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mechanical design an integrated approach Part 5

Tham khảo tài liệu 'mechanical design an integrated approach part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 220 PART í Fundamentals This results in a t -32PL3Ịfti El. The beam deflection is found by inserting the value of am obtained into Eq 5 60 . Comments At the free end x L retaining only the first three terms of the solution we have the value of the maximum deflection vmax PL3 3X QỈEỈ. The exact solution owing to bending is PL3 3EỈ. I 5.10 THE RAYLEIGH-RITZ METHOD The Rayleigh-Ritz method is a convenient approach for determining solutions by the principle of minimum potential energy. This technique was suggested by Lord Rayleigh in 1877 and extended by w. Ritz in 1909. Here we discuss application of the method to beam and plate bending problems. The procedure also is used in determining beam and plate buckling loads in the next chapter. In a beam bending problem the Rayleigh-Ritz method may be described as follows. First choose an expression such as polynomials or trigonometric series for the deflection curve containing undetermined coefficients am m 1 2 . . This function must satisfy the geometric deflection V and slope Ỡ boundary conditions. Static moment M and shear V boundary conditions need not be fulfilled. Obviously a proper choice of the deflection expression is important to ensure good accuracy for the final solution. Next using the selected displacement expression obtain the potential energy function n in terms of am. Note that the values of am govern the variation of the potential energy. Inasmuch as the potential energy must be minimum at equilibrium the Rayleigh-Ritz method is stated in the following form an 5.62 The preceding represents a set of algebraic equations solved to give the coefficients of am. Finally carrying these values into the assumed function for deflection we find the solution for a given problem. The advantages of the Rayleigh-Ritz technique lie in the relative ease with which mixed edge conditions can be treated. This method is among the simplest for solving plate and shell deflections by hand calculation. EXAMPLE 5.13 I Determining .