528 STRUCTURAL MECHANICS. CHAP. XVI. 



Elastic Limit. The elastic limit of a material is the highest unit stress to which that material 

 may be subjected and still return to its original shape when the stress is removed, and is the 

 limit within which the stresses are directly proportional to the deformations. 



Yield Point. In testing materials a point is reached beyond the elastic limit where unit 

 elongations increase very rapidly without any or with a very slight increase in unit stress. This 

 point is indicated by the drop of the scale beam of the testing machine. In steel the yield point 

 is from three to six thousand pounds per square inch above the elastic limit. 



Modulus of Elasticity. The modulus of elasticity of a material is the constant, which within 

 the elastic limit expresses the ratio between the unit stress and unit strain or deformation. If 

 E = modulus of elasticity, P = an axial force; A = cross sectional area of the bar, / = unit 

 stress = Pf A; A = deformation produced by P in a length /, and 5 = A//; then 



E = (PM)/(A//) or E = f/8. 



The modulus of elasticity may be defined as that force, were Hooke's law applicable without 

 limit, which would produce in a bar with a cross section of one square inch a deformation equal 

 to its original length. 



The modulus of elasticity of steel is very closely E = 30,000,000 Ib. per sq. in.; the modulus 

 of elasticity of timber is approximately E = 1,500,000 Ib. per sq. in.; while the modulus of elas- 

 ticity of concrete varies from E = 1,500,000 Ib. per sq. in. to E = 3,000,000 Ib. per sq. in. with 

 an average value of E = 2,000,000 Ib. per sq. in. 



Shearing Modulus of Elasticity. The shearing modulus of elasticity, also called the modulus 

 of rigidity, is the modulus expressing the ratio between unit shearing stress and unit shearing 

 strain. The value of shearing modulus of elasticity for steel is about f of the value of E, or 

 G = 12,000,000 Ib. per sq. in. 



Poisson's Ratio. Direct stress produces a strain in its own direction and an opposite kind 

 of strain in every direction perpendicular to its own. For example a bar under tensile stress 

 extends longitudinally and contracts laterally. Poisson's ratio is the ratio of lateral strain to 

 longitudinal strain, and is a constant below the elastic limit. For steel Poisson's ratio is ^ to , 

 while for concrete it is from | to xV 



Rupture Strength. In testing steel the cross sectional area rapidly decreases beyond the 

 ultimate stress and if the rupture stress be divided by the original cross sectional area the unit 

 stress at rupture will be less than the ultimate stress. 



Ultimate Deformation. The ultimate deformation is the total deformation in a prescribed 

 length, commonly 8 inches, or 2 inches. It is usually expressed in per cent for a length of 8 inches, 

 or of 2 inches. 



Work or Resilience in a Bar. The amount of work that can be stored up in a body under 

 stress within the elastic limit is called resilience or " internal work." When the external force 

 has been gradually applied all the work may be recovered when the force is removed. 



From the law of conservation of energy the external work due to the force is equal to the 

 resilience or internal work. If a load P is supported at the lower end of a bar without weight, hav- 

 ing a length / and a cross sectional area A ; then the external work will be |P-A, where A = the 

 total deformation, and the internal work or resilience will be 



when/ = elastic limit of the material then I/V-E is termed the Modulus of Resilience. 



Stresses due to Sudden Loads. In a bar acted on by a static load, P, gradually applied, 

 the total resilience will be K = fA.P. If the load P is suddenly applied we will have K = A.P, 

 from which it is seen that the stress produced by a sudden load is twice that produced by a load 

 gradually applied. 



