Tham khảo tài liệu 'the behavior of structures composed of composite materials part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 261 To utilize the Theorem of Minimum Potential Energy the stress-strain relations for the elastic body are employed to change the stresses in Equation to strains and the strain-displacement relations are employed to change all strains to displacements. Thus it is necessary for the analyst to select the proper stress-strain relations and straindisplacement relations for the problem being solved. Although this text is dedicated to composite material structures of all types it is best to introduce the subject using isotropic monocoque beams a much simpler structural component to first illustrate energy principles. Analysis of a Beam Using the Theorem of Minimum Potential Energy As the simplest example of the use of Minimum Potential Energy consider a beam in bending shown in Figure . To make it more simple consider a beam of an isotropic material. In this section Minimum Potential Energy methods are used to show that if one makes beam assumptions one obtains the beam equation. However the most useful employment of the Minimum Potential Energy Theorem is through making assumptions for the dependent variables the deflection and using the theorem to obtain approximate solutions. From Figure it is seen that the beam is of length L in the x-direction width b and height h. It is subjected to a lateral distributed load q x in the positive z-direction in units of force per unit length. The modulus of elasticity of the isotropic beam materials is E and the stress-strain relation is ơx Efx 262 FIGURE . Beam in bending The corresponding strain displacement relation is since in the bending of beams u -z dw dx only as discussed in Chapter 4. Looking at Equations through and remembering that in elementary beam theory ơy ơz ơỵy Exz yz ơxy 0 then Therefore the strain energy U which is the volume integral of the strain energy density function w is 1 T L J 3 J 1 1 n 1 pp p 1 p 1 where - M 12 the flexural stiffness for a beam of rectangular .
