SECTION XI 



FLAT PLATES 



83. Theory of flat plates. The analysis of stress in flat plates is 

 at present the most unsatisfactory part of the strength of materials. 

 Although flat plates are of frequent occurrence in engineering con- 

 structions (as, for example, in manhole covers, cylinder ends, floor 

 panels, etc.), no general theory of such plates has as yet been given. 

 Each form of plate is treated by a special method, which in most 

 cases is based upon an arbitrary assumption either as to the danger- 

 ous section or as to the reactions of the supports, and therefore 

 leads to questionable results. 



Although the present theory of flat plates is plainly inadequate, 

 it is nevertheless of value in pointing out the conditions to which 

 such plates are subject, and in furnishing a rational basis for the 

 estimation of their strength. The formulas derived in the following 

 paragraphs, if used in this way, with a clear understanding of their 

 approximate nature, will be found to be invaluable in designing, 

 or in determining the strength of flat plates. 



The following has come to be the standard method of treatment 

 and is chiefly due to Bach.* 



84. Maximum stress in homogeneous circular plate under uni- 

 form load. Consider a flat, circular plate of homogeneous ma- 

 terial, which bears a uniform load of amount w per unit of area, 

 and suppose that the edge of the plate rests freely on a circular 

 rim slightly smaller than the plate, every point of the rim being 

 maintained at the same level. The strain in this case is greater 

 than it would be if the plate was fixed at the edges, and conse- 

 quently the formula deduced will give the maximum stress in 

 all cases. 



* For an approximate method of solution see article by S. E. Slocum entitled " The 

 Strength of Flat Plates, with an Application to Concrete-Steel Floor Panels," Engineer- 

 ing News, July 7, 1904. 



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