Pressure Vessel Design Manual - PVManagepvmanage.com/wp-content/uploads/2019/01/...VESSEL-DESIGN-MANUAL-7-1.pdf · Procedure 7-4: Stresses in Cylindrical Shells from External Local

7Local Loads

Procedure 7-1 Stresses in Circular Rings 437

Procedure 7-2 Design of Partial Ring Stiffeners 446

Procedure 7-3 Attachment Parameters 448

Procedure 7-4 Stresses in Cylindrical Shells fromExternal Local Loads 449

Procedure 7-5 Stresses in Spherical Shellsfrom External Local Loads 465References 472

435

Stresses caused by external local loads are a majorconcern to designers of pressure vessels The techniques foranalyzing local stresses and the methods of handling theseloadings to keep these stresses within prescribed limits hasbeen the focus of much research Various theories andtechniques have been proposed and investigated byexperimental testing to verify the accuracy of the solutions

Clearly the most significant findings and solutions arethose developed by Professor P P Bijlaard of CornellUniversity in the 1950s These investigations were spon-sored by the Pressure Vessel Research Committee of theWelding Research Council His findings have formed thebasis of Welding Research Council Bulletin 107 aninternationally accepted method for analyzing stresses dueto local loads in cylindrical and spherical shells The ldquoBij-laard Curvesrdquo illustrated in several sections of this chapterprovide a convenient and accurate method of analysis

Other methods are also available for analyzing stressesdue to local loads and several have been included hereinIt should be noted that the methods utilized in WRCBulletin 107 have not been included here in theirentirety The technique has been simplified for ease ofapplication For more rigorous applications the reader isreferred to this excellent source

Since this book applies to thin-walled vessels only thedetail included in WRC Bulletin 107 is not warrantedNo distinction has been made between the inside andoutside surfaces of the vessel at local attachments Forvessels in the thick-wall category these criteria would beinadequate

Other methods that are used for analyzing local loadsare as follows The designer should be familiar with thesemethods and when they should be applied

1 Roark Technical Note 8062 Ring analysis as outlined in Procedure 7-13 Beam on elastic foundation methods where the

elastic foundation is the vessel shell4 Bijlaard analysis as outlined in Procedures 7-4

and 7-55 WRC Bulletin 1076 Finite element analysis

These methods provide results with a varying degree ofaccuracy Obviously some are considered ldquoball parkrdquotechniques while others are extremely accurate The use ofone method over another will be determined by howcritical the loading is and how critical the vessel isObviously it would be uneconomical and impractical toapply finite element analysis on platform support clips It

would however be considered prudent to do so on thevessel lug supports of a high-pressure reactor Finiteelement analysis is beyond the scope of this book

Another basis for determining what method to usedepends on whether the local load is ldquoisolatedrdquo from otherlocal loads and what ldquofixrdquo will be applied for overstressedconditions For many loadings in one plane the ring-typeanalysis has certain advantages This technique takes intoaccount the additive overlapping effects of each load onthe other It also has the ability to superimpose differenttypes of loading on the same ring section It also providesan ideal solution for design of a circumferential ringstiffener to take these loads

If reinforcing pads are used to beef up the shell locallythen the Bijlaard and WRC 107 techniques provide idealsolutions These methods do not take into account closelyspaced loads and their influence on one another Itassumes the local loading is isolated This technique alsoprovides a fast and accurate method of distinguishingbetween membrane and bending stresses for combiningwith other principal stresses

For local loads where a partial ring stiffener is to beused to reduce local stresses the beam on elastic foun-dation method provides an ideal method for sizing thepartial rings or stiffener plates The stresses in the shellmust then be analyzed by another local load procedureShell stresses can be checked by the beam-on-elastic-foundation method for continuous radial loads about theentire circumference of a vessel shell or ring

Procedure 7-3 has been included as a technique forconverting various shapes of attachments to those whichcan more readily be utilized in these design proceduresBoth the shape of an attachment and whether it is of solidor hollow cross section will have a distinct effect on thedistribution of stresses location of maximum stresses andstress concentrations

There are various methods for reducing stresses atlocal loadings As shown in the foregoing paragraphsthese will have some bearing on how the loads areanalyzed or how stiffening rings or reinforcing plates aresized The following methods apply to reducing shellstresses locally

1 Increase the size of the attachment2 Increase the number of attachments3 Change the shape of the attachment to further

distribute stresses4 Add reinforcing pads Reinforcing pads should not be

thinner than 075 times nor thicker than 15 times the

436 Pressure Vessel Design Manual

thickness of the shell to which they are attached Theyshould not exceed 15 times the length of the attach-ment and should be continuously welded Shellstresses must be investigated at the edge of theattachment to the pad as well as at the edge of the pad

5 Increase shell thickness locally or as an entire shellcourse

6 Add partial ring stiffeners7 Add full ring stiffeners

The local stresses as outlined herein do not apply to localstresses due to any condition of internal restraint such asthermal or discontinuity stresses Local stresses as definedby this section are due to external mechanical loads Themechanical loading may be the external loads caused bythe thermal growth of the attached piping but this is nota thermal stress For an outline of external local loads seeldquoCategories of Loadingsrdquo in Chapter 1

Procedure 7-1 Stresses in Circular Rings [1ndash6]

Notation

Rm frac14 mean radius of shell inR1 frac14 distance to centroid of ring-shell inM frac14 internal moment in shell in-lbMc frac14 external circumferential moment in-lbMh frac14 external longitudinal moment (at clip or

attachment only) in-lbML frac14 general longitudinal moment on vessel

in-lbFT frac14 tangential load lb

F1F2 frac14 loads on attachment lbfafb frac14 equivalent radial load on 1-in length of

shell lbf1 frac14 resultant radial load lbPr frac14 radial load lbP frac14 internal pressure psiPe frac14 external pressure psiT frac14 internal tensioncompression force lb

KmKTKr frac14 internal moment coefficientsCmCTCr frac14 internal tensioncompression coefficients

S1-8 frac14 shell stresses psi

Z frac14 section modulus in3

t frac14 shell thickness insx frac14 longitudinal stress psisf frac14 circumferential stress psie frac14 length of shell which acts with attachment

inq frac14 angular distance between loads or from

point of consideration degreesW frac14 total weight of vessel above plane under

consideration lbA frac14 ASME external pressure factorAs frac14 metal cross-sectional area of shell in2

Ar frac14 cross-sectional area of ring in2

B frac14 allowable longitudinal compressionstress psi

E frac14 joint efficiencyE1 frac14 modulus of elasticity psip frac14 allowable circumferential buckling

stress lbinI frac14 moment of inertia in4

S frac14 code allowable stress tension psi

Clockwise ( + )Counterclockwise ( ndash )

Due to localized moment Mc

Mc

θθ

θ θ

FT

PrPr

Due to radial load Pr

Due to tangential force FT

As shown ( + )

Outward ( + ) Inward ( ndash )Opposite shown ( ndash )

Figure 7-1 Moment diagrams for various ring loadings

Local Loads 437

Allowable Stresses

Longitudinal tension lt15SE frac14Longitudinal compression Factor ldquoBrdquo frac14Circumferential compression lt05Fy frac14Circumferential buckling p ndash lbin

p frac14 3E1I

4R3

(Assumes 41 safety factor)Circumferential tension lt15SE frac14Factor ldquoBrdquoDo

tfrac14 frac14 005 min

LDo

frac14 frac14 50 max

Enter Section II Part D Subpart 3 Fig G ASMECode

A frac14 frac14 01 max

Enter applicable material chart in ASME Code Section II

B frac14 psi

For values of A falling to left of material line

B frac14 AE1

2

Table 7-1Moments and forces in shell M or T

Due to Internal Moment M TensionCompression Force T

Circumferential moment Mc M frac14 P(KmMc) T frac14

PethCmMcTHORNRm

Tangential force FT M frac14 P(KTFT)Rm T frac14 P

(CTFT)

Radial load Pr M frac14 P(KrFr)Rm T frac14 P

(CrFr)

Substitute R1 for Rm if a ring is used Values of Km KT Kr Cm CT and Cr are from Tables 7-4 7-5 and 7-6

fa

fa =

fa =

fb =

fb =

f1 = fa

f1 = fa + fb

f1 = fb

f1 =

fa + fb fb

d

d

f1

f1

Mh Mh

e

d

e

Mh = aF2 + bF1

F1 = F cos θ

F2 = F sin θ

Case 1 Case 2 Case 3 Case 4

Continuous rings

ea

b

θ

e

e e

e = 078 e = 078 e = 078

t F

F1F1

F2

F1

F1

6Mh

6Mh Mh

d + e

d + e

(d + e)(d + 2e)

(d + e)(d + 2e)

Rm

Rm t Rm t Rm t

d d

Figure 7-2 Determination of radial load f1 for various shell loadings


Procedure

External localized loads (radial moment or tangential)produce internal bending moments tension andcompression in ring sections The magnitude of thesemoments and forces can be determined by this procedurewhich consists essentially of the following steps

1 Find moment or tension coefficients based onangular distances between applied loads at eachload from Tables 7ndash4 7-5 and 7-6

2 Superimpose the effects of various loadings byadding the product of coefficients times loads aboutany given point

Notes

1 Sign convention It is mandatory that sign conven-tion be strictly followed to determine both themagnitude of the internal forces and tension orcompression at any pointa Coefficients in Tables 7-4 7-5 and 7-6 are for

angular distance q measured between the point

Table 7-2Shell stresses due to various loadings

Stress Due To

Stress

Direction

Without

Stiffener

With

Stiffener

f1 or Pr

As = t times 1 in

1 in

t

Rm

R1

Rm t

f1 or Pr

As

e

ee = 078

Centroid

Internal pressure P sx S1 frac14 PRm

2tS1 frac14 PRm

2t

sf S2 frac14 PRm

tS2 frac14 PRm

t

As

As thorn Ar

Tensioncompression force T sf S3 frac14 T

AsS3 frac14 T

As thorn Ar

(thorn)tension (e)compression (thorn)tension (e)compression

Local bending moment M sf S4 frac14 6M

t2S4 frac14 M

Z

M can be (thorn) or (e) M can be (thorn) or (e)

External pressure Pe sx S5 frac14 PeRm

2tS5 frac14 PeRm

2t

sf S6 frac14 PeRm

tS6 frac14 2PeRme

As thorn Ar

Longitudinal moment ML sx S7 frac14 ML

pR2mt

S7 frac14 ML

pR2mt

Dead load W sx S8 frac14 W

2pRmtS8 frac14 W

2pRmt

Table 7-3Combined stresses

Type Tension Compression

Longitudinal sx sxfrac14 S1 thornS7 S8 sxfrac14 ethTHORNS5 S7 S8

Circumferential sf sffrac14 S2 thornS3 thornS4 sffrac14 ethTHORNS3 S6 S4

Local Loads 439

on the ring under consideration and loads Signsshown are for q measured in the clockwisedirection only

b Signs of coefficients in Tables 7-4 7-5 and 7-6 arefor outward radial loads and clockwise tangentialforces andmoments For loads andmoments in theopposite direction either the sign of the load or thesign of the coefficient must be reversed

2 In Figure 7ndash4 the coefficients have already beencombined for the loadings shown The loads must be

of equal magnitude and equally spaced Signs ofcoefficients Kr and Cr are given for loads in thedirection shown Either the sign of the load or thesign of the coefficient may be reversed for loads inthe opposite direction

3 The maximum moment normally occurs at the pointof the largest load however for unevenly spaced ormixed loadings moments or tension should beinvestigated at each load ie five loads require fiveanalyses

Figure 7-3 Sample ring section with various loadings

Notes (Cont)

(Continued)


Table 7-4Values of coefficients

q

Localized Moment Mc Tangential Force FT

q


Km Cm KT CT Km Cm KT CT

0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773

Reprinted by permission R I Isakower Machine Design Mar 4 1965

Local Loads 441

Table 7-5Values of coefficient Kr due to outward radial load Pr

q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188


Table 7-6Values of coefficient Cr due to radial load Pr

q Cr q Cr q Cr q Cr q Cr q Cr

0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915

2e358 +02555 33e327 +04225 64e296 +03945 95e265 +02144 126e234 e00175 157e203 e019453e357 +02650 34e326 +04250 65e295 +03904 96e264 +02075 127e233 e00250 158e202 e01985

4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380

24e336 +03940 55e305 +04214 86e274 +02775 117e243 +00490 148e212 e01550 179e181 e0238425e335 +03983 56e304 +04180 87e273 +02710 118e242 +00400 149e211 e01605 180 e02387

26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443

Figure 7-4 Values of coefficients Kr and Cr for various loadings


048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048

0 90 180 270 360Angledegrees

Valu

es o

f coe

ffici

ents

θ

K r

KT

Km

Figure 7-5 Graph of internal moment coefficients Km Kr and KT

048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048

0 90 180 270 360Angle degrees

CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-

Figure 7-6 Graph of circumferential tensioncompression coefficients Cm Cr and CT

Local Loads 445

4 This procedure uses strain-energy concepts5 The following is assumed

a Rings are of uniform cross sectionb Material is elastic but is not stressed beyond

elastic limitc Deformation is caused mainly by bendingd All loads are in the same plane

e The ring is not restrained and is supported alongits circumference by a number of equidistantsimple supports (therefore conservative for useon cylinders)

f The ring is of such large radius in comparisonwith its radial thickness that the deflection theoryfor straight beams is applicable

Procedure 7-2 Design of Partial Ring Stiffeners [7]

Notation

ML frac14 longitudinal moment in-lbM frac14 internal bending moment shell in-lbFb frac14 allowable bending stress psifb frac14 bending stress psi

f or fn frac14 concentrated loads on stiffener due to radial ormoment load on clip lb

Fx frac14 function or moment coefficient(see Table 7-7) frac14 ebx (cos bx ndash sin bx)

Ev frac14 modulus of elasticity of vessel shell at designtemperature psi

Es frac14 modulus of elasticity of stiffener at designtemperature psi

e frac14 log base 271I frac14 moment of inertia of stiffener in4

Z frac14 section modulus of stiffener in3

K frac14 ldquospring constantrdquo or ldquofoundation modulusrdquolbin3

x frac14 distance between loads inb frac14 damping factor dimensionlessPr frac14 radial load lb

Formulas

1 Single load Determine concentrated load on eachstiffener depending on whether there is a radial loador moment loading single or double stiffener

f frac14bull Calculate foundation modulus K

K frac14 Evt

R2

bull Assume stiffener size and calculate Z and IProposed size _____

I frac14 bh3

12

Z frac14 bh2

6

bull Calculate damping factor b based on proposedstiffener size

b frac14ffiffiffiffiffiffiffiffiffiK

4EsI4

r

bull Calculate internal bending moment in stiffener M

M frac14 f4b

bull Calculate bending stress fb

fb frac14 MZ

If bending stress exceeds allowable (Fb frac14 06Fy)increase size of stiffener and recalculate I Z b Mand fb

Table 7-7Values Of Function Fx

bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)


2 Multiple loads (see Figure 7-8) Determine concen-trated loads on stiffener(s) Loads must be of equalmagnitude

f frac14 fl frac14 f2 frac14 frac14 fn

bull Calculate foundation modulus K

K frac14 Evt

R2

bull Assume a stiffener size and calculate I and ZProposed size _____

I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r

bull Calculate internal bending moment in stiffener

Step 1 Determine bx for each load (bx is in radians)Step 2 Determine Fx for each load from Table 7-7 or

calculate as follows

Fx frac14 ebx cos bx sin bx

Step 3 Calculate bending moment M

M frac14 f4b

XFx


fb frac14 MZ

Notes

1 This procedure is based on the beam-on-elastic-foundation theory The elastic foundation is thevessel shell and the beam is the partial ring stiffenerThe stiffener must be designed to be stiff enough totransmit the load(s) uniformly over its full length

Figure 7-7 Dimensions forces and loadings for partial ring stiffeners

bx0frac14 0 F1 frac14 1

bx1frac14 __ F2 frac14 __


bxnfrac14 __ Fn frac14 __PFxfrac14 __

Local Loads 447

The flexibility of the vessel shell is taken intoaccount The length of the vessel must be at least 49ffiffiffiffiffiRt

pto qualify for the infinitely long beam theory

2 The case of multiple loads uses the principle ofsuperposition That is the effect of each load may bedetermined independent of the other loads and thetotal effect may be determined by adding the indi-vidual effects

3 This procedure determines the bending stress in thestiffener only The stresses in the vessel shell shouldbe checked by an appropriate local load procedureThese local stresses are secondary bending stressesand should be combined with primary membraneand bending stresses

Procedure 7-3 Attachment Parameters

This procedure is for use in converting the area ofattachments into shapes that can readily be applied in designprocedures Irregular attachments (not round square orrectangular) can be converted into a rectangle which has

bull The same moment of inertiabull The same ratio of length to width of the originalattachment

In addition a rectangular load area may be reduced to anldquoequivalentrdquo square area

Bijlaard recommends for non-rectangular attachmentsthe loaded rectangle can be assumed to be that which hasthe same moment of inertia with respect to the moment axisas the plan of the actual attachment Further it should beassumed that the dimensions of the rectangle in the longi-tudinal and circumferential directions have the same ratioas the two dimensions of the attachment in these directions

Dodge comments on this method in WRC Bulletin 198ldquoAlthough the lsquoequivalent moment of inertia procedurersquo issimple and direct it was not derived by any mathematicalor logical reasoning which would allow the designer torationalize the accuracy of the resultsrdquo

Dodge goes on to recommend an alternative procedurebased on the principle of superposition This methodwould divide irregular attachments into a composite ofone or more rectangular sub-areas

Neither method is entirely satisfactory and each ignoresthe effect of local stiffness provided by the attachmentrsquosshape An empirical method should take into consider-ation the ldquoarea of influencerdquo of the attachment whichwould account for the attenuation length or decay lengthof the stress in question

Studies by Roark would indicate short zones of influ-ence in the longitudinal direction (quick decay) anda much broader area of influence in the circumferentialdirection (slow decay larger attenuation) This would alsoseem to account for the attachment and shell acting asa unit which they of course do

Since no hard and fast rules have yet been determinedit would seem reasonable to apply the factors as outlinedin this procedure for general applications Very large orcritical loads should however be examined in depth

Notes

1 b frac14 tc thorn 2tw thorn 2ts where tw frac14 fillet weld size andts frac14 thickness of shell

2 Clips must be closer thanffiffiffiffiffiRt

pif running

circumferentially or closer than 6 in if runninglongitudinally to be considered as a singleattachment

xn

x2

x1

f2 f3f1 fn

Figure 7-8 Dimensions and loading diagram for beam onelastic foundation analysis

Rectangle to circlero = 0875 C1 C2

23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro

For radial load C = 07

Rectangle to square

2C1

2C2

C

C =

C 2C 123C =

For circumferential moment

For longitudinal moment

Figure 7-9 Attachment parameters for solid attachments


Procedure 7-4 Stresses in Cylindrical Shells from External Local Loads [791011]

Notation

Pr frac14 radial load lbP frac14 internal design pressure psi

ML frac14 external longitudinal moment in-lbMc frac14 external circumferential moment in-lbMT frac14 external torsional moment in-lbMx frac14 internal circumferential moment

in-lbin

Mf frac14 internal longitudinal momentin-lbin

VL frac14 longitudinal shear force lbVc frac14 circumferential shear force lbRm frac14 mean radius of shell inro frac14 outside radius of circular attachment

inr frac14 corner radius of attachment in

KnKb frac14 stress concentration factors

See Note 1 See Note 2

hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l

Figure 7-10 Attachment parameters for nonsolid attachments

C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML

Radial loadndashmembrane stress is compressive forinward radial load and tensile for outward load

Circumferential moment Longitudinal moment

Figure 7-11 Loadings and forces at local attachments in cylindrical shells

Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx

Figure 7-12 Stress indices of local attachments

2CSQ

2C2

2C1

rc

Figure 7-13 Load areas of local attachments Forcircular attachments use C frac14 0875ro

Figure 7-14 Dimensions for clips and attachments

Figure 7-15 Stress concentration factors (Reprinted by permission of the Welding Research Council)


KcKLK1K2 frac14 coefficients to determine b for rectan-gular attachments

Nx frac14 membrane force in shell longitudinallbin

Nf frac14 membrane force in shell circumferen-tial lbin

sT frac14 torsional shear stress psiss frac14 direct shear stress psisx frac14 longitudinal normal stress psisf frac14 circumferential normal stress psiC frac14 one-half width of square attachment in

CcCL frac14 multiplication factors for rectangularattachments

C1 frac14 one-half circumferential width ofa rectangular attachment in

C2 frac14 one-half longitudinal length of a rect-angular attachment in

h frac14 thickness of attachment indn frac14 outside diameter of circular attachment

inte frac14 equivalent thickness of shell and re-

inforcing intp frac14 thickness of reinforcing pad int frac14 shell thickness in

gbb1b2 frac14 ratios based on vessel and attachmentgeometry

Local Loads 451

Geometric Parameters

g frac14 Rm

t

b frac14 CRm

or for circular attachments

0875roRm

For rectangular attachments

b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure

To calculate stresses due to radial load Pr longitudinalmoment ML and circumferential moment Mc on a cylin-drical vessel follow the following steps

Step 1 Calculate geometric parametersa Round attachments

g frac14 Rm

t

b frac14 0875roRm

b Square attachments

g frac14 Rm

t

b frac14 CRm

c Rectangular attachments

g frac14 Rm

t

b values for radial load longitudinal moment andcircumferential moment vary based on ratios ofb1b2 Follow procedures that follow these steps tofind b values

Step 2 Using g and b values from Step 1 enter appli-cable graphs Figures 7-21 through 7-26 to

dimensionless membrane forces and bending momentsin shell

Step 3 Enter values obtained from Figures 7-21 through7-26 into Table 7-11 and compute stresses

Step 4 Enter stresses computed in Table 7-11 for variousload conditions in Table 7-12 Combine stresses inaccordance with sign convention of Table 7-12

Computing b Values for Rectangular Attachments

b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2

b Values for Radial LoadFrom Table 7-8 select values of K1 and K2 and compute

four b values as follows

Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN

ffiffiffiffiffiffiffiffiffiffib1b2

p

Loadarea

2C2

2C1

Figure 7-16 Dimensions of load areas

Table 7-8b Values of radial loads

K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125

Reprinted by permission of the Welding Research Council


Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p

b Values for Longitudinal MomentFrom Table 7-9 select values of CL and KL and

compute values of b as follows

For Nx and Nf b frac14ffiffiffiffiffiffiffiffiffiffib1b

22

3

q

For Mf b frac14 KL

ffiffiffiffiffiffiffiffiffiffib1b

22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx

Table 7-9Coefficients for longitudinal moment ML

b1b2 g CL for Nf CL for Nx KL for Mf KL for Mx

15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ

Figure 7-17 Graph of coefficients CL for values Nf amp NX

from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15

Figure 7-18 Graph of coefficients KL for values Mf amp MX

from Table 7-9

Local Loads 453

b Values for Circumferential MomentFrom Table 7-10 select values of Cc and Kc and


For Nx and Nf b frac14ffiffiffiffiffiffiffiffiffiffib21b2

3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100

Figure 7-20 Graph of coefficients Kc amp CC for valuesNX amp MX from Table 7-10

5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300

Figure 7-19 Graph of coefficients Kc amp CC for valuesNf amp Mf from Table 7-10

Table 7-10Coefficients for circumferential moment Mc

b1b2 g Cc for Nf Cc for Nx Kc for Mf KC for Mx

15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



Stresses caused by external local loads are a majorconcern to designers of pressure vessels The techniques foranalyzing local stresses and the methods of handling theseloadings to keep these stresses within prescribed limits hasbeen the focus of much research Various theories andtechniques have been proposed and investigated byexperimental testing to verify the accuracy of the solutions

Clearly the most significant findings and solutions arethose developed by Professor P P Bijlaard of CornellUniversity in the 1950s These investigations were spon-sored by the Pressure Vessel Research Committee of theWelding Research Council His findings have formed thebasis of Welding Research Council Bulletin 107 aninternationally accepted method for analyzing stresses dueto local loads in cylindrical and spherical shells The ldquoBij-laard Curvesrdquo illustrated in several sections of this chapterprovide a convenient and accurate method of analysis

Other methods are also available for analyzing stressesdue to local loads and several have been included hereinIt should be noted that the methods utilized in WRCBulletin 107 have not been included here in theirentirety The technique has been simplified for ease ofapplication For more rigorous applications the reader isreferred to this excellent source

Since this book applies to thin-walled vessels only thedetail included in WRC Bulletin 107 is not warrantedNo distinction has been made between the inside andoutside surfaces of the vessel at local attachments Forvessels in the thick-wall category these criteria would beinadequate

Other methods that are used for analyzing local loadsare as follows The designer should be familiar with thesemethods and when they should be applied

1 Roark Technical Note 8062 Ring analysis as outlined in Procedure 7-13 Beam on elastic foundation methods where the

elastic foundation is the vessel shell4 Bijlaard analysis as outlined in Procedures 7-4

and 7-55 WRC Bulletin 1076 Finite element analysis

These methods provide results with a varying degree ofaccuracy Obviously some are considered ldquoball parkrdquotechniques while others are extremely accurate The use ofone method over another will be determined by howcritical the loading is and how critical the vessel isObviously it would be uneconomical and impractical toapply finite element analysis on platform support clips It

would however be considered prudent to do so on thevessel lug supports of a high-pressure reactor Finiteelement analysis is beyond the scope of this book

Another basis for determining what method to usedepends on whether the local load is ldquoisolatedrdquo from otherlocal loads and what ldquofixrdquo will be applied for overstressedconditions For many loadings in one plane the ring-typeanalysis has certain advantages This technique takes intoaccount the additive overlapping effects of each load onthe other It also has the ability to superimpose differenttypes of loading on the same ring section It also providesan ideal solution for design of a circumferential ringstiffener to take these loads

If reinforcing pads are used to beef up the shell locallythen the Bijlaard and WRC 107 techniques provide idealsolutions These methods do not take into account closelyspaced loads and their influence on one another Itassumes the local loading is isolated This technique alsoprovides a fast and accurate method of distinguishingbetween membrane and bending stresses for combiningwith other principal stresses

For local loads where a partial ring stiffener is to beused to reduce local stresses the beam on elastic foun-dation method provides an ideal method for sizing thepartial rings or stiffener plates The stresses in the shellmust then be analyzed by another local load procedureShell stresses can be checked by the beam-on-elastic-foundation method for continuous radial loads about theentire circumference of a vessel shell or ring

Procedure 7-3 has been included as a technique forconverting various shapes of attachments to those whichcan more readily be utilized in these design proceduresBoth the shape of an attachment and whether it is of solidor hollow cross section will have a distinct effect on thedistribution of stresses location of maximum stresses andstress concentrations

There are various methods for reducing stresses atlocal loadings As shown in the foregoing paragraphsthese will have some bearing on how the loads areanalyzed or how stiffening rings or reinforcing plates aresized The following methods apply to reducing shellstresses locally

1 Increase the size of the attachment2 Increase the number of attachments3 Change the shape of the attachment to further

distribute stresses4 Add reinforcing pads Reinforcing pads should not be

thinner than 075 times nor thicker than 15 times the







Notation




















Mc

θθ

θ θ

FT

PrPr



As shown ( + )



Local Loads 437

Allowable Stresses


p frac14 3E1I

4R3



LDo





B frac14 psi


B frac14 AE1

2




PethCmMcTHORNRm


(CTFT)


(CrFr)


fa

fa =

fa =

fb =

fb =

f1 = fa

f1 = fa + fb

f1 = fb

f1 =

fa + fb fb

d

d

f1

f1

Mh Mh

e

d

e

Mh = aF2 + bF1

F1 = F cos θ

F2 = F sin θ


Continuous rings

ea

b

θ

e

e e

e = 078 e = 078 e = 078

t F

F1F1

F2

F1

F1

6Mh

6Mh Mh

d + e

d + e

(d + e)(d + 2e)

(d + e)(d + 2e)

Rm

Rm t Rm t Rm t

d d



Procedure




Notes




Stress Due To

Stress

Direction

Without

Stiffener

With

Stiffener

f1 or Pr

As = t times 1 in

1 in

t

Rm

R1

Rm t

f1 or Pr

As

e

ee = 078

Centroid


2tS1 frac14 PRm

2t

sf S2 frac14 PRm

tS2 frac14 PRm

t

As

As thorn Ar


AsS3 frac14 T

As thorn Ar



t2S4 frac14 M

Z



2tS5 frac14 PeRm

2t

sf S6 frac14 PeRm

tS6 frac14 2PeRme

As thorn Ar


pR2mt

S7 frac14 ML

pR2mt


2pRmtS8 frac14 W

2pRmt





Local Loads 439







Notes (Cont)

(Continued)



q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098








Notation




















Mc

θθ

θ θ

FT

PrPr



As shown ( + )



Local Loads 437

Allowable Stresses


p frac14 3E1I

4R3



LDo





B frac14 psi


B frac14 AE1

2




PethCmMcTHORNRm


(CTFT)


(CrFr)


fa

fa =

fa =

fb =

fb =

f1 = fa

f1 = fa + fb

f1 = fb

f1 =

fa + fb fb

d

d

f1

f1

Mh Mh

e

d

e

Mh = aF2 + bF1

F1 = F cos θ

F2 = F sin θ


Continuous rings

ea

b

θ

e

e e

e = 078 e = 078 e = 078

t F

F1F1

F2

F1

F1

6Mh

6Mh Mh

d + e

d + e

(d + e)(d + 2e)

(d + e)(d + 2e)

Rm

Rm t Rm t Rm t

d d



Procedure




Notes




Stress Due To

Stress

Direction

Without

Stiffener

With

Stiffener

f1 or Pr

As = t times 1 in

1 in

t

Rm

R1

Rm t

f1 or Pr

As

e

ee = 078

Centroid


2tS1 frac14 PRm

2t

sf S2 frac14 PRm

tS2 frac14 PRm

t

As

As thorn Ar


AsS3 frac14 T

As thorn Ar



t2S4 frac14 M

Z



2tS5 frac14 PeRm

2t

sf S6 frac14 PeRm

tS6 frac14 2PeRme

As thorn Ar


pR2mt

S7 frac14 ML

pR2mt


2pRmtS8 frac14 W

2pRmt





Local Loads 439







Notes (Cont)

(Continued)



q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



Allowable Stresses


p frac14 3E1I

4R3



LDo





B frac14 psi


B frac14 AE1

2




PethCmMcTHORNRm


(CTFT)


(CrFr)


fa

fa =

fa =

fb =

fb =

f1 = fa

f1 = fa + fb

f1 = fb

f1 =

fa + fb fb

d

d

f1

f1

Mh Mh

e

d

e

Mh = aF2 + bF1

F1 = F cos θ

F2 = F sin θ


Continuous rings

ea

b

θ

e

e e

e = 078 e = 078 e = 078

t F

F1F1

F2

F1

F1

6Mh

6Mh Mh

d + e

d + e

(d + e)(d + 2e)

(d + e)(d + 2e)

Rm

Rm t Rm t Rm t

d d



Procedure




Notes




Stress Due To

Stress

Direction

Without

Stiffener

With

Stiffener

f1 or Pr

As = t times 1 in

1 in

t

Rm

R1

Rm t

f1 or Pr

As

e

ee = 078

Centroid


2tS1 frac14 PRm

2t

sf S2 frac14 PRm

tS2 frac14 PRm

t

As

As thorn Ar


AsS3 frac14 T

As thorn Ar



t2S4 frac14 M

Z



2tS5 frac14 PeRm

2t

sf S6 frac14 PeRm

tS6 frac14 2PeRme

As thorn Ar


pR2mt

S7 frac14 ML

pR2mt


2pRmtS8 frac14 W

2pRmt





Local Loads 439







Notes (Cont)

(Continued)



q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



Procedure




Notes




Stress Due To

Stress

Direction

Without

Stiffener

With

Stiffener

f1 or Pr

As = t times 1 in

1 in

t

Rm

R1

Rm t

f1 or Pr

As

e

ee = 078

Centroid


2tS1 frac14 PRm

2t

sf S2 frac14 PRm

tS2 frac14 PRm

t

As

As thorn Ar


AsS3 frac14 T

As thorn Ar



t2S4 frac14 M

Z



2tS5 frac14 PeRm

2t

sf S6 frac14 PeRm

tS6 frac14 2PeRme

As thorn Ar


pR2mt

S7 frac14 ML

pR2mt


2pRmtS8 frac14 W

2pRmt





Local Loads 439







Notes (Cont)

(Continued)



q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098









Notes (Cont)

(Continued)



q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098




q


q



0 +05 0 0 e05 180 0 0 0 0

5 +04584 e00277 e00190 e04773 185 +00139 +00277 e00069 e00208

10 +04169 e00533 e00343 e04512 190 +00275 +00553 e00137 e0044215 +03759 e00829 e00462 e04221 195 +00407 +00824 e00201 e00608

20 +03356 e01089 e00549 e03904 200 +00533 +01089 e00261 e00794

25 +02960 e01345 e00606 e03566 205 +00651 +01345 e00345 e00966

30 +02575 e01592 e00636 e03210 210 +00758 +01592 e00361 e0112035 +02202 e01826 e00641 e02843 215 +00854 +01826 e00399 e01253

40 +01843 e02046 e00625 e02468 220 +00935 +02046 e00428 e01363

45 +01499 e02251 e00590 e02089 225 +01001 +02251 e00446 e01447

50 +01173 e02438 e00539 e01712 230 +01050 +02438 e00453 e0150255 +00865 e02607 e00475 e01340 235 +01080 +02607 e00449 e01528

60 +00577 e02757 e00401 e00978 240 +01090 +02757 e00433 e01522

65 +00310 e02885 e00319 e00629 245 +01080 +02885 e00405 e0148470 +00064 e02991 e00233 e00297 250 +01047 +02991 e00366 e01413

75 e00158 e03075 e00144 +00014 255 +00991 +03075 e00347 e01308

80 e00357 e03135 e00056 +00301 260 +00913 +03135 e00257 e01170

85 e00532 e03171 +00031 +00563 265 +00810 +03171 e00189 e0099990 e00683 e03183 +00113 +00796 270 +00683 +03183 e00113 e00796

95 e00810 e03171 +00189 +00999 275 +00532 +03171 e00031 e00563

100 e00913 e03135 +00257 +01170 280 +00357 +03135 +00056 e00301

105 e00991 e03075 +00347 +01308 285 +00158 +03075 +00144 e00014110 e01047 e02991 +00366 +01413 290 e00064 +02991 +00233 +00297

115 e01079 e02885 +00405 +01484 295 e00310 +02885 +00319 +00629

120 e01090 e02757 +00433 +01522 300 e00577 +02757 +00401 +00978

125 e01080 e02607 +00449 +01528 305 e00865 +02607 +00475 +01340

130 e01050 e02438 +00453 +01502 310 e01173 +02438 +00539 +01712

135 e01001 e02251 +00446 +01447 315 e01499 +02251 +00590 +02089

140 e00935 e02046 +00428 +01363 320 e01843 +02046 +00625 +02468

145 e00854 e01826 +00399 +01253 325 e02202 +01826 +00641 +02843

150 e00758 e01592 +00361 +01120 330 e02575 +01592 +00636 +03210

155 e00651 e01345 +00345 +00966 335 e02960 +01345 +00606 +03566

160 e00533 e01089 +00261 +00794 340 e03356 +01089 +00549 +03904

165 e00407 e00824 +00201 +00608 345 e03759 +00829 +00462 +04221

170 e00275 e00553 +00137 +00442 350 e04169 +00533 +00343 +04512

175 e00139 e00277 +00069 +00208 355 e04584 +00277 +00190 +04773


Local Loads 441


q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098




q Kr q Kr q Kr q Kr

0e360 e02387 46e314 +00533 92e268 +00883 138e222 e00212

1e359 e02340 47e313 +00567 93e267 +00868 139e221 e00237

2e358 e02217 48e312 +00601 94e266 +00851 140e220 e002683e357 e02132 49e311 +00632 95e265 +00830 141e219 e00284

4e356 e02047 50e310 +00663 96e264 +00817 142e218 e00307

5e355 e01961 51e309 +00692 97e263 +00798 143e217 e003306e354 e01880 52e308 +00720 98e262 +00780 144e216 e00353

7e353 e01798 53e307 +00747 99e261 +00760 145e215 e00382

8e352 e01717 54e306 +00773 100e260 +00736 146e214 e00396

9e351 e01637 55e305 +00796 101e259 +00719 147e213 e0041810e350 e01555 56e304 +00819 102e258 +00698 148e212 e00438

11e349 e01480 57e303 +00841 103e257 +00677 149e211 e00459

12e348 e01402 58e302 +00861 104e256 +00655 150e210 e00486

13e347 e01326 59e301 +00880 105e255 +00627 151e209 e0049814e346 e01251 60e300 +00897 106e254 +00609 152e208 e00517

15e345 e01174 61e299 +00914 107e253 +00586 153e207 e00535

16e344 e01103 62e298 +00940 108e252 +00562 154e206 e0055317e343 e01031 63e297 +00944 109e251 +00538 155e205 e00577

18e342 e00960 64e296 +00957 110e250 +00508 156e204 e00586

19e341 e00890 65e295 +00967 111e249 +00489 157e203 e00602

20e340 e00819 66e294 +00979 112e248 +00464 158e202 e0061721e339 e00754 67e293 +00988 113e247 +00439 159e201 e00633

22e338 e00687 68e292 +00997 114e246 +00431 160e200 e00654

23e337 e00622 69e291 +01004 115e245 +00381 161e199 e00660

24e336 e00558 70e290 +01008 116e244 +00361 162e198 e0067325e335 e00493 71e289 +01014 117e243 +00335 163e197 e00686

26e334 e00433 72e288 +01018 118e242 +00309 164e196 e00697

27e333 e00373 73e287 +01019 119e241 +00283 165e195 e0071528e332 e00314 74e286 +01020 120e240 +00250 166e194 e00719

29e331 e00256 75e285 +01020 121e239 +00230 167e193 e00728

30e330 e00197 76e284 +01020 122e238 +00203 168e192 e00737

31e329 e00144 77e283 +01019 123e237 +00176 169e191 e0074632e328 e00089 78e282 +01017 124e236 +00145 170e190 e00760

33e327 e00037 79e281 +01013 125e235 +00116 171e189 e00764

34e326 +00015 80e280 +01006 126e234 +00090 172e188 e00768

35e325 +00067 81e279 +01003 127e233 +00070 173e187 e0077236e324 +00115 82e278 +00997 128e232 +00044 174e186 e00776

37e323 +00162 83e277 +00989 129e231 +00017 175e185 e00787

38e322 +00209 84e276 +00981 130e230 e00016 176e184 e00789

39e321 +00254 85e275 +00968 131e229 e00035 177e183 e0079140e320 +00299 86e274 +00961 132e228 e00061 178e182 e00793

41e319 +00340 87e273 +00950 133e227 e00087 179e181 e00795

42e318 +00381 88e272 +00938 134e226 e00113 180 e0079643e317 +00421 89e271 +00926 135e225 e00145

44e316 +00460 90e270 +00909 136e224 e00163

45e315 +00497 91e269 +00898 137e223 e00188




0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098





0e360 +02387 31e329 +04175 62e298 +04010 93e267 +02280 124e236 e00040 155e205 e01870

1e359 +02460 32e328 +04200 63e297 +03975 94e266 +02225 125e235 e00018 156e204 e01915


4e356 +02775 35e325 +04266 66e294 +03875 97e263 +02000 128e232 e00325 159e201 e02025


7e353 +02960 38e322 +04315 69e291 +03740 100e260 +01774 131e229 e00550 162e198 e02110

8e352 +03040 39e321 +04325 70e290 +03688 101e259 +01700 132e228 e00620 163e197 e02140


11e349 +03240 42e318 +04332 73e287 +03540 104e256 +01480 135e225 e00804 166e194 e02220

12e348 +03310 43e317 +04335 74e286 +03490 105e255 +01394 136e224 e00870 167e193 e02240


15e345 +03492 46e314 +04332 77e283 +03325 108e252 +01150 139e221 e01050 170e190 e02303


18e342 +03655 49e311 +04308 80e280 +03150 111e249 +00925 142e218 e01230 173e187 e02345

19e341 +03720 50e310 +04301 81e279 +03090 112e248 +00840 143e217 e01280 174e186 e02351


22e338 +03855 53e307 +04248 84e276 +02900 115e245 +00627 146e214 e01450 177e183 e02375

23e337 +03900 54e306 +04231 85e275 +02837 116e244 +00550 147e213 e01500 178e182 e02380


26e334 +04025 57e303 +04160 88e272 +02650 119e241 +00335 150e210 e01651

27e333 +04060 58e302 +04130 89e271 +02560 120e240 +00250 151e209 e0169028e332 +04100 59e301 +04100 90e270 +02500 121e239 +00175 152e208 e01745

29e331 +04125 60e300 +04080 91e269 +02430 122e238 +00105 153e207 e01780

30e330 +04151 61e299 +04040 92e268 +02360 123e237 +00025 154e206 e01825

Local Loads 443



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098





048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



048044

040036032028024

020016012008004

0-004-008-012

-016

-020-024-028-032

-038-040-044-048


Valu

es o

f coe

ffici

ents

θ

K r

KT

Km


048044

040036032028024

020016012008004

0

-004-008-012

-016

-020-024-028-032

-038-040-044

048


CT

Cm

Cr

Valu

es o

f coe

ffici

ents

θ

-


Local Loads 445







Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098









Notation










Formulas



K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r


M frac14 f4b


fb frac14 MZ



bx Fx bx Fx

0 10 055 01903

005 09025 06 01431

01 08100 065 00997

015 07224 07 00599

02 06398 075 00237

025 05619 08 (e)00093

03 04888 085 (e)00390

035 04203 09 (e)00657

04 03564 095 (e)00896045 02968 10 (e)01108

05 02415

Notes (Cont)





K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098






K frac14 Evt

R2


I frac14 bh3

12

Z frac14 bh2

6



4EsI4

r






M frac14 f4b

XFx


fb frac14 MZ

Notes







Local Loads 447















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098

















Notes



pif running


xn

x2

x1

f2 f3f1 fn



23 C 41 C234

C 1C 223

2C2

2C1

ro

C = 0875ro

Circle to square

C

ro


Rectangle to square

2C1

2C2

C

C =

C 2C 123C =






Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098




Notation



in-lbin






hty

pica

l tc

bb b b

C1 05b

04h

03b

04h

025b

04h

03b

04hC2

C1 04b

05h

05b

05h

03b

05h

02b

04hC2

bbbb

hty

pica

l


C

t

Pr

NMM

N

R m

Mc

N

MM

NN

MM

N

ML




Local Loads 449

90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



90deg

180deg

270deg

0deg

0deg

90deg

180deg

0deg90deg

180deg

270deg

270deg

φσ

σx


2CSQ

2C2

2C1

rc
















Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098














Local Loads 451


g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098




g frac14 Rm

t

b frac14 CRm


0875roRm


b1 frac14 C1

Rm

b2 frac14 C2

Rm

Procedure



g frac14 Rm

t

b frac14 0875roRm


g frac14 Rm

t

b frac14 CRm


g frac14 Rm

t







b1 frac14 C1

Rm

b2 frac14 C2

Rm

b1

b2



Ifb1

b2 1 then b

frac141 1

3

b1

b2 1

eth1 K1THORN


p

Loadarea

2C2

2C1



K1 K2 b

Nf 091 148

Nx 168 12

Mf 176 088

Mx 12 125



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



Ifb1

b2lt 1 then b

frac141 4

3

1 b1

b2

eth1 K2THORN


p




22

3

q

For Mf b frac14 KL


22

3

q

For Mx b frac14 KL


22

3

q

CL KL b

Nf

Nx

Mf

Mx



15 075 043 180 124

50 077 033 165 116

025 100 080 024 159 111

200 085 010 158 111

300 090 007 156 111

15 090 076 108 104

50 093 073 107 103

05 100 097 068 106 102

200 099 064 105 102

300 110 060 105 102

15 089 100 101 108

50 089 096 100 107

1 100 089 092 098 105

200 089 099 095 101

300 095 105 092 096

15 087 130 094 112

50 084 123 092 110

2 100 081 115 089 107

200 080 133 084 099

300 080 150 079 091

15 068 120 090 124

50 061 113 086 119

4 100 051 103 081 112

200 050 118 073 098

300 050 133 064 083


251 2 3 5 6 7 8 9 10 11 12 13 14 15 164

1 2 3 5 6 7 8 9 10 11 12 13 14 15 164

5

75

10

15

β 1β 2

20

25

3

35

40

200300100

100

50

50

200300

300200 100 50 15

15

50100

200

300

15

15

CL for NX

CL for Nφ


from Table 7-9

25

6 7 8 9 10 11 12 13 14 15 16 17

5

75

10

15

β 1β 2

20

25

3

35

40

KL for MX

KL for MX

KL for Mφ

KL for Mφ

300

300

300

200

100

50

15

200

100

100200300

100200

50

50 50

15 15

15


from Table 7-9

Local Loads 453




3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098






3

q

For Mf b frac14 Kc


3

q

For Mx b frac14 Kc


3

q

Cc Kc b

Nf

Nx

Mf

Mx

5375

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for MX

CC for NX

β1β

2

300

300

300

200

200

200

100

100

50

50

50

15

1515

100


5

375

25

125

625

75

875

1

1125

125

1375

1625

175

1875

15

5

10 15 20 25 30 35 40

KC for Mφ

CC for Nφ

β1β

2

300

200

100

50

15

15 50

100

200

300




15 031 049 131 184

50 021 046 124 162

100 015 044 116 145

025 200 012 045 109 131

300 009 046 102 117

15 064 075 109 136

50 057 075 108 131

05 100 051 076 104 126

200 045 076 102 120

300 039 077 099 113

15 117 108 115 117

50 109 103 112 114

1 100 097 094 107 110

200 091 091 104 106

300 085 089 099 102

15 170 130 120 097

50 159 123 116 096

2 100 143 112 110 095

200 137 106 105 093

300 130 100 100 090

15 175 131 147 108

50 164 111 143 107

4 100 149 081 138 106

200 142 078 133 102

300 136 074 127 098



Pressure Vessel Design Manual - PVManagepvmanage.com/wp-content/uploads/2019/01/...VESSEL-DESIGN-MANUAL-7-1.pdf · Procedure 7-4: Stresses in Cylindrical Shells from External Local

Documents