Text

Rept on Evaluation of Torsion Shear Stress Components in Design of Pipe Supports
	ML20235N327
Person / Time
Site:	Grand Gulf
Issue date:	02/22/1989
From:	; SYSTEM ENERGY RESOURCES, INC.
To:	;
Shared Package
ML20235N321	List: AECM-89-0043, Forwards Grand Gulf Nuclear Station Rept on Evaluation of Torsion Shear Stress Components in Design of Pipe Supports; ML20235N327;
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_ _ _, - _ _,. _ - - - - - - _ _ _ _ _ - - - - - - -. _ _ - _ - - - - - - -, - - -. - _. -,. - -. -

f ATTACHMENT 2 GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS i

P O 000 26 PNu

j GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Table of Contents

1. DISCUSSION & BACKGROUND:

I 2. M ET H O DO L OG Y:..............................................................................................

2.1 Parametric Study :..............................................................................................7 3. E V A L U A TI O N D E T A I LS:..........................................................

.... l o 3.1 C as e w 1 : C a n t i l e v e r................................................................................................

3.2 Case #2 : Bea m wit h H i n ged E n d s................................................................. - l 0 3.3 Case w 3 : B ea m wit h Fi x ed E n ds............................................................................. 2 6 28

4. CONCLUSIONS:.....................................................................................................................35 5.

REFERENCES:

.............................................................................................................3G l

l e

t GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSloN SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS

1. DISCUSSION & BACKGROUND:

In a recent audit of the ADHRS system on the Grand Gulf Nuclear Station, a question was raised regarding the computation of shear due to torsion on wide flange members of several pipe supports. These supports were part of the additions to the plant for the new system. It was noted by the auditor that the St. Venant shear (or pure shear) term was computed but the" warping shear" term was missing.

It is rare that loads are transferred from member to member as a primary torsional moment.

Pipe supports must meet either a stiffness value (dependent upon pipe size) or a deflection criteria (1/16 inch maximum under all loads). Since the relative stiffness of open sections is very low, in general the support cannot meet this deflection / stiffness criteria if loads are transferred through open sections. For this reason they are not used to transfer large torsional loads. Closed sections are much more efficient in this regard than open sections. When closed sections are used to transfer primary loads by torsion, the analysis considers all appropriate terms when selected for use in pipe support design. Open sections are subject only to small incidental torsional loads. These include the effects of minor eccentricities, such as situations where it is not practical to transfer load from member to member directly through their shear center, or the transfer of pipe thermal friction loads to a plane frame in the out-of-plane direction.

Shear stresses in an open member such as a wide flange, in general consist of two terms, St.

Venant and warping shear. If the flanges are not restrained from warping the shear consists solely of St. Venant shear. This shear can be expressed as:

T x = M t/J Eq.1 m

( Reference 3, Page 405, eq. 8.2.11) where T.x = St. Venant shear m

M=

total torque applied to the section t-thickness of the element l

(flange or web) in the i

member J=

Di'/3 where b = width of element The shear stress pattern due to St. Venant shear is shown in Figure 1. It offers resistance from twisting that can be simulated by the resistance to twisting from the three elements (the two flanges and the web) acting individually. St. Venant is not very efficient in transferring shear and for this reason is not used to transfer other than incidental torsion.

Page I

1 4

GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Warping shear, conversely, develops when resistance to flange warping is provided. It occurs as transverse shear across each of the flanges, with the orientation opposite on the top and bottom flanges to produce a couple as shown in Figure 1. The maximum value of warping t

shear can be expressed (for a wide flange) as:

T,=[Eb h/16]d'$/dz Eq.2 8

8 (Reference 3, Page 406, eq.

8.5.19) where E = Young's Modulus h = height of the section d'$/dz'- third derivative of the twist angle At any section the total shear will consist of a portion of each of these stress components.

The afect cf each component can be thought of as an effective spring. The St. Venant contribution, which is the result of twisting from the individual flange and web elements, is a relatively soli :.7 ing, and therefore transfers thaar only when resistance to warping is not offered. The warping shear, which is the result of resistance to flange warping, is the stiffer of the two springs and therefore attracts the greater portion of the total section torque where resistance to flange warping is offered. This is most vividly shown in Figure 3 which represents the shear stress in a cantilever beam with the torque applied at the free end. The upper bound simplified St. Venant shear shown as a maximum constant value along the length of the member represents the stress in the flange assuming all the torsional loads is transferred as St. Venant shear. The T, value shown is the actual St. Venant shear. Since the warping shear effect is a much more efficient transfer mechanism the stress levels necessary to transfer its portion of the torque is much lower than that required for St. Venant shear. Therefore the total shear stress is lower than required if the total shear where transferred solely by St. Venant shear.

To evaluate these two stress components, it is instructive to rearrange the expressions of Eqs.

I and 2 into similar dimensional terms.

St. Venant Shear:

J -(2b,tj + b,t')/3 The web thickness, t, is generally less than the flange thickness, t. Substituting tras t in the t

eq. I and tg as t, in the above equation of J, an upper bound for the St. Venant shear is achieved. Therefore:

T,- Att,/[(2b,tj + b t')/3]

- 3 Af /(26, + b,)tj Eq. 3 Page h

s.

GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Note that although it is generally conservative to express stress as an upper bound, in this case the effort is to show that the St. Venant shear is significantly larger than the Warping shear, so that a lower bound expression for St. Venant shear is desired. This can be obtained i

by assuming that the b, term is as large as the bf term, so that:

T,-At/bIj Eq.3a f

i Warping Shear:

As an alternate to the expression in Eq. 2 the warping shear expression can similarly be developed from the physical representation shown in Figure 1. The warping shear results in equal but opposite shear stresses on the two flanges. Summing up these shear stresses and introducing the moment arm h, the torque imposed on the section can be closely approxi-mated by:

Af = V h y

or V =At/h f

where h = height of the section

/I t (Reference 3, Page 405, eq. 8.5.18) but 7, = V Qf ff f

upon combining T,=[ Af /h][bjt /8]/[f bj/12][t ]

f f

f or 1

T,=3Af/2b ht, Eq. A f

Comparing Equations 3a & 4 and recognizing that h is significantly larger than 3/2 tr, it is j

obvious that the value for St. Venant shear is always larger than that for warping shear.

In the preceeding discussion it was assumed that the total torsional shear at a given section is completely due either to St. Venant or warping shear. However, in most practical occurances I

j the total torsional moment, M at a given section is the sum of the St. Venant torsion, M1 and I

warping torsion, M2, or M = MI + M2 1

Therefore, combined shear stresses, T due to torsional moment, M is given by:

e l

i l

T, = T + T, i

i j

1 l

Psge }

l I

M

, GRAND GULF NUCLEAR STATION.

REPORT ON '

EVALUATION OF TORSION SIIEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS -

T,= M1lb tj + 3M2/2b ht, f

f T,= T,-i.il /b tj = 3M 2/2b ht,.

Eq. 5 i

f Using traditional approach, T,,, = M /b t j i

7,,, = (M 1 + M2)/b tj f

T,,, = M 1/b tj + M2/b tj f

i Therefore, equivalent warping shear stress based on the traditional formula can be given as follows:

T,- T,,,- M l /b tj = M2/b tj Eq. 6 f

i Based'on equations 5 and 6, the approximate conservatism in the assessment of warping shear stress in the traditional formula can be expressed as follows:

(M2/b tj)/(3M2/2b ht,)= h/(3f /2) f f

f Since M is either greater than or equal to either Mi or M2 and as before h is significantly

. larger than tr, it becomes obvious that using the maximum torsional stress as St. Venant shear

- results in a conservative and upper bound shear stress due to torsional load.

' A similar' expression can be developed for channel' sections. The result is the same i.e., St..

Venant shear is always greater than warping shear where they both transfer a given torsional moment. However, the difference is not quite as severe. Channels and wide flange members are the only two structural shapes used on Grand Gulf pipe supports that are subject to

~

warping shear.

Since the incidental torsion loads are generally small compared to the capacity of the section it has became the practice to conservatively express all the shear as St. Venant shear. This represen 3 a1 upper bound value that always will envelop the sum of the two shear compo-nents for the torsional moments at any given section. This practice is employed primarily to minimize time expenditure for analysis and can be utilized since the supports generally have significant margins (in part due to satisfying deflection / stiffness requirements) and because the incidental torsion represents a small portion of the total support loads.

Page d

GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Tables 1 through 19 and figures 2 through 12 demonstrate in numerical and graphical form that for common boundary conditions and representative member shapes St. Venant shear l

stress always envelopes warping shear.

Te ni i i !i O

._ u el a r

~

W

,a

ev5W

. puAk T085iONAL SHE AR 5 TRESSES WARPING Sr(AR STRCS$l5 ST VEN ANT TOR $ ION Figure 1 1

i-l 1

1 P.. 3 l

1

i:

GRAND GULF NUCLEAR STATION

. REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 2. METHODOLOGY:

Beam torsion at any point along the length of a flange type structure member is resisted by two types of resisting torsions simultaneously. They are pure torsion (MI) and warping torsion (M2). M = Mi + M2.

The actual torsional shear stress at any section is equal to T plus T,. The definitions of pure torsion, warping torsion, T, and T, are provided as follows:

1.

Pure torsion - It is also known as Saint-Venant's torsion. It assumes that a cross-sectional plane remains a plane prior to and after the application of torsion and only element rotation occurs during torsion. Shear stress induced by pure torsion can be expressed as

'. (G and d' are shear modulus and first derivative of rotation with T - G x tr x i

respect to distance on member respectively) [Ref.1) 2.

Warping torsion - It is caused by the out of plane effect that arises when the flanges are laterally displaced during the twisting. Shear stress induced by warping torsion can be expressed as T - -E x S x "'/ tr. (E, S, and @"'are modulus of elasticity, Warping statical moment and third derivative of rotation with respect to distance on member respectively) [Ref.1]

The traditional formula widely used by the industry to calculate the member shear stress due to torsion is M x tr/ J which is independent of member length. The formulas provided above for T and T., on the other hand, are dependent on overall member length and the location of the cross-section under investigation. It is therefore instructive to evaluate the adequacy of the traditional formula for beams made of various shapes, sizes and with different end conditions to demonstrate that the traditional formula provides an upper bound value of shear stress due to torsional moment.

l l

Page $

CRAND Gt1LF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 2.1 Parametric Study :

The following three cases are chosen as representatives of various load and end conditions for this extensive analysis.

1 Case 1. A Cantilever subjected to an unit torsion (1 in-kip) at the free end:

u. e Af hB

]

L Seven W-shapes, one S-shape and three C-shape 9 with lengths ranging from 10 inches to 100 inches with increment of 10 inches are selected for this study.

1 Shear stress components ( ie. Pure torsional shear stress and warping shear stress

) at both ends are calculated using the following formulas based on Ref. [4],

Page 305, case Ib (Note: the denotations used in the calculations are different).

8 A

g ?]

Tg n I

'sp'" throughout the thaneus as e distance 4 *

+

fram cesnen.4 and D 4

,A + 64 A + 64 g,

Te = 'c# ' ** 'h* *='k err-har' Hd o

C3' 4bL I

Wide Ranged beam wuh l

M 8880

l Tw = -( E("'h'*ach*=' 'h* 'hda*= **

Point midway between.4 and 8 e-.A T = tC[as the surface everywhere

.h_,, h t

h L

Fbi

, c. u ('Co.h A.J M Co h As i,,,

C,E Cook pl, Psse]

.s cRAND CULF NUCLEAR STATION d

REPORT ON.

EVALUATION OF TORSION SHEAR STRESS COMPONENTS

.{

IN DESIGN OF PIPE SUPPORTS j

l Where: E = Modulus of Elasticity,29000 kSi, G = Shear Modulus of elasticity,11200 ksi.

J = Torsiona! Constant from Ref. [1].

C, = Warping Constant from Ref. [1].

L = Length of the Cantilever in Inches.

X = Distance Measured from the Free End in Inches.

(X-0 at free end & X = L at fixed end) a = A Constant Equal to (E x C, / G x J)t/s from Ref. (1).

j g = 1/a i

l l

l Page _6 i

s CRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS

)

IN DESIGN OF PIPE SUPPORTS l

1 Case 2. A torsional simply supported beam is subjected to an unit torsion applied at 3 l

distance of 0.lL,0.3L or 0.5L from a beam end (L is the length of the i

beam).

y,;eE A

O C

A-15 rehrr

-*.c C = 0 I, 0.3 or,0.5 L

Three W shapes and one C-shape with two different lengths per shape are selected for this study Charu from Ref. [1](case 3) are utilized to determinate the shear stress compo-nents at loading point and two ends.

Case 3. A beam with torsional fixed ends is subjected to an unit torsion applied at a distance of 0.ll,0.3L and 0.5L from a beam end.

C Lj{

F Msl

A!

U+ ^

t g

tg

,iC i

_CL L

+ EXCF.PT C = 01 C: 0.I, 0.3 cm 0 5 Three W-shapes and one C-shape with two different lengths per shape are selected for this study Charts from Ref [1](case 6) are utilized to determinate the shear stress compo-nents at loading point, two ends and an additional point if high stress exists between loading point and the end.

Page j l

J

g.

GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Ma l'"

3. EVALUATION DETAILS:

0 A

\\

3.1 Case #1 : Cantilever L

L Based on the formulas shown in Case #1 of Section 2.1, eleven tables ( Tables # 1 to #11) and seven figures (figure # 2 thru 8) are generated for various I-shapes and C-shapes which provide the actual torsional shear stress and warping shear stress as well as the maximum torsional shear stress computed by Equation M x tr / J. These table values and graphs clearly indicate that the stresses computed with the traditional formula envelop actual combined shear stressesc i

NOMENCLATURE (for those not previously defined):

l d=

Depth of the member Twe=

Warping shear stress at fixed end, ksi Tre =

Pure torsional shear stress at fixed end, ksi Two-Warping shear stress at free end, ksi Tto =

Pure torsional shear stress at free end, ksi Tmax =

Shear stress computed by Equation M x tr / J br =

Width of the flange t=

Flange thickness f

t, =

Web thickness

)

Page L0 f

\\

GRAND GULF NUCLEAR STATION REPORT ON

- EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 4'

g.l A

c L

CAuT8 LEVE R Table 1 Beaa =tf6X20 NOTE:

FOR NOMENCLATURE.

Cw=

113.000

,d= 6.200, bf= 6.020 SEE SHT.10 -

E=

29000.00, Ga 11200.00, J=

.2400 tf=.3650, Square root of (ECw/GJ) 34.90

=

Length Tte Two Tte+Two Tto Two Two+Tto Tsar 10.000

.000

.117

.060

.112

.173 1.521 20.000

.000

.117

.219

.100

.319 1.521 30.000

.000

.117

.428

.084

.512 1.521 40.000

.000

.117

.642

.068

.710 1.521 50.000

.000

.117

.833

.053

.886 1.521 60.000

.000

.117

.992

.041 1.032 1.521 70.000

.000

.117

.117 1.118

.031 1.149 1.521 80.000

.000

.117

.117 1.216

.023 1.239 1.521 90.000

.000

.117

.117 1.290

.018 1.308 1.521 100.000

.000

.117

.117 1.347

.013 1.360 1.521 l

Figure 2 TORSIONAL SNEAR STRESS (ksi) EVALUATION FOR CANTILEVER BEAM - W6X20 DUE TO 1 inch-kip TORSION FYW

,,, C C

C C

C 1.4 T o+Iwo 1.2 t

1 c.s 0.4 Itc tTwg

/x o.2 x

x x

x o*

10 20 30 60 60 to 79 80 90 100 LINGTH (inches)

Pase !!

i CRAND CULF NUCLEAR STATION REPA RT ON EVALUATION OF TOR $!ON SHEAR STRESS COMPONENTS -

IN DESIGN OF PIPC SUPPORTS

)

i NOTE:

FOR NOMENCLATURE SEE SHT.10 Warping study

)

W 6 x 20 free end 1.6 L

L L

L 1.5 Tw. SIMPLIFIED UPPtk SOUNb ET.VENANT g,4 I.3 t'2 f

i

//

09 0.0 Yo b o

0.7 0.6

\\

-T (7mst anal _turAn snusech i

/s/

0-0.3

- //

1" (WARPING S H E.A t STRr M 0

10.000 20.000 30.000 40.000 50.000 60.000 70.000 00.000 90.000 100.00 length in Figure 3 Page /,2

CRAND OULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS cim hB A

L 4

CAufl LEVE R NOTE:

Beta =W4X13 FOR NOMENCLATURE SEE SHT.10 Cw=

14.000

,d= 4.160, bf= 4.060 E=

29000.00, G=

11200.00, J=

.1500 tf=.3450, Square root of (ECw/GJ) 15.50

=

Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.201

.281

.405

.231

.636 2.300 20.000

.000

.281

.281 1.117

.144 1.260 2.300 30.000

.000

.281

.281 1.640

.079 1.719 2.300 40.000

.000

.281

.281 1.942

.042 1.984 2.300 50.000

.000

.281

.281 2.105

.022 2.127 2.300 60.000

.000

.281

.281 2.191

.012 2.203 2.300 70.000

.000

.281

.281 2.237

.006 2.243 2.300 80.000

.000

.281

.281 2.260

.003 2.264 2.300 90.000

.000

.281

.281 2.273

.002 2.274 2.300 100.000

.000

.281

.281 2.279

.001 2.280 2.300 4

Figure 4 TORSIONAL SHEAR STRESS (ksi) EVALUATION FOR CANTILEVER SEAM - W4X13 DUE TO 1 inch-kip TORSION f

T,,,

C C

C

_C 2

1.5 f

+T t0 g

1 T

+T te we

/

o.s X

X X

X

,t i

to 20 30 60

$1 to 70 so 90 100 LINGTH (inches)

Psp @

i j

CRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS i

l M 4 "'

hB A

L L

CA5JTI LEVE R Table 3 NOTE:

han =W8X21 FOR NOMENCLATURE, SEE SHT.10 cv=

152.000

,d= 8.280, bf= 5.270 E=

29000.00, G=

11200.00, J=

.2800 tf=.4000, square root of (ECv/GJ)

=

37.30 Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.090

.049

.087

.136 1.429 20.000

.000

.090

.181

.078

.260 1.429 30.000

.000

.090

.360

.067

.427 1.429 40.000

.000

.090

.548

.055

.403 1.429 50.000

.000

.090

.721

.044 l

60.000

.000

.090

.570

.035

. 35 1.429

.904 1.429 70.000

.000

.090

.991

.027 1.018 1.429 80.000

.000

.090

.090 1.087

.021 1.108 1.429 90.000

.000

.090

.090 1.163

.016 1.179 1.429 100.000

.000

.090

.090 1.221

.012 1.233 1.429 1

Figure 5 I

TORSIONAL SHEAR STRESS (ksi) EVALUATION FOR l

CANTILEVER BEAM - W8X21 DUE TO 1 inch-kip TORSION Tmax 9.4 C

C C

C 1.2 s

Teo + T.0 t

e.s o.e a*

T

\\

eePh o.2,

X X

X to 2o so to so ao ro ao

,o see LENUTH (inches)

Pege ],f L_____________-.__.._____.--.

\\

.. 's l

CRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS I

go{"K thB A

L L

C AWTI LEVE R Table 4 NOTE:

Beam =S3X5.7 FOR NOMENCLATURE, SEE SHT.10 Cw=

.854

,da 3.000, bf= 2.330 E=

29000.00, G=

11200.00, J=

.0400 tf=.2600, Square root of (ECw/GJ)

=

7.43

~

Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000 1.089 1.089 3.324

.531 3.859 6.500 20.000

.000 1.089 1.089 5.621

.147 5.768 6.500 30.000

.000 1.089 1.089 6.270

.038 6.308 6.500 40.000

.000 1.089 1.089 6.440

.010 6.450 6.500 50.000

.000 1.089 1.089 6.484

.003 6 487 6.500 f

60.000

.000 1.089 1.089 6.496

.001 6.496 6.500 1

70.000

.000 1.089 1.009 6.499

.000 6.499 6.500 l

80.000

.000 1.089 1.089 6.499

.000 6.500 6.500 l

90.000

.000 1.089 1.089 6.500

.000 6.500 6.500

(

100.000

.000 1.089 1.089 6.500

.000 6.500 6.500 1

Figure 6 i

TORSIONAL SHEAR STRESS (ksi) EVAWATION FOR CANTILEVER BEAN - S3X5.7 DUE TO 1 inch-kip TORSION T

7 m,

b C

C C

w w

e T

5 to wo

'd 3

2 T.+T XjX t

l X

X X

X i

0 10 20 30 40 50 60 70 to 90 100 l

LENGTH (inches)

Psge LV 1

L_________________------__--_-_

_ _ - _ _ _. - _ _ _ - _ _ _ _ _ _ _ - _ _ _ _ _ _ = _ - _ _ _ -. _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ - _ _ _ _ _ _ _ _ _ _

's GRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS OOMPONENTS IN DESIGN OF PIPE SUPPORTS Me l #

ghB A

L L

C ANTI LEVE R Table 5 NOTE:

FOR NOMENCLATURE.

Beaa =C4X5.4 SEE SHT.10 cv=

.92

,d= 4.000, bf= 1.584 E=

29000.00, G=

11200.00, J=

.0400, tw=

.1840 tf=.2960, Square root of (ECw/GJ)

=

7.72 Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.936

.936 3.632

.477 4.108 7.400 20.000

.000

.936

.936 6.298

.139 6.437 7.400 30.000

.000

.936

.936 7.097

.038 7.135 7.400 40.000

.000

.936

.936 7.317

.011 7.327 7.400 50.000

.000

.936

.936 7.377

.003 7.380 7.400 60.000

.000

.936

.936 7.394

.001 7.395 7.400 70.000

.000

.934

.936 7.398

.000 7.399 7.400 80.000

.000

.936

.936 7.400

.000 7.400 7.400 90.000

.000

.936

.936 7.400

.000 7.400 7.400 100.000

.000

.936

.936 7.400

.000 7.400 7.400 Figure 7 TORSIONAL SHEAR STRESS (ksi) EVAIDATION FOR CANTII.EVER BEAM - C4X5.4 DUE TO 1 inch-kip TORSION

/

Nx

,h C

U C

C C

7 T

e to wo s

4 3

2 T

+T f

te we

'p X

X X

0 10 20 30 60 to 40 70 to 90

00 LENGTH (inches)

Page lf.o

's GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Me l NK hB A

^

L CAMTl!. EVE R NOTE:

Table 6 FOR NOMENCLATURE.

SEE SHT.10 s

Beaa =W8X31 Cw=

530.000

,d= 8.000, bf= 7.995 Em 29000.00, G=

11200.00, J=

.5400 tf=.4350, Square root of (ECw/GJ)

=

50.50 Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.057

.016

.056

.072

.806 20.000

.000

.057

.060

.053

.112

.806 30.000

.000

.057

.124

.048

.173

.806 40.000

.000

.057

.201

.043

.244

.806 50.000

.000

.057

.281

.037

.318

.806 60.000

.000

.057

.357

.032

.389

.806 70.000

.000

.057

.428

.027

.455

.806 80.000

.000

.057

.490

.022

.513

.806 90.000

.000

.057

.544

.019

.562

.806 100.000

.000

.057

.589

.015

.605

.806 Table 7 Beam =W10X19 Cw=

104.000

,d=10.240, bf= 4.020 Em 29000.00, G=

11200.00, J=

.2300 tf=.3950, Square root of (ECw/GJ) 34.00

=

Length Tte Tve Tte+Two Tto Two Two+Tto Taax 10.000

.000

.096

.071

.092

.162 1.717 20.000

.000

.096

.256

.081

.337 1.717 30.000

.000

.096

.497

.068

.565 1.717 40.000

.000

.096

.741

.054

.795 1.717 50.000

.000

.096

.955

.042

.997 1.717 60.000

.000

.096

.096 1.131

.032 1.163 1.717 70.000

.000

.096

.096 1.270

.024 1.294 1.717 80.000

.000

.096

.096 1.376

.018 1.394 1.717 90.000

.000

.096

. 096 1.457

.013 1.470 1.717 100.000

.000

.096 096 1.517

.010 1.527 1.717 l

l Page 3.7

1 GRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS

g. l"X hB A

l L

CAMTl LEVE R NOTE:

Table 8 FOR NOMENCLATURE.

SEE SHT.10 Beaa =W10X49 cv= 2070.000

,d= 9.980, bf=10.000 E=

29000.00, G=

11200.00, J=

1.3900 62.10 I

tf=.5600, square root of (ECv/C7)

=

IAngth Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.028

.005

.028

.033

.403 20.000

.000

.028

.020

.027

.047

.403 30.000

.000

.028

.043

.025

.068

.403 40.000

.000

.028

.071

.023

.095

.403 50.000

.000

.028

.103

.021

.124

.403 60.000

.000

.028

.135

.019

.154

.403 70.000

.000

.028

.167

.017

.183

.403 80.000

.000

.028

.196

.015

.211

.403 90.000

.000

.028

.224

.013

.236

.403 100.000

.000

.028

.248

.011

.259

.403 Table 9 Beaa =W12X40 Cw= 1440.000

,d=11.940, bf= 8.005 E=

29000.00, G=

11200.00, J=

.9500 tf=.5150, Square root of (ECw/GJ) 62.50

=

Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.032

.007

.031

.038

.542 20.000

.000

.032

.026

.030

.057

.542 30.000

.000

.032

.057

.028

.085

.542 40.000

.000

.032

.094

.026

.121

.542 50.000

.000

.032

.136

.024

.160

.542 60.000

.000

.032

.179

.021

.200

.542 70.000

.000

.032

.221

.019

.240

.542 80.000

.000

.032

.261

.016

.277

.542 90.000

.000

.032

.297

.014

.312

.542 100.000

.000

,032

.01 2

.330

.012

.343

.542 Pase l@

GRAND CULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF P!PE SUPPORTS m.,""

hB l

A L

L C A6JTl LEVE R NOTE:

FOR NOMENCLATURE, Table 10 SEE SHT 10 Beam =C6X8.2 Cw=

4.72

,d= 6.000, bf= 1.920 E=

29000.00, G=

11200.00, J=

.0800, tw=

.2000 tf=.3430, Square root of (ECv/GJ) 12.36

=

Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.446

.446 1.101

.332 1.433 4.287 20.000

.000

.446

.446 2.852

.170 2.822 4.287 30.000

.000

.446

.446 3.536

.078 3.615 4.287 40.000

.000

.446

.446 2.951

.035 3.986 4.287 50.000

.000

.446

.446 4.137

.016 4.153 4.287 60.000

.000

.446

.446 4.221

.007 4.228 4.287 70.000

.000

.446

.445 4.258

.003 4.261 4.287 80.000

.000

.446

.446 4.274

.001 4.276 4.287 90.000

.000

.446

.446 4.282

.001 4.282 4.287 100.000

.000

.446

.446 4.285

.000 4.285 4.287 Table 11 Beam =C8X11.5 Cw=

16.50

,da 8.000, bf= 2.260 E=

29000.00, G=

11200.00, J=

.1300, tw=

.2200 tf=.3900, Square root of (ECw/GJ)

=

18.13 Length Tte Two Tte+Two Tto Two Two+Tto Taax 10.000

.000

.250

.405

.217

.622 3.000 20.000

.000

.250

.250 1.207

.150 1.356 3.000 30.000

.000

.250

.250 1.894

.092 1.986 3.000 40.000

.000

.250

.250 2.347

.054 2,402 3.000 50.000

.000

.250

.250 2.621

.032 2.653 3.000 l

60.000

.000

.250

.250 2.781

.018 2.799 3.000 70.000

.000

.250

.250 2.874

.011 2.884 3.000 I

80.000

.000

.250

.250 2.927

.006 2.933 3.000 90.000

.000

.250 0.25' 2.958

.003 2.962 3.000 I

100.000

.000

.250

.250 2.976

.002 2.978 3.000 Pase l9 i

-c CRAND GULF NUCLEAR STATION A AT N OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS NOTE:

FOR NOMENCLATURE SEE SHT.10 TORSIONAL SHEAR ETRESS (ksi) FOR CANTILEVER BEAMS WITH LENGTH = 100 INCHES AND 1 inch-kip TOR $10N 8

7 6

5 Tmax 4

Tto+T,3, 3

_nB W10x49 W12x40 W8x31 W8x21 W6x20 W10x19 W4x13 c8x11.5 c6x8.2

$3x5.7 c4x5.4 VARIOUS BEAM $12ES Figure 8 i

Page EO

CRAND CULF NUC1. EAR STATION REPORT ON EVALUATION OF TORStON SNEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 3.2 Case #2 : Beam with Hinged Ends Msl'"-

A O

c 7A-

'S 4

0L The following formulas are used:

O p 0.5 T = Pi x (M x tr / J)

T = P2 /(M x Sw / tr x Cw)

Tm, = M x tr / J Where P1 and P2 are the coefficients provided on appropriate charu of Ref. [1] for pure shear, Q' (GJ/M), and warping shear. 0"'(GJa2/M), respectively.

Using the formulas provided above, four tables (Tables 12 to 15) and two figures (Figures 9 &

10) are generated for three I-shapes and one C-shapes which provide the actual torsional shear stress and warping shear stress as well as the maximum torsional shear stress computed by Equation M x tr / J. These table values also indicate that the stresses computed with the traditional formula envelop actual combined shear stresses.

1 Page _2/

g, ORAND GULF NUCLEAR STATIEN REP 2RT CN EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIC. : OF PIPE SUPPORTS Mal""

A D

A f5 k

_ OL L

i SlWPLY %Pp0t3D ggy

? = 0.I, a s at a f NOTE:

Tt = Pure torsion sheer stress Tw = Warplag shear stress Tsar = Maximum tosloc shear stress Table 12 For defloation of other terms.

see sheets 10 & 21 Beam =W4X13 Cw=

14.000

,d= 4.160, bf= 4.060 E=

29000.00, G=

11200.00, J=

.1500 tf=.3450, Square root of (ECw/GJ) 15.50

=

Length C

POINT P1 P3 Tt Tw Tt+Tw Tmax 48.000

.100 A

170

.730

.391

.206

.597 2.070 48.000

.100 B

.130

.770

.299

.217

.516 2.070 48.000

.100 C

.070

.030

.161

.008

.169

.230 48.000

.300 A

.310

.390

.713

.110

.323 1.610 48.000

.300 B

.130

.570

.299

.161

.460 1.610 48.000

.300 C

.200

.110

.460

.031

.491

.690 48.000

.500 A

.300

.200

.690

.056

.746 1.150 48.000

.500 B

.000

.500

.000

.141

.141 1.150 48.000

.500 C

.300

.200

.690

.056

.746 1.150 100.000

.100 A

.380

.530

.874

.150 1.024 2.070 100.000

.100 B

.270

.630

.621

.178

.799 2.070 100.000

.100 C

.100

.000

.230

.000

.230

.230 100.000

.300 A

.580

.130 1.334

.037 1.371 1.610 l

100.000

.300 B

.210

.500

.483

.141

.624 1.610 l

100.000

.300 C

.290

.010

.667

.003

.570

.690 100.000

.500 A

.490

.030 1.127

.008 1.135 1.150 100.000

.500 B

.000

.500

.000

.141

.141 1.150 100.000

.500 C

.490

.030 1.127

.008 1.135 1.150 l

Psee _Zt-l l

I

GULr NUCLEAR STATION nTny^!!! "r%13s s#!^^'""

NOTE:

Tt = Pure torsion shear stress Tw = Warping shear stress Tmax = Maximum tosion shear stress 1

For defination of other terms, j

see sheets 10 & 21 TORSIONAL SMEAR STRESS (ksi) EVALUATION FOR SIMPLE BEAM W4X13 FOR LENGTH = 100 inches AND 1 inch kip TOR $10N l

0.1 L 0.3 L

0. 5 L T + Tu t

i l

A B

C A

B C

A B

C STRE$$ EVALUATION LOCATION Figure 9 Page 9

y I

CRAND GULF NUCLEAR STATION REPORT CN EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 1

Msl'"

A O

C rA-

>S 4

_OL L

SIWP(,Y "yP90tT1D gg

= 0.I, 0,3 at o g NOTE:

Tt = Pure torsion shear stress Tw = Warpin8 shear stress Tsar = Maximum tosion shear stress Table 13 For definatlos of other terms.

see sheets 10 & 21 Beam =W6X20 Cw=

113.000

,d= 6.200, bf= 6.020 E=

29000.00, G=

11200.00, J=

.2400 l

tf=.3650, Square root of (ECw/GJ) 34.90

=

Length C

POINT P1 P3 Tt Tw Tt+Tw Tmax 48.000

.100 A

.050

.840

.076

.098

.174 1.369 48.000

.100 8

.040

.860

.061

.101

.162 1.369 48.000

.100-C

.030

.070

.046

.008

.054

.152 48.000

.300*

A

.100

.590

.152

.069

.221 1.065 48.000

.300 B

.050

.650

.076

.152 1.065 48.000

.300 C

.080

.230

.122

.027

.149

.456 48.000

.500 A

.110

.390

.167

.046

.213

.760 48.000

.500 B

.000

.500

.000

.059

.760 48.000

.500 C

.110

.390

.167

.046

.213

.760 100.000

.100 A

.150

.750

.228

.088

.316 1.369 100.000

.100 B

.120

.780

.183

.091

.274 1.369 100.000

.100 C

.070

.030

.106

.004

.110

.152 100.000

.300 A

.290

.410

.441

.048

.489 1.065 100.000

.300 B

.120

.580

.183

.068

.250 1.065 100.000

.300 C

.190

.110

.289

.013

.302

.456 100.000

.500 A

.280

.220

.426

.026

.452

.760 100.000

.500 B

.000

.500

.000

.059

.760 100.000

.500 C

.280

.220

.426

.026

.452

.760 Paee M

GRAND QULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Msl'"

A O

c A-

'S a

OL L

SlWPLY Suppegno mg C = 0.I, 0.3 a af NOTE:

Tt = Pure torsion shear stress Tw = Warpin8 shear stress Table 14 Tanx = Maximum tosion shear stress For defloation of other terms, see sheets 10 & 21 Beam =W8X21 i

Cv=

152.000

,d= 8.280, bf= 5.270 E=

29000.00, G=

11200.00, J=

.2800

=

37.30 tf=.4000, Square root of (ECw/GJ) l Length C

POINT P1 P3 Tt Tw Tt+Tv Tmax 48.000

.100 A

.050

.850

.071

.077

.149 1.286 48.000

.100 B

.040

.860

.057

.078

.135 1.286 48.000

.100 C

.030

.080

.043

.007

.050

.143 48.000

.300' A

.090

.600

.129

.055

.183 1.000 48.000

.300 B

.040

.660

.057

.060

.117 1.000 48.000

.300 C

.070

.230

.100

.021

.121

.429 48.000

.500 A

.100

.400

.143

.036

.179

.714 48.000

.500 B

.000

.500

.000

.045

.714 48.000

.500 C

.100

.400

.143

.036

.179

.714 100.000

.100 A

.140

.750

.200

.068

.268 1.286 100.000

.100 B

.110

.790

.157

.072

.229 1.286 100.000

.100 C

.060

.040

.086

.004

.089

.143 100.000

.300 A

.260

.420

.371

.038

. 4 ?.0 1.000 100.000

.300 8

.110

.580

.157

.053

.210 1.000 100.000

.300 C

.180

.110

.257

.010

.267

.429 100.000

.500 A

.250

.357

.023

.380

.714 100.000

.500 B

.000

.500

.000

.045

.714 100.000

.500 C

.250

.357

.023

.380

.714 Psee ! 5

i CU F NUCLEAR STATION f!ni^oi o",%i.'U#"ni^^ **" " "" """"

NOTE:

Tt = Pure torsion shear stress Tw = Warping shear stress Tsax = Maximum tosion shear stress For defination of other terms, see sheets 10 & 21 TOR $10NAL SHEAR STRESS (ksi) EVALUATION FOR $!MPLE BEAN C6x8.2 FOR LENGTH = 60 inches AND 1 inch kip TOR $10N 0.1 L 03 L 05L 35 T +Tw 3

A B

C A

B C

A B

C STRESS EVALUATION LOCATION Figure 10 Page [d

'e GRAND CULF NUCLEAR STATION REPORT CN EVALUATION OF TORSION SREAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS Ms l "

A O

C rA-13

,-4n.

OL L

=

SlWPLY GuPPORTED 864W 0 8 0.1, 0.3 of.0.5 NOTE:

Tt = Pure torslos shear stress Tw = Warpla8 shear stress Tanx = Maximum tostos shear stress For defloation of other terms, see s eets 10 & 21 Table 15 Be2a =C6X8.2 Cw=

4.720

,d= 6.000, bf= 1.920 E=

29000.00, G=

11200.00, J=

.0800, Tw=.2000 tf=.3430, Square root of [ECw/GJ) 12.36

=

Length C

POINT P1 P3 Tt Tw Tt+Tw Tmax 24.000

.100 A

.080

.820

.343

.366

.709 3.859 24.000

.100 B

.060

.830

.257

.370

.628 3.859 24.000

.100 C

.040

.060

.171

.027

.198

.429 24.000

.301 A

.160

.540

.686

.241

.927 3.001 24.000

.300 B

.060

.630

.257

.281

.538 3.001 24.000

.300 C

'.110

.190

.472

.085

.556 1.286 24.000

.500 A

.160

.340

.686

.152

.838 2.144 24.t.93

.500 B

.000

.500

.000

.223

.223 2.144 24.000

.500 C

.160

.340

.686

.152

.838 2.144 60.000

.100 A

.280

.620 1.201

.277 1.477 3.859 60.000

.100 B

.210

.690

.900

.308 1.208 3.859 60.000

.100 C

.090

.010

.386

.004

.390

.429 60.000

.300 A

.450

.250 1.929

.112 2.041 3.001 1

60.000

.300 B

.160

.530

.686

.237

.923 3.001 60.000

.300 C

.260

.040 1.115

.018 1.133 1.286 60.000

.500 A

.400

.100 1.715

.045 1.760 2.144 60.000

.500 B

.000

.500

.000

.223

.223 2.144 1

60.000

.500 C

.400

.100 1.715

.045 1.760 2.144 Page J7

l

-c a

GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS C W*

3.3 Case #3 : Beam with Fixed Ends H

wi*

Ad 3i I.

+ $1CTM C s 0.l

'O

,P A

Cs0.1,0.3on05 Using the same formulas provided in Section 3.2, four additional tables ( Tables 16 to 19) and two figures ( Figures 11 & 12 ) are generated. The table values also indicate that the stresses computed with the traditional formula envelop actual combined shear stresses.

l l

l i

l Page $$

l l

CRAND GULF NUCLEAR STATISM REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS

)

C '/ *

., Ms ( "'

g Dn I

h O

IO B

_ C'l L

+ EXCEPT C

0.1 FIXED BEM4 C : 0.I, 0.3 $R 0.5 NOTE:

Table is Tt = Pure torsion shear stress Tw = Warping shear stress Beam =W4x13 Tmax = Maximum tosion shear stress For defination of other terms, see sheets 10 & 21 Cv=

14.000

,da 4.160, bf= 4.060 E=

29000.00, G=

11200.00, Ja

.1500 tf=.3450, square root of (ECw/GJ) 15.50

=

kngth C

POINT P1 P3 Tt Tw Tt+TW Tmax 48.000

.100 A

.000

.970

.000

.274

.274 2.070 48.000

.100 B

.026

.937

.060

.265

.324 2.070 48.000

.100 C

.000

.003

.000

.001

.230 48.000

.300' A

.000

.780

.000

.220

.220 1.610 48.000

.300 3

.c65

.715

.149

.202

.351 1.610 48.000

.300 C

.000

.225

.000

.064

.690 48.000

.330 D

.110

.660

.253

.186

.439 1.610 48.000

.500 A

.000

.500

.000

.141

.141 1.150 48.000

.500 5

.000

.500

.000

.141

.141 1.150 44.000

.500 C

.000

.500

.000

.141

.141 1.150 48.000

.500 D

.120

.380

.276

.107

.383 1.150 100.000

.100 A

.000

.970

.000

.274

.274 2.070 100.000

.100 5

.096

.860

.221

.243

.464 2.070 100.000

.100 C

.000

.040

.000

.011

.230 100.000

.300 A

.000

.760

.000

.215

.215 1.610 100.000

.300 3

.165

.580

.380

.164

.543 1.610 100.000

.300 C

.000

.240

.000

.063

.068

.690 100.000

.300 D

.320

.420

.736

.119

.855 1.610 100.000

.500 A

.000

.500

.000

.141

.141 1.150 100.000

.500 5

.000

.500

.000

.141

.141 1.150 100.000

.500 C

.000

.500

.000

.141

.141 1.150 100.000

.500 D

.320

.190

.736

.054

.790 1.150 Paee [

GRAND GULF NUCLEAR STATION IG OF P S RS C-Qf Msl"'

l A 4 00 f

3 r3

,c C=L

~

1

+ EXCf.PT C

01 DEO E t = Pure torsion shear stress

)

C2 0.i, 0 3 cm 0.5 Tw - Warping shear stress Tmax = Maximum tosion shear stress For defination of other terms, see sheets 10 & 21 TOR $10NAL $NEAR STRESS (ksi) EVALUATION FOR FIXED BEAM - W4X13 FOR LENGTH = 100 inches AND Ifnch kip TOR $10N 0.1 L 0.3 L 0.5 L

,,3 Te + Tw l

1 2

Iffl0X J 1l}

A B

C A

B C

D A

B C

D STRESS EVALUATION LOCATIONS rigure 11 Page_30

1

'\\

}

r GRAND GL%F NUCLEAR 8TATION J

- REPORT ON EVALUATION OF TORJION SHYJJt STRESS COMPONENTS D( DEJIGN OF PIPE SUPPORTS C~Lpz+

b---

g. c c k

O C

B CL

~

L

+ EXCEPT C = 01 FIXED BEAM Cso1,0.:non0.5 T821e 17 NOTE:

Tt = Pure torsion shear stress Tw = Warping shear stress Tmax = Maximum tosion shear stress seam =w4x20 For defination of other terms, see sheets 10 & 21 Cvw 113.000

,de 6.200, bf= 6.020 Z=

29000.00, G=

11200.00, Je

.2400

=

34.90 tf=.3650, Square root of (ICw/GJ) 14ngth C

POINT P1 P3 Tt Tw Tt+Tw Tuax 40.000

.100 A

.000

.970

.000

.114

.114 1.369 48.000

.100 5

.030

.960

.046

.112

.158 1.369 48.000

.100 C

.000

.030

.000

.004

.152 48.000

.300 A

.000

.750

.000

.091

.091 1.065 48.000

.300 B

.017

.770

.026

.090

.116 1.065 48.000

.300 C

.000

.215

.000

.025

.454 48.000

.300 0

.030

.755

.046

.088

.134 1.065 48.000

.500 A

.000

.500

.000

.059

.059 760 48.000

.500 5

.000

.500

.000

.059

.760 48.000

.500 C

.000

.500

.000

.059

.760 48.000

.500 D

.025

.465

.038

.054

.092

.760 100.000

.100 A

.000

.970

.000

.114

.114 1.369 100.000

.100 5

.026

.940

.040

.110

.150 1.369 100.000

.100 C

.000

.030

.000

.004

.152 100.000

.300 A

.000

.780

.000

.091

.091 1.065 100.000

.300 5

.060 720

.091

.084

.176 1.065 100.000

.300 C

.000

.220

.000

.026

.456 100.000

.300 D

.100

.670

.152

.078

.231 1.065 100.000

.500 A

.000

.500

.000

.059

.059 760 100.000

.500 B

.000

.500

.000

.059

.059 760 100.000

.500 C

.000

.500

.006

.059

.059 760 100.000

.500 D

.105

.395

.160

.046

.206

.760 Psee h

-a.

CRAND GULF NUCLEAR STATION I

REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS C '/g+

b

+ Ms;"

l l

Al DA C

l j

"B I

CL L

+ DCEPT C = 0.1 FIXED BEM4 C

0,i, ca ost0.5 Table 13 NOTE:

Tt = Pure torsion shear stress Tw = Warplag shear stress seas =wSx21 Tmax = Maximum tostos shear stress For defination of other terms, ev. 152.000

,d-8.280, br= 5.270 E=

29000.00, G=

11200.00, J.

.2800 see sheets 10 & 21 i

37.30 t.f=.4000, Square root of (ECv/GJ)

=

14ngth C

POINT P1 P3 Tt TV Tt+TV Tsax 48.000

.100 A

.000

.970

.000

.088

.088 1.286 48.000

.100 8

.060

.960

.086

.087

.173 1.286 48.000

.100 C

.000 030

.000

.003

.143 48.000

.300 A

.000

.700

.000

.071

.071 1.000 48.000

.300 3

.017

.770

.024

.070

.094 1.000 48.000

.300 C

.000

.215

-.000

.020

.429 48.000

.300 D

.030

.755

.043

.069

.111 1.000 48.000

.500 A

.000

.500

.000

.045

.714 48.000

.500 5

.000

.500

.000

.045

.714 48.000

.500 C

.000

.500

.000

.045

.714 48.000

.500 0

.040

.465

.057

.042

.099

.714 100.000

.100 A

.000

.970

.000

.088

.088 1.286 100.000

.100

.026

.945

.037

.086

.123 1.286 100.000

.100 C

.000

.030

.000

.003

.143 100.000

.300 A

.000

.780

.000

.071

.071 1.000 100.000

.300 5

.050

.725

.071

.064

.137 1.000 100.000

.300 C

.000

.230

.000

.021

.429 100.000

.300 D

.090

.680

.129

.042

.190 1.000 100.000

.500 A

.000

.500

.000

.045

.714 100.000

.500 5

.000

.500

.000

.045

.714 100.000

.500 C

.000

.500

.000

.045

.714 100.000

.500 D

.095

.405

.136

.037

.173

.714 Page [

'r d'

! ~.

. GRAND GULF NUCLEAR STATION REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 1

C \\f

~ " ", Ms l ""

A4 Dn IC j

"B E

_ C'L,

t

+ EXCEPT C s 01 AXf.D BEAb4 Cs0.I,0.3om0.5 NOTE:

Table 19 Tt = Pure torslos shear stress Tw = Warping shear stress Tsaar = Maximum tosion shear stress neaa =C6X8.2 For deflaation of other terms, see sheets 10 & 21 Cve 4.720

,d= 6.000, hf= 1.920 E=

29000.00, G=

11200.00, J=

.0800, tw=.2000 tf=.3430, Square root of (ECw/GJ}

=

12.36 Length C

POINT P1 P3 Tt Tw Tt+Tw Taax 24.000

.100 A

.000

.970

.000

.433

.433 3.859 24.000

.100 3

.012

.955

.051

.424

.478 3.859 24.000

.100 C

.000

.030

.000

.013

.429 24.000

.300*

A

.000

.750

.000

.348

.344 3.001 24.000

.300 5

.030

.755

.129

.337

.466 3.001 24.000

.300 C

.000

.220

.000

.094

.098 1.286 24.000

.300 D

.050

.730

.214

.326

.540 3.001 24.000

.b00 A

.000

.500

.000

.223

.223 2.144 24.000

.500 5

.000

.500

.000

.223

.223 2.144 24.000

.500 C

.000

.500

.000 223

.223 2.144 24.000

.500 0

.055 440

.236

.196

.432 2.144 40.000

.100 A

.000 970

.000

.433

.433 3.859 60.000

.100 3

.000

.890

.257

.397

.655 3.059 60.000

.100 C

.000

.035.

.000

.016

.429 60.000

.300 A

.000

.770

.000

.344

.344 3.001 60.000

.300 3

.120

.650

.514

.290

.805 3.001

(

60.000

.300 C

.000 220

.000

.098

.098 1.286 60.000

.300 D

.300

.565

.858

.252 1.110 3.001 60.000

.500 A

.000

.500

.000

.223

.223 2.144 40.000

.500 8

.000 500

.000

.223

.223 2.144 60.000

.500 C

.000

.500

.000

.223

.223 2.144 60.000

.500 D

.220 275

.943

.123 1.066 2.144 Pn.23 l

e GU r NUCLEAR STATION BreiV^!J o"r%I,"J#"n?^^ ' ' "" "*""

CLpz+

i b

Ms l"%

A 4 U^

,C g

a3 CL

+ EicEPT C = 01 FIXED EEAM NOTE:

C 0.I, 0.3 on 0.5 Tt = Pure torsion shear stress Tw = Warping shear stress Tmax = Maximum tosion shear stress For defination of other terms, see sheets 10 & 21 l

TOR $10NAL SHEAR STRESS (ksi) EVALUATI FOR FIXE BEAM - C6x8.2 FOR LENGTH e 60 inches 0.1 L 0.3 L 0.5L

'5 1-Te + L I

Tmax

,3 A

B C

A B

C D

A B

C D

STRESS EVALUAfl0N LOCATIONS i

l Figure 12 1

Psee _A$

D'

^ < '

GRAND GULF NUCLEAR STATION '

REPORT ON EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 4. CONCLUSIONS:

Based on evaluation presented in the calculation, it is concluded that the value derived from M x t / J formula for torsional shear stress envelopes the combination of actual torsion shear stress and warping shear stress.

Page _3[

Q b'

GRAND GULF NUCLEAR STATION REPORT ON.

EVALUATION OF TORSION SHEAR STRESS COMPONENTS IN DESIGN OF PIPE SUPPORTS 5.

REFERENCES:

[1] " Torsional Analysis of Steel Members" Published in'1983 by AISC.

[2] " Manual of Steel Construction" of AISC,8th Edition.

[3] Salmon, C.G., and Johnson, J.E.: $1g11 Structures Design and Behavior,2nd Edition, Harper and Row, New York,1980

[4] " Formulas for Stress and Strain" by Roark and Young,5th Edition.

l l

Page 36

t.-

I--

ATTACHMENT 3 EVALUATION OF TORSION SHEAR STRESS COMPONENTS EXAMPLE CALCULATIONS EXAMPLE 1: Reference Calculation No. NPE-Q1E12G011C01, Rev. 1 i

I Member Size:

W8 x 48, Torque = M = 13.2 in-k 1.

Torsional Shear Stress Per AISC (Ref. 1) a.

St. Venant Shear Stress, Tt = G x tf x 9' Use Case 3, Length = 89.56 in, a = 35 in,M=.05 4

L/a = 2.6, G = 10385 ksi. J = 1.96 in, tf = 0.683 in

-4 0.27=9'[GJ) 9' = 0.27 M = 1.75 x 10

\\M /

GJ Tt = 10385 ksi x 0.683 in x (1.75 x 10~4) = 1.24 Esi b.

Warping Shear Stress, Tw = -ESwe'"

(Ref. 1) tf Use Case 3, Length = 89.56 in, a = 35 in,o4= 0.5 4

'L/a = 2.6, G = 10385 ksi, J = 1.96 in Sw = 22.0 in, tf = 0.683 in 0.5 = 9 GJa 9'" = 0.5 M

9"' = 2.65 x 10~

4 Tw = -27000 ksi x 22.0 in x (2.65 x 10 ~7)

= 0.23 ksi 0.683 in l

l

\\

l NRDGREP EVAL TORSION COMPONENT

{

el 2.

Torsional Shear Stress Using Traditional Formula (Ref. 2)

St Venant Shear Stress, Tt = Mtf J

Torque = M = 13.2 in-k, tf = 0.638 in, J = 1.96 in' Tt = 13.2 in-k (0.683 in)/1.96 in' Tt - 4.60 ksi 3.

Comparison

- Summation of Torsional Shear Stress Utilizing AISC Methodology:

1.24 kai

+

0.23 ksi

= 1.47 kai (St. Venant Shear)

(Warping Shear)

- Torsional Shear Stress per Traditional Formula:

4.60 ksi NRDGREP EVAL TORSION COMPONENT 2L

,m, y EVALUATION OF TORSION SHEAR STRESS COMPONENTS

/-

EXAMPLE CALCULATIONS EXAMPLE 2:

Reference Calculation No. NPE-AQ1G41G006C02, Rev. C Member Size: W6 x 20 Torque = M = 5.5 in-k 1.

Torsional Shear Stresses per AISC a)

St. Venant Shear Stress, Tt = G x tf x 9' (Ref. 1)

Use Case 3, Length = 36.2 in, a = 34.9 in, M = 0.5 L/a = 1.04, G = 10385 ksi, J = 0.24 in', tf = 0.367 in

-4 0.07-9'[GJ) 9'=0.07[M 1.5 x 10

=

\\M /

\\ GJ Tt = 10385 ksi x 0.367 in x (1.54 x 10-0) = 0.60 ksi b)

Warping Shear Stress, Tw = -ESw9'"

t f.'

Use Case 3, Length = 36.2 'in, a = 34.9 in, c' = 0.5 0

~

L/a = 1.04, G = 10385 ksi, J = 0.24 in.

tf = 0.367 in Sw = 4.82 in' O.5 - 9'" GJ x a 9'" = 0.5 M (Ref. 1)

M GJa

-7 9'" = 9.06 x 10 4

Tw - 27000 ksi x 4.82 in x (9.06 x 10

= 0.32 kai 0.367 in NRDGREP EVAL TORSION COMPONENT 3

1:

l 2.

Torsional Shear Stress Using Traditional Formula St.'Venant Shear Stress,.Tt = Mtf (Ref. 2)

J Torque = M = 5.5 in-k, tf = 0.367 in, J = 0.24 in' It = 5.5 in-k (0.367 in) 4 0.24 in Tt = 8.41 ksi 3.

Comparison

- Summation of torsional shear stress utilizing AISC Methodology:

0.60 kai

+

0.32 ksi 0.92 kai

=

(St. Venant Shear)

(Warping Shear)

- Torsional Shear Stress per Traditional Formula:

8.41 kai i

NRDGREP EVAL TORSION COMPONENT 4

y..,,

EVALUATION OF TORSION SHEAR STRESS COMPONENTS EXAMPLE CALCULATIONS EXAMPLE 3: Reference Calculation No. NPE-N1P44G004H35, Rev. 1 MEMBER SIZE: W4 x 13, Torque - M = 0.13 in-k 1.

Torsional Shear Stress per AISC a)

St. Venant Shear Stress, Tt = G x tf x O' (Ref. 1)

Use Case 3 Length = 13 in, a = 15.5 in, A = 0.5 4

L/a = 0.84, G = 10385 ksi J = 0.15 in, tf = 0.345 in

-6 0.05=0'[GJ O'=0.05[M

= 4.17 x 10

\\M

\\ GJ Tt = 10385 x 0.345 in (4.17 x 10-6) = 0.01 ksi b)

Warping Shear Stress, Tw - -ESw0'"

(Ref. 1) t Use Case 3, Length = 13 in, a = 15.5, o<.= 0.5 L/a = 0.84, G = 10385 ksi, J = 0.15 in', tf = 0.345 in 4

Sw = 1.36 in 0.5 = 0" GJ x a O'" = 0.5 M

GJa

~

O'" = 1.74 x 10 Tw = -27000 X 1.36 in' (1.74 x 10-) = 0.02 kai 0.345 in l

NRDGREP EVAL TORSION COMPONENT 5

g.,.

o lfic" A

'l 1

2.

Torsional Shear Stress.Using Traditional Formula

]

l St. Venant Shear Stress, Tt = Mtf (Ref. 2)'

.l 4'

j J-Torque'= M = 0.13 in-k, tf = 0.345 in J = 0.15 in 0.13 in-k (0.345 in),

Tt

=

0.15 in

j 4

Tt

= -0.30 ksi I

I

'3.

Comparison-

.q

- Summation of torsional shear stress utilizing AISC Methodology:

O.01 kai

+

0.02 kai

= 0.03 ksi

'(St Venant Shear)

(Warping Shear)

- Torsional Shear Stress per Traditional Formula 0.30 ksi I

References:

1.

AISC " Torsional Analysis to Steel Members", 1983 2.

Steel Structures Design and Behavior by Salmon & Johnson, 1971 NRDGREP EVAL TORSION COMPONENT 6

. _ -. -_ __ d.

.

ML20235N327

Text

REFERENCES:

REFERENCES:

References:

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