The Behavior of Structures Composed of Composite Materials Part 6 pptx

for a plate simply supported on allfour edges made of a material whose flexural stiffness properties are given asfollows and whose thickness is 1 inch.. Consider a composite material pla

Proceeding as before for the mid-plane symmetric rectangular plate of Section 3.3, theresulting three coupled equations using classical plate theory, i.e no transverse sheardeformation, have the following form:

Because of the bending-stretching coupling not only are lateral displacements,

w(x,y), induced but in-plane displacements, and as well; hence, three coupledequations (3.165) through (3.167)

3.23 Governing Equations for a Composite Material Plate With Bending-Twisting Coupling

Looking at Equation (2.66), the moment curvature relations for a rectangular plane symmetric plate with bending-twisting coupling are:

mid-Of course if transverse shear deformation is ignored, i.e classical theory then thecurvatures are given by (3.23), and the moment curvature relations become:

Substituting these into (3.19), provides the following governing differential equation

Comparing (3.170) with (3.29), it is seen that due to the presence of the andbending-twisting coupling terms, odd numbered derivatives appear in the governingdifferential equation That precludes the use of both the Navier approach of Section 3.5,and the use of the Levy approach of Section 3.7 in obtaining solutions for plate withbending-twisting coupling With these complications one may want to obtain solutionsusing the Theorem of Minimum Potential Energy discussed in Chapter 6 below

3.24 Concluding Remarks

It appears that there is no end in trying to more adequately describe

mathematically the behavior of composite materials utilized in structural components.Unfortunately, the more sophisticated one gets in such descriptions the more difficult themathematics becomes, as is evidenced in the increasing difficulty observed as oneprogresses through the sections of Chapter 3

One additional complication that is important in some composite materialstructures is that the stiffness (and other properties) are different in tension than they are

in compression This occurs because (1) sometimes the tensile and compressivemechanical properties of both fiber and matrix materials, differ and (2) sometimes itoccurs because the matrix material is very weak compared to the fiber (that is

such that the fibers buckle in compression under a small load so that for

the composite the stiffness in compression differs markedly than the stiffness in tension.Hence, one can idealize a little and say that one has one set of elastic properties in tensionand another set of elastic properties in compression Bert [25] has termed this a

bimodular material, typical of some composites, certainly typical of aramid (Kevlar)fibers in a rubber matrix as used in tires, and also typical of certain biological tissuesmodeled in biomedical engineering In this context

and all of the complications that result therefrom are too difficult to treat in this text forfirst year students trying to learn the fundamentals of composite materials

Lastly, time dependent effects in the stresses, deformation and strains of

composite materials are becoming more important design considerations Viscoelasticityand creep are respected disciplines about which entire books have been written Theseeffects have been deemed important in some composite material structures Crossman,Flaggs, Vinson and Wilson have all commented thereupon Wilson and Vinson [26] haveshown that the effects of viscoelasticity on the buckling resistance of polymer matrixcomposite material plates is very significant Similarly, the effect of viscoelasticity onthe natural vibration frequencies will also be significant Many of these effects have beenincluded in a survey article by Reddy [27] who has focused primarily on plates composed

Vinson, J.R (1974) Structural Mechanics: The Behavior of Plates and Shells,

Wiley-Interscience, John Wiley and Sons, New York

Vinson, J.R and Chou, T.W (1975) Composite Materials and Their Use in

Structures, Applied Science Publishers, London.

Levy, M (1899) Sur L’equilibrie Elastique d’une Plaque Rectangulaire, Compt Rend

129, pp 535-539.

Timoshenko, S and Woinowsky-Krieger, S (1959) Theory of Plates and Shells,

McGraw-Hill Book Co Inc., edition, New York

Vinson, J.R (1961) New Techniques of Solutions for Problems in Orthotropic Plates,

Ph.D Dissertation, University of Pennsylvania.

Vinson, J.R and Brull, M.A (1962) New Techniques of Solutions of Problems in

Orthotropic Plates, Transactions of the Fourth United Stated Congress of Applied Mechanics, Vol 2, pp 817-825.

Vinson, J.R (1989) The Behavior of Thin Walled Structures: Beams, Plates and Shells, Kluwer Academic Publishers, Dordrecht, The Netherlands.

Whitney, J.M (1987) Structural Analysis of Laminated Anisotropic Plates,

Technomic Publishing Co Inc., Lancaster, Pa

Vinson, J.R (1999) The Behavior of Sandwich Structures of Isotropic and Composite Materials, Technomic Publishing Co Inc., Lancaster, Pa.

Dobyns, A.L., (1981) The Analysis of Simply-Supported Orthotropic Plates

Subjected to Static and Dynamic Loads, AIAA Journal, May, pp 642-650 Leissa, A.W (1973) The Free Vibration of Rectangular Plates, Journal of Sound and Vibration, Vol 31, No 3, pp 257-293.

Nashif, A.D., Jones, D.I.G and Henderson, J (1985) Vibration Damping, Wiley

Interscience

Inman, D.J (1989) Vibration with Control Measurement and Stability, Prentice Hall,

Englewood Cliffs, New Jersey

Warburton, G The Vibration of Rectangular Plates, Proceedings of the Institute of Mechanical Engineers, 1968 (1954), pg 371-384.

Young, D and Felgar, R., Jr (1944) Tables of Characteristic Functions Representing

Normal Modes of Vibration of a Beam, The University of Texas Publication Number 4913.

Felgar, R., Jr (1950) Formulas for Integral Containing Characteristic Functions of a

Vibrating Beam, Bureau of Engineering Research, The University of Texas Publication.

Moh, J-S and Hwu, C (1997) Optimization for Buckling of Composite Sandwich

Plates, AIAA Journal, Vol 35, pp 863-868.

Kerr, A.D (1964) Elastic and Viscoelastic Foundation Models, Journal of Applied Mechanics, Vol 31, pp 491-498.

Paliwal, D.N and Ghosh, S.K (1944) Stability of Orthotropic Plates on a Kerr

Foundation, AIAA Journal, Vol 38, pp 1993-1997.

Zenkert, D (1995) An Introduction to Sandwich Construction, EMAS Publications,

West Midlands, UK

Sierakowski, R.L and Mukhopadhyay, A.K (1990) On Sandwich Beams WithLaminate Facings and Honeycomb - Cores Subjected to Hygrothermal Loads:

Part I – Analysis, Journal of Composite Materials, Vol 24, No 4, pp 382-400.

Sierakowski, R.L and Mukhopadhyay, A.K (1990) On Sandwich Beams WithLaminate Facings and Honeycomb – Cores Subjected to Hygrothermal Loads:

Part II – Application, Journal of Composite Materials, Vol 24, No 4, pp

Sierakowski, R.L and Mukhopadhyay, A.K (2000) On Thermoelastic and

Hygrothermal Response of Sandwich Beams With Laminate Facings and

Honeycomb – Cores: Part IV – A Dynamic Theory, Journal of Composite Materials, Vol 34, pp 174-199.

Bert, C.W., Reddy, J.N Reddy, V.S and Chao, W.C (1981) Analysis of Thick

Rectangular Plates Laminated of Bimodulus Composite Materials, AIAA Journal,

Vol 19, No 10, October, pp 1342-1349

Wilson, D.W and Vinson, J.R (1984) Viscoelastic Analysis of Laminated Plate

Buckling, AIAA Journal, Vol 22, No 7, July, pp 982-988.

Reddy, J.N (1982) Survey of Recent Research in the Analysis of Composite Plates,

Composite Technology Review, Fall.

3.26 Problems and Exercises

3.1 Find the critical buckling load, in lbs./in for a plate simply supported on allfour edges made of a material whose flexural stiffness properties are given asfollows and whose thickness is 1 inch

(a) If a = 30 inches and b = 20 inches.

(b) If a = 50 inches and b=12 inches.

3.2 Find the fundamental natural frequency in Hz (cps) for each of the plates of Problem3.1, if the mass density of the material is

3.3 The following material properties are given for a unidirectional, 4 ply laminate,

h = 0.020”

the mass density (corresponding to

Consider a plate made of the above material with dimensions a = 20”, b = 30”, h =

0.020” For the first perturbation method of Section 3.8 determine and Is aproper value to use this perturbation technique?

3.4 Consider the plate of problem 3.3 If it is simply supported on all four edges, what

is its fundamental natural frequency in cycles per seconds neglecting transverseshear deformation?

3.5 For a box beam whose dimensions are b = 4”, h = 2”, L = 20”, composed of

T300/934 graphite/epoxy, whose properties are given in Appendix 2, determine theextensional stiffness, EA; the flexural stiffness, EI, and the torsional stiffness, GJ, ifthe box beam is made of a 4 laminae, unidirectional composite, with a laminathickness of 0.0055”, all fibers being in the length direction

3.6 Consider a composite material plate of dimensions of thickness

h, composed of an E Glass/epoxy, which is modeled as being simply supported on

all four edges It is part of a structural system, which is subjected to a hydraulic load

as shown below

The load is where is the weight density of the water

(a) To utilize the Navier approach determine which is given by

(b) At what value of x will the maximum deflection occur?

(c) At what value of x will the maximum stress occur?

3.7 Consider a square plate in which a = b = 20” made of a unidirectional Kevlar/epoxy

composite, whose properties are:

(a) Determine the flexural stiffness matrix [D].

(b) In the first perturbation technique of Section 3.8, calculate and

(c) Can this perturbation technique be used for this problem?

(d) What is the total weight of this plate?

(e) If this plate is simply supported on all four edges at what location (i.e., x = ?and y = ?) will the maximum deflection occur?

(f) For the plate in e above at what location will the maximum bending-stressoccur?

3.8 Consider a plate composed of aluminum, an isotropic material of modulus ofelasticity E, shear modulus G, and Poisson’s ratio The plate is of thickness h.

Analogous to the stiffness matrix of Equation (2.66) determine the values of

and for this construction

3.9 Consider a plate measuring 16” x 16” in planform of [ 0°, 90°, 90°, 0° ], of total

thickness 0.022” The [D] matrix for this construction is

If the plate is subjected to an in-plane compressive load in the direction, what

is the critical buckling load per inch of the edge distance, using classical platetheory?

3.10 What is the fundamental natural frequency of the plate of Problem 3.9 in Hz (i.e.cycles per second), using classical plate theory? The weight density of the

composite is

3.11 In designing a test facility to demonstrate the buckling of the plate of Problem 3.9,what load cell capacity (force capability) is needed to attain the loads necessary tobuckle the plate?

3.12 (a) The plate of Problems 3.9 through 3.11 above will be used in an environment

in which it will be exposed to a sinusoidal frequency of 6 Hz Is it likely therewill be a vibration problem requiring detailed study? Why?

(b) What about 12 Hz? Why?

3.13 Could the first perturbation solution technique of Section 3.8 be used to obtain

solution for the plate of Problem 3.9 subjected to a static lateral load, p(x,y)?

3.14 Consider a rectangular plate of composite materials which is part of a space vehiclestructure Its dimensions are 10” x 10” It is composed of Kevlar 49/epoxy

(properties are given in Appendix 2 of the text) It is composed of six laminae,

unidirectional (all 0°), with ply thickness The density of thematerial is

The plate is simply supported on all four edges

(a)

(b)

(c)

(d)

What are the flexural stiffnesses and

Because the panel is part of a large space vehicle structure, care must be taken

to identify all natural frequencies in the 0-1.5 Hz Range What, if any naturalfrequencies fall in this range?

If the plate is subjected to a uniform in-plane compressive load in the

x-direction, what is the critical buckling load,

Will the plate buckle before it is overstressed or will it be overstressed before itbuckles?

3.15 For a plate or panel, what are the four ways in which it may fail or become

subjected to a condition which may terminate its usefulness?

3.16 A panel simply supported on all four edges, measuring a = 30”, b = 10”, composed

of T300-5208 graphite epoxy, composed of laminae with the following properties:

In the October 1986 issue of the AIAA Journal, M.P Nemeth discuses the

conditions in which one can ignore and in determining the buckling load

for a composite plate He defines:

If both of these ratios are less than 0.18, one can use Equation (3.149) to determinethe buckling load within 2% of the correct value for a plate simply supported on allfour edges If either of the ratios is greater than 0.18 one must replace the left handside of Equation (3.146) with the left hand side of Equation (3.170), which negatesthe use of the Navier and Levy methods being used, thus complicating the solution.For a four ply panel with stacking sequence of [+45°, – 45°, – 45°, + 45°],determine and to see if the simpler solution can be used

Determine the fundamental natural frequency in Hertz (cycles per second) for thepanel of Problem 3.16 made of four plys, unidirectionally oriented (all 0º plys).Determine the critical buckling load, for the same panel as in Problem3.16

For the panel of Problems 3.16 and 3.17, could the perturbation method of Section3.8 be used to solve for deflections and stresses, i.e., is

Consider a plate of dimensions a = 18” and b = 12”, composed of a laminated

composite material whose lamina properties are:

3.17

3.18

3.19

3.20

The stacking sequence of the plate is [ 0°, 90°, 90°, 0° ] in which each lamina is

The plate is simply supported on each edge

What are and for this plate?

For the plate of Problem 3.20, at what values of x and y will the maximum

deflection occur if the plate is subjected to a uniform lateral load (aconstant)?

For the plate of Problems 3.20 and 3.21 above at which values of x and y would

maximum ply stresses occur?

For the plate of Problem 3.20, calculate the critical buckling load per unit width,

3.21

3.22

3.23

and

if the plate is subjected to a uniform compressive load in the x direction.What is the fundamental natural vibration frequency in Hz for the plate of Problem3.20 Assume a weight density for the composite to be

Suppose the plate of Problem 3.24 were designed to be subjected to a continuingharmonic forcing function at:

(a) 38 to 48 Hz(b) 10 Hz

Would there be a problem structurally with this due to dynamic effects? Why?Consider a Kevlar 49/epoxy composite, whose properties are given in Appendix 2

of the text, and whose weight density is A plate whosestacking sequence is [0°, 90°, 90°, 0° ] is fabricated wherein each ply is 0.0055”thick The plate is in planform dimensions, and is simply supported onall four edges

Determine and

Could the perturbation solution technique of Section 3.8 be used to solve problemsfor the plate of Problem 3.26?

If the plate of Problem 3.26 is subjected to an in-plane compressive load in the

x-direction only, what is the critical buckling load per inch of edge distance,

using classical plate theory?

What is the fundamental natural frequency of the plate of Problem 3.26 in Hz.,using classical plate theory?

If the fundamental natural frequency were calculated including the effects oftransverse shear deformation, would that frequency be higher, lower or equal to thefrequency calculated in Problem 3.29 above?

Consider a Kevlar 49/3501-6 epoxy composite with the following properties:

For a unidirectional composite of thickness 0.1 inches, calculate

A square plate, simply supported on all four edges is composed of GY70/339

graphite epoxy If this square plate is made of four plys with the A and D matrix

values shown on the accompanying page, which stacking sequence would youchoose for a design to have the highest fundamental natural frequency? Whichstacking sequence has the lowest fundamental natural frequency?

3.32

GY70/339 graphite epoxy composite

For a plate simply supported on all four edges that is 6 inches wide and 15 incheslong made up of the unidirectional four ply graphite epoxy described in (a) above

of Problem 3.32, what is the critical buckling load, if the compressive load isapplied parallel to the longer direction of the plate?

If the plate of Problem 3.33 were subjected only to a uniform lateral load,

where would the maximum value of the tensile stress be located (i.e.where would you place a strain gage to measure the largest tensile strain, x = ?, y =

a Unidirectional four ply

b Crossply [0°, 90°, 90°, 0° ] four ply

c Angle ply four ply

What is

3.36 Consider a boron-epoxy material with the following properties:

Consider a two ply laminate wherein lamina 1 is oriented at 0°, and lamina 2 isoriented at 90° Each ply is 0.01” thick Calculate and

3.37 For a two ply laminate of materials and ply thickness of Problem 3.36 above,wherein lamina 1 is + 45° and lamina 2 is oriented at – 45°, calculate

3.38 Consider a square plate with length and width of 12 inches, and thickness of h=

0.020”, composed of graphite/epoxy whose stiffness matrix properties are given inProblem 3.32a Calculate the natural frequency in cycles per second (i.e., m =

2, n = 3,

3.39 (a) Does a natural frequency of vibration of a plate clamped on all four edges,

subjected to a lateral distributed load where is a constant,depend on the value of the load ?

(b) Does a plate of lamina stacking sequence [+45°, – 45°, – 45°, + 45° ], where

each ply has the same material and same thickness have a non-zero [B] matrix?

(c) Does bending-twisting coupling involve the and terms?

3.40 You have been asked to replace an existing aluminum plate structure by a

unidirectional Kevlar/epoxy structure using the material properties given inProblem 3.7 The loading on the aluminum plate is all in one direction, both an in-plane tensile load and a bending moment as shown below, and the structure isstiffness critical Therefore, you must design a unidirectional fiberglass structure

to have an extensional stiffness, and a flexural stiffness, that equals orexceeds those values for the aluminum structure The aluminum properties are

and the aluminum plate is 0.101 inchesthick

In the existing structure what is the flexural stiffness per unit width,

If you replace the aluminum structure with the Kevlar/epoxy structure, what

thickness h is required of your composite plate to have equal the

extensional stiffness of the aluminum structure?

What thickness h is required to your composite plate to have equal theflexural stiffness of the aluminum structure?

Which h must your composite design be to achieve the stated design

requirement?

Will your composite design be heavier or lighter than the aluminum structureand by what percentage?

3.41 Consider a rectangular panel simply supported on all four edges The panel

measure a = 25” , b = 10”, where The laminated plate is

composed of unidirectional boron/aluminum with the following properties:

(a)

(b)

(c)

Determine and for a lamina (ply) of this material

Determine the flexural stiffness and for a plate made offour ply, unidirectionally oriented (all 0° plys)

Determine the fundamental natural frequency in Hertz (cycles per second) forthe panel

(e)

Determine the critical buckling load, for the panel, when it is subjected

to a uniform compressive load in the x-direction.

If the panel were made of one ply with the fibers oriented at what is

3.42

3.43

Consider a laminated plate composed of the first graphite/epoxy composite whoseproperties are given in Appendix 2 of the text The laminate consists of fourlaminae, stacked as [ 0°, 90°, 90°, 0° ], where each ply is 0.0055 inches thick.Calculate a) b) c) and d)

Consider a rectangular composite plate whose stiffness matrices are given inProblem 3.3 above The plate is 15 inches wide, 60 inches long, simply supported

on all four edges, is 0.020 inches thick, whose weight density is andthe fibers are in the long direction

If an in-plane compressive load , is applied in the direction parallel to thelonger dimension, what is the critical buckling load, using classical platetheory?

Using classical plate theory what is the fundamental natural frequency in Hz.?

If transverse shear deformation effects were included in the above calculationswould the buckling load and fundamental natural frequency be higher, thesame, or lower?

(a) Calculate the axial stiffness, EA, of this box beam

(b) Calculate the flexural stiffness, EI, of this box beam

(c) Calculate the torsional stiffness, GJ, of this box beam.

3.45 Given a Kevlar/epoxy rectangular plate, with the unidirectional material propertiesgiven in Problem 3.7 above, for a plate of dimensions 16” x 12”, and a thickness of0.1”, as shown below, simply supported on all four edges

What is the critical load per unit inch, to cause plate buckling of theplate

What is the stress in the load direction at buckling?

Will the plate be overstressed before it could buckle?

What is the total weight of this plate?

What thickness would this plate have to be to have the buckling stress equal tothe compressive strength of the composite material?