The Behavior of Structures Composed of Composite Materials Part 10 pdf

To utilize the Theorem of Minimum Potential Energy, the stress-strain relationsfor the elastic body are employed to change the stresses in Equation 6.4 to strains, andthe strain-displace

To utilize the Theorem of Minimum Potential Energy, the stress-strain relationsfor the elastic body are employed to change the stresses in Equation (6.4) to strains, andthe 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 isbest to introduce the subject using isotropic monocoque beams, a much simpler structuralcomponent, 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 abeam in bending, shown in Figure 6.1 To make it more simple consider a beam of anisotropic material In this section, Minimum Potential Energy methods are used to showthat if one makes beam assumptions, one obtains the beam equation However, the mostuseful employment of the Minimum Potential Energy Theorem is through makingassumptions for the dependent variables (the deflection) and using the theorem to obtainapproximate 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 beammaterials is E, and the stress-strain relation is

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 elementarybeam theory

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:

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 lateralload 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 thebeam bending equation However, it can be emphasized that if a nonclassical-shapedelastic structure were being analyzed, by using physical intuition, experience or someother reasoning to formulate stress-strain relations, and strain-displacement relations forthe body, then through the Theorem of Minimum Potential Energy one can formulate thegoverning differential equations for the structure and load analogous to Equation (6.13).Incidentally, the resulting governing differential equations derived from the Theorem ofMinimum 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 classicalboundary conditions,; including simple supported, clamped and free edges are contained

in the above “natural boundary conditions.” This is a nice byproduct from using thevariational approach for deriving governing equations for analyzing any elastic structure.The above discussion shows that if in using The Theorem of Minimum PotentialEnergy 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 boundaryconditions given in (6.12) as discussed above

Equally or more important the Theorem of Minimum Potential Energy provides ameans to obtain an approximate solution to practical engineering problems by assuminggood deflection functions which satisfy the boundary conditions As the simplestexample consider a beam simply supported at each end subjected to a uniform lateral loadper unit length a constant As shown in Figure 4.4, if the exactsolution 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.

This is not the exact solution, but should lead to a good approximation because (6.14) is acontinuous 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

In determining maximum stresses the accuracy of the energy solution is less,because bending stresses are proportional to second derivatives of deflection By takingderivatives the errors increase (conversely, integrating is an averaging process and errorsdecrease) so the stresses from the approximate solution differ more from the exactsolution than do the deflections.

To continue this example for a one lamina composite beam, simply supported ateach 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 manyapplications

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, takingvariations with respect to and which provides three algebraic equations fordetermining 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 theload discontinuity can be accommodated, with the result that there are two fourth orderdifferential equations and eight boundary conditions Not so with the case of MinimumPotential Energy to obtain an approximate solution, as follows Again assume (6.14) asthe approximate deflection because it is single valued, continuous and satisfies theboundary conditions at the end of the beam There,

Again, instead of (6.14) one could have chosen (6.21) as the trial function to use insolving this problem.

Thus, the Theorem of Minimum Potential Energy can be used easily forcomplicated laterally distributed loads, concentrated lateral loads, any boundaryconditions, 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

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 shownbelow 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 trialfunction be:

therefore,

If the beam has a uniform taper, then the beam diameter can be written as:

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 isseen 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

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 4P,

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

at

In this case

of material used in the design

Design 2 Straight tower, constant diameter and wall thickness

Following the same procedures as noted earlier,

Again in (6.46) because K involves an iteration is necessary The result is

of material used

Summary: It is seen that Design 3 is the best, being 21.8% lighter than Design 1 and

34.6% lighter than Design 2

6.5 Minimum Potential Energy for Rectangular Plates

Turning now to the more complicated problem of rectangular plates, andemploying the Theory of Minimum Potential Energy, the strain-energy-density function,

W, for a three-dimensional solid in rectangular coordinates is given by Equation (6.4).

The assumptions associated with the classical plate theory of Chapter 3 are nowemployed in Equation (6.4) for a rectangular plate If transverse shear deformation isneglected, then If transverse normal strains are neglected thenStresses are written in terms of strains, such that for the classical plate

Therefore, Equation (6.4) becomes

From this the strain energy is found for an isotropic classical rectangularplate.

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 thefirst term is proportional to the square of the average plate curvature, while the secondterm is known as the Gaussian curvature If Equation (6.49) is used for the usualcomposite plate one may use the following for the extensional stiffness

and the flexural stiffness as discussed previously

For the plate the total work term due to surface loadings, p(x,y), and

is seen to be

Hence, in Equations (6.49) and (6.50) if one considers a plate subjected only to a lateral

only (except for buckling) assume w(x, y) = p(x, y) = 0 If one is looking for buckling

loads, assume The rationale for each of these will be discussed insubsequent sections

6.6 A Rectangular Composite Material Plate Subjected to Lateral and

Hygrothermal Loads

A detailed study by Sloan [4] shows clearly what is involved in analyzingcomposite panels to accurately account for the effects of anisotropy, transverse sheardeformation, thermal and hygrothermal effects

The stress-strain equations to be considered are given by Equation (2.43), whereinthe are given by Equation (2.44) The strain-displacement equations are given byEquation (2.48), the form of the panel displacements by Equation (2.49), from whichEquation (2.50) results By neglecting and the constitutive relations for thelaminate reduce to Equation (2.66)

Employing the Theorem of Minimum Potential Energy, Equation (6.1), for theplate under discussion it is seen that summing the strain-energy-density functions for

each lamina across the N laminae that comprise the plate gives the total potential energy

as

and It is noted that the strains used in the energy relations are the isothermal strains, hence one notes the differences between totalstrain and the thermal and hygrothermal strains in Equation (6.51).

strain-Now, substituting the constitutive Equations (2.66) and the strain-displacementrelations (2.50) into Equation (6.51) results in

In the above, all quantities (except displacements) are defined by Equations (2.56)through (2.61), Equations (2.63) through (2.65) and the following (note that theand below are the coefficients of thermal and hygrothermal expansion):

As written Equation (6.52) provides the expression to use in the analysis ofmonocoque or composite rectangular plates of constant thickness, wherein one uses the

appropriate values of the [A], [B] and [D] stiffness matrix quantities discussed in Chapter

2

Equation (6.52) is the most general formulation and it is seen that without thesurface load term there are 30 terms to represent the strain energy in the compositematerial panel Referring to Equation (2.66) and the ensuing discussion it is seen that ifthe laminate has no stretching-shearing coupling then two terms would

be dropped; if no twisting-stretching coupling two more would bedropped; likewise two more are dropped if there were no bending-twisting coupling

If the laminate were symmetric about the mid-plane, a very commonconstruction, then five terms would be canceled out, because there would be nobending-stretching coupling Analogous to (6.52) the corresponding equation for thepotential energy for a circular cylindrical shell of a composite material is given byEquation (22.15) pp 454-456 in Reference [5]

6.7 In-Plane Shear Strength Determination of Composite Materials in Laminated Composite Panels

In this example problem using the Theorem of Minimum Potential Energy,consider a simple test procedure to determine the in-plane shear strength of laminatedcomposite materials, as well as other orthotropic and isotropic advanced materialsystems; see Vinson [6] The test apparatus shown in Figure 6.3 is simple, inexpensive,and the flat rectangular plate test specimen is not restricted in size or aspect ratio Inaddition to its use for laminated composite materials, the test can also be used for foamcore sandwich panels In sandwich panels the tests can be used to determine the in-planeshear 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 overthe entire planform area