ASME Section VIII div 2 2010. ASME Boiler and Pressure Vessel Code. Alternative Rules

2010 SECTION VIII, DIVISION 2

5-99

5.E.4.4 The elasticity matrix

[]E for the triangular and square patterns that are used in the stress analysis

of the effective solid plate are provided in Tables 5.E.8 and 5.E.9, respectively.

5.E.5 Pressure Effects in Tubesheet Perforations

5.E.5.1 The effect of pressure in the perforations is not directly included in the stress analyses of the

effective solid plate. Pressure in the tubesheet perforations tends to expand the perforated portion of the

plate. This expansion is resisted by the solid rim and by the ligaments themselves. The effect of this

pressure on the local stresses in the ligaments can be included in the stress analysis results by use of the

superposition principle.

5.E.5.2 In order to include the effect of pressure, an additional analysis needs to be performed. In this

analysis, the boundary conditions are shown in Figure 5.E.2 (c). These boundary conditions can be achieved

by superposing the results from the boundary conditions shown in Figures 5.E.2 (a) and 5.E.2 (b),

respectively. The solution for the hydrostatic compression boundary condition case, see Figure 5.E.2 sketch

(a) is

***

rzs

p

θ

σσσ

===− for tubes welded on the tubeside or

***

rzt

p

θ

σσσ

=

==− for tubes welded on the

shellside. The solution for the boundary conditions of Figure 5.E.2 (b) can be obtained from the stress

analyses considering the loading conditions shown.

5.E.5.3 A further modification to the boundary conditions used in paragraph 5.E.2 can be used to obtain

more exact results. The modified boundary conditions are shown in Figure 5.E.3.

5.E.5.4 The stress results obtained using the boundary conditions shown in Figures 5.E.2 and Figure 5.E.3

must be superposed before

m

P ,

Lb

PP+ , and

Lb

PPQ

+

+ stresses are evaluated and before transforming

the cylindrical coordinate stresses to the local x-y-z coordinate system for computing

Lb

PPQF

+

++

stresses.

5.E.6 Protection Against Plastic Collapse

5.E.6.1 Uniform Hole Pattern – The following equations shall be used for a perforated plate with a uniform

hole pattern. The equivalent stresses

m

P ,

Lb

PP

+

, and

Lb

PPQ

+

+ are determined from the in-plane stress

components determined by stress analysis of the equivalent solid plate. The loads to be considered in the

design shall include, but not be limited to, those given in Table 4.1.1. The load combinations that shall be

considered for each loading condition shall include, but not be limited to those given in Table 4.1.2.

a) General Primary Membrane Equivalent Stress

()

m

P – evaluate the through thickness stress distribution

from the numerical analysis of the equivalent plate for mechanical plus pressure loading. The membrane

stress is determined by linearization of the through thickness stress distribution. The maximum stress

intensity,

m

P , shall satisfy the following:

*

m

P

S

μ

≤ (5.E.5)

b) Primary Membrane (General or Local) Plus Primary Bending Equivalent Stress

()

LB

PP+ – determine

the linearized surface stresses from the numerical analysis of the equivalent plate for mechanical plus

pressure loading. The maximum stress intensity,

Lb

PP

+

, shall satisfy the following:

2010 SECTION VIII, DIVISION 2

5-100

()

*

1.5

LbPS

m

PPK

S

μ

+

≤

(5.E.6)

where

0.5

2

23

1.07802 0.342503 1.50452

1 0.0706632 1.15182 0.158343

PS

K

ββ

ββ β

⎡⎤

−+

=

⎢⎥

−++

⎣⎦

(5.E.7)

*

**

11

r

or where

θ

σ

ββ

σσ

=−≤≤ (5.E.8)

5.E.6.2 Primary Membrane (General or Local) Plus Primary Bending Plus Secondary Equivalent Stress

()

LB

PPQ++ – if a fatigue analysis is required, determine the linearized surface stresses from the numerical

analysis of the equivalent plate for mechanical, pressure, and thermal loading. The maximum primary plus

secondary equivalent stress,

LB

PPQ++, shall satisfy the following where

P

S

K is computed using Equation

(5.E.7).

*

()

LB PS

P

S

PPQK

S

μ

++

≤

(5.E.9)

5.E.7 Protection Against Cyclic Loading

5.E.7.1 Stress Components From The Equivalent Plate – The stress components

*

x

σ

,

*

y

σ

,

*

z

σ

,

*

x

y

τ

,

*

x

z

τ

,

and

*

yz

τ

, are determined from the stress analysis of the equivalent solid plate.

a) Triangular Hole Pattern – For material constants evaluated using Option A, B, or C (see paragraph

5.E.4.3), the perforated plate displacements and stresses are minimally affected by the orientation with

respect to the hole pattern. Therefore, the numerical stress results can be used directly in the

evaluation.

b) Square Hole Pattern – The determination of the stress results depends on the material property model.

1) For material constants evaluated using Option A or Option C (see paragraph 5.E.4.3), the

perforated plate displacements and stresses are minimally affected by the orientation with respect

to the hole pattern. Therefore, the numerical stress results can be used directly in the evaluation.

2) For material constants evaluated using Option B, the square pattern displacements do not

dependent strongly on the orientation with respect to the hole pattern; therefore, the axisymmetric

approximation provides sufficiently accurate strain results. However, the stress components have

directional dependence with the stiffness properties. Therefore, the stress components shall be

determined using the equations in Table 5.E.10 where the elasticity matrix is dependent on the

orientation of the hole pattern with respect to the equivalent plate cylindrical coordinate system.

Using the axisymmetric strains in the plate, the non-axisymmetric nominal stresses in the pitch and

diagonal directions for the square pattern perforations can be obtained from the equations below:

() () ()()

** ****

1

2

rr pdr

pa

GG

θ

σ

σεε

=−− − (5.E.10)

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2010 SECTION VIII, DIVISION 2

5-101

() () ()()

** ****

1

2

pdr

pa

GG

θ

θθ

σ

σεε

=+− − (5.E.11)

() () ()()

** ****

1

2

rr pdr

da

GG

θ

σ

σεε

=+− −

(5.E.12)

() () ()()

** ****

1

2

pdr

da

GG

ϑ

θθ

σ

σεε

=−− − (5.E.13)

()

**

zz

a

σσ

= (5.E.14)

()

**

rz rz

a

σσ

=

(5.E.15)

c) For both the triangular and square pattern perforated plates, the hydrostatic compression stress must be

superposed prior to transformation to the x-y-z Cartesian coordinate system (see Figure 5.E.4 or 5.E.5)

for computing the peak stresses.

5.E.7.2 Stress Components For Fatigue Assessment – The stress components for the fatigue assessment

are computed using the following equations.

a) Stress Results In The Perforated Region Of The Plate – the stress multipliers

x

K ,

y

K ,

x

y

K ,

x

z

K , and

yz

K

for triangular and square hole patterns are given in Tables 5.E.11 through 5.E.15 and 5.E.16

through 5.E.18, respectively. The stress orientation associated with the stress multipliers for triangular

and square hole patterns are shown in Figures 5.E.4 and 5.E.5, respectively. The stress results in

paragraph 5.E.7.1 must be transformed to the local x-y-z Cartesian coordinate system before the stress

multipliers can be applied.

()

***

11

*

1

x

xyyxyxy

KKK

θ

σ

σσστ

μ

== + + (5.E.16)

()

**

12

*

1

zxzxzyzyz

KK

θ

σ

τττ

μ

== + (5.E.17)

13

0.0

σ

= (5.E.18)

At the tubeside surface of the plate,

22 t

p

σ

=− (5.E.19)

At the shellside surface of the plate,

22

s

p

σ

=− (5.E.20)

Between the surfaces of the plate through the thickness,

()

***

22 11

*

hzxy

z

E

p

E

σνσ σνσσ

⎛⎞

⎡⎤

=−+ −+

⎜⎟

⎣⎦

⎝⎠

(5.E.21)

and

2010 SECTION VIII, DIVISION 2

5-102

23

0.0

σ

= (5.E.22)

33 h

p

σ

=− (5.E.23)

b) Stress Results At The Rim – the region of the perforated plate outside the diameter

o

D is called the

plate rim (see Figure 5.E.1). The stress components for the fatigue assessment at the outer most hole

shall be computed using the following equations. The stress results in paragraph 5.E.7.1 must be

transformed to the local x-y-z Cartesian coordinate system before the stress multipliers can be applied

11, , , , , ,rim rim rim rim r rim r rim

KK

θθθ

σ

σσσ

== + (5.E.24)

12, , , 12,rim z rim rz rim rim

K

θ

σ

ττ

==

(5.E.25)

13,

0.0

rim

σ

= (5.E.26)

At the tubeside surface of the plate,

22,rim t

p

σ

=−

(5.E.27)

At the shellside surface of the plate,

22,rim s

p

σ

=− (5.E.28)

Between the surfaces of the plate through the thickness,

22, 11, , , ,

()[( )]

rim rim h z rim r rim rim

p

θ

σ

νσ σ νσ σ

=−+−+ (5.E.29)

and

23,

0.0

rim

σ

=

(5.E.30)

33,rim h

p

σ

=− (5.E.31)

The stress multipliers are computed using the information in Tables 5.E.11 through 5.E.18, as applicable,

with the following modifications.

,

*1.0

rim x

K K evaluated at

θ

μ

== (5.E.32)

,

*1.0

rrim y

K K evaluated at

μ

== (5.E.33)

,

*1.0

rz rim xz

K K evaluated at

μ

==

(5.E.34)

c) Thermal Skin Stresses – the temperature gradient through the thickness of a perforated plate can be

closely approximated by a step change in the metal temperature near the surface of the plate.

Significant thermal stresses develop only in the skin layer of the plate at the surface where the

temperature change occurs and the thermal stresses in the remainder of the plate are negligible. The

thermal skin stresses at any location on the surface of the equivalent solid plate are given by Equation

(5.E.35). The thermal skin stresses should be added to the stresses determined from the numerical

analysis

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2010 SECTION VIII, DIVISION 2

5-103

()

**

1

ms

rr skin

ET T

K

α

σσ

ν

⎡⎤

−

==

⎢⎥

−

⎢⎥

⎣⎦

(5.E.35)

with

**2

**2*3

9.43983 421.179 6893.05

1 4991.39 6032.92 1466.19

skin

K

μμ

μ

μμ

−+

=

++ −

(5.E.36)

d) Stress Results In The Solid Rim And Other Locations – the stress components from the solid rim and

other locations that do not contain perforations can be used directly in the fatigue assessment.

5.E.7.3 Stress Range For Fatigue Assessment – The stress range to be used in the fatigue assessment is

determined in accordance with paragraph 5.5.

5.E.8 Nomenclature

A

unit cell area

0

A tube open area

α

coefficient of thermal expansion

β

biaxiality stress factor

*d effective tube hole diameter

t

d nominal outside diameter of tubes

o

D equivalent diameter of outer tube limit circle

E

modulus of elasticity for tubesheet material at tubesheet design temperature

t

E modulus of elasticity for tube material at tube design temperature

*

E effective Young's modulus for the perforated plate for a triangular hole pattern

*

p

E effective Young's modulus for the perforated plate for a square hole pattern – pitch direction

*

d

E effective Young's modulus for the perforated plate for a square hole pattern – diagonal

direction

*

z

E

effective Young's modulus for thickness direction loading for perforated plate

11