This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.
Now to proceed to the calculation of the resistance of this wedge. It is evident that whatever may be the external strain on the beam at the section A B C D, the beam will owe whatever resistance it has at that point to the resistances of the fibres of the section or wedge to compression and tension.
Now considering the right-hand side of the beam as rigid, and the section A B C D as the point of fulcrum of the external forces, we have only one external force p, tending to turn the left-hand side of the beam upwards around the section A B C D, its total tendency, effect or moment mat A B C D, we know is m=p. x (law of the lever).
Now to resist this we have the opposition of the fibres in the wedge A B A, B, M N to compression and the opposition to tension of the fibres in the wedge D C D1C1 M N. For the sake of convenience, we will still consider these wedges, as wedges but so infinites-imally thin that we can safely put down the amount of their contents as equal to the area of their sides, so that - if AB =b (the width of beam) and AD = d (the depth of beam) - we can safely call each wedge as equal to b. d/2 .
Now as the centre of gravity of a wedge is at 1/3 of the height from its base, or § of the height from its apex (and as the height of each wedge is= d/2 ) it would be = 2/3 . d/2 = d/3 from axis M - N. The moment of a wedge at any axis M-N is equal to the contents of the wedge multiplied by the distance of its centre of gravity from the axis, the whole multiplied by the stress of the fibres, (that is their resistance to tension or compression). Now the contents of each wedge being = b. d/2, the distance of centre of gravity from M-N = d/3, and the stress being say = s, we have for the resistance of each wedge
= b. d/2. d/3. s
= b.d2/6. s
Now if the stress on the fibres along the extreme upper or lower edges = k (or the modulus of rupture), it is evident that the average stress on the fibres in either wedge will = k/2, or s = k/2 (for the stress on each fibre being directly proportionate to the distance from the neutral axis the stress on the average will be equal to half that on the base). Now inserting k/2 for s in the above formula, and multiplying also by 2, (as there are two wedges resisting), we have the total resistance to rupture or bending of the section ABCD (A1 B1 C1 D1)
= b.d2/6 . k/2. 2
= b.d2/6. k
Now, by reference to Table I, section No. 2, we find that b.d2/g =
Moment of Resistance for the section ABCD; therefore, we have proved the rule, that when the beam is at the point of rupture at any point of its length the bending moment at that point is equal to the moment of resistance of its cross-section at said point multiplied by the modulus of rupture.
Where girders or beams are of wood, it becomes of the highest importance that they should be sound and perfectly dry. The former that they may have sufficient strength, the latter that they may resist decay for the longest period possible.
Every architect, therefore, should study thoroughly the different kinds of timber in use in his locality, so as to be able to distinguish their different qualities. The strength of wood depends, as we know, on the resistance of its fibres to separation. It stands to reason that the young or newly formed parts of a tree will offer less resistance than the older or more thoroughly set parts. The formation of wood in trees is in circular layers, around the entire tree, just inside of the bark. As a rule one layer of wood is formed every year, and these layers are known, therefore, as the "annular rings," which can be distinctly seen when the trunk is sawed across. These rings are formed by the (returning) sap, which., in the spring, flows upwards between the bark and wood, supplies the leaves, and returning in the fall is arrested in its altered state, between the bark and last annular ring of wood. Here it hardens, forming the new annular ring. As subsequent rings form around it, their tendency in hardening is to shrink or compress and harden still more the inner rings, which hardening (by compression) is also assisted by the shrinkage of the bark. In a sound tree, therefore, the strongest wood is at the heart or centre of growth. The heart, however, is rarely at the exact centre of the trunk, as the sap flows more freely on the side exposed to the effects of the sun and wind; and, of course, the rings on this side are thicker, thus leaving the heart constantly, relatively, nearer to the unexposed side.
From the above it will be readily seen that timber should be selected from the region of the heart, or it should be what is known as "heart-wood." The outer layers should be rejected, as they are not only softer and weaker, but, being full of sap, are liable to rapid decay. To tell whether or no the timber is "heart-wood" one need but look at the end, and see whether it contains the centre of the rings. No bark should be allowed on timber, for not only has it no strength itself, but the more recent annular rings near it, are about as valueless.
In some timbers, notably oak, distinct rays are noticed, crossing the annular rings and radiating from the centre. These are the "medullary rays," and are elements of weakness. Care should be taken that they do not cross the end of the timber horizontally, as shown at A in Fig. 12G, but as near vertically as possible, see B in Fig. 127. The beautiful appearance of quartered oak and other woods is obtained by cutting the planks so that their surfaces will show slanting cuts through these medullary rays.
All timber cracks more or less in seasoning, nor need these cracks cause much worry, unless they are very deep and long. They are, to a certain extent, signs of the amount of seasoning the timber has had. They should be avoided, as much as possible, near the centre of the timber, if regularly loaded, or near the point of greatest bending moment, where the loads are irregular. If timber without serious cracks cannot be obtained, allowance should be made for these, by increasing its size.