The Behavior of Structures Composed of Composite Materials Part 5 pps

determination of lateral deformations, wx, y , and stress couples, andFor isotropic plate monocoque plates several textbooks such as Timoshenko andWoinowsky-Krieger [4] and Vinson [7] ha

determination of lateral deformations, w(x, y ), and stress couples, and

For isotropic plate monocoque plates several textbooks such as Timoshenko andWoinowsky-Krieger [4] and Vinson [7] have provided expressions for the maximum

deflection, and the maximum stress couple, M, a plate attains when subjected to a

constant laterally distributed load such as,

where a is the smaller plate dimension; b is the longer plate dimension, E is the

modulus of elasticity of the plate; h is the plate thickness; and

The dimensionless constants and are given in tabular form for various

boundary conditions, and these are repeated herein for completeness in Tables 3.1

through 3.4 Table 3.5 also provides information for the case wherein the plate is

subjected to a hydraulic head These tables and procedures are well known and well

used

Of course, for the isotropic plate, the flexural stiffness is given by

and the maximum bending stress, which occurs on the top and bottom surfaces of theplate, is

Also for Tables 3.1 through 3.5, the numerical coefficients correspond to aPoisson’s ratio of wherein Therefore, for materials with other

Poisson ratios, ,, Equation (3.71) must be changed to

With Equations (3.75) and (3.72), the well-used Tables 3.1 through 3.4 formonocoque plates can be used to analyze quasi-isotropic composite plates as well Itmust be remembered that for these classical theory solutions no transverse-shear

deformation effects are included Table 3.5 can be used to analyze and design compositematerial plates subjected to a hydraulic head

It is seen that for a monocoque quasi-isotropic plate design,

The plate must not be overstressed, i.e., the maximum stress is determined from the

use of Equation (3.72) to determine the maximum stress couple, M, and the

procedures described earlier to determine the stresses in each lamina The

determined maximum stress cannot exceed some allowable stress, defined bythe material’s ultimate stress or yield stress divided by a factor of safety on ultimatestress or yield stress, whichever is smaller This requires a certain value of plate

thickness, h.

The monocoque plate must not be over deflected determined by Equation (3.75).This is sometimes specified, but in other cases the plate deflection cannot exceedthe plate thickness or some fraction thereof If the maximum plate deflectionreaches a value of the plate thickness, h, the equations discussed herein becomeinapplicable because the plate behavior becomes increasingly nonlinear whichrequires that other equations be used Again, to prevent over deflection, a plate

thickness, h, is required as determined by Equation (3.75).

(1)

(2)

Therefore, in monocoque plate design, the plate thickness, h, is determined either from a

strength or stiffness requirement, whichever requires the larger thickness

3.10 A Static Analysis of Composite Material Panels Including Transverse Shearr Deformation Effects

The previous derivations have involved "classical" plate theory, i.e., they haveneglected transverse shear deformation effects Because in many composite materiallaminated plate constructions, transverse-shear deformation effects are important, a morerefined theory will now be developed However, because of its simplicity, and thenumber of solutions available, classical theory is still useful for preliminary design andanalysis to size the structure required in minimum time and effort

In the simpler classical theory, the neglect of transverse shear deformation effectsmeans that To include transverse shear deformation effects, one uses

Now substituting the admissible forms of the displacement for a plate or panel,Equation (2.49) into Equations (3.76) and (3.77), shows that

No longer are the rotations and explicit functions of the derivatives of the

lateral deflection w, as shown by Equation (3.23) for classical plate theory The result is

that for this refined theory there are five geometric unknowns, andinstead of just the first three in classical theory

Now one needs to look again at the equilibrium equations, the constitutiveequations (stress-strain relations), the strain-displacement relations and the compatibilityequations For the plate, the equilibrium equations are given by Equations (3.9) through(3.15), because they do not change from classical theory The constitutive equations for acomposite material laminated plate and sandwich panel are given by Equations (2.58)through (2.66) The new cogent strain-displacement (kinematic) relations are givenabove in Equation (3.78) and (3.79) Because the resulting governing equations are interms of displacements and rotations, any single valued, continuous solution will bydefinition satisfy the compatibility equations

As an example consider a plate that is mid-plane symmetric and has nocoupling terms the constitutive equations for this orthotropicplate can be written as follows, where is a transverse shear coefficient to be discussedlater

Because the plate is mid-plane symmetric there is no bending-stretching coupling,hence the in-plane loads and deflections are uncoupled (separate)from the lateral loads, deflections and rotations Hence, for the lateral distributed static

loading, p(x, y), Equations (3.16) through (3.18) and Equations (3.83) through (3.87) are

utilized: 8 equations and 8 unknowns

Substituting Equations (3.83) through (3.87) into Equations (3.16) through (3.18)and using Equation (2.66) results in the following set of governing differential equationsfor a laminated composite plate subjected to a lateral load, with

and no applied surface shear stresses (for simplicity)

The inclusion of transverse shear deformation effects results in three coupled

partial differential equations with three unknowns, and w, contrasted to having one partial differential equation with one unknown, w, in classical plate (panel) theory; see

Equation (3.29) Incidentally if one specified that and substitutingthat into Equations (3.88) through (3.90) reduces the three equations to Equation (3.29),the classical theory equation The symbol with no subscript in (3.88) through (3.90) is

a transverse shear deformation shape factor which varies from 1 to 2 depending upon thegeometry

The classical plate theory governing partial differential equation is fourth order in

both x and y, and therefore requires two and only two boundary conditions on each of the

four edges, as discussed in Section 3.4 This refined theory, including transverse shear

deformation, is really sixth order in both x and y, and therefore requires three boundary

conditions on each edge as discussed in Section 3.11 below

If the laminated plate is orthotropic but not mid-plane symmetric, i.e.,

the governing equations are more complicated than Equations (3.88) through (3.90) andare given by Whitney [8], Vinson [9] and are discussed briefly in Section 3.23 below

3.11 Boundary Conditions for a Plate Using the Refined Plate Theory Which Includes Transverse Shear Deformation

3.11.1 SIMPLY-SUPPORTED EDGE

Again Equation (3.30) holds, but now a third boundary condition is required forthe plate bending because it can be shown that Equations (3.88) through (3.90) are sixth

order in w with respect to x and y In addition, since the in-plane and lateral behavior are

coupled, a fourth boundary condition enters the picture as well This has resulted in theuse of two different simply supported boundary conditions, both of which aremathematically admissible as natural boundary conditions (to be discussed later) and arepractical structural boundary conditions By convention the simply supported boundaryconditions are given as follows:

where is the mid-surface displacement in the x-direction and is the mid-surface

displacement in the y-direction.

Whether one uses S1 or S2 boundary conditions is determined by the physicalaspects of the plate problem being studied

3.11.3 FREE EDGE

The free edge requires three boundary conditions on each edge; therefore, it is nolonger necessary to resort to the difficulties of the Kirchhoff boundary conditions for thebending of the plate needed for classical plates The boundary conditions for the bending

of the plate are simply:

where n and t are directions normal to and tangential with the edge Again, the in-plane

boundary conditions for the free edge are

3.11.4 OTHER BOUNDARY CONDITIONS

In addition to the above boundary conditions, which are widely used toapproximate the actual structural boundary conditions, sometimes it is desirable toconsider an edge whose lateral deflection is restrained, whose rotation is restrained orboth The means by which to describe these boundary conditions is given for example in[7, pp 20-21]

3.12 Composite Plates on An Elastic Foundation

Consider a composite material plate that is supported on an elastic foundation Inmost cases an elastic foundation is modeled as an elastic medium with a constant

foundation modulus, i.e., a spring constant per unit planform area, of k in units such as

Therefore, the elastic foundation acts on the plate as a force in the negative

direction proportional to the local lateral deflection w(x,y) The force per unit area is -kw, because when w is positive the foundation modulus is acting in a negative direction, and

vice versa In order to incorporate the effect of the elastic foundation modeled as above

one simply adds another force to the p(x,y) load term The results are, that for classical

theory, Equation (3.29) is modified to (3.94), and for the refined theory, Equation (3.90)

is modified to Equation (3.95):

3.13 Solutions for Plates of Composite Materials Including Transverse-Shear Deformation Effects, Simply Supported on All Four Edges

Some solutions are now presented for the equations in Section 3.10 and 3.11,using the governing differential equations (3.88) through (3.90) In the following with

no subscript is a transverse shear factor, often give as or 5/6

Dobyns [10] has employed the Navier approach to solving these equations for acomposite plate simply supported on all four edges subjected to a lateral load, using thefollowing functions:

It is seen that Equations (3.96) through (3.98) satisfy the simply supported boundaryconditions on all edges given in Equation (3.91)

Substituting these functions into the governing differential equations (3.88)through (3.90) results in the following:

if and is the lateral load coefficient of (3.99) above, thenthe operators are given by the following:

Solving Equation (3.100), one obtains

where det is the determinant of the [L] matrix in Equation (3.100).

Having solved the problem to obtain and w, the curvatures

substituted back into Equations (3.20) through (3.22) to obtain the stress couples

and to determine the location where they are maximum, to help in determining

where the stresses are maximum

For a laminated composite plate, to find the bending stresses in each lamina one

must use the above equations to find the values for and in Equation (3.23)

Finally, for each lamina the bending stresses can be found using:

The stresses in each lamina in each direction must be compared to the strength of

the lamina material in that direction Keep in mind that quite often the failure occurs in

the weaker direction in a composite material

Looking at the load p(x,y) in Equation (3.99), if the lateral load p(x,y) is

distributed over the entire lateral surface, then the Euler coefficient, is found to be

If that load is uniform then,

For a concentrated load located at and

where P is the total load.

For loads over a rectangular area of side lengths u and v whose center is at and

as shown in Figure 3.8, is given as follows:

where P is the total load Note that when n/b = 1/v, m/a = 1/u, then Of course,any other lateral load can be characterized by the use of Equation (3.105).

3.14 Dynamic Effects on Panels of Composite Materials

Seldom in real life is a structure subjected only to static loads More oftenproducts and structures are subjected to vehicular, impact, crash, earthquake, handling, orfabricating dynamic loads In the linear-elastic range, dynamic effects can be dividedinto two categories: natural vibrations and forced vibrations, and the latter can be furthersubdivided into one-time events (an impact) or recurring loads (such as cyclic loading).These will be discussed in turn

Physically every elastic continuous body has an infinity of natural frequencies,only a few of which are of practical significance When a structure is excited cyclically

at a natural frequency, it takes little input energy for the amplitude to grow until one offour things happens:

The amplitude of vibration grows until the ultimate strength of a brittle material isexceeded and the structure fails

Portions of the structure exceed the yield strength, plastically deform and the

behavior changes drastically

The amplitude grows until nonlinear effects become significant, and there is nonatural frequency

Due to damping or other mechanism the amplitude is limited, but as the naturalvibration continues, fatigue failures may occur

be avoided Also in complex structures of course the structural natural frequencies can

be coupled involving all components

Mathematically, natural vibration problems are called eigenvalue problems Theyare represented by homogeneous equations, for which nontrivial solutions only occur at

certain characteristic (eigen, in German) values of a parameter, from which the natural

frequencies are determined In a natural vibration the displacement field comprises a

normal mode At any two different natural frequencies the corresponding normal modesare orthogonal to each other hence the compartmentalization of the energy) The normalmodes comprise the solutions to the homogeneous governing differential equations,which are now zero only at the eigenvalues for those equations and boundary conditions.

If there is a forcing function, then the particular solution for the specific forcingfunction (which can be cyclical or a one time dynamic impact load) is added onto thehomogeneous solution, which involves the natural frequencies and mode shapes

Physically, any dynamic load excites each and every one of the normal modes andcorresponding natural frequencies Usually, only a relatively few are large enough to be

of concern The largest amplitude of response will be in those mode shapes whosenatural frequency is closest to the oscillatory component of the forcing functions.When there are no natural frequencies close to the oscillatory portion of thedynamic load, then the structure will respond in deflection and stresses correspondingonly to the magnitude and spatial distribution of the load Such a condition results in

solving the worst-case static problem in which the biggest load at some time, t, is applied.

This is termed a quasistatic case However, if the dynamic load oscillatory component isclose to one of more natural frequencies, then the structural response can be much largerthan the value obtained from a quasistatic calculation, and that increase can be

represented by a dynamic load factor

In what follows natural frequencies are treated first, then forced linear vibrationsand finally nonlinear large-amplitude vibrations are discussed

3.15 Natural Flexural Vibrations of Rectangular Plates: Classical Theory

Consider a rectangular composite material plate that is mid-plane symmetric suchthat If this plate is isotropic, i.e., then the governingdifferential equation is given by Equation (3.29) for the classical theory, i.e., no

transverse-shear deformation, and repeated here as

For dynamic loads, using d’Alembert’s Principle, the equation is written as

where the last term is the mass per unit planform area times the acceleration For naturalvibrations the load (the forcing function) can be ignored and the resulting homogeneousequation is

This equation yields all of the natural frequencies for plates with any boundary

conditions For the easiest case, consider the composite material plate to be simplysupported on all four edges In that case, one may guess the following form for thedeflection function because it satisfies all boundary conditions:

where is the natural circular frequency in radians per unit time Note that instead of

will be the same

Substituting Equation (3.112) into Equation (3.111) one sees that the nontrivialsolution exists only when

In the above m and n are integers only In this case, the lowest natural frequency occurs when m = n= 1, and this lowest frequency is always termed the fundamental natural

frequency To obtain the natural frequency in terms of cycles per second (Hz)

Equation (3.113) is more accurate at low values of m and n, but as these integers

increase, i.e., higher mode shapes, the calculated values increasingly exceed measuredvalues This is because transverse shear deformation effects increase with increased

values of m and n, i.e., with the increased ratio of plate thickness to the wavelength of the m-nth mode of vibration One major reference on the free vibrations of rectangular

isotropic plates is Leissa [11]

If the composite plate is specially orthotropic and mid-plane symmetric, then forthe natural vibration problem the governing differential equation is written as in Equation(3.29),

Again, if all four edges are simply supported then the mode shapes are give by Equation

(3.112) with the result that the natural circular frequency in radians per second is given

by

Again, the natural frequency in Hz is given by Equation (3.114)

Keep in mind that for Equation (3.116) and other equations for the frequencies for

natural vibrations to accurately describe the motion, the maximum deflection must be a

fraction of the plate thickness since the theory is linear Above that level of motion,

nonlinear effects become increasingly significant

3.16 Natural Flexural Vibrations of Composite Material Plates Including

Transverse-Shear Deformation Effects

The governing partial differential equations for a composite plate or panel that is

specially orthotropic and mid-plane symmetric subjected to a lateral static load p(x,y) are

given in Equations (3.88) through (3.90) If one now wishes to find the natural

frequencies of this composite plate, that has mid-surface symmetry no other

couplings but includes transverse shear deformation,

then one sets p(x,y) = 0 in Equation (3.90), but adds to theright-hand side In addition, because and are both dependent variables that are

independent of w, there will be an oscillatory motion of the lineal element across the plate

thickness about the mid-surface of the plate This results in the last term on the left-hand

side of Equations (3.88) and (3.90) becoming and

respectively, as shown below:

where is the mass density of the kth lamina, and here I is

In Equations (3.117) through (3.119) the ’s are transverse-shear deformation

parameters, as discussed earlier

Similar to the Navier procedure used in previous analyses and following Dobyns[10] for the simply supported plate let

Substituting these into the dynamic governing equations above results in a set of

homogeneous equations that can be solved for the natural frequencies of vibration

where the unprimed L quantities were defined below Equation (3.100) and