s before mentioned, when a material is subjected to an in creasing load the deformation increases faster than the load beyond the elastic limit, and much faster near the stage of rupture. Not only do tension' bars and compression blocks elongate and shorten respectively, but their cross-sectional areas change also; tension bars thin down and compression blocks "swell out" more or less. The value of the ultimate strength for any material is ascertained by subjecting a specimen to a gradually increasing tensile, com pressive, or shearing stress, as the case may be, until rupture oc curs, and measuring the greatest load. 77th breaking load divided by the area of the original cross-section sustaining the stress, is the value of the ultimate strength.
Example. Suppose that in a tension test of a wrought-iron rod inch in diameter the greatest load was 12,540 pounds. What is the value of the ultimate strength of that grade of wrought iron? The original area of the cross-section of the rod was 0.7854 ( X 0.1964 square inches; hence the ultimate strength equals 12,540÷0.196463,850 pounds per square inch.
ii. Diagram. A "test" to determine the elastic limit, ultimate strength, and other information in re gard to a material is conducted by applying a gradually increasing load until the specimen is broken, and noting the deformation cor responding to many values of the load. The first and second col umns of the following table are a record of a tension test on a steel rod one inch in diameter. The numbers in the first column are the values of the pull, or the loads, at which the elongation of the specimen was measured. The elongations are given in the sec ond column. The numbers in the third and fourth columns are the values of the unit-stress and unit-elongation corresponding to the values of the load opposite to them. The numbers in the third column were obtained from those in the first by dividing the latter by the area of the cross-section of the rod, 0.7854 square inches. Thus, 3,930-0.7854=5,000 7,850÷0.7851=10,000, etc.
The numbers in the fourth column were obtained by dividing those in the second by the length of the specimen (or rather the length of that part whose elongation was measured), 8 inches. Thus, Looking at the first two columns it will be seen that the elonga tions are practically proportional to the loads up to the ninth load, the increase of stretch for each increase in load being about 0.00135 inch, but beyond the ninth load the increases of stretch are much greater. Hence the elastic limit was reached at about the ninth load, and its value is about 45,000 pounds per square inch. The greatest load was 65,200 pounds, and the corresponding unit-stress, 83,000 pounds per square inch, is the ultimate strength.
Nearly all the information revealed by such a test can be well represented in a diagram called a stress-deformation diagram. It is made as follows: Lay off the values of the unit-deformation (fourth column) along horizontal line, according to some con venient scale, from some fixed point in the line. At the points on
the horizontal line representing the various unit-elongations, lay off perpendicular distances equal to the corresponding unit-stresses. Then connect by a smooth curve the upper ends of all those dis tances, last distances laid off. Thus, for instance, the highest unit elongation (0.20) laid off from o (Fig. 4) fixes the point a, and a distance to represent the highest unit-stress (83,000) fixes the point b. All the points so laid off give the curve ocb. The part oc, within the elastic limit, is straight and nearly vertical while the remainder is curved and more or less horizontal, especially toward the point of rupture b. Fig. 5 is a typical stress-defor mation diagram for timber, cast iron, wrought iron, soft and hard steel, in tension and compression.
Ia. Working Stress and Strength, and Factor of Safety.
The greatest unit-stress in any part of ,a structure when it is sus taining its loads is called the working stress of that part. If it is under tension, compression and shearing stresses, then the corresponding highest unit stresses in it are called its work ing stress in tension, in com pression, and in shear respect ively; that is, we speak of as many working stresses as it has kinds of stress.
By working strength of a material to be used for a certain purpose is meant the highest unit-stress to which the material ought to be subjected when so used. Each material has a working strength for tension, for compression, and for shear, and they are in general different.
By factor of safety is meant the ratio of the ultimate strength of a material to its working stress or strength. Thus, if S,„ denotes ultimate strength, denotes working stress or strength, and f denotes factor of safety, then f — ; also S„ (3) When a structure which has to stand certain loads is about to be designed, it is necessary to select working strengths or fac tors of safety for the materials to be used. Often the selection is a matter of great importance, and can be wisely performed only by an expefienced engineer, for this is a matter where hard-and fast rules should not govern but rather the judgment of the expert. But there are certain principles to be used as guides in making a selection, chief among which are: 1. The working strength should be considerably below the elastic limit. (Then the deformations will be small and not per manent.) 2. The working strength should be smaller for parts of a structure sustaining varying loads than for those whose loads are steady. (Actual experiments have disclosed the fact that the strength of a specimen depends on the kind of load put upon it, and that in a general way it is less the less steady the load is.) 3. The working strength must be taken low for non-uniform material, where poor workmanship may be expected, when the loads are uncertain, etc. Principles 1 and 2 have been reduced to figures or formulas for many particular cases, but the third must remain a subject for display of judgment, and even good guessing in many cases.