The Behavior of Structures Composed of Composite Materials Part 4 potx

The properties of graphite fibers and a polyimide matrix are as follows:a Find the modulus of elasticity in the fiber direction, of a laminate of polymide composite with 60% fiber volume

Sloan, J.G (1979) The Behavior of Rectangular Composite Material Plates Under

Lateral and Hygrothermal Loads, MMAE Thesis, University of Delaware (also

AFOSR-TR-78-1477, July 1978)

Tsai, S.W and Pagano, N.J (1968) Invariant Properties of Composite Materials,

Tsai, S.W et al., eds., Composite Materials Workshop, Technomic Publishing

Co., Inc., Lancaster, PA, pp 233-253

Reissner, E (1950) On a Variational Theorem in Elasticity, J Math Phys., 29, pp.

90

Mindlin, R.D (1951) Influence of Rotatory Inertia and Shear on Flexural Motions of

Isotropic Elastic Plates, Journal of Applied Mechanics, pg 73.

Lee, C.K and Moon, F.C (1990) Modal Sensors/Actuators, Journal of Applied Mechanics, Vol 57, pp 434-441.

Yu., Y.Y (1977) Dynamics for Large Deflection of a Sandwich Plate With Thin

Piezoelectric Face Layers, Ed G.S Simitses, Analysis and Design Issues for Modern Aerospace Vehicles, ASME AD-Vol 55, pp 285-292.

Leibowitz, M and Vinson, J.R (1991) Intelligent Composites: Design and Analysis

of Composite Material Structures Involving Piezoelectric Material Layers Part

A-Basic Formulation, Center for Composite Materials Technical Report 91-54,

University of Delaware, November

Larson, P.H (1993) The Use of Piezoelectric Materials in Creating Adaptive Shell

Structures, Ph.D Dissertation, Mechanical Engineering, University of

Delaware

Newill, J.F (1996) Composite Sandwich Structures Incorporating Piezoelectric

Materials, Ph.D Dissertation, Mechanical Engineering, University of Delaware.

Hilton, H.H., Yi, S and Vinson, J.R (1998) Probabilistic Structural Integrity of

Piezoelectric Viscoelastic Composite Structures, Proceedings of the 39th AIAA/ASME/ASCE/AHS/ASC SDM Conference, April.

2.12 Problems

1 Consider a laminate composed of boron-epoxy with the following properties:

If the laminate is a cross–ply with [0°/90°/90°/0°], with each ply being 0.25 mm(0.11") thick, and if the laminate is loaded in tension in the x direction (i.e., the 0°direction):

(a)

(b)

(c)

What percentage of the load is carried by the 0 plies? The 90° plies?

If the strength of the 0° plies is MPA(198,000 psi), and the strength

of the 90° plies is 44.8 MPA (6,500psi) which plies will fail first?

What is the maximum load, that the laminate can carry at incipient failure?What stress exists in the remaining two plies, at the failure load of the other twoothers?

The properties of graphite fibers and a polyimide matrix are as follows:

(a) Find the modulus of elasticity in the fiber direction, of a laminate of polymide composite with 60% fiber volume ratio

graphite-(b) Find the Poisson’s ratio, ?

(c) Find the modulus of elasticity normal to the fiber direction,

(d) What is the Poisson’s ratio, ?

Consider a laminate composed of GY70/339 graphite epoxy whose properties aregiven above in Problem 2 For a lamina thickness of 0.127 mm (0.005"), calculatethe elements of the A, B and D matrices for the following:

(a) [0°, 0°, 0°, 0°] (unidirectional);

(b) [0°, 90°, 90°, 0°] (across-ply);

(c) i.e [+45°/-45°/-45°/+45°] (an angle-ply);

(e) Compare the various stiffness quantities for the four laminates above

7 Consider a laminate composed of GY70/339 graphite epoxy whose properties aregiven above in Problem 2 For a lamina thickness of 0.127 mm (0.005") cited inProblem 6, calculate the elements of the [A], [B] and [D] matrices for the followinglaminates:

Compare the forms of the A, B and D matrices between laminate type.

What type of coupling would you expect in the (B) matrix for (a) and (b) below:(a) 0º/90° laminate

(b) laminate

Given a composite laminate composed of continuous fiber laminate laminae of HighStrength Graphite/Epoxy with properties of Table 2.2, if the laminate architecture is[0°, 90°, 90°, 0°], determine if and each ply thickness is 0.006".Consider a plate composed of a 0.01" thick steel plate joined perfectly to analuminum plate, 0.01" thick Using the properties of Table 2.2 calculate if thePoisson’s Ratio of each material is

Consider a unidirectional composite composed of a polyimide matrix and graphitefibers with properties given in Problem 5 above In the fiber direction, what volumefraction is required to have a composite stiffness of psi to match analuminum stiffness

A laminate is composed of ultra high modulus graphite epoxy with properties given

in Table 2.2 below Determine the elements of the [A], [B] and [D] matrices for a

two ply laminate [+45°/-45°], where each ply is 0.006" thick For the material

0.31

A laminate is composed of boron-epoxy with the properties of Problem 2.1 and a

stacking sequence of [0/+45°/–45°/0°], and a ply thickness of 0.006" Determine

the elements of the A, B and D matrices

GRAPHITE POLYIMIDE

Finder the modulus of elasticity in the fiber direction, of a lamina ofgraphite – polyimide composite with 70% fiber volume ratio

Find the Poisson’s Ratio,

Finder the modulus of elasticity normal to the fiber direction,

What is the Poisson’s Ratio,

Compare these properties with those obtained for the same material systembut with in problem 2.5

In a given composite, the coefficient of thermal expansion for the epoxy and thegraphite fibers are in/in/°F and in/in/°F respectively Forspace application where no thermal distortion can be tolerated what volumefractions of each component are required to make zero expansion and contraction inthe fiber direction for an all 0° construction? (Hint: Use the Rule of Mixtures)

Find the A, B and D matrices for the following composite: 50% volume Fraction Boron- Epoxy Composite

Stacking Sequence (each lamina is 0.0125" thick)

Three composite plates are under uniform transverse loading All the conditions,such as materials, boundary conditions and geometry, etc are the same except the18

(a) Determine the A, B, and D matrix component.

(b) What if any are the couplings in this cross-ply construction that are decidedbelow Equation (2.62)?

(c) If only in-plane loads are applied, is the plate stiffer in the x direction or ydirection, or are they the same?

(d) If only plate bending is considered, is the plate stiffer in the x direction, the ydirection, or are they equally stiff?

Given the following fiber and matrix properties for HM-S/epoxy compositecomponents:

Epoxy HM-S/Graphite

Determine each of the following properties for a unidirectional composite:

and for the fiber volume fractions of30%, 60% Which properties increase linearly with volume fraction?Which do not increase linearly with volume fraction?

Given a cross-ply construction of four lamina of the same composite material systemoriented as 0°, 90°, 90°, 0°, each lamina being equally thick, which elements of the[A], [B] and [D] matrices of Equation (2.66) will be equal to zero

Given an angle-ply construction of five plies of the same composite material oriented

as each of equal thickness, which elements of the [A], [B] and[D] matrices of Equation (2.66) will be equal to zero

Determine the elements of the matrix analogous to the of Equation (2.10)through (2.12) for orthotropic materials (Hint: start with Equation (2.17) and solvefor the

A laminate is composed of graphite epoxy (GY70/339) with the following properties:

and a stacking sequence of [0°, +45°, -45°, 0°], and a ply thickness of 0.006 inches,determine the elements of the A, B and D matrices What would the elements be ifthe ply thickness were 0.0055 inches?

Determine how the A, B, D matrices are populated for the following two stackingsequences and The subscript QS mean symmetric

What type of couplings, as discussed below Equation (2.66) would you expect in the

B matix for (a) and (b) below: (that is, identify the non-zero terms)

In problem 2.28 which laminate is stiffest and which is the least stiff for

(a) In-plane loads in the 0° direction

(b) In-plane loads in the 90° direction

(c) Bending in the 0° direction

(d) Bending in the 90° direction

Consider a laminate composed of GY 70/339 graphite/epoxy with the followingproperties,

29

30

Compare the forms of the [A], [B] and [D] matrices between laminate types.

A composite material has stiffness matrix as follows,

31

Determine the state of stress if the strains are given by,

Consider the stress acting on an element of a composite material to be as shownbelow The material axes 1,2 are angle with respect to the geometry loading axesfor the element Taking the material properties as noted below, find the m-planedisplacements u(x,y), v(x,y)

33 A 50% boron-epoxy orthotropic material is subjected to combined stress as shownbelow

32

Find the stress on the material element for a 45° rotation about the z-axis in apositive sense.

If the strain components in the non-rotated system are given by:

Find the corresponding strains in the rotated system

Comment on the corresponding stresses and strains in the rotated system.(a)

(b)

as seen in Equation (2.66) These will be utilized with the displacement relations of Equations (2.48) and (2.50) and the equilibrium equations to bedeveloped in Section 3.2 to develop structural theories for thin walled bodies, theconfiguration in which composite materials are most generally employed Plates andpanels are discussed in this chapter Beams, rods and columns are discussed in Chapter

strain-4 Shells will be the subject of Chapter 5 The use of energy methods for solvingstructures problems is discussed in Chapter 6 However to study any of the structuralequations it is necessary to first develop suitable equilibrium equations

3.2 Plate Equilibrium Equations

The integrated stress resultants (N), shear resultants (Q) and stress couples (M),

with appropriate subscripts, are defined by Equation (2.54), and their positive directions

are shown in Figure 2.13, for a rectangular plate, defined as a body of length a in the x direction, width b in the y -direction, and thickness h in the z -direction, where h<< b, h<< a, i.e a thin plate.

-In mathematically modeling solid materials, including the laminates of Chapter 2,

a continuum theory is generally employed In doing so, a representative material pointwithin the elastic solid or lamina is selected as being macroscopically typical of allmaterial points in the body or lamina The material point is assumed to be infinitelysmaller than any dimension of the structure containing it, but infinitely larger than thesize of the molecular lattice spacing of the structured material comprising it Moreover,the material point is given a convenient shape; and in a Cartesian reference frame that

convenient shape is a small cube of dimensions dx, dy, and dz as shown in Figure 3.1

below

This cubic material point of dimension dx, dy and dz is termed a control element.

The positive values of all stresses acting on each surface of the control element are shown

in Figure 3.1, along with how they vary from one surface to another, using the positivesign convention consistent with most scientific literature, and consistent with Figure 2.1.Details of the nomenclature can be found in any text on solid mechanics, includingVinson [1,2], In addition to the surface stresses acting on the control element shown inFigure 3.1, body force components and can also act on the body These bodyforce components such as gravitational, magnetic or centrifugal forces are proportional tothe control element volume, i.e., its mass

A force balance can now be made in the x, y and z directions resulting in three equations of equilibrium For instance, a force balance in the x-direction would yield

Canceling terms and dividing the remaining terms by the volume results in the following

Similarly, equilibrium in the y and z directions yields:

Using Einsteinian notation, these three equations can be consolidated to the following:

These three equations comprise the equilibrium equations for a three dimensionalelastic body However, for beam, plate and shell theory, whether involving compositematerials or not, one must integrate the stresses across the thickness of the thin walledstructures to obtain solutions

Recalling the definitions of Chapter 2, the stress resultants and stress couplesdefined previously are:

The first form of each is applicable to a single layer plate, while the second form

is necessary for a laminated plate due to the stress discontinuities associated withdifferent materials and/or differing orientations in the various plies

Turning now to (3.1), neglecting the body force term, for simplicity of thisexample, integrating term by term across each ply, and summing across the plateprovides

In the first two terms integration and differentiation can be interchanged, hence:

i, j = x, y, z

Equation (3.7) can then be written as:

Similarly, integrating the equilibrium equation in the y-direction provides

where

Likewise equilibrium in the z-direction upon integration and summing provides

where

In addition to the integrated force equilibrium equations above, two equations of

moment equilibrium are also needed, one for the x-direction and one for the y-direction Multiplying equation 3.1 through by zdz, integrating across each ply and summing across

all laminae results in the following

Again, in the first two terms integration and summation can be interchanged with

differentiation with the result that the first two terms become

However, the third term must be integrated by parts as follows:

Here the last term is clearly Again in the first term on the right, clearly the

moments of all the interlaminar stresses between plies cancel each other out, and the only

non-zero terms are the moments of the applied surface shear stresses hence that term

becomes

Using the former expression, the equation of equilibrium of moments in the

x-direction is

Similarly in the y-direction the moment equilibrium equation is

where all the terms are defined above Thus, there are five equilibrium equations for a

rectangular plate, regardless of what material or materials are utilized in the plate: (3.9),

(3.10), (3.12), (3.14) and (3.15)

3.3 The Bending of Composite Material Laminated Plates: Classical Theory

Consider a plate composed of a laminated composite material that is mid-plane

shear stresses and no hygrothermal effects The plate equilibrium equations for the

bending of the plate, due to lateral loads given by Equations (3.14), (3.15) and (3.12)

become:

where Equations (3.16) and (3.17) can be substituted intoEquation (3.18) with the result that:

The above equations are derived from equilibrium considerations alone FromEquation (2.66) and for the case of mid-plane symmetry and no andterms, the constitutive relations are:

where from Equation (2.49)

It is well known that transverse shear deformation (that is

effects are important in plates composed of composite material plates in determiningmaximum deflections, vibration natural frequencies and critical buckling loads.However, it is appropriate to use a simpler stress analysis involving classical theorywhich neglects transverse shear deformation for preliminary design to determine a “firstcut” for stresses, the required overall stacking sequence and required plate thickness isappropriate

If, in fact, transverse shear deformation is ignored, then from Equation (2.48)

hence, from the above,

So, substituting Equation (3.23) into Equations (3.20) through (3.22) results in thefollowing for the case of no transverse shear deformations, i.e., classical plate theory:

Substituting these three equation in turn into Equation (3.19) results in:

The above coefficients are usually simplified to:

This is the governing differential equation for the bending of a plate composed of

a composite material, excluding transverse shear deformation, with no coupling terms(that is and no hygrothermal terms (that is, subjected

to a lateral distributed load p(x, y) As stated previously, neglecting transverse shear

deformation and hygrothermal effects can lead to significant errors, as will be shown, butwith the result that (3.27) becomes