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 (6.4) 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 strain- displacement 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. 6.3 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 6.1. 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 6.1 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 262 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 (6.4) through (6.6) and remembering that in elementary beam theory then Therefore, the strain energy, U, which is the volume integral of the strain energy density function, W, is where, the flexural stiffness for a beam of rectangular cross-section. 263 Similarly, from the surface traction work term in Equation (6.1) it is seen that Equation (6.1) then becomes Following Equation (6.2) and remembering Equation (6.3) then The variation can be included under the integral, because the order of variation and integration can be interchanged. Also, there is no variation of E, I or q(x) because they are all specified quantities. Integrating by parts the first term on the right-hand side of Equation (6.10). Substituting Equation (6.11) into (6.10) and rearranging, it is seen that: 264 For this to be true, the following equation must be satisfied for the integral above to be zero: This is obviously the governing equation for the bending of a beam under a lateral load. So, it is seen that if one considers a beam-type structure, uses beam assumptions, and uses proper stress-strain relations and strain-displacement relations, the result is the beam bending equation. However, it can be emphasized that if a nonclassical-shaped elastic structure were being analyzed, by using physical intuition, experience or some other reasoning to formulate stress-strain relations, and strain-displacement relations for the body, then through the Theorem of Minimum Potential Energy one can formulate the governing differential equations for the structure and load analogous to Equation (6.13). Incidentally, the resulting governing differential equations derived from the Theorem of Minimum Potential Energy are called the Euler-Lagrange equations. Note also for Equation (6.12) to be true, each of the first two terms must be zero. Hence, at x = 0 and x = L (at each end) either or (dw / dx) must be specified (that is, its variation must be zero), also either or w must be specified. These are the natural boundary conditions. All of the classical boundary conditions,; including simple supported, clamped and free edges are contained in the above “natural boundary conditions.” This is a nice byproduct from using the variational approach for deriving governing equations for analyzing any elastic structure. The above discussion shows that if in using The Theorem of Minimum Potential Energy one makes all of the assumptions of classical beam theory, the resulting Euler- Lagrange equation is the classical beam equation (6.13) and the natural boundary conditions given in (6.12) as discussed above. Equally or more important the Theorem of Minimum Potential Energy provides a means to obtain an approximate solution to practical engineering problems by assuming good deflection functions which satisfy the boundary conditions. As the simplest example consider a beam simply supported at each end subjected to a uniform lateral load per unit length a constant. As shown in Figure 4.4, if the exact solution for this problem is given by Equation (4.49), which is seen to be a polynomial. Here, an example, assume a deflection which satisfies the boundary conditions for a beam simply supported at each end, where A is a constant to be determined. 265 This is not the exact solution, but should lead to a good approximation because (6.14) is a continuous single valued function which satisfies the boundary conditions of the problem. Proceeding, Substituting (6.14) into (6.9) results in Therefore, The exact solution is (4.50) The difference is seen to be 0.386%. So the Minimum Potential Energy solution is seen to be almost exact in determining the maximum deflection. 266 In determining maximum stresses the accuracy of the energy solution is less, because bending stresses are proportional to second derivatives of deflection. By taking derivatives the errors increase (conversely, integrating is an averaging process and errors decrease) so the stresses from the approximate solution differ more from the exact solution than do the deflections. To continue this example for a one lamina composite beam, simply supported at each end, subjected to a constant uniform lateral load per unit length of it is clear that the maximum stress occurs at x = L /2 . From (4.26) and (4.49) the exact value of the maximum stress is Likewise, for the Minimum Potential Energy solution, using (4.26) and (6.14) The difference between the two is 3.2%, so the energy solution is quite accurate for many applications. If one wishes to increase the accuracy, instead of using (6.14) one could use If N were chosen to be three, for example, the expression for w ( x ) is given by and one would proceed as before, taking variations with respect to and which provides three algebraic equations for determining the three Of course as N increases, the accuracy of the solution increases until as N approaches infinity it is another form of the exact solution. As a second example, examine the same beam, this time subjected to a concentrated load P at the mid-length, x = L/2. From Chapter 4, to obtain an exact solution, one must divide the beam into two parts, as discussed in Section 4.3, so that the load discontinuity can be accommodated, with the result that there are two fourth order differential equations and eight boundary conditions. Not so with the case of Minimum Potential Energy to obtain an approximate solution, as follows. Again assume (6.14) as the approximate deflection because it is single valued, continuous and satisfies the boundary conditions at the end of the beam. There, 267 Again, instead of (6.14) one could have chosen (6.21) as the trial function to use in solving this problem. Thus, the Theorem of Minimum Potential Energy can be used easily for complicated laterally distributed loads, concentrated lateral loads, any boundary conditions, and/or variable or discontinuous beam thicknesses. One only needs to select a form of the lateral displacement such as the following examples. Clamped Clamped Beam Clamped-Simple Beam Cantilevered Beam 268 6.4 Use of The Theorem of Minimum Potential Energy for Designing a Composite Electrical Transmission Tower Consider as an example a tapered beam of hollow circular cross-section as shown below in Figure 6.2. This beam can be tapered or not, can have a varying wall thickness or not, the material has an axial modulus of elasticity E, is subjected to a load P at as shown, is clamped at x = 0 and free at x = L. A factor of safety is applied to the maximum stress, and a maximum tip deflection at x = L is specified. For the case shown above the potential energy, V, can be written as: where I( x ) is included in the integral because it may be a function of the length coordinate x. For a uniform cantilever beam with an end load P the exact beam solution is In the Theorem of Minimum Potential Energy, the analyst may choose any trial function he wishes as long as it is single valued, continuous and satisfies the boundary conditions. In this case the following is chosen rather than (6.24). From (6.26) above let the trial function be: therefore, If the beam has a uniform taper, then the beam diameter can be written as: 269 where is the base diameter and C is a constant describing the taper. At any axial location, the cross-sectional area can be written as: For a constant cross-sectional area of substituting (6.29) into (6.30), it is seen that the beam wall thickness is For the case described above, then the area moment of inertia is: Using (6.32) and (6.28), (6.25) can be written as: After manipulating the above, Setting the variation of the potential energy to zero, by varying A, results in where 270 So and This is the expression to use for displacement restrictions. The maximum stress occurs at x = 0 for this cantilevered beam and can be written as This is the expression to use for a strength requirement. If the strength requirement has a factor of safety of 4, consider the load applied in Figure 6.2 to be 4 P, in which case (6.39) becomes Equations (6.38) and (6.40) provide two equations with which to determine and The results are: Because (6.41) involves in the K term, an iteration is necessary to determine As a design example, let P = 475 l bs, psi, psi and when or when at [...]... represents a simple example of the power of this technique The first application is the development of the governing equations for the static deformations of moderately thick rectangular beams, including the effects of transverse-shear deformation and transverse normal stress The second application involves the use of the Theorem, together with Hamilton’s Principle, to develop a theory of beam vibrations including... Substituting these into Equation (6.47) results in the following: From this the strain energy is found for an isotropic classical rectangular plate It is seen that the first term is the extensional or in-plane strain energy of the plate, and the second is the bending strain energy of the plate In the latter, it is seen that the first term is proportional to the square of the average plate curvature, while the. .. shear modulus of the material, and is given by the in-plane Equation (6.60) provides an easy way to calculate the shear strength of the failed material simply by measuring the load P at failure, and the location z, i.e., the distance from the midsurface of the panel, of the initial failure site Likewise, if one is only interested in overall panel in-plane shear strength, then knowing the load P at... to the threedimensional equations of elasticity and is, therefore, established Consider now some typical applications of the Theorem to the static and dynamic deformations of beams 6.11 Static Deformation of Moderately Thick Beams As a first illustration, consider the development of the theory for the static deformations of moderately thick beams in which the effects of transverse shear deformation and... 290 In the above, M is the beam bending moment, and Q is the beam transverse shear resultant It should be noted that the form of the stress components and is identical to that of classical theory The form of the transverse normal stress may easily be derived from the stress equation of equilibrium in the thickness direction, as a consequence of the assumptions made above for and The expression shown... free on the other side 6 .10 Reissner’s Variational Theorem and Its Applications Because of its broad usage in the analysis of isotropic or anisotropic thin-walled structures accounting for transverse shear deformation and transverse-normal stress, Reissner’s Variational Theorem is discussed herein A general discussion of the Variational Principle is presented, followed by a treatment of the theory of moderately... lamina across the N laminae that comprise the plate gives the total potential energy as 275 Here A refers to the planform area of the plate whose dimensions are and It is noted that the strains used in the strainenergy relations are the isothermal strains, hence one notes the differences between total strain and the thermal and hygrothermal strains in Equation (6.51) Now, substituting the constitutive... core material: For the adhesive material: If the adhesive being used is isotropic such that one can assume the shear strength in the x-y plane is the same as the shear strengths in the x-z and y-z planes, i.e., the transverse planes, then Equation (6.67) can be used to determine the shear strength of the adhesive material where the subscript a refers to the adhesive material 282 In the above, after... rail and many other structural applications The plate is subjected to a uniform compressive in-plane load of (lbs./in of width) in the x direction as shown in the Figure 6.4 The Theorem of Minimum Potential Energy is used to determine the critical buckling load, From Equation (6.52), and using classical theory i.e the potential energy expression is: 283 In this expression the effect of the in-plane load... panels the tests can be used to determine the in-plane shear strengths of the faces, the core and/or the adhesive bond between face and core The shear stresses developed vary linearly in the thickness direction and are constant over the entire planform area 277 Consider a panel of the material to be tested to be rectangular in planform, with dimensions a in the x-direction and b in the y-direction The . energy of the plate, and the second is the bending strain energy of the plate. In the latter, it is seen that the first term is proportional to the square of the average plate curvature, while the. using The Theorem of Minimum Potential Energy one makes all of the assumptions of classical beam theory, the resulting Euler- Lagrange equation is the classical beam equation (6.13) and the natural. the following (note that the and below are the coefficients of thermal and hygrothermal expansion): As written Equation (6.52) provides the expression to use in the analysis of monocoque or composite