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EQUILIBRIUM, BUCKLING AND VIBRATION OF A 30°-60°-90°-TRIANGULAR PLATE SIMPLY SUPPORTED AT THE EDGES

PAN LIH 一CHOW
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Abstract

In this paper a triangular plate is considered as a portion of a rectangular plate by using the method of images in a special manner. Thus the expression for the deflection of this triangular plate is the same as that for the deflection of the rectangular plate, in spite of of the difference between the regions in which the expressions are defined. Since the solutions of the problems of a simply supported rectangular plate can be obtained without difficulty, the bending problem, as well as the problem of bending combining with tension or compression, of a 30° -60° -90° - triangular plate is solved at once. Their results are expressed in the forms of trigonometric series, single or double, in this paper. The bucking problem and the vibration problem of such a triangular plate are also discussed. The main results are as follows: The smallest critical value of the compressive force per unit length is Where D is the flexural rigidity of the plate and b is the length of the side opposite the 60° angle of the triangular plate. The relation between the fundamental natural frequency w and the tensile (N＜0),or compressive (N＞0) force per unit length acting along the boundaries of the plate is Where is the mass per unit area of the plate. The method of images applied, in the same manner as mentioned above, to the torsion problem of a prismatical bar with 30°-60°-90°- triangular cross section gives more practical results than those obtained by other authors. The torsional rigidity numerically calculated coincides with that obtained by G.E.Hay in 1939. In this paper a triangular plate is considered as a portion of a rectangular plate by using the method of images in a special manner. Thus the expression for the deflection of this triangular plate is the same as that for the deflection of triangular plate, in spite of the difference between the regions in which the expressions are defined. Since the solutions of the problems of a simply supported rectangular plate can be obtained without difficulty, the bending problem, as well as the problem of bending combining with tension or compression, of a 30 -60 -90 - triangular plate is solved at once. Their results are expressed in the forms of trigonometric series, single or double, in this paper. The bucking problem and the vibration problem of such a triangular plate are also discussed. The main results are as follows: The smallest critical value of the compressive force per unit length is Where D is the flexural rigidity of the plate and b is the length of the side opposite the 60 angle of the triangular plate. The relation between the fundamental natural frequency w and the tensile (N＜0),or compressive (N＞0) force per unit length acting along the boundaries of the plate is Where is the mass per unit area of the plate. The method of images applied, in the same manner as mentioned above, to the torsion problem of a prismatical bar with 30°-60°-90°- triangular cross section gives more practical results than those obtained by other authors. The torsional rigidity numerically calculated coincides with that obtained by G.E.Hay in 1939.

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出版历程
• 收稿日期:  2013-09-01
• 修回日期:  2013-09-01

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