The Behavior of Structures Composed of Composite Materials Part 8 pps

Consider the beam to be made of an all layup, of 30 laminae, each thick,such that the total beam thickness is The beam is one inch wideand twelve inches long If the beam is simply suppor

Equation (4.179) has been termed the “reduced” or “apparent” flexural stiffness.

Gere, J and Timoshenko, S (1984) Mechanics of Materials, Second Edition,

PWS-Kent Publishing Company, Boston, MA, Appendix G

Warburton, G (1968) “The Vibration of Rectangular Plates,” Proceedings of the Institute of Mechanical Engineers, Institute of Mechanical Engineers, London,

U.K., 371

Young, D and Felgar, R.F Jr., “Tables of Characteristic Functions RepresentingNormal Modes of Vibration for a Beam,” The University of Texas EngineeringResearch Series Report No 44, July 1, 1949

Felgar, R.P Jr (1950) “Formulas for Integrals Containing Characteristic Functions of

a Vibrating Beam,” The University of Texas Bureau of Engineering ResearchCircular No 14

Vinson, J.R and Dee, A.T (1998) Use of Asymmetric Sandwich Construction to

Minimize Bending Stresses, Sandwich Constructions 4, ed K.-A Olson, Vol 1,

EMAS Publishers, Ltd UK, pp 391-402

Whitney, J.M (1969) The Effect of Transverse Shear Deformation on the Bending of

Laminated Plates, Journal of Composite Materials, Vol 3, pp 534-547.

Timoshenko, S.P and Gere, J.M (1961) Theory of Elastic Stability, McGraw-Hill

Book Co., Inc., 2nd Edition

Bleich, H.H (1952) Buckling of Metal Structures, McGraw-Hill Book Co., Inc.

determine explicitly the expression w(x).

Given a beam made of T300/5208 graphite epoxy with the following mechanicalproperties at

4.2

70º F

Consider the beam to be made of an all layup, of 30 laminae, each thick,such that the total beam thickness is The beam is one inch wideand twelve inches long

If the beam is simply supported at each end, and subjected to a uniformlateral load of 10 lbs./in of length, what is the maximum deflection?What is its maximum stress?

What is its fundamental natural frequency in bending?

If the beam were subjected to a compressive end load what would thecritical buckling load be?

Consider a beam of length width of the material of Problem 4.2 If the

beam is simply supported at each end, what minimum beam thickness, h, is

necessary to insure that the beam is not overstressed when it is subjected to auniform lateral load of

For a beam of the material of Problem 4.2, with and

what is the fundamental natural frequency in cycles per second (Hz) if the beam issimply supported at each end? Neglect transverse shear deformation

For the beam of Problem 4.4 above, what is the critical buckling load,

neglecting transverse shear deformation?

For a clamped-clamped composite beam of uniform cross-section but mid-planeasymmetric subjected to mechanical and hygrothermal loads, determine explicit

expressions for the lateral (transverse) deflection, w(x), and the in-plane stresses.

For a composite beam simply supported at each end, made of a laminate of

i.e., [0/90/90/0] whose ply thickness is 0.01 inches, subjected to a uniform lateralload determine the following stress profiles of

(a)

(b)

The material properties are:

4.8 Consider the following simply supported beam subjected to the loading shown,where the beam is mid-plane asymmetric

So from the equilibrium equations, prove that

4.9 Consider a composite material beam, with constant flexural stiffness, simplysupported at each end, and subjected to a lateral load per unit length given by

4.13

4.14

For the same beam as in Problem 4.11 above, what is the critical compressive

buckling load, neglecting transverse shear deformation

A beam is made of unidirectional Spectra 900 fibers in a Metton matrix The

properties are:

The beam is made by injection molding, hence is uniform in construction

What is the longitudinal stiffness in the fiber direction?

If the beam is thick, wide, long what is its fundamental natural

frequency if it is simply supported at each end?

What is its critical buckling load when subjected to an axial compressive

Consider a composite beam of T300/5208 graphite/epoxy, composed of 4

unidirectional laminae each of thickness whose properties are given in

Problem 4.2 The beam is wide and long, simply supported at the end x =

0 and clamped at the end x = L A later load of q = 10 lbs./in of length is

4.16

4.17

(c) If so, how many plys are needed for an adequate design so that the

maximum stress is not greater than the strength of the material?

What is the fundamental natural frequency of the beam of Problem 4.14 if theweight density is 0.06 lbs./cubic inch, and the gravitational constant is 386in./second squared?

For a clamped-clamped beam, the axial buckling load is four times that of a beamsimply supported at each end For the beam of Problem 4.14, clamped at each end,what is the axial buckling load?

A cantilever beam of length L, composed of a graphite/epoxy composite material

T300/5208 with the stacking sequence shown is subjected to a uniform mechanicalload For the beam shown plot the stress distribution at the section ofmaximum moment Compare the shape of this stress distribution with that of anisotropic beam

Calculate the bending and shear stresses at x = L/2 for a simply supported

[0/90/90/0], composite beam composed of the following materials,whose properties are given in Appendix 2:

(b) Triangular transverse load

4.19 A load P is supported by three vertical bars as shown below The horizontal bar

remains horizontal during deformation Consider the middle bar to be made from acomposite material consisting of an aluminum matrix with boron fibers continuousand aligned parallel to the load Assume a 50% volume fraction of fibers to matrixand a structurally contiguous bond between fibers and matrix

(a) Find the fiber and matrix stresses for the middle bar

(b) If all bars were made of aluminum, what would the middle load stress be?Compare to the composite bar

4.20 Consider a simply supported laminated beam subjected to a single concentratedload as shown

For this beam find:

If the beam consists of a

laminate, find the stress distribution at x - a Consider the beam to be made of a

Kevlar-epoxy composite Use properties from Appendix 2 Each ply thickness is0.010 inches

Find the stress distribution in the cantilever beam shown below at thesection of maximum moment The beam is made of T300/5208 Graphite-epoxywith the following properties

4.21

Compare the shape of the stress distribution with the case of an isotropic beam

Consider a beam simply supported at each end under a constant (uniform) load

as shown below Find the stresses and plot the stress distribution at the mid-span

L/2 and at L/4

4.22

Kevlar/Epoxy Composite

[0/+45/-45/90/90/-45/+45/0],

Stacking sequence with

Consider a simply supported beam under a transverse ramp loading Find thelateral deflection and in-plane displacement

The governing differential equations can be reduced to

4.24 A graphite-epoxy structure with fiber orientation 30° from the x-axis is loaded by a

4.23

triangular loading as shown below Assume for this problem that it is a beam.

Obtain the expressions for the in-plane displacement normal displacement w

and the rotation Find the maximum deflection w and identify the shear

4.25 Consider a simply supported beam with two equal concentrated loads

symmetrically placed as shown below, i.e., four point bending

For a composite laminated stacking sequence of [0/90/0/90] with each ply ofthickness and properties given by

Find the maximum stresses in the beam for each ply.

For a clamped-clamped beam, the axial buckling load is 4 times that of a beamsimply supported at each end Consider a composite beam made of T300/5208 (seeProblem 4.2) graphite/epoxy composed of 4 unidirectional laminae each of

thickness the beam being wide and in length, find the axialbuckling load

Find the variation in the flexural stress for the beam shown below and

compare the solution with that of a homogeneous/isotropic beam Use the materialproperties given in Problem 4.17

4.26

4.27

Find the variation in the flexural stress for the laminated beam cross-section

shown below, subjected to a bending moment M.

4.28

Take each ply as thick and consider two composite materials, with theproperties indicated below

4.29 A laminated [0/90/0/90/90/0/90/0], composite beam simply supported ateach end, made from graphite/epoxy is 1mm thick, 20mm wide and has plies of

0.125mm thickness The lamina properties are as follows, where the S values are strength, the L and T refer to fiber and transverse directions, and + and – refer to

tension and compression values, respectively

The beam is subjected to a uniformly distributed mechanical Find thefollowing:

The bending stress and the shear stress through the thickness of thebeam

For a constant temperature gradient and a constant hygrometric gradienteach applied separately, find the and through the thickness.Compare the temperature and hygrometric loading stress distributions withthe mechanical loading case

(a)

(b)

(c)

4.30 Consider a simply supported composite beam of length L with a symmetric

stacking sequence given by Find the transverse deflection at L/4 and L/2 for each of the following loadings.

(a)

(b)

A constant thermal load

A constant hygrometric load

Consider the following two composite materials, whose properties are given inAppendix 2

E-Glass/Epoxy

T300/5208 Graphite/Epoxy

4.31 Find the flexural stresses and maximum displacement in a composite beam withclamped boundaries subjected to a uniform mechanical load a constant thermalgradient and a constant moisture gradient

4.32 Consider a 4 ply composite laminated beam with geometry as shown below Thebeam is simply supported on elastic springs of equal spring stiffness K For plysequences of with a mid-span load P, find the stress distribution

at the quarter span and at the center of the beam For the same system if the beam

is subjected to a hygrometric load only (1% moisture), what would the

corresponding stress distribution be at the quarter and mid-span of the beam Usethe governing equations for simple laminated beam theory For a compositematerial use AS4/3501 Graphite whose properties are given in Appendix 2

4.33.(a) Design a simply supported polymer matrix composite beam of laminateconstruction such that the maximum allowable deflection due to a uniform lateralload of 12 lb./in is less than 0.10 inches The beam length is 6 feet Consider thebeam to be initially designed for the indicated mechanical load only Variables forthe design include

Material Type

Stacking Sequence

Beam Depth and Breadth

(b) Ignoring moisture effects, how would the design change if the beam was

subjected to both the specified mechanical load and a temperature of aboveambient room temperature, taken as

Consider a beam of length width of the following material

If the beam is simply supported at each end, what minimum beam thickness, h, is

necessary to insure that the beam is not overstressed when it is subjected to auniform lateral load of

Consider a beam on an elastic foundation as shown below with two rigid endsupports The beam is subjected to a uniform mechanical load in lb./in Take

the elastic foundation as Q = Kw(x) lb./in.

Discuss approaches to solving the problem

Write the governing differential equations

Find the deflection w(x).

Find the deflection at the center of the beam

Discuss the effects of the end supports

Plot the first five vibration mode shapes for a beam simply supported at each endwith a stacking sequence and a ply thickness of

Assume the material is E-Glass/Epoxy

4.36

4.37 Consider a composite beam made of T300/5208 Graphite/epoxy (see Problem 4.2),

25º F75º F

frequencies and mode shapes for the flexural vibrations.

Make the following assumptions:

Classical beam theory

No coupling between the extensional and flexural deformation

Effects of rotatory inertia is negligible

No hygrothermal loads

Uniform beam

COMPOSITE MATERIAL SHELLS

5.1 Introduction

A shell is a thin walled body, just as a beam or plate is, whose middle surface is curved in

at least one direction For instance a cylindrical shell and a conical shell have only one direction

in which the middle surface is curved On the other hand in a spherical shell there is curvature in both directions Such mundane shells as a front fender of a car and an eggshell are examples of shells with double curvature.

Shell theory is greatly complicated, compared to beam and plate theory, because of this curvature The treatment of shell theory in its proper detail is the subject of a graduate level full semester or full year course, and hence, well beyond the scope of this book One recent textbook dealing with shells of composite materials as well as isotropic materials is that of Reference [1] Even to derive the governing differential equations for a shell of general curvature from first principles require several lectures in topology.

Then, to complicate shell theory all of the material complexities associated with laminated composite materials makes the shell theory of composite materials very complicated, and a great challenge.

In this text, the shell geometries will be restricted to circular cylindrical shells; for other geometries see Reference [1].

5.2 Analysis of Composite Material Circular Cylindrical Shells

5.2.1 GENERAL EQUATIONS

The simplest of all shell geometries is that of the circular cylindrical shell shown below

in Figure 5.1 The positive directions of the displacements u, v, and w are shown, as well as the positive directions of the coordinates x and The remaining coordinate is the circumferential coordinate The positive value of all stress resultants and stress couples are shown in Figure 5.2 below.

In the classical shell theory discussed in this section, all of the assumptions used in classical plate theory of Chapters 2 and 3 are utilized:

In addition, there is one other assumption known as Love’s First Approximation, which is consistent with the neglect of transverse shear deformation:

It is true that the accurate analysis of shells of composite materials should include transverse shear deformation because of the fact that the modulus of elasticity in the fiber direction is a fiber dependent property, while the transverse shear modulus is a matrix dominated property However, for preliminary design, one can often neglect transverse shear deformation effects with resulting simplifications in the governing equations.

To derive the governing differential equations for cylindrical shells, one begins withintroducing the elasticity equations in a curvilinear coordinate system, just as the elasticityequations were derived using a Cartesian coordinate system in Chapter 3 for plates One canproceed to develop the governing equations for a shell of general shape, then specialize the shell

to any particular configuration such as a circular cylindrical shell This would require significantspace, for a complete derivation and thus too involved for the scope of this text, however, it isavailable in Reference [1] Therefore, the governing equations are presented below withoutderivation The equilibrium equations are:

where

These equilibrium equations are independent of the material system The circumferential

terms can also be written in terms of ds, the arc distance where The quantities

and are functions of the surface shear stresses on the outer and inner surfaces ofthe composite shell wall, and is the laterally distributed load per unit area, positive in thepositive direction