E L A 



Since the force is perpendicular to the . ^_ ^^ 13 



beam, and the beam is nearly a straight ' 



line, we may (by Cor. last Art.) suppose 



the neutral point coincident with the axis. 



Let A M E represent the axis bent by a 



force acting- perpendicularly to A D its original position ; and let F be a 



length of the beam equivalent to the force/, I length b breadth, and 



a = thickness of the beam, E the modulus of elasticity, then the whole 



deflexion S 



- 



or if for F we put its value o^-i 



a 



Cor. 1. Hence for a given breadth and thickness, the deflexion is as the 

 force and cube of the length ; and for a given weight and length, the de- 

 flexion is inversely as the breadth and cube of the thickness. 



Cor. 2. Let the direction of the tangent at E make an ^ 6 with the tan- 

 gent at A ; then 6 may be called the angular deflexion, and we have 



The angular deflexion is as the force and square of the length. 



4. When a rectangular prismatic beam in a horizontal position is bent 

 by its own weight j (its thickness being vertical) to find the deflexion. 



The same notation being retained, the whole deflexion 



3ft 

 8Ea*' 

 Cor. In this and the last Art. 3 being objerved, E may be found. 



5. A rectangular prismatic beam is compressed by a given force acting 

 in a direction parallel to the axis j to find the deflexion. 



Let a be | the thickness of the beam, I - f the length, h distance 

 of the force from the axis j then if E be very large compared with F, 

 we have the deflexion 



7 



= h (eec. - - 1). 

 aVE 



Cor. If the force act at the extremities of the axis, h o, and there 

 will be no deviation except 



100 



