratio of the height of the beam to the span is relatively large. It is not common in small clear specimens. It is often due to shake or season checks, common in large timbers, which reduce the actual area resisting the shearing action considerably below the calculated area used in the formulæ for horizontal shear. ([See page 98] for this formulæ.) For this reason it is unsafe, in designing large timber beams, to use shearing stresses higher than those calculated for beams that failed in horizontal shear. The effect of a failure in horizontal shear is to divide the beam into two or more beams the combined strength of which is much less than that of the original beam. [Fig. 18] shows a large beam in which two failures in horizontal shear occurred at the same end. That the parts behave independently is shown by the compression failure below the original location of the neutral plane.

Figure 18

Failure of a large beam by horizontal shear. Photo by U. S, Forest Service.

Table XI gives an analysis of the causes of first failure in 840 large timber beams of nine different species of conifers. Of the total number tested 165 were air-seasoned, the remainder green. The failure occurring first signifies the point of greatest weakness in the specimen under the particular conditions of loading employed (in this case, third-point static loading).

TABLE XI
MANNER OF FIRST FAILURE OF LARGE BEAMS
(Forest Service Bul. 108, p. 56)
COMMON NAME OF SPECIES Total number of tests Per cent of total failing by
Tension Compression Shear
Longleaf pine:



green 17 18 24 58
dry 9 22 22 56
Douglas fir:



green 191 27 72 1
dry 91 19 76 5
Shortleaf pine:



green 48 27 56 17
dry 13 54
46
Western larch:



green 62 23 71 6
dry 52 54 19 27
Loblolly pine:



green 111 40 53 7
dry 25 60 12 28
Tamarack:



green 30 37 53 10
dry 9 45 22 33
Western hemlock:



green 39 21 74 5
dry 44 11 66 23
Redwood:



green 28 43 50 7
dry 12 83 17
Norway pine:



green 49 18 76 6
dry 10 30 60 10
NOTE.—These tests were made on timbers ranging in cross section from 4" × 10" to 8" × 16", and with a span of 15 feet.