ARTICLE A-8000 STRESSES IN PERFORATED FLAT PLATES

ARTICLE A-8000STRESSES IN PERFORATED FLAT PLATES

A-8100 INTRODUCTIONA-8110 SCOPE(a) This Article contains a method of analysis for flat

perforated plates when subjected to directly applied loadsor loadings resulting from structural interaction with adja-cent members. This method applies to perforated plateswhich satisfy the conditions of (1) through (5).

(1) The holes are in an array of equilateral triangles.(2) The holes are circular.(3) There are 19 or more holes.(4) The ligament efficiency is greater than 5%

(η ≥ 0.05).(5) The plate is thicker than twice the hole pitch

(t/P ≥ 2). If only in‐plane loads or thermal skin stressesare considered, this limitation does not apply.(b) Credit may be taken for the stiffening effect of the

tubes in the perforations. The extent to which the tubesstiffen the perforated plate depends on the materials, themanufacturing processes, operating conditions, and de-gree of corrosion. This stiffening effect may be includedin the calculations by including part or all of the tube wallsin the ligament efficiency used to obtain the effective elas-tic constants of the plate. Such stiffening may either in-crease or decrease stresses in the plate itself and in theattached shells.(c) Credit may be taken for the staying action of the

tubes where applicable.

A-8120 NOMENCLATURE18

c = radius of ring load (Figure A-8132.2-1)E = Young’s modulus for plate material

E* = effective Young’s modulus for perforated plate(Figure A-8131-1)

Et = Young’s modulus for tube materialh = nominal width of ligament at the minimum cross

sectionK = stress multiplier for stresses averaged across the

width of the ligament but not through the thick-ness (Figure A-8142-1)

Km = ratio of peak stress in reduced ligament to thepeak stress in normal ligament

Kr = stress multiplier for circumferential stress in theplate rim (Figure A-8142-6)

Ksk in = stress multiplier for thermal skin stress(Figure A-8153-1)

ln = loge

M = radial moment acting at edge of plate, in.-lb/in.(N · mm/mm) of circumference

P = nominal distance between hole center lines,pitch

p1 , p2 = pressures acting on surfaces of the platepi = pressure inside tubesps = pressure on surface where stress is computed,

p1 or p2

Q = radial force acting at edge of plate, lb/in. (N/mm) of circumference

r = designation of radial location in plateR* = the effective radius of the perforated plate

= ro + 1/4 (P − h)ro = radial distance from center of plate to center of

outermost holeS = stress intensity (A-8142)t = thickness of plate exclusive of cladding or corro-

sion allowanceTm = mean temperature averaged through the thick-

ness of the plateTs = temperature of the surface of the platet t = tube wall thicknessW = t o t a l r i n g l o a d a c t i n g o n p l a t e

(Figure A-8132.2-1), lb (N)w = radial displacement of plate edgex = axis of symmetry of hole pattern through the

smaller ligament thickness (Figures A-8142-3through A-8142-5)

y = axis of symmetry of hole pattern, perpendicularto x axis

Y = stress multiplier for peak ligament stresses(Figure A-8142-1)

Δp = differential pressure across the plateα = coefficient of thermal expansion, in./in.-°F (mm/

mm-°C)β = biaxiality ratio (σ r/σθ or σθ/σ r) or (σ1/σ2 or

σ2/σ1), where −1 ≤ β ≤ 1η = ligament efficiency

= h/Pθ = rotation of plate edge, radν = Poisson’s ratio

ν* = effective Poisson’s ratio for perforated plate(Figure A-8131-1)

ρ = radius of holes in the plateσ1 , σ2 = principal stress in the plane of the equivalent so-

lid plate (A-8142.2)σave = larger absolute value of σ r or σθ [A-8142.1(b)]σ r = radial stresses in the equivalent solid plate

2013 SECTION III APPENDICES

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Delete endnote 18, which says "Express metric values in exponential form"

σ r im = nominal circumferential stress in solid rimσ s k i n = thermal skin stress

σθ = tangential stress in the equivalent solid plate= radial stress averaged through the depth of the

equivalent solid plate

A-8130 ANALYSIS OF CIRCULAR PERFORATEDAREA

A-8131 Procedure

(a) The analysis method for perforated plates presentedin this Article utilizes the concept of the equivalent solidplate. In this method, the perforated plate is replaced bya solid plate which is geometrically similar to the perfo-rated plate but has modified values of the elasticconstants.

(b) The elastic modulus E and Poisson’s ratio ν are re-placed by the effective elastic modulus E* and effectivePoisson’s ratio ν* of the perforated plate, and conventionalequations for plates are used to determine the deforma-tions and nominal stresses for the equivalent solid plate.The deformations so computed may be used directly inevaluating interaction effects. The actual values of thestress intensities in the perforated plate are determinedby applying multiplying factors to the nominal stressescomputed for the equivalent solid plate.

(c) The effective elastic constants are functions of the li-g amen t e f f i c i ency η . The va lue s ar e g i ven inFigure A-8131-1 for the range of 0.05 ≤ η ≤ 1.0 in the formof ν* vs. η for a material with ν = 0.3, and E*/E vs. η . Thestress multipliers are given in Figures A-8142-1 throughA-8142-6. The Y factors presented in Figures A-8142-3and A-8142-4 represent the largest values occurringthrough the thickness at the given angular position.

(d) The region of the perforated plate outside the effec-tive radius R* is called the plate rim. This unperforatedportion of the plate may be considered as a separate con-necting member, a ring or cylinder, and the structure maybe analyzed in accordance with the procedures of A-6000.

A-8132 Analysis of Equivalent Solid Plate

In the following subparagraphs, equations are given forthe nominal stresses and edge displacements for theequivalent solid circular plate under various axisymmetricload conditions.

A-8132.1 Edge Loads (see Figure A-8132.1-1).(a) Stresses at any location on the surface of the equiva-

lent solid plate.

ð1Þ

When double signs are used, the upper sign applies tothe top surface as shown in Figure A-8132.1-1.

(b) Edge displacements of midplane at R*:

ð2Þ

A-8132.2 Ring Loads Transverse to the Plane of thePlate (See Figure A-8132.2-1).

(a) Stresses at any radial location r on the surfaces ofthe equivalent solid plate: for r ≤ c ,

ð4Þ

for r > c ,

ð5Þ

ð6Þ

Figure A-8120-1


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Figure A-8131-1


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(b) Edge displacements of midplane at r = R*:

ð7Þ

ð8Þ

A-8132.3 Uniformly Distributed Pressure Loads (SeeFigure A-8132.3-1).

(a) Stresses at any location r on the surfaces of theequivalent solid plate:

ð9Þ

ð10Þ

(b) Edge displacement of midplane at r = R*:

ð11Þ

ð12Þ

A-8132.4 Pressure in Tubes or Perforations (SeeFigure A-8132.4-1).

(a) Stresses at any location in the equivalent solid plate:

ð13Þ

(b) Edge displacements of midplane at r = R*:

ð14Þ

where E*/E and v* should be evaluated for the ligamentefficiency:

ð15Þ

using Figure A-8131-1;

Figure A-8132.1-1

Figure A-8132.2-1

Figure A-8132.3-1

Figure A-8132.4-1


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Uppercase "P"

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A-8140 STRESS INTENSITIES AND STRESSLIMITS FOR PERFORATED PLATES

A-8141 Equations for Stress Intensities

In A-8140 equations are given for the stress intensitiesin a perforated plate using the stresses determined for theequivalent solid plate.

A-8142 Typical Ligaments in a Uniform PatternA-8142.1 Mechanical and Pressure Loads on Circu-

lar Plates.(a) The stress intensity based on stresses averaged

across the minimum ligament width and through thethicknesses of the plate is l imited according toNB‐3221.1 and is computed from the larger of:

ð16Þ

or

ð17Þ

where only the positive root is used. The first term underthe radical reflects the effect of the transverse shear stressdue to the mechanical and pressure loads. It is a maximumin the outermost ligament of the perforated region, but itmay be determined for any radius, larger than c , by substi-tuting r for R* in the expression. For r < c , the W/πtR*term should be omitted. is the stress resulting from ap-plied in‐plane loading averaged through the thickness ofthe equivalent solid plate. It includes the stresses due topressure in the tubes or perforations given in A-8132.4.No bending stresses are included.(b) The stress intensity based on stresses averaged

across the minimum ligament width but not through thethickness of the plate is limited according to NB‐3221.3and is computed from

ð18Þ

where

K = stress multiplier from Figure A-8142-1σave = larger value of σ r or σθ , psi (MPa), caused by me-

chanical loading and structural interaction withadjacent members, computed as the sum of thesurface stresses in the equivalent solid plate,using the applicable equations in A-8130

However, supporting interactions from adjacent mem-bers may only be considered if the primary plus secondarystresses in such members are limited to 1.5Sm . Effects oftemperature are not included.

A-8142.2 Combined Mechanical and Thermal Ef-fects.(a) The range of the stress intensity based on stresses

averaged across the minimum ligament width but notthrough the thickness of the plate is limited according toNB‐3222.2 and is computed from

ð19Þ

where

K = stress multiplier from Figure A-8142-1σ1 = larger absolute value of σ r or σθ , psi (MPa), caused

by mechanical loading or structural interaction withadjacent members, computed as the sum of the sur-face stresses in the equivalent solid plate using theapplicable equations in A-8130 and A-8150

The effects of temperature are included in the consid-eration of the structural interaction with adjacentmembers.

Figure A-8142-1


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(b) The peak stress intensity due to all loadings is lim-ited by cumulative fatigue considerations as described inNB‐3222.4 and is given by

ð20Þ

where

ps = pressure on the surface where the stress is beingcomputed, psi (MPa)

Ymax = stress multiplier given in Figure A-8142-2 as afunction of the biaxiality ratio β = σ2/σ1

σ1 = principal stress being the largest absolute valuein the plane of the equivalent solid plate, psi(MPa)

σ2 = principal stress having the smallest absolute va-lue in the plane of the equivalent solid plate, psi(MPa) (Equivalent solid plate stresses due to var-ious loads shall be superimposed in order to ob-tain σ1 and σ2 before any multipliers are applied,and the s i gn s o f σ 1 and σ 2 shou ld bemaintained.)

The solid curves in Figure A-8142-2 give the maximumstress multipliers for the worst angular orientation of σ1

and σ2 with respect to the axes of symmetry x and y ofthe hole pattern. In some cases, the worst orientationmay not exist anywhere in the plate, and the use of lowerstress multipliers is justified. An important case concerns

the thermal stress produced by a temperature gradientacross the diametral lane in a perforated plate. Such a gra-dient causes a uniaxial stress oriented parallel to the dia-metral lane. If the diametral lane is parallel to the y axis asshown in Figure A-8142-3, the stress multiplier given bythe dashed line in Figure A-8142-2 may be used.

(c) Equation (b)(20) will give the maximum stress in-tensity for any loading system. Equation (b)(20) is notadequate for more complex cyclic histories where the an-gular orientation of the maximum stress intensity variesduring the cycle. In such cases, it is necessary to computethe stress history at each angular orientation ϕ usingeq. (21)

ð21Þ

where

Sϕ = peak stress intensity at the angular orientation ϕY1, Y2 = stress multipliers in Figures A-8142-3 through

A-8142-5 for various orientations of the princi-pal stresses σ1 and σ2 computed for the equiva-lent solid plate

Note that these figures give stress multipliers for parti-cular angular orientation only. The graph for the angularorientation closest to the actual angular orientation shouldbe used. This is sufficiently accurate since the maximumpossible difference between the actual orientation andthe nearest orientation given in Figures A-8142-3 throughA-8142-5 is only 7.5 deg. Examples for the computation ofSϕ are given as follows.

Example 1

The combined stresses in the equivalent solid plate for aperforated plate of 0.05 ligament efficiency were com-puted at a point as:

and σ r is rotated 12 deg, measured from the y axis of thehole pattern. To determine the value of σϕ at 40 deg fromthe y axis, use the following procedure: let σ 1 = σr ,

σ2 = σθ . Since the angular orientation of 12 deg is closestto 15 deg, use Figure A-8142-5 for the stress multipliers.Read at ϕ = 40 deg on Scale A: Y1 = + 1.65; on Scale B:Y2 = −0.70. Then from eq. (21), the peak stress intensityis computed as

Figure A-8142-2


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Figure A-8142-3


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Example 2

For the same plate as above at another point, the direc-tion of σ r coincides with the x axis. Let σ r = σ2 , σθ = σ1 .Read at ϕ = 40 deg from Figure A-8142-3, Y1 = + 2.75 andfrom Figure A-8142-4, Y2 = –1.80. Then

(d) The peak stress intensity at the outermost hole iscomputed from

ð22Þ

where

Kr = a stress multiplier from Figure A-8142-6σ r im = the nominal circumferential stress in the rim, psi

The stresses given by eqs. (b)(20) and (c)(21) and byeq. (22) are limited by cumulative fatigue considerations,as described in NB‐3222.4.

A-8143 Irregular Ligament Patterns or ThinLigaments in a Nominally UniformPattern

For irregular ligament patterns or thin ligaments in anominally uniform pattern, the stresses are determinedas given in the following subparagraphs.

A-8143.1 Average Stress Intensity. The stress inten-sity based upon the ligament stresses averaged across theligament width and through the plate thickness due topressure plus other mechanical loads is limited to 3.0Smin accordance with Table NB‐3217‐1. The appropriate va-lue is computed according to A-8142.1(a), where ha (theactual width of the thin ligament) is used in place of thenominal width h .

A-8143.2 Peak Stress Intensity. The peak stress in-tensity in the thin ligament due to mechanical loadingand structural interaction with adjacent members, includ-ing thermal effects, is limited by cumulative fatigue con-siderations. This peak stress intensity is computed bymultiplying the peak stress intensity for a nominal thick-ness ligament by the Km value given in Figure A-8143.2-1.

(a) The peak stress intensity in nominal ligament is cal-culated as indicated in A-8142.2(b).

A-8150 THERMAL SKIN EFFECTA-8151 General Considerations

In certain cases, the temperature gradient through thethickness of a perforated plate can be closely approxi-mated by a step change in the metal temperature nearthe surface of the plate. In such a case, significant thermalstresses develop only in the skin layer of the plate at thesurface where the temperature change occurs and thethermal stresses in the remainder of the plate arenegligible.

A-8152 Maximum Thermal Skin Stress

The maximum thermal skin stress on the surface of aperforated plate can be computed from the relation:

ð23Þ

where

E, α, ν = modified material propertiesh = ligament widthP = pitch, in. (mm)

Tm = mean temperature of the plateTs = temperature of the plate at the surface under

considerationYmax = stress multiplier from Figure A-8142-2, for

β = +1

A-8153 Peak Stress Intensities When ThermalSkin Stresses Are Included

(a)When thermal skin stresses are to be combined withother stresses to obtain the peak stress intensity,eq. A-8152(23) may not be used. In such a case the ther-mal stresses at any location on the surface of the equiva-lent solid plate are given by:

ð24Þ

where

E, ν = unmodified material properties (since K s k i n in-cludes the consideration for E* and ν*)

Ksk in = stress ratio from Figure A-8153-1Tm = mean temperature of the plateTs = surface temperature of the plate

(b) The equivalent solid plate stresses given byeq. A-8152(23) can then be combined with other solidplate stresses and the method given in A-8142.2(b) canbe used to obtain the peak stress intensity.


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ARTICLE A-8000 STRESSES IN PERFORATED FLAT PLATES