frp analysis in caesar-ii

8/9/2019 FRP Analysis in Caesar-II

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Technical Discussions

CAESAR II User's Guide 929

TSO=Torsional Stress, Outside

Pipe Stress Analysis of FRP Piping

Underlying TheoryThe behavior of steel and other homogeneous materials has been long understood, permittingtheir widespread use as construction materials. The development of the piping and pressurevessel codes (Reference 1) in the early part of this century led to the confidence in their use inpiping applications. The work of Markl and others in the 1940’s and 1950’s was responsible forthe formalization of today’s pipe stress methods, leading to an ensuing diversification of pipingcodes on an industry by industry basis. The advent of the digital computer, and with it theappearance of the first pipe stress analysis software (Reference 2), further increased theconfidence with which steel pipe could be used in critical applications. The 1980’s saw the widespread proliferation of the microcomputer, with associated pipe stress analysis software, whichin conjunction with training, technical support, and available literature, has brought stressanalysis capability to almost all engineers. In short, an accumulated experience of close to 100years, in conjunction with ever improving technology has led to the utmost confidence on the

part of today’s engineers when specifying, designing, and analyzing steel, or other metallic,pipe.

For fiberglass reinforced plastic (FRP) and other composite piping materials, the situation is notthe same. Fiberglass reinforced plastic was developed only as recently as the 1950’s, and didnot come into wide spread use until a decade later (Reference 3). There is not a large base ofstress analysis experience, although not from a lack of commitment on the part of FRP vendors.Most vendors conduct extensive stress testing on their components, including hydrostatic andcyclic pressure, uni-axial tensile and compressive, bending, and combined loading tests. Theproblem is due to the traditional difficulty associated with, and lack of understanding of, stressanalysis of heterogeneous materials. First, the behavior and failure modes of these materialsare highly complex and not fully understood, leading to inexact analytical methods and a generallack of agreement on the best course of action to follow. This lack of agreement has slowed thesimplification and standardization of the analytical methods into universally recognized codes

BS 7159 Code Design and Construction of Glass Reinforced Plastics Piping (GRP) Systems forIndividual Plants or Sites and UKOOA Specification and Recommended Practice for the Use ofGRP Piping Offshore being notable exceptions. Second, the heterogeneous, orthotropicbehavior of FRP and other composite materials has hindered the use of the pipe stress analysisalgorithms developed for homogeneous, isotropic materials associated with crystallinestructures. A lack of generally accepted analytical procedures has contributed to a generalreluctance to use FRP piping for critical applications.

Stress analysis of FRP components must be viewed on many levels. These levels, or scales,have been called Micro-Mini-Macro levels, with analysis proceeding along the levels accordingto the "MMM" principle (Reference 4).


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930 CAESAR II User's Guide

Micro-Level Analysis

Stress analysis on the "Micro" level refers to the detailed evaluation of the individual materialsand boundary mechanisms comprising the composite material. In general, FRP pipe ismanufactured from laminates, which are constructed from elongated fibers of a commercial

grade of glass, E-glass, which are coated with a coupling agent or sizing prior to beingembedded in a thermosetting plastic material, typically epoxy or polyester resin.

This means, on the micro scale, that an analytical model must be created which simulates theinterface between these elements. Because the number and orientation of fibers is unknown atany given location in a FRP sample, the simplest representation of the micro-model is that of asingle fiber, extending the length of the sample, embedded in a square profile of matrix.

Micro Level GRP Sample -- Single Fiber Embedded in Square Profile of Matrix

Evaluation of this model requires use of the material parameters of:

1. the glass fiber

2. the coupling agent or sizing layer normally of such microscopic proportion that it may beignored

3. the plastic matrix

It must be considered that these material parameters might vary for an individual material basedupon tensile, compressive, or shear applications of the imposed stresses, and typical valuesvary significantly between the fiber and matrix (Reference 5):

Young's Modulus Ultimate Strength Coefficient of Thermal Expansion

Material tensile (MPa) tensile (MPa) m/m/ºC

Glass Fiber 72.5 x103 1.5 x 103 5.0 x 10-6

PlasticMatrix

2.75 x 103 .07 x 103 7.0 x 10-6

The following failure modes of the composite must be similarly evaluated to:

§ failure of the fiber

§ failure of the coupling agent layer

§ failure of the matrix


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§ failure of the fiber-coupling agent bond

§ failure of the coupling agent-matrix bond

Because of uncertainties about the degree to which the fiber has been coated with the couplingagent and about the nature of some of these failure modes, this evaluation is typically reducedto:

§ failure of the fiber

§ failure of the matrix

§ failure of the fiber-matrix interface

You can evaluate stresses in the individual components through finite element analysis of thestrain continuity and equilibrium equations, based upon the assumption that there is a goodbond between the fiber and matrix, resulting in compatible strains between the two. For normalstresses applied parallel to the glass fiber:

ef = em = saf / Ef = sam / Em

saf = sam Ef / Em

Where:

ef = Strain in the Fiber

e = Strain in the Matrixsaf = Normal Stress Parallel to Fiber, in the Fiber

Ef = Modulus of Elasticity of the Fiber

sam = Axial Normal Stress Parallel to Fiber, in the Matrix

Em = Modulus of Elasticity of the Matrix

Due to the large ratio of the modulus of elasticity of the fiber to that of the matrix, it is apparentthat nearly all of the axial normal stress in the fiber-matrix composite is carried by the fiber.Exact values are (Reference 6):

saf = sL / [f + (1-f)Em/Ef ]

sam

= sL

/ [fEf

/Em

+ (1-f)]Where:

sL = nominal longitudinal stress across composite

f = glass content by volume


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The continuity equations for the glass-matrix composite seem less complex for normal stressesperpendicular to the fibers, because the weak point of the material seems to be limited by theglass-free cross-section, shown below:

Stress Intensification in Matrix Cross-Section

For this reason, it would appear that the strength of the composite would be equal to that of thematrix for stresses in this direction. In fact, its strength is less than that of the matrix due tostress intensification in the matrix caused by the irregular stress distribution in the vicinity of thestiffer glass. Because the elongation over distance D1 must be equal to that over the longerdistance D2, the strain, and thus the stress at location D 1 must exceed that at D2 by the ratioD2/D1. Maximum intensified transverse normal stresses in the composite are:

Where:

sb = intensified normal stress transverse to the fiber, in the composite

s = nominal transverse normal stress across composite

nm = Poisson's ratio of the matrix

Because of the Poisson effect, this stress produces an additional s'am equal to thefollowing:

s'am = Vm sb


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Shear stress can be allocated to the individual components again through the use of continuityequations. It would appear that the stiffer glass would resist the bulk of the shear stresses.However, unless the fibers are infinitely long, all shears must eventually pass through the matrixin order to get from fiber to fiber. Shear stress between fiber and matrix can be estimated as

Where:

tab = intensified shear stress in composite

T = nominal shear stress across composite

Gm = shear modulus of elasticity in matrix

Gf = shear modulus of elasticity in fiber

Determination of the stresses in the fiber-matrix interface is more complex. The bonding agenthas an inappreciable thickness, and thus has an indeterminate stiffness for consideration in thecontinuity equations. Also, the interface behaves significantly differently in shear, tension, andcompression, showing virtually no effects from the latter. The state of the stress in the interfaceis best solved by omitting its contribution from the continuity equations, and simply consideringthat it carries all stresses that must be transferred from fiber to matrix.

After the stresses have been apportioned, they must be evaluated against appropriate failurecriteria. The behavior of homogeneous, isotropic materials such as glass and plastic resin,under a state of multiple stresses is better understood. Failure criterion for isotropic material

reduces the combined normal and shear stresses (sa, sb, sc, tab, tac, tbc) to a single stress, anequivalent stress, that can be compared to the tensile stress present at failure in a materialunder uniaxial loading, that is, the ultimate tensile stress, Sult.

Different theories, and different equivalent stress functions f(sa, sb, sc, tab, tac, tbc) have beenproposed, with possibly the most widely accepted being the Huber-von Mises-Hencky criterion,which states that failure will occur when the equivalent stress reaches a critical value theultimate strength of the material:

seq = Ö{1/2 [(sa - sb)2 + (sb - sc)2+ (sc - sa)2 + 6(tab2+ tac2+ tbc2)} £ Sult

This theory does not fully cover all failure modes of the fiber in that it omits reference to directionof stress, that is, tensile versus compressive. The fibers, being relatively long and thin,predominantly demonstrate buckling as their failure mode when loaded in compression.


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The equivalent stress failure criterion has been corroborated, with slightly non-conservativeresults, by testing. Little is known about the failure mode of the adhesive interface, althoughempirical evidence points to a failure criterion which is more of a linear relationship between thenormal and the square of the shear stresses. Failure testing of a composite material loaded onlyin transverse normal and shear stresses are shown in the following figure. The kink in the curveshows the transition from the matrix to the interface as the failure point.

Mini-Level Analysis

Mini-Level Analysis Fiber Distribution Models

Although feasible in concept, micro level analysis is not feasible in practice. This is due to theuncertainty of the arrangement of the glass in the composite the thousands of fibers that mightbe randomly distributed, semi-randomly oriented, although primarily in a parallel pattern, and ofrandomly varying lengths. This condition indicates that a sample can truly be evaluated only on

a statistical basis, thus rendering detailed finite element analysis inappropriate.For mini-level analysis, a laminate layer is considered to act as a continuous hence the commonreference to this method as the "continuum" method, material, with material properties andfailure modes estimated by integrating them over the assumed cross-sectional distribution,which is, averaging. The assumption regarding the distribution of the fibers can have a markedeffect on the determination of the material parameters. Two of the most commonly postulated


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distributions are the square and the hexagonal, with the latter generally considered as being abetter representation of randomly distributed fibers.

The stress-strain relationships, for those sections evaluated as continua, can be written as:

eaa = saa/EL - (VL/EL)sbb - (VL/EL)scc

ebb = -(VL/EL)saa + sbb/ET - (VT/ET)scc ecc = -(VL/EL)saa - (VT/ET)sbb + scc/ET

eab = tab / 2 GL

ebc = tbc / 2 GT

eac = tac / 2 GL

Where:

eij = strain along direction i on face j

sij, tab = stress (normal, shear) along direction i on face j

EL = modulus of elasticity of laminate layer in longitudinal direction

VL = Poisson’s ratio of laminate layer in longitudinal direction

ET = modulus of elasticity of laminate layer in transverse direction

VT = Poisson’s ratio of laminate layer in transverse direction

GL = shear modulus of elasticity of laminate layer in longitudinal direction

GT = shear modulus of elasticity of laminate layer in transverse direction

These relationships require that four modules of elasticity, EL, ET, GL, and GT, and two Poisson’sratios, VL and V, be evaluated for the continuum. Extensive research (References 4 - 10) hasbeen done to estimate these parameters. There is general consensus that the longitudinal termscan be explicitly calculated; for cases where the fibers are significantly stiffer than the matrix,they are:

EL = EF f + EM(1 - f)

GL = GM + f/ [ 1 / (GF - GM) + (1 - f) / (2GM)]

VL = VFf + VM(1 - f)

You cannot calculate parameters in the transverse direction. You can only calculate the upperand lower bounds. Correlations with empirical results have yielded approximations (Reference 5and 6):

ET = [EM(1+0.85f 2) / {(1-VM2)[(1-f)1.25 + f(EM/EF)/(1-VM2)]}

GT = GM (1 + 0.6Öf) / [(1 - f)1.25 + f (GM/GF)]

VT = VL (EL / ET)

Use of these parameters permits the development of the homogeneous material models thatfacilitate the calculation of longitudinal and transverse stresses acting on a laminate layer. Theresulting stresses can be allocated to the individual fibers and matrix using relationshipsdeveloped during the micro analysis.


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Macro-Level Analysis

Macro to Micros Stress Conversion

Where Mini-level analysis provides the means of evaluation of individual laminate layers,Macro-level analysis provides the means of evaluating components made up of multiplelaminate layers. It is based upon the assumption that not only the composite behaves as a

continuum, but that the series of laminate layers acts as a homogeneous material withproperties estimated based on the properties of the layer and the winding angle, and that finally,failure criteria are functions of the level of equivalent stress.

Laminate properties may be estimated by summing the layer properties (adjusted for windingangle) over all layers. For example

Where:

ExLAM = Longitudinal modulus of elasticity of laminate

tLAM = thickness of laminate

Ek = Longitudinal modulus of elasticity of laminate layer k

Cik = transformation matrix orienting axes of layer k to longitudinal laminate axis

C jk = transformation matrix orienting axes of layer k to transverse laminate axis

tk = thickness of laminate layer k

After composite properties are determined, the component stiffness parameters can bedetermined as though it were made of homogeneous material that is, based on componentcross-sectional and composite material properties.

Normal and shear stresses can be determined from 1) forces and moments acting on thecross-sections, and 2) the cross-sectional properties themselves. These relationships can bewritten as:

saa = Faa / Aaa ± Mba / Sba ± Mca / Sca

sbb = Fbb / Abb ± Mab / Sab ± Mcb / Scb

scc = Fcc / Acc ± Mac / Sac ± Mbc / Sbc

tab = Fab / Aab ± Mbb / Rab


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in each direction. In other words, the allowable stresses for the two equivalent stresses abovewould be (ed ELAMX) and (ed ELAMH ) respectively. In lieu of test data, system design strain isselected from Tables 4.3 and 4.4 of the Code, based on expected chemical and temperatureconditions.

Actual stress equations as enumerated by BS 7159 display below:

1. Combined stress straights and bends:

sC = (sf 2+ 4sS2)0.5 ed ELAM

or

sC = (sX2 + 4sS2)0.5 ed ELAM

Where:

ELAM = modulus of elasticity of the laminate; in CAESAR II, the first equation uses themodulus for the hoop direction and in the second equation, the modulus for the longitudinaldirection is used.

sC = combined stress

s• = circumferential stress = s•P + s•B

sS = torsional stress

= MS(Di + 2td) / 4I

sX = longitudinal stress

= sXP + sXB

s•P = circumferential pressure stress

= mP(Di + td) / 2 td

s•B = circumferential bending stress

= [(Di + 2td) / 2I] [(Mi SIF•i)2 + Mo SIF•o)2] 0.5 for bends, = 0 for straights

MS = torsional moment on cross-section

Di = internal pipe diameter

td = design thickness of reference laminate

I = moment of inertia of pipe

m = pressure stress multiplier of component

P = internal pressure

Mi = in-plane bending moment on cross-section

SIF•i= circumferential stress intensification factor for in-plane moment

M = out-plane bending moment on cross-section

SIF•o = circumferential stress intensification factor for out-plane moment

sXP = longitudinal pressure stress

= P(Di + td) / 4 t d


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BS 7159 also dictates the means of calculating flexibility and stress intensification (k- and i-)factors for bend and tee components, for use during the flexibility analysis.


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BS 7159 imposes a number of limitations on its use, the most notable being: the limitation of asystem to a design pressure of 10 bar, the restriction to the use of designated design laminates,and the limited applicability of the k- and i- factor calculations to pipe bends (that is, mean wallthickness around the intrados must be 1.75 times the nominal thickness or less).

This code appears to be more sophisticated, yet easy to use. We recommend that its calculationtechniques be applied even to FRP systems outside its explicit scope, with the followingrecommendations:

§ Pressure stiffening of bends should be based on actual design pressure, rather thanallowable design strain.

§ Design strain should be based on manufacturer’s test and experience data whereverpossible (with consideration for expected operating conditions).

§ Fitting k- and i- factors should be based on manufacturer’s test or analytic data, if available.

UKOOA

The UKOOA Specification is similar in many respects to the BS 7159 Code, except that itsimplifies the calculation requirements in exchange for imposing more limitations and moreconservatism on the piping operating conditions.


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Rather than explicitly calculating a combined stress, the specification defines an idealizedenvelope of combinations of axial and hoop stresses that cause the equivalent stress to reachfailure. This curve represents the plot of:

(sx / sx-all)2 + (shoop / shoop-all)2 - [sx shoop / (sx-all shoop-all)] £ 1.0

Where:

sx-all = allowable stress, axial

shoop-all = allowable stress, hoop

The specification conservatively limits you to that part of the curve falling under the line between

sx-all (also known as sa(0:1) ) and the intersection point on the curve where shoop is twice sx-(anatural condition for a pipe loaded only with pressure), as shown in the following figure.

An implicit modification to this requirement is the fact that pressure stresses are given a factor ofsafety (typically equal to 2/3) while other loads are not. This gives an explicit requirement of:

P des £ f 1 f 2 f 3 LTHP

Where:

P des = allowable design pressure

f 1 = factor of safety for 97.5% lower confidence limit, usually 0.85

f 2 = system factor of safety, usually 0.67

f 3 = ratio of residual allowable, after mechanical loads

= 1 - (2 sab) / (r f 1 LTHS)

sab = axial bending stress due to mechanical loads

r = sa(0:1)/sa(2:1)

sa(0:1) = long term axial tensile strength in absence of pressure load

sa(2:1) = long term axial tensile strength under only pressure loading

LTHS = long term hydrostatic strength (hoop stress allowable)

LTHP = long term hydrostatic pressure allowable


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This has been implemented in the CAESAR II pipe stress analysis software as:

Code Stress Code Allowable

sab (f 2 /r) + PDm / (4t) £ (f 1 f 2 LTHS) / 2.0

Where:

P = design pressure

D = pipe mean diameter

t = pipe wall thickness

K and i-factors for bends are to be taken from the BS 7159 Code, while no such factors are tobe used for tees.

The UKOOA Specification is limited in that shear stresses are ignored in the evaluation process;no consideration is given to conditions where axial stresses are compressive; and most requiredcalculations are not explicitly detailed.


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FRP Analysis Using CAESAR II

Practical Applications

CAESAR II has had the ability to model orthotropic materials such as FRP almost since its

inception. It also can specifically handle the requirements of the BS 7159 Code, the UKOOASpecification, and more recently ISO 14692. FRP material parameters corresponding to those ofmany vendors’ lines are provided with CAESAR II. You can pre-select these parameters to bethe default values whenever FRP piping is used. Other options, as to whether the BS 7159pressure stiffening requirements should be carried out using design strain or actual strain can beset in CAESAR II’s configuration module as well.


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Selecting material 20 — Plastic (FRP) – activates CAESAR II’s orthotropic material model andbrings in the appropriate material parameters from the pre-selected materials. The orthotropicmaterial model is indicated by the changing of two fields from their previous isotropic values:Elastic Modulus (C) changes to Elastic Modulus/axial and Poisson's Ratio changes to Ea/Eh*Vh/a. These changes are necessary because orthotropic models require more materialparameters than do isotropic. For example, there is no longer a single modulus of elasticity for

the material, but now two: axial and hoop. There is no longer a single Poisson’s ratio, but againtwo: V h/a (Poisson’s ratio relating strain in the axial direction due to stress-induced strain in thehoop direction) and V a/h (Poisson’s ratio relating strain in the hoop direction due tostress-induced strain in the axial direction). Also, unlike isotropic materials, the shear modulusdoes not follow the relationship G = 1 / E (1-V ), so that value must be explicitly input.

To minimize input, a few of these parameters can be combined due to their use in the program.Generally, the only time that the modulus of elasticity in the hoop direction or the Poisson’sratios is used during flexibility analysis is when calculating piping elongation due to pressure(note that the modulus of elasticity in the hoop direction is used when determining certain stress

allowables for the BS 7159 code):

dx = (sx / Ea - Va/h * shoop / Eh) L

Where:

dx = extension of piping element due to pressure


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sx = longitudinal pressure stress in the piping element

E = modulus of elasticity in the axial direction

Va/h = Poisson’s ratio relating strain in the axial direction due to stress-induced

strain in the hoop direction

shoop = hoop pressure stress in the piping element

Eh = modulus of elasticity in the hoop direction

L = length of piping element

This equation can be rearranged, to require only a single new parameter, as:

dx = (sx - Va/h shoop * (Ea / Eh )) * L / Ea

In theory, that single parameter, Vh/a is identical to (Ea / Eh * Va/h) giving: dx = (sx -Vh/ashoop) * L / Ea

The shear modulus of the material is required in ordered to develop the stiffness matrix. InCAESAR II, this value, expressed as a ratio of the axial modulus of elasticity, is brought in fromthe pre-selected material, or can be changed on a problem-wise basis using the SpecialExecution Parameter (see "Special Execution Parameters" on page 287) dialog boxaccessed by the Environment menu from the piping spreadsheet (see figure). This dialog boxalso shows the coefficient of thermal expansion (extracted from the vendor file or user entered)for the material, as well as the default laminate type, as defined by the BS 7159 Code:

§ Type 1 – All chopped strand mat (CSM) construction with an internal and an externalsurface tissue reinforced layer.

§ Type 2 – Chopped strand mat (CSM) and woven roving (WR) construction with an internaland an external surface tissue reinforced layer.

§ Type 3 – Chopped strand mat (CSM) and multi-filament roving construction with an internal

and an external surface tissue reinforced layer.

The latter is used during the calculation of flexibility and stress intensification factors for pipingbends.

You can enter bend and tee information by using the auxiliary spreadsheets.


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You can also change bend radius and laminate type data on a bend by bend basis, asshown in the corresponding figure.

Specify BS 7159 fabricated and molded tee types by defining CAESAR II tee types 1 and 3respectively at intersection points. CAESAR II automatically calculates the appropriate flexibilityand stress intensification factors for these fittings as per code requirements.

Enter the required code data on the Allowables auxiliary spreadsheet. The program providesfields for both codes, number 27 – BS 7159 and number 28 – UKOOA. After selecting BS 7159,CAESAR II provides fields for entry of the following code parameters:

SH1 through SH9 = Longitudinal Design Stress = ed ELAMX

Kn1 through Kn9 = Cyclic Reduction Factor (as per BS 7159 paragraph 4.3.4)

Eh/Ea = Ratio of Hoop Modulus of Elasticity to Axial Modulus of Elasticity

K = Temperature Differential Multiplier (as per BS 7159 paragraph 7.2.1)


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After selecting UKOOA, CAESAR II provides fields for entry of the following code parameters:

SH1 through SH9 = hoop design stress = f 1 * LTHS

R1 through R9 = ratio r = (sa(0:1) / sa(2:1))

f 2 = system factor of safety (defaults to 0.67 if omitted)

K = temperature differential multiplier (same as BS 7159)

These parameters need only be entered a single time, unless they change at some point in thesystem.

Performing the analysis is simpler than the system modeling. evaluates the operatingparameters and automatically builds the appropriate load cases. In this case, three are built:

§ Operating includes pipe and fluid weight, temperature, equipment displacements, and

pressure. This case is used to determine maximum code stress/strain, operationalequipment nozzle and restraint loads, hot displacements, and so forth.

§ Cold (same as above, except excluding temperature and equipment movements). This caseis used to determine cold equipment nozzle and restraint loads.

§ Expansion (cyclic stress range between the cold and hot case). This case may be used toevaluate fatigue criteria as per paragraph 4.3.4 of the BS 7159 Code.

After analyzing the response of the system under these loads, CAESAR II displays a menu ofpossible output reports. Reports may be designated by selecting a combination of load case andresults type (displacements, restraint loads, element forces and moments, and stresses). Fromthe stress report, you can determine at a glance whether the system passed or failed the stresscriteria.


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For UKOOA, the piping is considered to be within allowable limits when the operating stress fallswithin the idealized stress envelope this is illustrated by the shaded area in the following figure.

Conclusion

A pipe stress analysis program with worldwide acceptance is now available for evaluation ofFRP piping systems as per the requirements of the most sophisticated FRP piping codes. Thismeans that access to the same analytical methods and tools enjoyed by engineers using steelpipe is available to users of FRP piping design.

References

1. Cross, Wilbur, An Authorized History of the ASME Boiler an Pressure Vessel Code, ASME,1990

2. Olson, J. and Cramer, R., "Pipe Flexibility Analysis Using IBM 705 Computer Pro\-gramMEC 21, Mare Island Report 277-59," 1959

3. Fiberglass Pipe Handbook, Composites Institute of the Society of the Plastics Indus\-try,1989

4. Hashin, Z., "Analysis of Composite Materials a Survey," Journal of Applied Mechanics, Sept.1983

5. Greaves, G., "Fiberglass Reinforced Plastic Pipe Design," Ciba-Geigy Pipe Systems

6. Puck, A. and Schneider, W., "On Failure Mechanisms and Failure Criteria ofFilament-Wound Glass-Fibre/Resin Composites," Plastics and Polymers, Feb. 1969

7. Hashin, Z., "The Elastic Moduli of Heterogeneous Materials," Journal of Applied Mechanics,March 1962

8. Hashin, Z. and Rosen, B. Walter, "The Elastic Moduli of Fibre Reinforced Materials," Journalof Applied Mechanics, June 1964

9. Whitney, J. M. and Riley, M. B., "Elastic Properties of Fiber Reinforced CompositeMaterials," AIAA Journal, Sept. 1966

10. Walpole, L. J., "Elastic Behavior of Composite Materials: Theoretical Foundations,"

Advances in Applied Mechanics, Volume 21, Academic Press, 198911. BS 7159: 1989 British Standard Code of Practice for Design and Construction of Glass

Reinforced Plastics GRP Piping Systems for Individual Plants or Sites.

12. UK Offshore Operators Association Specification and Recommended Practice for the Use ofGRP Piping Offshore., 1994

frp analysis in caesar-ii

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