# Mathematical Questions and Solutions Volume 40

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## Description

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1884 edition. Excerpt: ...the function dx2 dy k. By, where k is any constant, satisfies, for all values of k, the " d2t d-tb general equation--2 ]--? = 0 throughout the entire extent of the lamina, dx-dy'2 and the particular conditions = 0 and T) = 0 at its outer boundary, and for the particular value k = ' that for its inner boundary also. It follows therefore from the aforesaid considerations that a = r2.--+--, and, by x-+ y' derivation from it in the usual manner, that v r2.--mx-which accord x2 + y'2 ingly are the potential and displacement functions, respectively, of the Btrain. Putting u = c and v = c, which represent, for different values of c, the potential and displacement line-systems, respectively, of the strain; we get, for the two systems of lines respectively, the equations (x' + y1) =--(lx + my) and (x2 + y'2) =--(ly--mx), e e which manifestly establish the two particulars of the question. To find the principal displacement of the strain at any point xy of the lamina. From the above value of u, by a first differentiation with respect to x and to y, we get at once from which we see that, throughout the entire extent of the strained mass, the principal displacement depends only on the distance from the centre of the plate, and varies from point to point inversely as the square of that distance. To find the principal dilatation of the strain at any point xy of the lamina. From the same, by a second differentiation with respect to x and to y, we get again at once r/ M_y_ dju Ati_ 2, _dxdy) dx1 dy-J (x2 + 2jj from which we see that, throughout the entire extent of the strained mass, the principal dilatation depends only on the distance from the centre of the plate, and varies from point to point inversely as the cube of that distance. To find...show more

## Product details

• Paperback | 28 pages
• 189 x 246 x 2mm | 68g
• United States
• English
• black & white illustrations
• 1236813693
• 9781236813695