The Behavior of Structures Composed of Composite Materials Part 9 ppt

What is the magnitude and exact location of the maximum stress that occurs in the bending boundary layer at the simply supported end?The same information near the clamped end?. If the ma

The terms are omitted in the above for brevity, and

Stresses in each lamina are calculated at two axial locations to show the stressdistributions that occur in the bending boundary layer

Stresses are calculated at an axial location where the deflection is on half of the imposededge displacement, i.e Secondly, stresses are calculated at an axial location of

x = l/2, where l is the axial distance at which w = 0, nearest to the end of the shell Results are

shown in Figure 5.6 It is seen that the stresses are tensile on the outer layers andcompressive on the inner layers It is also seen that the circumferential stresses, aremaximum in the layers, which is logical because the circumferential stiffness is greatest

in these layers and hence the circumferential load is carried in the 90° layers analogous to a stiffspring in a collection of parallel springs Lastly, even though the deflections and stresses areaxially symmetric, in-plane shear stresses, do exist because of the stacking sequence of thelaminate

5.6 Sample Solutions

5.6.1 EFFECTS OF SIMPLE AND CLAMPED SUPPORTS

Consider the circular cylindrical shell composed of a transversely isotropic composite

shown in Figure 5.7, typical of many RTM fabric composites today The end of the shell x = 0 is simply supported, the end x = L is clamped The ends of the shell are assumed to be rigid plates.

The internal pressure is and is written simply as v in this example The following

information is desired:

What is the magnitude and exact location of the maximum stress that occurs in the bending boundary layer at the simply supported end?

The same information near the clamped end?

What is the magnitude of the maximum stress occurring outside the bending boundary

layers, say at x = L/2?

If the maximum principal stress failure theory is used, would the shell be structurally sound if in the design the thickness had been determined by membrane shell theory? (Membrane shell theory neglects all bending effects, see Section 5.10).

a

b

c

d

From the external axial force equilibrium, The boundary conditions at x =

0 are w(0) = 0 and M(0) = 0 From Equations (5.56) and (5.59), the boundary conditions are:

of x where Since therefore the extrema occur at FromEquation (5.64) and (5.65), the fact that and since the shell is long, the effects ofand are negligible Equation (5.60) becomes

exponential decay with increasing x, the largest extreme value, the maximum, occurs at

Hence,

Equations (5.35) and (5.36) can be used instead of (5.62) since the shell is made of a transverselyisotopic composite material, such as many RTM fabric composites

From Equations (5.64), (5.65), (5.56) and (5.59)

Therefore from Equations (5.18), (5.36), (5.62), (5.68) and (5.69)

For v = 0.3, this reduces to

Extreme values occur for the condition which requires that

The + requirement occurs for the negative when Of these two, is

a maximum in the former case The maximum value of in the range is

At the clamped end, the boundary conditions are From Equations (5.56) and(5.58) and making use of the fact that the shell is long

Hence,

However, occurs away from the end of the shell Analogous to the procedures used at the previous end, it is found that

At x = L/2, which is outside either bending boundary layer, it is seen from Equation

(5.35) that hence,

which is recognized as the membrane solution.

Likewise,

This is also the membrane solution.

It is seen that the maximum principal stress occurring in the boundary layer at the simply supported end is, from Equation (5.72) correspondingly, at the clamped end it is

[see (5.75)] The maximum stress predicted by membrane theory is in the hoop direction seen in (5.78) Hence, stresses greater than the membrane stresses occur in both boundary layers; 10% higher in the simply supported area, and 104% higher in the clamped edge Thus, this shell if designed on a basis of membrane shell theory would be woefully inadequate.

In the above example, the location of the maximum axial stress and the maximum circumferential stress at each end of the shell have been determined It should be remembered that a biaxial stress state exists everywhere Hence, when dealing with a material which follows yield or fracture criteria such as a maximum distortion energy criteria, maximum shear stress, or one of several others, then that criteria must be used to find the location of the maximum value

of the equivalent uniaxial stress state as discussed in Chapter 7 Quite often a very simple digital

computer routine can be employed to do the arithmetic, using the analytical solution, to determine the location and magnitude of the maximum stress state.

As pictorial examples of the stress couples, are the transverse shear resultants, that exist in the bending boundary layers, Figures 5.8 and 5.9 illustrate that for a simply

supported edge at x = 0 the stress couple is zero at the edge and peaks a short distance away,

while the maximum value of the shear resultant occurs at the edge Both and go to zero

in the bending boundary layer, and at greater distances from the edge only membrane stresses and deflections occur.

Figures 5.10 and 5.11 show analogous results for the clamped edge at x = L = 200 inches

where it is seen that both the stress couple, and the transverse shear resultant, are maximum at the clamped edge, but diminish to zero in the bending boundary layer.

Figures 5.8 through 5.11 are from a paper by Preissner [3] in which the values of aare defined by (5.88) in the next section, and was defined also in Chapter 4 for an asymmetric composite beam.

5.7 Mid-Plane Asymmetric Circular Cylindrical Shells

In many applications there are good reasons to have the shell structure be mid-plane asymmetric with respect to the materials used The exterior environment may differ markedly from the interior environment to which the shell is exposed the outer environment may have extremes of temperature and humidity, blast damage, etc while the inner environment may have chemical, abrasive, esthetic or other considerations Also stealth considerations may play a role

in some shell structures.

5.7.1 GOVERNING DIFFERENTIAL EQUATIONS

To study the shell with mid-plane asymmetry under axially symmetric loads, the equilibrium equations and the strain displacement relations remain the same as in Section 5.2.2,

in Equation (5.19) through (5.28).

However, the constitutive equations for the mid-plane asymmetric composite, which is specially orthotropic so that the material axes 1-2-3 coincide with the axes are found from Equation (2.66) to be

As before, substituting derivatives of these results into the equilibrium equations and using thestrain-displacement relations results in the following equation analogous to Equation (5.34).

The second governing differential equation analogous to Equation (5.33) is

Defining a reduced or effective flexural stiffness as:

and introducing the following definition

Equation (5.84) can now be written as:

Except for the second term on the left-hand side of Equation (5.87), this is the usualgoverning equation for an axially symmetric cylindrical shell of flexural stiffness subjected

to axially symmetric loads as given in Equation (5.33) It is also noted that the second term ofEquation (5.87) varies directly with the asymmetry quantities, and

Recall that a mid-plane asymmetry parameter was defined previously in Chapter 4 as

It is convenient to solve this relationship for and substitute the result into the second term inEquation (5.87), with the result that

where:

Thus, Equations (5.83) and (5.89) are the two governing differential equations to solve for thecircular cylindrical shell of a mid-plane asymmetric composite subjected to axially symmetriclateral and axial loads

Note the important feature of this perturbation approach If the problems are solved

successively from n = 0, it is first seen that the differential equation for n = 0 is the customary

axially- and mid-plane symmetric cylindrical shell equation There are many solutions availablefor this problem For it is seen that the term involving the second derivative is actually

known as it involves the w-solution for n-1 Thus these terms can be moved to the right-hand

side and the Equations (5.93) can be written as:

Thus, by canceling the terms it is found for n = 0:

and for

The forcing function of the n = 0 equation, (5.95), is the actual applied load to the shell.

For the forcing function involves derivatives of the (n-1) equation solution, and is

therefore a known quantity

It has been shown previously by Vinson and Brull [4] that whenever Equation(5.91) is another form of the exact solution to equations such as Equation (5.87) Also, inpractice, it is seldom necessary to utilize more than the first two terms of the perturbationsolution, i.e., Through this perturbation technique, the solution ofEquation (5.87) can be treated as a successive set of solutions of the customary axiallysymmetric, mid-plane symmetric, cylindrical shell There are many solutions available for suchproblems

5.7.3 EDGE LOAD SOLUTIONS AND THE BENDING BOUNDARY LAYER FOR A PLANE ASYMMETRIC SHELL

MID-It has been shown that for the mid-plane asymmetric shell the boundary layer exists andhas the same length as it is for the usual mid-plane symmetric shell as given by Equation (5.55)

As in the case of beams and plates, when loads applied to circular cylindrical shells arecompressive, are in beam type bending, involve an external pressure or are torsional about thelongitudinal axis, buckling can occur Such buckling can result in shell failure, and if thebuckling stress is lower than the allowable stress from a strength point of view, then that value isthe limit to the useful load carrying capability of the shell

5.8 Buckling of Circular Cylindrical Shells of Composite Materials Subjected to Various Loads

5.8.1 APPLIED LOADS

Consider a circular cylindrical shell as shown in Figure 5.12 below of mean radius, R, wall thickness, h and length, L, subjected to a compressive load, P, a beam-type bending moment, M, a torque, T and an external pressure, p For ease of presentation the compressionstress resultants resulting from some of these loads will be denoted as positive quantities, asopposed to the usual conventions used previously throughout this book

where is the axial load per unit circumference

Each of the above results represent the applied loads If for these applied loads, the

external pressure p or the applied torque T, equal or exceed a critical value, buckling will result,

which for most practical purposes is synonymous to collapse and thus failure of the shell

Shell structures have little respect for solid mechanicists trying to developanalytical solutions for shells buckling under most loads! Physically, shell structures arevery sensitive to initial imperfections when it comes to their stability Also, in any shellthere are slight variations in the shell wall thickness, the value of the radius, any seamsresulting from their manufacturing and such things as the inability to have a compressiveload, for example, be uniformly distributed around the entire circumference As a resultthe experimentally determined buckling loads are often significantly diminished from theanalytically determined predictions.

Hence, in any analyses, the analytical values must have an empirical factorrelating the analytical prediction to the experimental values of test data Without suchempirical factors the analytical predictions are not conservative

This means that any new buckling analyses methods should also have anempirical “knockdown” factor in order to relate analyses to the same experimental data

So from this very practical point of view the NASA methods presented here arecurrent, and have been validated over time

5.8.2 BUCKLING DUE TO AXIAL COMPRESSION

Assumptions

Special anisotropy [that is

Prebuckled deformations are not taken into account

Ends of the cylindrical shell are supported by rings rigid in their planes, but offer no resistance to rotation or bending out of their plane.

a.

b.

c.

5.8.2.1 General Case, No Mid-Plane Symmetry, and n > 4

where

critical compressive load per unit circumference

L cylinder length

R cylinder radius

m number of buckle half waves in the axial direction

n number of buckle waves in the circumferential direction

Here, (5.106) through (5.108) are repeated from Chapter 2 for the reader’s convenience.

It is seen that the comprise the extensional stiffness matrix of the composite shell,comprise the bending – extension coupling matrix, and the comprise the flexural orbending stiffness matrix of the shell material This notation is discussed previously in Chapter 2

It should be noted that if the fiber orientation is circumferential, then indetermining the values of and in Equations (5.106) through (5.108)

To determine the critical load for a cylindrical shell with given dimensions and a

given material system, one determines those integer values of m and n which make a

minimum If choices can be made regarding the ply orientation and number of plys, then anoptimization analysis can be performed to determine the construction that provides the highestbuckling load per unit weight

After the buckling load has been determined, a check must be made to see that the finalconstruction is not overstressed at a load below the critical buckling load, because if that is thecase the cylinder is limited to a load that will result in overstressing rather than buckling For alaminated composite construction, the most general constitutive equation for the cylinder isgiven by the following, which is a repeat of Equation (2.66)

In this case of buckling due to axial compression,

one can obtain the and matrices from the above Then, using these matriceseach stress component in each lamina and ply can be calculated using

Remember that if the shell wall is one ply or is unidirectional, then and only then can (5.35) through (5.37) be used instead of (5.110) to determine the shell stresses.

These stresses can then be compared to the allowable or failure stress in each ply as discussed in Chapter 7 below.

5.8.2.2 Special Case, Mid-Plane Symmetry

For this case the appropriate equation to use is

where

Here is an empirical (knockdown) factor that insures that the calculated buckling load will be conservative with respect to all experimental data that are available to date.

5.8.4 BUCKLING DUE TO EXTERNAL LATERAL PRESSURE AND HYDROSTATIC PRESSURE

For a lateral external pressure, the critical value of pressure, that will cause buckling

is determined by:

where the are defined by (5.100) through (5.105).

In this case m = 1, and one varies the integer to find the minimum value of for a given construction because that is the physical buckling load.

For long cylinders subjected to a lateral pressure, the critical buckling pressure is given by: