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On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions

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Abstract

The present paper describes the advancement of displacement potential approach in relation to solution of plane problems of structural mechanics with mixed mode of boundary conditions. Both the conditions of the plane stress and the plane strain are considered for analyzing the displacement and stress fields of the structural problem. Using the finite difference technique based on the present displacement potential approach for the case of the plane stress and the plane strain conditions, firstly an elastic cantilever beam subjected to a pure shear at its tip is solved and these two solutions (plane stress and plane strain) are compared with Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951); secondly the above-mentioned displacement potential approach for the case of the plane stress and the plane strain conditions are applied to solve a one-end fixed square plate subjected to a combined loading at its tip. Effects of plane stress and plane strain on the elastic field of the plate are discussed in a comparative fashion. Limitations of Timoshenko and Goodier cantilever beam bending solutions (Theory of elasticity, 2nd edn. McGraw-Hill, New York, 1951) over the displacement potential approach for the case of the plane stress and the plane strain conditions are not only discussed but also the superiority of the present displacement potential approach for the case of the plane stress and the plane strain conditions are reflected in the present research work.

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Abbreviations

u x (x, y):

Axial displacement component

u y (x, y):

Lateral displacement component

σ xx (x, y):

Axial stress component

σ yy (x, y):

Lateral stress component

σ xy (x, y):

Shear stress component

σ n :

Normal stress

σ t :

Lateral stress

PN:

Plane strain condition

PS:

Plane stress condition

r :

x/b

E :

Young’s modulus

ν :

Poisson’s ratio

l :

Length of the cantilever beam

2c :

Width of the cantilever beam

b :

Length of the plate

2a :

Width of the plate

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Authors and Affiliations

  1. Division of Mechanical and Automotive Engineering, Kongju National University, Cheonan, Republic of Korea

    S. K. Deb Nath & Sung-Gaun Kim

  2. Department in Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka, 1000, Bangladesh

    S. Reaz Ahmed

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  1. S. K. Deb Nath
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Nath, S.K.D., Ahmed, S.R. & Kim, SG. On the displacement potential solution of plane problems of structural mechanics with mixed boundary conditions. Arch Appl Mech 80, 1125–1147 (2010). https://doi.org/10.1007/s00419-010-0428-5

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