Strength Of Beams, Floor And Roofs - Including Directions For Designing And Detailing Roof Trusses, With Criticism Of Various Forms Of Timber Construction. Frank E. Kidder
the square of its depth. A 2 × 8 inch beam will be four times as strong as a 2 × 4 inch beam, and a 2 × 12 inch beam will be nine times as strong as a 2 × 4 inch beam, the square of four being 16, and of twelve 144, or nine times as great.
It follows from the second and third paragraphs that the strength of a rectangular beam is in proportion to the product of the breadth by the square of the depth if the span remains the same. A knowledge of these facts is very important for the wise use of timber.
A beam 8 × 8 contains 64 square inches in cross section, and a beam 6 × 10 contains 60 square inches, yet their strength will be in the proportion of 512 (8 × 8 × 8) to 600 (6 × 10 × 10), the 6 × 10 beam being the stronger. The strength of a 6 × 8 inch beam on edge in proportion to the strength of the same beam laid flat wise is as 6 × 8 × 8 to 8 × 6 × 6, or 384 to 288.
Deep beams are also very much stiffer than shallow beams, the resistance of a beam to bending increasing in proportion to the cube of the depth. The stiffness therefore of a 2 × 12 inch beam and a 2 × 10 inch beam is in the proportion of the cube of 12 to the cube of 10, or 1728 to 1000. This property of stiffness is very important in floor joists, where the span in feet is usually greater than the depth in inches, but for shorter beams it need not be considered.
In speaking of the strength or stiffness of beams the breadth of the beam always refers to the thickness measured horizontally, and the depth to the height of the beam as it sets in place, without regard to which is the larger dimension. When a beam is supported at each end the distance between supports is called the span. The distance which the ends rest on their support is called the bearing.
MEASURE OF BREADTH, DEPTH AND SPAN.
In the rules hereinafter given the breadth and depth of the beam are always supposed to be measured in inches and the span in feet. The meaning of the terms referred to is clearly shown in Fig. 1. Beams are also sometimes supported at three or more points, in which case they are called continuous beams. These will be considered in their proper place. There is also the cantilever beam, or a beam fixed at one end. The cantilever portion of the beam is that which projects beyond the support. The other end may be fixed in a wall, as at A, Fig. 2, or it may be held down by its own weight and the load on it, as at B. A beam supported at the center only, as at C, is a double cantilever, each side being considered as a cantilever. All three cases are met with in building construction, although that shown at B is the most common.
There are also different ways of loading a beam, although loads are usually classed either as distributed or concentrated. A distributed load is one that is applied over the entire length of the span, and when the load is uniform, as in the case of a plain brick wall of uniform height, the load is called uniformly distributed. Floor loads, although as a matter of fact not absolutely uniform, are generally considered as such. Floor joists resting on a girder may be considered as a uniformly distributed load, when the joists are not spaced more than 2 feet on centers. When they are spaced 4 feet or more on centers they should be considered as a series of concentrated loads.
A concentrated load is one that is applied at a single point of a beam, although in practice the “point” may be perhaps 3 feet long. An iron safe resting on the center of a beam 10 feet or more in length would be considered as a concentrated load. The end of a header framed to a trimmer is also a concentrated load, as is also a partition extending across a series of beams or joists.
The effect of a concentrated load applied at the center of a beam is just twice as great as if the load were uniformly distributed. When the load is applied between the center and the end the effect may be greater or less than that of a distributed load, according as the point of application is nearer to the center or to the support.
LIVE AND DEAD LOADS.
Loads are also spoken of as “live” and “dead” loads. A dead load is one that does not move of itself, such as the weight of any kind of material or a brick wall, for instance. A live load is one that is constantly moving and quickly applied. Live loads that produce a decided impact or vibrations are nearly twice as destructive as those that remain perfectly still. The principal live loads met with in building construction are moving crowds of people, particularly if they move in regular time, as in dancing or marching; machinery and wind pressure.
RULES FOR THE STRENGTH OF BEAMS.
The strength of a beam subject to almost any of the different variations of loading may be determined with about the same degree of accuracy as if simply loaded at the center, but the calculations require a considerable knowledge of mathematics, so that only a few of the more common cases can be covered by simple rules. These we will now consider.
When considering the strength of beams we usually have either one of two problems to solve—namely, to find the strength of a given beam or to determine the necessary size of beam to support a given load. The same algebraic formula really answers for both, but for the benefit of those not proficient in algebraic equations we will give a simple rule for each question, and also for each of the common conditions of support and loading. When we have to determine the strength of a given beam all of the conditions are known, but when we wish to determine the size of beam to carry a given load we must guess at or assume one dimension of the beam and solve for the other. If our first guess gives a badly proportioned beam we must guess again, and do the problem over again a second time. The quantity which represents the strength of the wood or the resistance of the fibers to breaking is now commonly designated as “fiber stress.” In text books written previous to the year 1885 the same quantity is called “modulus of rupture.” This quantity, of course, varies with different woods, and has been determined by numerous experiments on beams of the different kinds of woods. For convenience in making calculations one-eighteenth of the modulus of rupture is generally used for determining the breaking strength of wooden beams, and one-third of this latter value for determining the safe strength.
In the following rules this quantity will be represented by the letter A, the values of this letter for the different woods used in construction being given in Table I:
Table I.—Values of A, Used in Determining the Safe Strength of Beams.
| Kind of Wood. | A, in Pounds. |
| Chestnut | 60 |
| Hemlock | 55 |
| Oak, white | 75 |
| Pine, Georgia yellow | 100 |
| Pine, Norway | 70 |
| Pine, Oregon | 90 |
| Pine, Texas yellow | 90 |
| Pine, common white | 60 |
| Redwood | 60 |
| Spruce | 70 |
| Whitewood (poplar) | 65 |
To find the strength of a rectangular beam, supported at both ends and uniformly loaded over its entire length.
Rule 1.—Multiply twice the breadth of the beam by the square of the depth and by the value of A in Table