CHAP. V.I SUPPLEMENT TO CHAP. XIV. 303 



CHAPTER V. 



INFLUENCE OF TEMPERATURE. 



21. General considerations. When the temperature of a per- 

 fectly free body, which possesses in every direction the same coefficient of 

 elasticity and expansion, changes equally at all points, there can be no 

 strains in the body. For were there such strains, then, as there are no 

 outer forces, there could be no equilibrium. 



If, however, the change of temperature is not the same at all points ; or 

 if the body is not free, so that it is possible for outer forces to act, there 

 are strains. 



In the following we assume the change of temperature to be everywhere 

 the same, but that the body is not free. 



We assume that at a certain temperature t 9 no strain exists in the body, 

 and call this the mean temperature. The deviation above or below the 

 mean temperature we call + t or t, and denote the coefficient of expan- 

 sion for one degree by e. 



The determination of the strain in a straight beam held at both ends, is 

 very simple. If the length is I, its relative change of length is e t. Since, 

 however, it cannot expand, the strain S per unit of area is precisely as great 

 as the force which would be required to produce this relative elongation, 



or from eq. (4) a = + E e t ........ (48) 



If the area of cross-section is A, then the strain at each end is 



In equation (48) it is assumed that a compressive strain, due to + 1, ia 

 positive, a tensile strain, due to t, is then negative. 



22. Influence of Temperature on the Arch. Since by a 

 change of temperature the length of the arch varies, while the span remains 

 always the same, the shape or curvature must change, which naturally must 

 give rise to strains and outer forces. In the following we have only to 

 determine these outer forces, since, as shown in Chap. XIV., these are all we 

 need to determine the strains themselves. 



The relative change of length is from eq. (8), Art. I., =-.- [ G + J. 



This change is caused by the outer forces. The relative change of length 

 due to temperature alone is e t. Hence the total relative change of length is 



Ids Qr + M 

 da ~ EAr "** ...... (49) 



