To find the value of a unit-deformation: Divide the whole deformation by the length over which it is distributed. Thus, if D denotes the value of a deformation, I the length, s the unit-deformation, then s= D/J , also D=ls. (2)

Both D and I should always be expressed in the same unit.

Example. Suppose that a 4-foot rod is elongated ½ inch. What is the value of the unit-deformation?

Here D=i inch, and l=4 feet=48 inches; hence s=½÷48=1/96 inch per inch.

That is, each inch is elongated 1/96 inch.

Unit-elongations are sometimes expressed in per cent. To express an elongation in per cent: Divide the elongation in inches by the original length in inches and multiply by 100.

8. Elasticity. Most solid bodies when deformed will regain more or less completely their natural size and shape when the deforming forces cease to act. This property of regaining size and shape is called elasticity.

We may classify bodies into kinds depending on the degree of elasticity which they have, thus:

1. Perfectly elastic bodies; these will regain their orig-. inal form and size no matter how large the applied forces are if less.than breaking values. Strictly there are no such materials, but rubber, practically, is perfectly elastic.

2. Imperfectly elastic bodies; these will fully regain their original form and size if the applied forces are not too large, and practically even if the loads are large but less than the breaking value. Most of the constructive materials belong to this class.

3. Inelastic or plastic bodies; these will not regain in the least their original form when the applied forces cease to act. Clay and putty are good examples of this class.

9. Hooke's Law, and Elastic Limit. If a gradually increasing force is applied to a perfectly elastic material, the deformation increases proportionally to the force; that is, if P and P' denote two values of the force (or stress), and D and D' the values of the deformation produced by the force, then P:P'::D:D'.

This relation is also true for imperfectly elastic materials, provided that the loads P and P' do not exceed a certain limit depending on the material. Beyond this limit, the deformation increases much faster than the load; that is, if within the limit an addition of 1,000 pounds to the load produces a stretch of 0.01 inch, beyond the limit an equal addition produces a stretch larger and usually much larger than 0.01 inch.

Beyond this limit of proportionality a part of the deformation is permanent; that is, if the load is removed the body only partially recovers its form and size. The permanent part of a deformation is called set.

The fact that for most materials the deformation is proportional to the load within certain limits, is known as Hooke's Law. The unit-stress within which Hooke's law holds, or above which the deformation is not proportional to the load or stress, is called elastic limit.

10. Ultimate Strength. By ultimate tensile, compressive, or shearing strength, of. a material is meant the greatest tensile, compressive, or shearing unit-stress which it can withstand.

As before mentioned, when a material is subjected to an increasing 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, compressive, or shearing stress, as the case may be, until rupture occurs, and measuring the greatest load. The 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 ½ 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 (diameter) 2=0.7854 x ¼ = 0.1964 square inches; hence the ultimate strength equals

12,540 ÷ 0.1964=63,850 pounds per square inch, (nearly).

11. Stress=Deformation 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 corresponding to many values of the load. The first and second columns 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 second 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 ÷ 0.7854 = 10,000, etc.
 Total Pull in pounds, P Deformation in inches, D Unit-Stress in pounds per square inch, S Unit-Deformation, s 3930 0.00136 5000 0.00017 7850 .00280 10000 .00035 11780 .00404 15000 .00050 15710 .00538 20000 .00067 19635 .00672 25000 .00084 23560 .00805 30000 .00101 27490 .00942 35000 .00118 31415 .01080 40000 .00135 35345 .01221 45000 .00153 39270 .0144 50000 .00180 43200 .0800 5.1000 .0100 47125 .1622 60000 .0202 51050 .201 65000 .0251 54980 .281 70000 .0351 58910 .384 75000 .048 62832 .560 80000 .070 65200 1.600 83000 .200

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,

 0.00136 + 8 = 0.00017, 0.0028 + 8 = .00035, etc.

Looking at the first two columns it will be seen that the elongations 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.