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One-Dimensional and Two-Dimensional Analytical Solutions for Functionally Graded Beams with Different Moduli in Tension and Compression

by Xue Li 1, Jun-yi Sun 1,2,*, Jiao Dong 1 and Xiao-ting He 1,2
1
School of Civil Engineering, Chongqing University, Chongqing 400045, China
2
Key Laboratory of New Technology for Construction of Cities in Mountain Area (Chongqing University), Ministry of Education, Chongqing 400045, China
*
Author to whom correspondence should be addressed.
Materials 2018, 11(5), 830; https://doi.org/10.3390/ma11050830
Received: 17 April 2018 / Revised: 13 May 2018 / Accepted: 15 May 2018 / Published: 17 May 2018
The material considered in this study not only has a functionally graded characteristic but also exhibits different tensile and compressive moduli of elasticity. One-dimensional and two-dimensional mechanical models for a functionally graded beam with a bimodular effect were established first. By taking the grade function as an exponential expression, the analytical solutions of a bimodular functionally graded beam under pure bending and lateral-force bending were obtained. The regression from a two-dimensional solution to a one-dimensional solution is verified. The physical quantities in a bimodular functionally graded beam are compared with their counterparts in a classical problem and a functionally graded beam without a bimodular effect. The validity of the plane section assumption under pure bending and lateral-force bending is analyzed. Three typical cases that the tensile modulus is greater than, equal to, or less than the compressive modulus are discussed. The result indicates that due to the introduction of the bimodular functionally graded effect of the materials, the maximum tensile and compressive bending stresses may not take place at the bottom and top of the beam. The real location at which the maximum bending stress takes place is determined via the extreme condition for the analytical solution. View Full-Text
Keywords: functionally graded beams; different moduli in tension and compression; bimodulus; analytical solution; neutral layer functionally graded beams; different moduli in tension and compression; bimodulus; analytical solution; neutral layer
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Li, X.; Sun, J.-Y.; Dong, J.; He, X.-T. One-Dimensional and Two-Dimensional Analytical Solutions for Functionally Graded Beams with Different Moduli in Tension and Compression. Materials 2018, 11, 830.

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