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The Behavior of Structures Composed of Composite Materials Part 7

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Tham khảo tài liệu 'the behavior of structures composed of composite materials part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 170 BEAM CLAMPED AT x 0 FREE AT x L Px3 0 x 4.60 6bDti P 2X 2M n X L 4.61 BEAM SIMPLY SUPPORTED AT EACH END These equations are quite general for a beam of uniform flexural stiffness bl J subjected to any concentrated load P acting at A - 5 . For instance Equation 4.60 can be used for a clamped-free beam with a load P acting at the tip of x L by letting f Ỉ. Also because the equations evolve from linear theory superposition can be used if there are two concentrated loads at two different locations but care must be used to accurately depict regions to the left and right of each load to insure correct solutions. After the solutions u I-Ọ are found Equations 4.14 and 4.17 are used to obtain bending moments and shear resultants and Equation 4.26 is used to determine stresses everywhere. 171 4.4 Solutions by Green s Functions From Section 4.3 consider a composite beam subjected to a unit concentrated load i.e. P 1. By way of example consider the beam simply supported at each end given by Equations 4.62 and 4.63 . In this case Note that G .rj is the deflection to the left of the load ÍịÍ.v to the right of the load. Green s function j and t. can be defined as the deflection at x due to a unit load at . It can be reasoned that any distributed load q x is in fact an infinity of concentrated loads which can be summed to obtain the solution of the response to a distributed load. Because of the infinity of concentrated loads the infinite summation can be replaced by an integration such that the following is correct. Ị G q dỉ Ị x q . 4.66 For a beam simply supported at each end Equations 4.66 and 4.67 would be inserted for G l and G- . and other analogous expressions from Section 4.3 would be used for beams with other boundary conditions. As an example solving Equations 4.68 4.66 and 4.67 with i . .I - i i. a uniform load over the entire length results in 4.67 which is the solution obtained through solving the governing differential equation 4.20 and satisfying the appropriate