5.0 Support for vertical vesselst

(A) Tall Cylindrical Process Columns\

t

• Supported on cylindrical or conical shells (ski s)• The support skirts are directly welded to the v ssel bottoms head or shell• The skirt base is stiffened by a continuous sti ening ring, which consists of top and

bottom annulus plates with intermediate verti al stiffeners, to reduce localizedbending stresses.

• They are designed as cantilever beams

(B) Small and medium sized vertical v ssels

• Supported on legs or lugs (brackets)• Provision of good access to the bottom dished end and any nozzles located there• Minimum thermal stresses arising from shell- upport temperature gradient

Minimum diameter = 6"Maximum HID = 5"Maximum LID = 2"

Number of legs:N = 3 for D < 3' 6"N = 4 (or more) for D > 3' 6"

Maximum operating temperature = 6500p

,,'D~--7 ,,,

H

1

• Supported on uniformly spaced leg supports• Four legs are usually used• Legs are normally fabricated from equal leg gle and T section shapes. They are

welded to the cylindrical shell wall, often usi g a reinforcing pad.• Some manufacturers prefer to use supports m de from pipe that is then welded to

the dished end.

5.1 Leg Supports

IIII'-'-'-'-r-'-'-'-'III

IIII.- ._.- .-r·- .-. - .- .III

IIII

'-'-'-'-i'-'-'-'-'III

Two possible ways of welding the angled beam and I-beam to the vessel. The choice is between"easy to weld" and "offering more flexural rigidity". Besides cold-formed beams, sometimes a roundpipe may be used as leg column, which has equal strength in all direction and has a high bendingrigidity.

2

Loads on the vessel

1. The wind load (Pw) is horizontal and acts at t e centroid of the projected exposedsurface

2. The earthquake load (Pe) acts horizontally on he center of gravity of the vessel3. Piping or other equipment loads are not consi ered

Stress Analysis (to determine design di ensions)

• Support-leg columns• Base plate• Leg-to-shell weld size• Leg-to-plate weld size• Stresses in the vessel shell at supports• Size of anchor bolts

Support-Leg Columns

In the case of 4-leg support

Over-turning moment (MD) at the base is about the diametral axis A-A

Vertical reaction (due to dead load) = WIN

MD R

~ 1Vertical reaction (R) due to turning moment = Mg/D,

(Db = Base diameter)

In the case of 3-leg support

Db Db' 3MD =R(-+-sm30)=-RDb224

=>

General expression:

J~( ~...::..b __ -7)1

R '"",,---'-," ," , 1200

.--.-~--/i

III

I

3

Maximum load on the leeward side (compreSSioj side) is:

Co = Wo + 4Mb (Operating condition)N NDb

C =WTo N (Test condition, no win loading)

Maximum axial load on the windward side (upli

TWo 4Mb=--+--o N ND b

(operating condi ion)

T =_We + 4Mbo N NDb

(empty)

The eccentric loads P1 and P2 at the column top are:

11 = ~ + ~ (operating condition)

11 = WT (test condition)N

D2 -__ Wo + 4M a ( . dition)1- operatmg con rtionN ND

P2 = - We + 4M a (empty)N ND

Lateral load (F) per column

It is derived based on equal deflection at the top edge of the leg support .

=>

For 4-column leg support, IIi = 2Ix + 2Iy

•......····•···•·

Column Stress (- designed in accordance with any structural codes)

4

Base plate

C . CompresslOn stress = -ab

B d· 11(d/2)·(a/2) 11(d/2)

en mg stress = -. = 2ba~/12 ba 16

Compressive stress must be always greater than e bending stress.

d

b

1< d 1<)1

Weld Size

Shear stress = PI 1 (2L1 + h)

Bending moment on the weld joint= C (d/2) + F (L/4)

Size of Anchor bolt

S ·Ab = (4Mb - WJa NDb N

Anchor bolts are designed to resist the uplift force.

a )1

hLIr:::-lLII ge;~etry I

If W > 4Mb, no uplift exists and the minimum bolt size % to 1 inch.Db

5

5.2 Bracket Supports (or Support Lug)

For vessels with small to medium diameters « 1 ft.) and height-to-diameter ratio 2.5

When the bracket or leg support is attached to th cylindrical shell, a longitudinalmoment arises. In each case the vessel wall is su ject to the extemallongitudinalmoment of 'Fd' where 'F' is the maximum react"on at the support and 'd' is thedistance from the centerline of the support to the shell outside surface.

The stresses in the shell induced by theBracket can be found by the local load method. Vessel

Wall ~

I

(d )

Forces and stresses on the bracket

: Top bar I~------~I--------~~ ta

TR

h GussetVplate ------

BaseplatedF F/2 ( )

( b )

a( )

F . d =R . d sin a2

R= F2sina

6

Maximum compressive stress (Sg) on the gusset (regarded as an eccentrically loadedplated) I

II

i

II

Iwhere the force eccentricity, e = (d - b)sina !

2

s = R + 6Reg (bsina).tg £t.tg

Force and stress on the Top Bar

The top bar is assumed to be a simply-supported earn with uniformly distributed loadFd/ha

Maximum moment occurs at the mid-span

M =(Fd).~=Fdamax ha 8 8h

d" 6MBen mg stress = 2fa . C

(2":::;c:::;8ta)

7

6.0 Saddle Supported Cylindrical Vess Is

The code design of saddle-supported horizontalL.P. Zick (1951, The Welding Journal Researchbeam and ring analysis so that the mathematicalresults he had available.

ylindrical vessel follows the work ofupplement) who used a modifiedodel agrees with the experimental

Most recent work has indicated that Zick's appr ch gives reasonable agreement onlywhen a flexible saddle support is employed. Wh n the saddle is rigid the simple Zick'sanalysis significantly underestimate the peak str s in the vessel by a factor as much as50%.

6.1 General considerations

(a) Saddle supports should be located to cause minimum stress in shell and withoutadditional reinforcement

(b) Most vessels are supported on two saddle supports. The saddles have anembracing angle between 120 and 150 degrees. Any relative settlement of thesupports does not change the support reactions, therefore, the stresses in theshell remain quite the same.

(c) In the case of very long vessel that rested on more than two supports, thesupport reactions are calculated based on continuous beam theory and increasedby 20 to 50% as a safety factor for relative support settlements.

(d) The support reactions are highly concentrated and they induce highly localizedstresses at the support regions. To reduce highly localized stresses, the saddlemust be designed to provide flexibility at the support-shell junctions. Extendedplates or wear plates may be used to provide a gradual transition of structuralrigidity between the support and the vessel's shell.

(e) One of the saddles should be designed at the base to provide free horizontalmovement, thereby avoiding restraint due to thermal expansion.

8

The Mathematical Model

4 HI ~II.: I -.I RorrII+-- I ~-j.-.-.-.-.-.-.-.-.-.-.-.-.-.-.~.-.-.-.-. _._._._._._.-.-._'I

- I -I,I

,,I

< A)

A< )

~ L ,<, -'

I

WR2/4 2Hw/3

~ ~/8

2Hw/3 )WR2/43H/8 ---. ~

w

Q QA A)1 1<< )L

><

Shear force diagram

Q

Bending moment diagram

9

(1) The support reaction is Q - the total weigh is 2Q(2) The dished head is replaced by equivalent ylindrical segment of length 2HJ3.

The weight of dished end is therefore = 2 w/3, acting at a distance 3HJ8 fromend of parallel

(3) The total length is = L + 4HJ3(4) The uniform load has a intensity w = 2Q/( +4H/3)(5) The hydrostatic pressure that acts normal t the dished end creates a couple

given by WR2/4

Points to note:

6.2 Longitudinal bending stresses in the she I

(a) The bending moment at the mid-span '

2 (3H L) wR2 wL L LM} =--Hw -+- +---(-)+Q(--A)3 82 4 24 2

= QL [1+ 2(R2 - H2)/ L2 _ 4A]4 1+(4HI3L) L

=K{~L)Bending stress at the mid-span:

The above expression assumes that the full vessel section is available in resistingbending stresses and the cross section remains circular. For very thin vessels it isfound that the cross section does not remain circular especially so during filling withliquid. Nevertheless, the expression gives satisfactory design dimensions for vesselswith D/t ratio up to 1250.

(b) The bending moment at the saddle-support

2 (3H ) wR2

wA2

M2 =:": S+A +-4---2-

= QA[I- 1-AI L + (R2

- H2)/(2AL)]

1+ (4H /3L)

10

The top portion of shell above the saddle support would feform under load and is deemed ineffectivein resisting longitudinal moment. So the moment of inerti at this cross section is reduced to that of aring with its top portion removed.

The effective arc is assumed to be: 2i1 = 2(8/2 + ~1 )

I

yJ:---<J0o

:: t,NA

.........................- ~+~C_~ Effectiveportion

The position of the neutral axis, N.A. and the second moment of area I about this axis can be found.

_ rsini1y=

~ (sini1 JCl = r ~ - cos A ,

[

. 2 A]3. sm D.I = r t i1 + sm i1cos i1 - 2 ~

Longitudinal bending stresses at the highest and lowest point of the effective cross section are:

(Highest point - tension)

. M)SI =--- ·C)

I -(Lowest point - compression)

Allowable stress limits

The tensile stress combined with the pressure stress (pr/2t) should not exceed theallowable tensile stress for the shell material.

11

6.3 Shear stresses in the plane of the saddle[

The distribution and magnitude of the shear stresses in the shell in the plane of thesaddle depend a great deal on how the shell is reinforced.

!!

The inner shear force, V=Q-w(A+ 2 H)= ~(L-2A)3 f + 4H 13

i

The outer shear force, v = w(A + 2 H) = 2Q( i + 2H 13)3 L 4H /3

Note: the inner shear force is greater then outer shear force when

Q(L - 2A) 2Q(A + 2H 13)=-:'---~ < orL+4H13 L+4H13

L> 4A +4H13

(A) Shell is stiffened by a ring at support region

If the shell is made rigid, the whole section is effective in resisting load-induced shearstresses.

The shear flow (shear force per unit arc length) is:

(~ is measured from the top of the cross section)

v . d,qo = -SIllynr

The maximum stress flow is when ~ =90 degree.

=> Max. shear stress = qo = Q [ L - 2A ]t nrt L + 4H 13

(B) Shell not stiffened by ring

When the shell is free to deform above the saddle, it is considered that the shear stressacts on a reduced cross section. The upper portion of the shell is consideredineffective.

The effective portion is assumed to be:

12

19 ( ())2~ = 2(e / 2 + jJ /20) = - J( --

20 2

iI

As a result, the shears in the effective portion will be increased by a factor:

Factor C = qo(unstiffened shell) = r sin2

¢d¢ = J(

, qo(stiffenedshell) fsin2¢d¢ J(-a+sinacosa

(Vsin~)The shear flow, qo = C nr

"qo (Vsin¢)The shear stress, S2 = - = c ---t srt

The maximum shear is now at the tip of the saddle, i.e. ~ = a

(C) Shell stiffened by heads (A <Rl2)

If the saddle is close to the end closure the shell is stiffened on the side of the head. Itis assumed that the shell above the horn (tip of saddle) is stiffened by the end closure.The shear distribution in this upper region is therefore similar to that for a stiffenedregion.

For the upper portion (0:::; ~:::; a): Shear stress = ~sin ~nrt

13

For the lower portion - in the saddle region, (a::;!~::; n) , the shear distribution can beI

found by summing the shears to one side of the saddle. The sum of vertical shear forcefor the upper portion is equal to the sum of verticrl shear force in the lower portion.

II

---------------------------------J -------

II

Shear force near the enr closure

------------------------------1 -------

II

I,II

~r--r------~--,,

The vertical shear force,

V = 21iL(sin~)t(sin~)rd~ = Q (a - sinacosa)o tirt n

The shear flow is assumed to be the same as that for the unstiffened shell, that is:

The shear flow, qo = c(Vsin~) = ~. Q (a - sin o.cosc.Isin enr nr n

qo Q a - smacosaThe shear stress, S = - = - . . . sin ~t tcrt n - a + smacosa

The maximum shear occurs at ~ = a

Allowable stress limits:I

The tangential stress should not exceed 0.8 of the allowable tensile stress.

14

6.4 Ring compression in the shell over the saddle

Assuming that the surface of the shell and saddle are in frictionless contact withoutattachment. Ring compression is caused by shear forces.

The ring compression in the region a :s; ¢ :s; J[

The shear flow is:

v . d.( 1[ )q3 = -smlf' .Jrr J[ - a + smacosa

The total shear force at any point on the shell arc above that point.

¢ Q(cos¢ - cosa)Total shear force = fq3rd¢ = .

J[ - a + smacosaa

The contact pressure between the saddle and shell would induce a tangentialcompression force similar to the above. That is:

Tangential compression force due to saddle force = Q(cos¢ - cos fJ)J[ - fJ + sin j3 cos j3

Therefore, the maximum tangential force is = _ Q(1 + cos fJ)J[ - fJ + sin f3 cos f3

The contact pressure can be deduced from the tangential force:

1 Q(cos d. - cos /3)Contact pressure = _. --='---'----'If'__ ~-'----r J[ - fJ + sin fJ cos fJ

The maximum contact pressure occurs at ¢ = 7r

Max. contact force (at ¢ = 7r )

1 Q(l + cosb)- - - . --=-'------'----'---r n - 0 + sinbcosf

15

The width of shell that resists this force was considered by Zick to be '5t' on each sideplus the width of the saddle, i.e. width = b + -.

In a follow-up paper, he suggested width = b + 1.16-JrlI

The tangential stress can be calculated, S5 = shea~ force / width

The stress S5 is important when concrete saddle J used. It should be checked for largediameter vessel. I

Recent experimental and theoretical work on sad les welded to the vessel have foundthat this tangential stress is very small, about 111 of that predicted by Zick'sapproach. However, for the saddle not welded to he shell, the Zick's approach givesthe correct order of stresses.

The ring compression may be reduced by attaching a wea plate somewhat larger than the saddlesurface area directly over the saddle.

Allowable stress limits

The compressive stress S5 should not exceed 1/2 of S, and is not additive to thepressure stress. If wear plate is used, the combined thickness of wear plate and theshell can be used to calculate S5, provided the wear plate extends r/I 0 inch beyond thehorn.

Despite the limitations of Zick's approach it does provide a workable design methodthat has been used extensively over many years. However, the very highcircumferential stresses known to exist at the saddle horn region when the vessel issupported on a rigid saddle at not predicted adequately by the analysis. Although thesepeak stresses do exist, they are very local to the saddle horn and are unlikely to causeplastic collapse of the support. However, their existence does cause concern when thevessel is subject to high cyclic stressing.

16

Local stresses in shell due to loads on attachmentI

Types of attachments: Nozzles, supporting lugs,i lifting brackets, etc.!

Main concerns - High concentrated stresses at tHe attachment due to combinedinternal pressure and external loads applied through the attachment can be a source offailure if proper reinforcement is not supplied. !

iDesign consideration:

• Opening in vessel shell must be reinforced or operating pressure• Reinforcement is usually a rectangular or s uare pad welded to the shell• Over-reinforcement may create 'hard spot' on the vessel and induce large

secondary stresses• Reinforcement material should be close to he opening for effectiveness, of

which 2/3 of the required material should b. within a distance d/4 from theopening, where d is the diameter of the opening.

• Sharp junctions should be avoided; fillets should be incorporated to reduce themagnitudes of stresses at the junctions.

"The best arrangement is the so-called balanced reinforcement, which consists of about 35-40% ofthe area on the inside and about 60-65% on the outside. On many designs, however, it is difficult toplace reinforcement on the inside. Balanced reinforcement is often used at manway and inspectionopening where no nozzle is attached"

Area Replacement for Nozzles

This method formed the basic design method in many design codes. The origin of thearea replacement idea is not entirely clear. Simply expressed one replaces the area cutaway by the cross section of the hole in the shell and relocates it around the hole closeto the cutout. Notice it is an area replacement rather than a volume replacement.

The disposition of the replaced area is important. To be effective it needs to be close tothe edge of the opening where the stress field is increased. The extent of thereinforcement is preferably equal the die-out distance of the peak stresses at the edgeof the opening. That is why in some codes the extent of reinforcement is expressed asa function of Jrl, the characteristic length parameter for the die-out distance of thediscontinuity stress. In any case one simply obeys the rules as stipulated and noexplanation is given. It should be noted that the distance for reinforcement is generallyquite shout.

17

Cylindrical vessel with local loads on a rectangular attachment

Assumptions:• Attachments are rectangular or square w th two edges parallel to the circular

profile• The radial force produces uniform press re over the attachment area• The moment loading produces a triangul r pressure distribution

External loads(a) Radialload, P(b) Longitudinal moment, ML(c) Tangential (or circumferential) moment,(d) Torque, T(e) Shears VL, Vt

The shear stress in shell due to the torque Tis: ri-, T-r------- 2Jrrot - 2Jrr;t

Maximum shear due to VL or Vt is: v Vr'=_t_ or -r,=_L_Jrrot Jrrj

The shear stresses rand t' are usually small enough to be disregarded.

Parameters for cylindrical shell:

Shell parameter: y = R I t or R I(t + tp)Square attachment; ~ = cl R where 'c' is the half-length of the loaded areaCylindrical attachment: ~ = O.875ro / RRectangular attachment: it can be converted into equivalent square loaded area. For

small side ratios with a I b ::;1.5 , the equivalent c = -Jab /2

General expression for stress in the shell

Circumferential (tangential) stress:

Longitudinal (axial) stress:

18

iFor different loadings, the circumferential and tlhe longitudinal stresses are expressedin different parametric forms as follows: I

(1) Radial load, P

0.=(;'XcP~;)y + 6;.]i

=Cp(Plt2) (outward force)

= C~ (P I t2) (inward force)

u, {Nep 6Mep}O"ep = t2 R~ (M

tI R2~)'Y + (M{ I R~)

=Ct (Mt 1(2R2B )

(2) Circumferential (tangential) moment, M,

(3) Longitudinal moment, ML

&

Design considerations

(a) lfthe maximum stress at the attachment is too high, the shell must be reinforcedby a reinforcing pad or the thickness of the reinforcing pad required for internalpressure must be increased. The width of the pad is such that stresses at theedges of the pad are below the allowable stress.

(b) If two local loads are too close to each another, i.e. within the stress die-outdistance, then their influence on each other must be considered.

Note: The analysis presented above for local loads applied on cylindrical shell is too simplistic. Moredetailed and accurate analyses for different types of attachments are available in the literature andrecommended by design codes, specifically for loads on the nozzle, and openings. For example, theWelding Research Council Bulletins 107 & 297 (WRC 107 & 297).

19

Design by analysis

Essentially Design by Analysis is based on the dea that if a proper stress analysis canbe conducted then a better, less conservative, a sessment of the design can be madecompared to the usual approach of Design by le. The philosophy was originated inthe 1960's in the US. The motivation was drive by the sophisticated design work inthe nuclear industry. There were many design £ atures that were not covered directlyby the existing Design by Rule methods.

In the early years, all design by analysis ideas ere developed based on thin shellanalysis and in particular the analysis of discon inuity effects including thermaldiscontinuities.

It was suggested that different types of stress h d different degrees of importance andthis led to the idea of categorization of stress. T e stresses are cast in the form of'stress intensities' to reflect the Tresca yield cri eria and then compared with specifiedstress limits that are set at different levels for the different stress categories. Thismethodology was first incorporated in ASME PV code Section III and Section VIIIDivision 2 in 1968 and later into BS 5500 as Appendix A. Many countries have nowadopted the same basic approach.

Multiaxial Stress States

In real world, all stresses are three-dimensional. It is the simplifying assumptions thatreduce the 3-D stresses into 2-D and I-D. Yielding in the presence of multiaxial stressstates is not governed by the individual component but by some combination of all thestress components. The two commonly used yield criteria are the Von-Mises criterionand the Tresca criterion.

Von Mises criterion (distortion energy theory) states that yielding will take placewhen;

Tresca criterion (maximum shear stress theory);

= +0" /2- y

20

Although it is generally agreed that the Mises criterion is better for common pressuresteel, ASME chose to use the Tresca criterion as a framework for the Design byAnalysis procedure. The reason is that Tresca is the more conservative and it is easierto apply. The later is longer true now since computer can perform complexcalculations at ease.

In order to avoid the unfamiliar (and unnecessary) operation of dividing bothcalculated and yield stress by two, a new term called 'stress intensity' was defined.The stress differences of the principal stresses are as follows:

The STRESS INTENSITY, S is the maximum absolute value of the stress difference.

That is:

So the Tresca criterion reduces to: S=(J' y

Throughout Design by Analysis procedure stress intensities are to be used.

Stress Categories

Certain types of stresses are more important than others and that these should beassigned to different categories with different levels of importance having differentstress limits. ASME chooses the following categories:

(A) Primary Stress(i) General Primary Membrane Stress, Pm(ii) Local Primary Membrane Stress, PL

(iii) Primary Bending Stress, P,

(B) Secondary Stress, Q

(C) Peak Stress, F

Primary stress is a stress developed by the imposed loading that is necessary tosatisfy the law of equilibrium between external and internal forces and moments. Thebasic characteristic of a primary stress is that it is not self-limiting.

Note: A stressed region may by considered as 'local' if the distance over which the stress intensity

exceed 1.1 Sm does not extend in the meridional direction more than 1.O-JRt . Local primary

membrane sources must be 2.5-J Rt apart. Examples of primary membrane sources are nozzle andsupport.

21

Secondary stress is stress developed by the self-constraint of a structure. It mustsatisfy an imposed strain pattern rather than being in equilibrium with an externalload. The secondary stress is self-limiting, its may cause local yielding and minordistortion resulting from discontinuity condition or thermal expansion.

Peak stress is the highest stress in the region under consideration. The basiccharacteristic of a peak stress is that it causes no significant distortion and isobjectionable mostly as a possible source of fatigue failure.

Failure modes1. Excessive elastic deformation incl ding elastic instability2. Excessive plastic deformation3. Brittle fracture4. Stress rupture and creep deformati n5. Plastic instability - incremental co lapse6. High strain - low cycle fatigue7. Stress corrosion8. Corrosion fatigue

In setting the stress limits, however, attention is concentrated in 3 areas. They are:(a) Avoidance of gross distortion or bursting, Pill' PL and P,(b) Avoidance of ratcheting, PL + P,(c) Avoidance of fatigue, P+ Q

Relationship between stress limits to the various categories

Stress Intensity Allowable Stress Equivalent YieldGeneral primary membrane, Pm Sm 2S

3 y

Local primary membrane, PL 1.5 Sm Sy

Primary membrane + bending, 1.5 Sm Sy(Pm+ Pb) or (PL + Pb)

Primary + secondary 3Sm 2Sy(PL + P, + Q) or (Pm+ P, + Q)Fatigue, 2Sa -(PL + Pb + Q + F) or (Pm+ Pb + Q + F) (allowable fatigue stress range)

The above limits are not always applicable; there are a number of special cases. In thecase of nuclear vessels the service loadings are classified into normal, upset,emergency and faulted conditions. This is formalized in ASME with k-factors appliedto the limits. For example, for earthquake loading, k = 1.2, for hydraulic test k =1.25,etc.

22

For attachments and supports the limits are:

The membrane stress intensity :S 1.2 Sm(0.8 Sy)Membrane + bending stress intensity s Sm(1.33 Sy)

I

For nozzles and openings:

Membrane + bending stress intensity s i .25 Sm(1.5 Sy)

Some cautionary words are necessary for the u wary. The manner in which thesymbolism is used can lead to confusion. For e ample a stress limit on somecombination of stress categories denoted as CPL + P, + Q) needs to be clearlyunderstood. It is the stress intensity evaluated om the principal stresses after thestresses for each category have been added tog ther in the appropriate way. It shouldnot be interpreted as the combination of stress ~ntensity from each category.

IIn summary: ONLY add stresses, DO NOT add stress intensities.

A trivial example of the wrong way of summing the stresses in given below:

Stresses Pm Q Pm+QSx= S] 10 25 35Sy= S2 10 -5 5Sz= S3 -2 0 -2

The maximum stress intensity for Pm= 12The maximum stress intensity for Q = 30The maximum stress intensity for (Pm+ Q) = 37 (from [Pm+ Q] column)It is wrong to add stress intensities of Pm and Q, that would give (12 + 30) = 42

When we add stresses, of course, they need to be in the same directions and at theappropriate locations for the identified combination of loads. The approach is toevaluate all the stresses for the different types of loading. These should be assigned tocategories as necessary. Then the stresses in the various categories should be summedand finally the stress intensities calculated for the particular combination of categoriesrequired.

23

FE Analysis for Pressure Vessel Desig

The Design by Analysis is closely rooted in thi shell discontinuity analyses. WhenFE (finite element) method is used, some diffi ulties in stress categorization occur.The FE gives accurate stress information for c mplex geometries. These stresses mayvary nonlinearly through the thickness. For ass ssment purposes it is necessary tolinearize the stress distribution and separate membrane and bending effects. In asimple case the procedure would be straightforward and membrane, bending and peakelements of the stress could be identifies. Unfo unately things are not always sosimple. Firstly the Linearization procedure is it elf subject to a number ofuncertainties. Secondly the bending componen in general may include primarybending as well as secondary bending.

In practice it tends to assume the membrane str ss intensity as primary and thebending stress intensity as secondary (which m y not be conservative). In criticalsituation the designer may wish to impose his dwn conservatism at this point.

Until today no entirely satisfactory solution has been found for the stress linearization.However, alterative methods may be forthcoming that would by-pass thecategorization problem or at least simplify its interpretation. The Standards allow thedesign to be based on limit load analysis with a suitable factor where the factor has tobe the same as the main shell (i.e. 1.5). Design may proceed directly with a factor onload without detailed consideration of the stresses. The approach seems promising if itcan be extended to complex loading situations it could provide a relatively simplealternative to the current classification route.

ASME identifies 8 modes of failure "which confront the pressure vessel designer."The evaluation of failure modes requires the computation of membrane and bendingstresses and their classification into certain categories - primary, secondary and peak -to which different design allowable stresses applied.

The original techniques for evaluating the stress limits were based on shell theory bywhich membrane and bending stresses are determined directly - so there is no muchconfusion in the classification of stresses.

With the advent of finite element (FE) techniques, the transition from the stressdistribution to the failure mode requires a different path.

The results of axisymmetric or 3-D solid FE analysis are not immediately in a formsuitable for the extraction of shell type membrane and bending stresses. Difficultiesare associated with linearization procedure used to obtain membrane and bendingstresses.

24

Unless we are dealing with well established ca es, as listed or referenced in codes,there has always been a problem with the cate orization of stresses into primary andsecondary.

The problems of assessing primary and second ry stress failure modes and theirrelationship to stress results from axisymmetri and 3D geometries were firstaddresses by Hechmer and Hollinger in 1986.

"3D stress criteria - a weak link in vessel design and a lysis", PVP Vo1.109, A Symposium onASME Codes and Recent Advances in PVP and Valve Technology including a Survey ofOperational Research Methods in Engineering, July 19 6, ASME, New York, NY.

Three approaches for determining the membra e and bending stresses were discussed:(i) stress-at-a-point(ii) stress-along-a-line, and(iii) stress-on-a-plane.

A quantitative comparison of the three approaches was presented in:

3D stress criteria - application of Code rules, "PVP Vo1.120, Design and Analysis of Piping, PressureVessels, and Components, July 1987, ASME, New York, NY.

The study shows that the 3 approaches can give substantially different results. Themost complex of the three approaches is stress-on-a-plane. The definition of the planefor 3D geometries is subjective and the resultant stresses and conclusions are merelyengineering judgement.

Some issues:

It should be emphasized that these issues actually arise from the nature of the Coderules, rather from any deficiency in the finite element solution.

In 2D axisymmetric analyses, the bending stress can be calculated using componentnormal and/or shear stresses or principal stresses. The distinction is that for a givenset of geometric reference axes, component stress directions remain constant withlocation and load application whereas principal stress directions vary with location andload application. This distinction is important when considering the pros and cons ofusing component versus principal stresses.

Which stresses are consistent with bending theory? Code implies that bending isapplicable only to normal stress components, because the Code links bending tobending moments. Mathematically, one can calculate linearized shear stresses and callit a bending stress. However, it is difficult to conceptualize a bending moment for anyshear stress in the realm of traditional engineering mechanics.

25

For 3D geometries, the issue is evaluation of s I esses along lines versus on planes.The code implies evaluation along a line. H01ever, the code does not preclude theuse of planes.

Two PVRC grants were established to investigate and document the need to update theASME B&PV and Piping Code criteria and re~uirements for relating 3D stressdistributions to failure criteria. The findings ar1 presented in the following paper.

J.,L. Hechmer and G.L. Hollinger, 3D Stress Criteria, JvP-Vol. 210-2, Codes and Standards andApplications for Design and Analysis of Pressure vess~l and Piping Components, ASME 1991.

Recommendations I

• The stresses for Pmcan and should be calculated by simple equilibrium equations.The same is true for Pb if Pmis small (for example, the plate structures). Stressesfor Pmneed only be evaluated in basic structural elements. Designer should applyhis ingenuity to calculate equilibrium stresses, not to extract stresses from a generalFE model.

• It is appropriate to calculate PL stresses in the vicinity of all discontinuities. Thereare discontinuities where PL stress exists, but need not be evaluated. Because coderule reinforcing rules ensure that PL limit is met.

o Nozzle-shell junctionso Formed heads to shellso Cones to shellso Tapered cylinders to shells

• Linearization algorithm calculates the net force distribution on the cross section.The average net force can be calculated from the total net force. The average netforce is then subtracted from the net force distribution that is used to calculate thebending moment. The bending moment is computed relative to the neutral axis.

• Calculate (PL + Pb) and (P + Q) in the basic structural elements (and not in thetransition elements). The reason is that plastic collapse and gross strainconcentration will not occur in the stiff transition elements. They will occur in themore flexible shell element.

If a fatigue analysis is to be performed within a transition elements due to highstress concentration, it may be appropriate to consider the (P + Q) in the adjacentstructural elements.

• For assessing the membrane stress limits (Pm+ Pb), all the stress components (3normal + 3 shear components) should be included. The average principal stressesmust be computed from the average stress components through the thickness andNOT from average principal stresses. That is: compute the average stresscomponents first then compute the principal stresses.

26

44 Page 2 of24

Conventional bolts are usually made to the specific project requirements by steel fabricators or they may bepurchased in standard sizes (diameters and lengths) from steel suppliers. The availability and cost ofconventional bolts are generally based on demand and fabrication requirements. The types of conventionalanchor bolts most often used are discussed below.

Headed Bolts. Square or hex-headed ASTM A 307 bolts are frequently used as anchor bolts due to theirwide availability and relatively low cost (see Figure 1). Higher strength bolts, such as ASTM A 325 bolts, areavailable and can be used, but are more expensive. A washer placed against the bolt head is often usedwith the intention of increasing the bearing area and thus increasing the anchor strength. However, theactual strength increase obtained by adding a washer is small, if any, and under certain conditions (smalledge distances), may actually decrease the tensile strength.

(::])A) HEX-HEAD

Bent Bar Anchors. Bent bar anchors, frequently used in masonry construction, are usually made in "J" or"L" shapes (see Fig. 2). Even though the "J" and "L" shapes are the more popular, a variety of shapes (seeFig. 3) is available since there currently is no standard governing the geometric properties of bent baranchors. These anchors are usually made from ASTM A 36 bar stock and are shop-threaded.

Headed Bolts

FIG. 1

http://www.bia.org/BIA/technotes/t44.htm 3115/2008

44 Page 3 of24

Tov

~ .,-_,,/_~ 1h TO 1% D ~

~ ,_~ ~IaI~0

~ _I TA) Bl" BOLT

B) "J" BOLT

"L" and "J" Bent Bar Anchors

FIG.2

A) EYE BOLT

S) "U" BOLT

C) ACUTE BEND '~" BOLT

http://www .bia. org/BIA/technotes/t44 .htm 311512008

44 Page 40f24

Other Bent Bar Anchors

FIG.3

Plate Anchors. Plate anchors are usually made by welding a square of circular steel plate perpendicular tothe axis of a steel bar that is threaded on the opposite end (see Fig. 4). There are no standards governingthe dimensions (length, width or diameter) of the plate. The American Institute of Steel Construction doeslimit the fillet weld size based on the plate thickness (see Table 1). Both the plate and bar are usually madefrom ASTM A 36 steel.

~~\ ~

~) ~'-- -"'Im

A) CIRCUlAR PLATE ANCHOR

B) SQUARE PLATE ANCHOR

Plate Anchors

FIG. 4

Through Bolts. As the name implies, through bolts extend completely through the thickness of themasonry and are composed of a threaded rod or bar with a bearing plate located on the surface oppositethe attachment (see Fig. 5). In the early 1900's, through bolts were used in loadbearing masonry structuresto tie floor and wall systems together. Often decorative cast bearing plates were used since through boltswere visible on the exterior masonry surfaces (see Fig. 6). Today, through bolts are primarily used inindustrial construction where aesthetics are not a principal concern, or in retrofitting existing structures.Through bolt rods are usually made from ASTM A 307 threaded rod or threaded ASTM A 36 bar stock.Bearing plates are typically made from ASTM A 36 steel plate.

http://WWVll.bia.org/BINtechnotes/t44.htm 311512008

44

Through Bolt

FIG. 5

Decorative Through Bolt Bearing Plate

FIG.6

http://www.bia.org/BIA/technotes/t44.htm

Page 5 of24

3115/2008

44

Proprietary Anchor Bolts

,-----_._--_.- ._--- --- .._--- - - .- --

* American Institute of Steel Construction

Page 60f24

Proprietary anchors are available through a number of manufacturers under numerous brand names.Although the style and physical appearance of the anchors differ between manufacturers, the basic theoriesbehind the anchors are very similar. For this reason, proprietary anchors can be divided into two genericcategories: expansion-type anchors and adhesive or chemical-type anchors.

Expansion Anchors. Two different types of expansion anchors are generally recommended by theirmanufacturers for use in brick masonry: the wedge anchor and the sleeve anchor (see Fig 7). Theseanchors develop their strength by means of expansion into the base material. Wedge anchors develop theirhold by means of a wedge or wedges that are forced into the base material when the bolt is tightened. Thewedges create large point bearing stresses within the hole; therefore, this anchor requires a solid basematerial to develop its full capacity. For this reason, voids formed by brick cores and partially filled mortarjoints in some brick masonry may make the construction unsuitable for wedge anchor installation.

A) WEDGE A.NCHOR B) SLEEVE ANCHOR

Proprietary Expansion Anchors

http://www .bia.org/B IA/technotes/t44. htm 3/15/2008

44

FIG. 7

Page 7 of24

Sleeve anchors develop their strength by the expansion of a cylindrical metal sleeve or shield into the basematerial as the bolt is tightened. The expansion of the sleeve along the length of the anchor provides alarger bearing surface than the wedge anchor, and is less affected by irregularities and voids in the basematerial than is the wedge anchor. For this reason, sleeve tnChors are recommended by theirmanufacturers for use in brick masonry more often than we ge anchors.

Drop-in and self-drilling anchors (see Fig. 8) are two other t. pes of expansion anchors available, but aretypically not recommended by their manufacturers for use i i masonry. The reason for this is due to theembedment and setting characteristics of the two anchors. Both anchors are produced to allow shallowembedment depths and are expanded or set by an impact setting tool. The combination of shallowembedment and high stresses imparted by the expansion tend to cause cracking or splitting in masonry.Depending on the extent of cracking or splitting, the anchor could experience a reduction in load-carryingcapacity or undergo complete failure during installation.

A)SELFORILllNG ANCHOR B) DROP4N ANCHOR.

Other Proprietary Expansion Anchors

FIG. 8

There are several considerations that should be examined when contemplating the use of expansion-typeanchors in brick masonry. These are: 1) Expansion anchors should not be used to resist vibratory loads.Vibratory loads tend to loosen expansion anchors. 2) Specific torques are required to set expansionanchors. Excessive torque can reduce anchor strength or may lead to failure as excessive torque isapplied. 3) Expansion anchors require solid, hard embedment material to develop their maximumcapacities. Some brick construction may not provide a good embedment material due to voids formed bybrick cores and partially filled mortar joints.

Adhesive Anchors. Two basic types of adhesive anchors are currently available. The major differencebetween the two is that one anchor is manufactured as a pre-mixed, self-contained system, whereas thesecond type requires measurement and mixing of the epoxy materials at the time of installation. The morepopular self-contained types use a double glass vial system (see Fig 9) to contain the epoxy. The outer vialcontains a resin and the inner vial contains a hardener and aggregate The glass vial is placed in a pre-drilled hole and a threaded rod or bar is driven into the hole with a rotary hammer drill, breaking the vials

http://www.bia.org/BIA/tcchnotes/t44.htm 3115/2008

44 Page 13 of24

In hollow brick construction, the units are laid so that the cells are aligned and provide continuous channelsfor reinforcing steel placement and for grouting. Depending on the design, every cell or intermittent cellsmay be reinforced and grouted (see Technical Notes 41 Revised). The anchor embedment detail willdepend on the reinforcing pattern used in the construction. Figure 15 shows typical embedment details forconventional anchors embedded between reinforcing cells. The anchor should be solidly surroundedvertically and horizontally by grout for a minimum distance of twice the embedment depth (1b) (Figs. 14 and15) for full tension cone development. The tension cone theory is discussed in following sections. This mayrequire that some cells be partially grouted. A wire mesh screen can be placed in the bed joint across cellsthat are to be partially grouted to restrict the grout flow beyond a certain point. Figure 16 shows typicalembedment details for conventional anchors embedded in reinforced cells. In this detail, the anchor may betied with wire to the reinforcing to secure the anchor during the grouting process Again, the anchor shouldbe solidly surrounded by grout to a minimum distance of tWife the actual anchor embedment depth, bothvertically and horizontally. I

if

P;; "L" socr s, ",I" SOLT CI HEA.OED ;BOLT

IF;::::~~·*I;>

O)PlA15 ANCHOR

DMIN.

Conventional Anchors in Reinforced Hollow Brick

FIG.15

http://www.bia.org/BIA/technotes/t44.htm 3/15/2008

44 Page 14 of24

0) P~JE P.fl-cHOR

Conventional Anchors in Partially Grouted Hollow Brick

FIG.16

Two typical embedment details for conventionally embedded anchor bolts installed in composite brick andconcrete block construction are shown in Fig. 17. As shown, anchor bolts may be placed in the collar jointbetween the brick and block wythes or placed into cells in the concrete block wythe and grouted into place.In details similar to Fig. 17(a), the anchor bolt type and diameter may be controlled by the width of the collarjoint. Collar joints should be a minimum of 1 in. (25 mm) wide when fine grout is used, or a minimum of 2 in.(50 mm) wide when coarse grout is used (see Technical Notes 7A Revised). When the collar jointdimension is in the 1 in. (25 mm) range, it may become difficult to position anchor bolts in the collar jointand maintain the recommended clear distance between the masonry and the anchor (Fig. 17). The practiceof using soaps to accommodate anchors larger than the collar joint is not recommended because thereduction in the brick masonry thickness around the anchor could lead to strength reductions. If the anchordimensions required are larger than the collar joint, a detail similar to that shown in Fig 17(b) should beconsidered.

http://www.bia.org/BIAIteclmotes/t44.htm 3115/2008

44

B) ANCHOR IN BLOCK WYTHE

Page 15 of24

GROUTSTOP

Conventional Anchors in Composite Brick/Block Masonry

FIG. 17

Through bolts are typically installed after construction and grouting by drilling through the completedmasonry work. When through bolts are to be installed after construction in reinforced brick masonry, careshould be taken during installation to avoid cutting or damaging reinforcement while drilling the through boltholes. Reinforcing bar locations can be identified by specially tooled joints or other marks made duringconstruction.

Proprietary Anchors

Proprietary expansion and adhesive anchors typically require special installation procedures andequipment. The manufacturer should be contacted to determine the appropriate anchor for a particularapplication, the correct installation procedure and if any special installation equipment is required. Improperapplication and installation of proprietary anchors may lead to less than satisfactory structural performance.

Typical proprietary anchor details are shown in Fig. 18. It is suggested that proprietary anchors beembedded in head joints when facing or building brick are used. This reduces the possibility of placinganchors in brick cores that occur within the thickness of the brick and adjacent to the bed joint surfaces.Anchors set in grouted hollow brick should be placed in holes drilled in the bed joints so that they intersectgrouted cells, or should be placed in holes drilled through the faces of the units into the grouted cells. Aswith conventional anchors, proprietary anchors should be solidly surrounded vertically and horizontally bygrout for a minimum distance of twice their embedment depth.

http://www.bia.org/BIA/technotes/t44.htm 3115/2008

44

] n "/11 ru

'~It-::ll>

Jf'( lu 'V IJ

A) GROUTED OOLL~R ,JOINT CONSTRUCnON

'b

J ~ ! ~ l.,

[II ~;;J~

II -. II'v

Typical Proprietary Anchor Details

FIG.18

ANCHOR BOLT DESIGN

Page 16 of24

Anchor bolts are used as a means of tying structural elements together in construction and therefore,provide continuity in the overall structure. In virtually all applications, anchor bolts are required to resist acombination of tension and shear loads acting simultaneously due to combinations of imposed dead loads,live loads, wind loads, seismic loads, thermal loads and impact loads. For this reason, and also to insuresafety, anchor bolt details should receive the same design considerations as would any other structuralconnection. However, due to a lack of available research and design guides, anchor bolt designs are basedlargely on past experience with very little engineering backup. This situation may lead to conservative,uneconomical designs at one extreme, or nonconservative designs at the other.

Recently, however, research investigating the strength of conventional and proprietary anchors in masonryhas been completed. Reports have been issued that evaluate anchor performance and suggest equationsto predict ultimate anchor strengths. By combining the research findings with design practices currentlyused in concrete design, equations for allowable tension, shear and combined tension/shear loads for plateanchors, headed bolts and bent bar anchors are under consideration for adoption in the proposed "BuildingCode Requirements for Masonry Structures" (ACIIASCE 530). These equations are outlined below.

Tension

The tensile capacity of an anchor is governed either by the strength of the masonry or by the strength of theanchor material. For example, if the embedded depth of an anchor is small relative to its diameter, a tensioncone failure of the masonry is likely to occur. However, if the embedded depth of the anchor is large relativeto its diameter, failure of the anchor material is likely. For these reasons, the allowable tensile load is based

http://www.bia.org/BIA/technotes/t44.htm 3115/2008

44

a

~~st°MIN.

OJ THROUGH BOLT--V-:;"Cr~.r-r-r;!;~:},-",... r-r-~"TT""

D MIN.

B) .~•• BOLT

o~~.~~~ ....~.~

C) HEADEOBOLT

Conventional Anchors in Grouted Collar Joints

FIG.12

Page 11 of24

Typical embedment details of conventional anchors in multi-wythe brick construction are shown in Fig. 13.A brick, or portion of a brick, is left out of the inner wythe to form a cell for the embedded anchor (Fig. 14).After the anchor is placed, the cell is filled with mortar or grout prior to placement of the next course.

http://V'lTWW.bia.org/BIA/teclmotes/t44.htm 311512008

44

----------------

Page 10 of24

aThe manufacturer should always be consulted when adhesive anchors are to be used in areas where contact with chemicals is likely.

~ ''']W.J~V)

ZW~ 50

" ~rl '1"1""t4_._-r--r-.-rl 'I T,'1-'--'---' -rf "--.",,,,, ,..r"Tf""f·T'"f~r"1'I,I_._i " T,'I"-'-"--'--'-'-TI "'"""!100 150 200 25G50

TEMPERATURE, 'F

Effect of Temperature on Ultimate Tensile Capacity

FIG.11

INSTALLATION DETAILS

Conventional Anchor Bolts

Typical embedment details for each type of conventional anchor used in grouted collar joint construction areshown in Fig_ 12_The conventional embedded anchors (headed bolts, bent bar and plate anchors) areusually placed at the intersection of a head joint and bed joint. By using this location, the brick unitsadjacent to the anchor can be chipped or cut to accept the anchor without altering the joint thickness.

http://www.bia_org/BIA/technotes/t44 .htm 3115/2008

44 Page 12 of24

Q.1..f"

D) PLATE ANCHOR

B) '~" BOLI

E) 1HROUGH sOLT··

Conventional Anchors in Multi-Wythe Brick Masonry

FIG.13

A) FUll BRICK OMITTED B) BRICK CUPPED

Plan View of Grout Cell in Multi-Wythe Brick Masonry

FIG. 14

http://www.bia.org/BIA/technotesIt44.htm 3/15/2008

44

MIX

Page 9 of24

There are special requirements and limitations. that should be considered when contemplating the use ofadhesive anchors in brick masonry. They are: 1) Specially designed mixing and/or setting equipment maybe required 2) Dust and debris must be removed from the pre-drilled holes to insure proper bond betweenthe adhesive and base material. 3) The adhesive mixture tends to fill small voids and irregularities in thebase material. 4) Large voids (due to brick cores, intentional air spaces and partially filled joints) may causereductions in anchor capacities. This is especially true with the self-contained adhesive anchors since alimited volume of epoxy is available to fill the voids and provide a bond to the anchor. 5) The adhesive bondstrength is reduced at elevated temperatures and may also be adversely affected by some chemicals (seeTable 2 and Fig. 11).

HARD!-ENER

RESIN

PLACE

Site-Mixed Adhesive Anchor

FIG. '10

http://www.bia.org/BIA/technotes/t44.htm 3115/2008

44 Page 8 of24

and mixing the adhesive components. The other type of adhesive anchor requires that the epoxycomponents be hand-measured and mixed before the epoxy is placed into a pre-drilled hole. A threadedrod or bar is then set into the epoxy mixture, as shown in Fig. 10. Adhesive epoxies usually vary slightlybetween manufacturers, but the steel rods or bars are typically ASTM A 307 or ASTM A 325 threaded rod,or ASTM A 36 shop-threaded bar.

A) EPOXY CAPSULE

a) THREADED ROD

C)INSTALLED ANCHOR

Self-Contained Adhesive Anchor

FIG.9

http://www.bia.org/BIA/technotes/t44.htm 3115/2008

44

TABLE 4Allowable Shear on Anchor Bolts - l-rcrn use

1985 Edition"

Page 22 of24

(a) ,A.LLOV\,I,ABLE SHE.A,R C)~\J.A.f",JCHOF.'BOLTS1 FOR CLAY .A.f",JDCOf\JCF.'ETE t"llASOf\JRY

Tot<lI AllowableDiameter Elllbedmenf Shear3

(inches) (inches) fibs}1/4 4 270:3/8 4 41D1(2 4 5505/B 4, ~~I-3/4 r- 1100~I

7/8 f3 1500"7 18504I

1-1 fa 0 22504u

'P',n anchor bo~ is eIbolt that h;",,, eIright elngle extension of elt leelst three dierneters.

,f!., standard machine bo~ i:::oacceptable.

"Of the total required embedment, a minimum of five bolt diameters must be

perpendicular to the masonrv surface.

1--10reduction in value" required for uninspected mesonrv.

",f!.,pplicable for unit:; ha"iing a net area strength of 2500 psi or more.

(b) ,A.LLOVVABLE ~:HE.A.R IJr\j EiOLn:; FOR D ...1F'1F.~ICALLY DE::::IC;r\JEDMA,Sm',JF~Y EXCEF'T Ur'"JBURf\JED cu~.,,( U~',JlTS

Solid GroutedDiameter Embedment' Masonry Masonry

Bolt (inches) (Shear in (Shear in(inches) Pounds! Pounds)

112 4 350 5505/8 4 500 750

3/4 5 750 1100na f3 1000 1500

7 1250 113502

1-1/8 e 1500 22502

',f!.,nadditional 2 inches of embedment shall be provided for anchor bolts located

in the top of columns: for buildinqs located in Seis:mic Zones: ~,los:.2, 3, and 4.

2Pennitted onl'i with not less than 2500 pounds per sq in. units

* Reproduced from the Uniform Building Code, 1985 Edition, Copyright 1985 with permission of the publisher, TheInternational Conference of Building Officials."

http://www .bia. org/BIA/technotes/t44 .htrn 3115/2008

44 Page 17 of24

on the smaller of the two loads calculated for the masonry and anchor material. Thus, the allowable load intension is the lesser of:

(Eq.l )

or

(Eq. 2)

where: TA= Allowable tensile load, Ib,

Ap = Projected area of the masonry tension cone, in2, fm = Masonry prism compression strength (Incomposite construction, when the masonry cone intersects different materials, fm should be basedon the weaker material), psi,

AB = Anchor gross cross-sectional area, in2,

fy = Anchor steel yield strength, psi.

The value of Ap in Eq. 1 is the area of a circle formed by a failure surface (masonry cone) assumed toradiate at an angle of 45° (see Fig. 19) from the anchor base. When an anchor is embedded close to a freeedge, as shown in Fig 20, a full masonry cone cannot be developed and the area Ap must be reduced soas not to over-estimate the masonry capacity. Thus, the area Ap, in Eq. 1 will be the lesser of:

(Eq 3)

or

(Eq. 4)

where: Ap = Projected area of the masonry tension cone, in.2,

1b = Effective embedded anchor length, in.,

1be = Distance to a free edge, in.

http://www .bia. orglBlA/technotes/t44 .htm 3115/2008

44

Full Masonry Tension Cone

FIG. 19

. '~"

G',~..----"~~

A) PROJECTED CONE

http://www.bia.org/BIA/technotes/t44.htm

Page 18 of24

3/15/2008

44 Page 19 of24

Reduced Masonry Tension Cone

FIG.20a

B) PROJECTED AREA

Reduced Masonry Tension Cone

FIG.20b

The effective anchor embedded length (1b) is the length of embedment measured perpendicular from thesurface of the masonry to the plate or head for plate anchors or headed bolts. The effective embeddedlength of bent bar bolts (1b) is the length of embedment measured perpendicular from the surface of themasonry to the bearing surface of the bent end minus one bolt diameter. Where the projected areas ofadjacent anchors overlap, Ap of each bolt is reduced by one-half of the overlap area. Also, any portion ofthe projected cone falling across an opening in the masonry (i.e., holes for pipes or conduits) should bededucted from the value of Ap calculated in Eqs. 3 or 4.

Shear

The allowable shear load is based on the same logic as the allowable tension load. That is, the anchorcapacity is governed by either the masonry strength or the anchor material strength. The distance betweenan anchor and a free masonry edge has an effect on the masonry shear capacity. Calculations have shownthat for edge distances less than twelve times the anchor diameter, the masonry shear strength controls theanchor capacity. (C. I ations based on masonry with f'm = 1000 psi and anchor steel yield strength with f .= 60 ksi. Therefore, where the edge IS ance u or exceeds 12 anc or diameters. the allowable shearlOad is the lesser of: ..-

'i -.",rJ41~f' A'i A. - ",01 ~J rn' '8 (Eq 5)

http://wvl/w.bia.org/BIA/technotes/t44.htm 3/15/2008

44 Page 20 of24

or

(Eq. G)

where: VA = Allowable shear load, lb.

When anchors are located less than 12 anchor diameters fro! a free edge, the allowable shear load isdetermined by linear interpolation from a value of VA obtained in Eq. 5 at an edge distance of 12 anchordiameters to an assumed value of zero at an edge distance 0 1 in. (25 mm). This takes into considerationthe reduction in the masonry shear capacity due to the edge d,istance.

Combined Tension and Shear

Allowable combinations of tensile and shear loads are based on a linear interaction equation between theallowable pure tension and pure shear loads calculated in Eqs. 1, 2, 5 and 6. Anchors subjected tocombinations of tension and shear are designed to satisfy the following equation:

T / TA + V / VA ~ 1.0 (Eq. 7)

where: T = Applied tensile load, lb..

V = Applied shear load, lb.

Proprietary Anchor Bolts

The allowable load equations previously presented are intended for use with plate anchors, headed boltsand bent bar anchors and have been proposed to the ACIIASCE 530 Committee on Masonry Structures.However, when the allowables from these equations are compared to test results for proprietary anchors,they appear to produce acceptable safety factors.

Allowable Loads. Average factors of safety are 4.0 for tensile tests and 5.0 for shear tests on proprietaryanchors. The combined tension/shear interaction equation produced an average safety factor of 7.0 whencompared to test results on proprietary anchors. Therefore, based on comparison to test results, theallowable load equations proposed in this Technical Notes are suggested for use in the design ofproprietary anchors in brick masonry. The embedment depth used to calculate the allowable load valuesshould be equal to the embedded depth of the proprietary anchor.

Edge Distance. Edge distance is of particular concern when expansion anchors are used in brick masonry,due to lateral expansion forces produced when the anchors are tightened. These forces are often largeenough to cause cracking or spelling of the brick when edge distances become small. To date, no researchhas been conducted in this area. Therefore, due to the lack of information, it is suggested that a minimumedge distance of 12 in. (300 mm) be maintained when expansion anchors are installed in brick masonry.

http://www.bia.org/BIA/technotes/t44.htm 3/15/2008

44 Page 21 0[24

Through Bolts

There are no known published reports available addressing the strength characteristics of through bolts inbrick masonry. However, based on the conservatism in the allowables for bent bar anchors and proprietaryanchors, the allowable load equations should provide acceptable allowable load values for through boltsused in brick masonry. The embedment depth used to calculate the allowable load values should be takenas equal to the actual thickness of the masonry.

Current Codes and Standards iAt the present time, one model code and one design standard contain provisions for anchor bolt design inbrick masonry. The BIA Standard, Building Code Requirements for Engineered Brick Masonry, and theUniform Building Code cover design allowables and embedment depths for anchors loaded in shear. Thereare no provisions for axial tensile loads or combined tension/shear loads in these documents. Tables 3 and4 show the allowable shear loads and minimum embedment depths from the two documents. The values inTable 4(a) are based on rational analysis and in Table 4(b) on empirical analysis. As can be seen, thetables are very similar and are generally more conservative than the allowable shear loads obtained fromEqs. 5 and 6 for the same embedment depths (Table 5).

!xl! L:J

• From Building Code Requirements for Engineered Brick Masonry, BrickInstitute of America, August 1969.

'In determining the stresses on brick masonry, the eccentricity due to loadedbolts and anchors shall be considered.

280lts and anchors shall be solidly embedded in mortar or grout

3No engineering or architectural inspection of construction and workmanship.

4Construction and workmanship inspected by engineer, architect or competentrepresentative.

http://www.hia.org/BINtechnotes/t44.htm 3/15/2008

44 Page 23 of24

'American Concrete Institute/American Society Of Civil Engineers Committee 530 onMasonry Structures.

1Assuming fm = 2,000 psi

ASTM A36 steel fy = 36 ksi

Edge Distance = 12 Bolt Diameters

SUMMARY

This Technical Notes is the first in a series on brick masonry anchors, fasteners and ties, It covers anchorbolt types, detailing and allowable loads for anchor bolts in brick masonry. Other Technical Notes in thisseries will address brick masonry fasteners and ties.

The information and suggestions contained in this Technical Notes are based on the available data and theexperience of the technical staff of the Brick Institute of America. The information and recommendationscontained herein should be used along with good technical judgment and an understanding of theproperties of brick masonry. Final decisions on the use of the information discussed in this Technical Notesare not within the purview of the Brick Institute of America and must rest with the project designer, owner orboth

REFERENCES

1. Manual of Steel Construction, 8th Edition, American Institute of Steel Construction, Inc., Chicago,Illinois, 1980.

2. Whitlock, A.R. and Brown, R.H., Strength of Anchor Bolts in Masonry, NSF Award No, PRF-7806095, "Cyclic Response of Masonry Anchor Bolts", August 1983,

http://www .bia. org/BIA/technotes/t 44 .htm 3115/2008

44 Page 24 of24

3. Brown, R.H. and Dalrymple, GA, Performance of Retrofit Embedments in Brick Masonry, NSFAward No. CEE-8217638, "Static and Cyclic Behavior of Masonry Retrofit Embedments (EarthquakeEngineering)", Report No.1, April 1985.

4. Hatzinikolas, M.; Lee, R.; Longworth, J. and Warwaruk, J., "Drilled-In Inserts in MasonryConstruction", Alberta Masonry Institute, Edmonton, Alberta, Canada, October 1983.

5. Building Code Requirements for Engineered Brick Masonry, Brick Institute of America, McLean,Virginia, August 1969.

6. Uniform Building Code, International Conference of Building Officials, Whittier, California, 1985.

7. Technical Notes on Brick Construction 17 Revised, "Reinforced Brick Masonry, Part I of IV", BrickInstitute of America, McLean, Virginia, October 1981.

8. Technical Notes on Brick Construction 41 Revised, "Hollow Brick Masonry-Introduction", BrickInstitute of America, McLean, Virginia, 1983.

9. Specification for the Design and Construction of Load-Bearing Concrete Masonry, NationalConcrete Masonry Association, McLean, Virginia, April 1971.

10. The BOCA Basic/National Building Code, 9th Edition, Building Officials and Code Administrators,International, Country Club Hills, Illinois, 1984.

11. Standard Building Code, Southern Building Code Congress, International, Inc .. Birmingham,Alabama, 1985.

12. Technical Notes on Brick Construction 7A Revised, "Water Resistance of Brick Masonry-Materials, Part 1\ of III", Brick Institute of America, Reston, Virginia, 1985.

http://www.bia.orgIBIA/technotes/t44.htm 311512008

-- 4Yi' -- 1 - 2x4 or 1 - 2x6--- 5W -- 1 - 3x4 or 1 - 3x6---- 6" -- 2 - 2x4 or 2 . 2x6

8" -- 2 - 3x4 or 2 . 3x6---- 8Yi -- 1 - 4x6 or 1 - 6x6_10W __ 1-4x8orl-6x8

12Y4_ 1 - 4xl0 or 1 - 6x10

ATS-AB anchor bolts are pre-assembled anchor bolts that have been designed foruse with the ATS system. They are available in 18", 24" and 36" lengths and match thestrength and material grade of the corresponding Strong-Rod connecting rods. Theheavy hex nuts are pressed onto the bolt to keep them in place.

Material: Standard (Model ABJ - ASTM A307, Grade AHigh strength (Model AB_H) - ASTM A449 or ASTM A193, Grade B7Higher strength (Model AB H150) - ASTM A434, Class BD or ASTM A354, Claks BD- i

Finish: None i

Naming Scheme:

ATS-AB5Hx24ATS =::J T L Length

Anchor DiameterBolt and Grade

* Units in Va" Increments(Ex: 9 = 'fa" or 1%")

Anchor Bolt Bolt Diameter Plate Washer Size 1, Component ColorModel No. - (in) (in) i. (in) Cod'e

ATS-AB5 S/s 3fsxl'hxlY, 1v. Blue

ATS-AB7 1's 3fs x 2v.. x2.'A ..... r 3reen·,> ...

ATS-AB9 1% 3/8 x 23iA x 2% PiA Orange

ATS-AB5H s/s 3/S xl 'h x 1'/2 . 1'.4 Blue

ATS-AB7H 'Is 3,4, X 2'.4 x 2V. 1'h Green

ATS-AS9H 11/8 3/ax2:v..x2%. '.'

j3iA .•...Orange....

ATS-AB9H150 1% 'hx3x3 1'/8 Orange

ATS-AB10H150 1% 1 x3'hx3'h 2'h Purple

1. Anchor rods are available in 18",24" and 36" lengths.2. Standard Anchor bolts are based on minimum Fu = 60,000 psi and Fy = 43,000 psi.3. High strenqtn anchor bolts are based on minimum Fu = 120,000 psi and Fy = 92,000 psi.4. H150 anchor bolts are based on minimum Fu = 150,000 psi and Fy = 130,000 psi.

ANCHOR BOLT LOCATIONSAnrnor bolts shall be specified by the Designer. 1 - 2x4 or 1 - 2x6 = 4Vi

1 - 3x4 or 1 - 3x6 = 5 W2 - 2x4 or 2 - 2x6 = 6'2 - 3x4 or 2 ' 3x6 = 8'1 - 4x6 or 1 - 6x6 = 8Y,1 - 4x8 or 1 - 6x8 = lOW1 - 4x10 or 1 . 6x10 = 12'1.

1-2x4or1-2x6=4'fl1 - 3x4 or 1 - 3x6 = 5 'h"2 - 2x4 or 2 - 2x6 = 6'2 - 3x4 or 2 - 3x6 = 8'1 - 4x6 or 1 - 6x6 = 8'h"1 - 4x8or 1 - 6x8= 10Y,1 - 4x10 or 1 - 6xl0 = 12Y;

ATS-ABAnchor Boll

~ COMPRESSION/ MEMBERS

Perpendicular-To-WallInstallation

Corner Installation~ COMPRESSION

MEMBERSMid-Wall Installation16

ATS: Anchor Bolts Page 4 of 4

Anchor Rodf.'odel No.

ATS·A85

ATS·,l"B7H

ATS-,4B9H

ATS·ABBH15G

1. IBC calculations are based on ACI 3113,Appendix D2. For UBC and IBC wind design, embedment de, is based on the design strength of the anchor per AISC. Embedment and

edge distances are calculated in order to attain a ductile steel failure mode.3. For IBC seismic design, concrete strength is reduced by a factor of 0.75 per ACI 318, Section D.3.3.3. Steel strength is based

on AISC calculations and does not include an 0.75 reduction factor. Embedment and edge distances meet the ductilerequirements of ACI 318, Section D.3.3.4.

4. For UBC design anchor design for 2500 psi minimum concrete assumes no special inspection and a multiplier of 2.0 on theconcrete per section 1923.3.2. For 3000 psi and 4500 psi concrete, special inspection is assumed and a multiplier of 1.3 isapplied.

5. Plate washers have been designed for plate bending.6. Alternate anchor bolt solutions may be provided by the Designer.7. Foundation dimensions are for anchorage only. The Designer is responsible for the foundation size and reinforcement for all

load conditions.

ReLated Catalog Pages (PDFs): "" top

C-ATS07 (AnchorTiedQwn$ysiem),page 16 (173k) Order frE?",_cat§Log~by mail

Printed March 15, 2008 from http://www.strongtie.com/products/ats/connectors/anchor-bolts.html© 2008 Simpson Strong-Tie®

http://www.strongtie.com/products/ats/connectors/anchor-bolts.html 3/15/2008

ATS: Anchor Bolts Page 3 of 4

Anchor RodModel Nu.

•• See footnotes below

Wind and Seismic Design 97 USC without Supplementary Reinforcing:

Anchor RodModel No.

ATS-ABS

ATS-AB7

ATS-'/\B9

Jl,TS-AB5H

.w.TS-AB7H

I\TS-AB9H

ATS·AB9H1 SO

• See footnotes below

Seismic Design All IBC Codes:

Anchor RodIWodel No.

A,TS-AB5H

ATS-AB7H

•• See footnoles below

Wind Design All IBC Codes:

http://www.strongtie.com/products/ats/connectors/anchor-bolts.html 3/15/2008

17 Sample Vessel 8 <- Vessel

19 Vessel Dimensions (Inch and Lbs):20 130.000 <- H, height21 80.000 <- L, center of gravity22 26.500 <- Is, leg free length23 42.000 <- Do, shell outside diameter24 44.500 <- ds, leg pitch diameter25 0.750 <- t, shell corroded thickness26 0.250 <- ws - leg weld size27 13.500 <- Iw - length of leg to shell weld28 16.000 <- Iwf - length of weld on foot29 12,300 <- W, Weight Ibs30 353.9 <- Pr, Pressure

32 Site Specific Seismic Information per NBC-95:35 1.000 <- I, occupation importance factor44 0.400 <- v, zonal velocity ratio45 6.000 <- Za, acceleration-related seismic zone46 5.000 <- Zv, velocity-related seismic zone47 1.300 <- Foundation Factor (F)

52 Leg Supports:53 Angles 4" x 5/8" <- Structural Description54 4 <- n, number of legs55 6.660 <- lx, for one leg56 6.660 <- Iy, for one leg57 1.200 <- fFactor, Least radius of Gyration58 4.610 <- A, Leg Cross Sectional Area59 4.000 <- 2cx, Beam Depth60 4.000 <- 2cy, Beam Width61 0.800 <- K1, Leg Anchor Factor

63 Material Properties:64 17,100 <- maximum leg bending stress (Sb)65 16,200 <- maximum shell stress (Sa)

67 Attachment Dimensions:68 5.657 <- 2C1, Width of rectangular loading69 13.500 <- 2C2, Length of rectangular loading

71 Static Deflection72 E = 30,000,00073 bc = 12.0 leg boundary condition based on fixed or loose leg74 y = (2*W*ls"3)/(bc*n*E*(lx + Iy» y = 0.02475 (2*12300*26.5"3)/(12*4*30000000*(6.66 + 6.66»

15 Vessel On Beams Ver2.24 27-Apr-07 Page 22 of 25

NBC-95www.pveng.com

Fv

~ __ +_ Fh \iiiJ Fhc.g.

Fv

f3

vds

o

---.u -:\.AI..--------- ... -CLRECTANGULAR ATTACHMENT

x

77 Period of Vibration78 g = 38679 T = 2*pi*sqrt(y/g)

84 Base Shear103 U = 0.6

104 R = r-=4__ -:-::--::-;:-,105 I 4.2001<- Seismic Response Factor (S)106 Ve = v*S*I*F*W = 0.4*4.2*1*1.3*12300/107 V = (Ve/R)*U = (26863.2/4)*0.6 .

=2 * 3.14 * sqrt(0.02/386) T = 0.049

Ve = 26863V = 14029

115 Sample Vessel 8 Vessel On Beams117 Horizontal Seismic Force at Top of Vessel118 Ftmax = 0.25*V = 0.25 * 4029119 Ftp = 0.07 * T * V = 0.07 * 0.049 * 4029120 Ft = if (T < 0.7, 0, min(O.07*T*V, Ftmax))

122 Horizontal Seismic Force at cg123 Fh = V - Ft

27-Apr-07

= 4029 - 0

125 Vertical force at cg126 Fv = W

128 Overturning Moment at Base129 Mb=L*Fh+H*Ft = 80 * 4029 + 130 * 0

131 Overturning Moment at Bottom Tangent LineMt = (L-ls)*Fh + (H-ls)*Ft = (80 - 26.5) * 4029 + (130 - 26.5) * 0132

"0

134 Maximum eccentric load135 f1 = Fv/n + 4*Mto/(n*Do) = 12300/4 + 4*215577/(4 * 42)

137 Axial Load138 f2 = Fv/n + 4*Mb/(n*ds) = 12300/4 + 4*322358/(4 * 44.5)

140 Leg Loads141 f3x = 0.5*V*lx/(lx+ly)142 f3y = 0.5*V*ly/(lx+ly)

144 Leg Bending Moments145 e= (ds-Do)/2146 Mx = f1 *e + f3x*ls147 My = t1 "e + t3y~ls

149 Leg Bending Stress150 Sbmax = Sb * 1.25151 fx = Mx*cx/lx152 fy = My*cy/ly

=0.5* 4029*6.66 I( 6.66+6.66)=0.5* 4029*6.66 /( 6.66+6.66)

=(44.5-42)/2=8208*1.25 + 1007*26.5=tl2otn.25 + 1UUr20.5

=17100 * 1.25=36955 * 2 f 6.66=36955 * 2 f 6.66

AcceptableAcceptable

154 Leg axial stressK1*Is/r = =0.8 * 26.5 I 1.2

Fa max = AISC code lookup based on K1*Islrfa = f2/A =10319 14.61

155

''156

Acceptable~57

159 Maximum Euler Stress160 Fe = 12*piA2*E/(23*(K1 *L1r)"2)161 = 12*piA2*30000000/(23*17.667A2)

163 Combined Stress164 Fc1 = fa/Famax + O.85*fxl«1-fafFe)*Sbmax) Acceptable165 2238/25675 + 0.85*11098/«1-2238/494954)*21375)166 Fe2 = fa/Famax + 0.85*fy/«1-fa/Fe)*Sbmax) Acceptable167 2238/25675 + 0.85*11098/«1-2238/494954)*21375)

Page 23 of 25

Ftmax = 1007Ftp = 13.94

Ft= 1L...:...9 __

Fh = 14,029

Fv = 112,300

Mb = 322,358

Mt = 215,577

f1 = 8,208

f2 = 10,319

f3x = 1,007f3y = 1,007

e = 1.25Mx = 36,955My = 30,955

Sbmax =fx = ,...,.,....,:-,.-........,......,

fy = ~'"'--"---'

K1*Islr = 17.667Fa max = 25,675

fa = 12,.23$

Fe = 494,954

Fe1 = 10.53

Fc2 = 10.53

171 Sample Vessel 8 Vessel On Beams 27-Apr-07

173 Beam to Shell Attachment Stresses

181 C dimensions for weld stress182 weld area = ws*lw wa = 3.375183 wex = Iw/2 wex = 6.750184 wez = ey + ws wez = 2.250185 wey = sqrt(wex"2 + wey"2)

187 Shear Force Distribution188 Vx = (V*lx)/( (n/2) "(Ix+ly)189 Vy= (V*ly)/( (n/2)*(lx+ly»190 Vg = Win gravity

175 Beam Dimensions176 ex = 2cx/2177 ey = 2cy/2178

179

ex = 2.000cy = 2.000

= sqrt(6. 5"2 + 7.115"2)

= (4029. 8*6.66)/«4/2)*(6= (4029 .. 8*6.66)/«4/2)*(6= 12300/4

192 Weld Moments of Inertias193 Iwx = (ws*lw"3/12)*2 = (0.25*13.5"3/12)*2194 Iwz = (lw*ws"3/12 + wa*(cy+ws/2)"2)*2195 (13.5*0.251\3/12 + 3.375*(2+0.25/2)1\2)*2196 Iwy = Iwx + Iwz = 103 + 31

198 Weld Moments199 Mx =200

201 My1 =202 Mz=

Vx*(ls+lw/2) + Vg*(ds-Do)/21007*(26.5+13.5/2) + 3075*(44.5-42)/2Vy*(ls+lw/2) = 1007*(26.5+13.5/2)Vy*(ds-Do)/2 = 1007*(44.5-42)/2

204 Weld Stresses205 Sx = Mx*wcx/lwx200 Sy = My1 *wcy/lwy207 Sz = Mz*wcz/lwz208 Sg = Vg/(wa*2)

210 Stress Limits and Ratios211 Slim = min(Sb,Sa)*OA9

213 SxR = Sx/Slim214 SyR = Sy/Slim215 SzR = SzlSlim216 SgR = Sg/Slim217

218 Foot Plate Attachment Stresses219 waf = ws*lwf220 Vv= V/n

222 Sv = Vv/waf223 Sgf = Vg/waf

225 SvRf= Sv/Slim226 SgRf= Sgf/Slim227

BendingTwistingTorisionGravity

= 36955*6.75/102.5= 33495*7.115/133= 1259*2.25/30.5= 3075/(3.375*2)

= min(17100,16200)*0.49

= 2433/7938= 1791/7938= 93/7938= 456/7938

Acceptable

weld area in foot = 0.25*16= 4029/4

= 1007/4= 3075/4

= 252/7938= 768.75/7938

Acceptable

Page 24 of 25Mz

ws:

: IV1x-rI

wey= 7.115

Vx = 1,007Vy= 1,007Vg = 3,075

Iwx = 102.5Iwz= 30.5

Iwy= 133.0

Mx = 37,339

My1 = 33,495Mz = 1,259

Sx = 2,433Sy = 1,791Sz = 93S9 = 456

Slim = 7,938

SxR = 0.307SyR = 0.226SzR= 0.012SgR = 0.057

total «1) 0.601

waf = 4.000Vv = 1,007

Sv = 252Sgf = 769

SvRf = 0.032SgRf = 0.097

total «1) O~129

27-Apr-07 Page 25 of 25

WRC 107 - shell local stress at supportTO>

'" Loads (psi and Ib)1,007,4

10,319.00.0

36,955.00.00.0

<- P, Axial Load (=vx)<- VL, Lon9itudinalload(=f2)<- Ve, Circumferential load<- ML, Moment (=My)<- Me, Moment-c, MT, T orisional

'"

242 ParametersMaxSPm;:;; Sa for Pm stresses

MaxSPmb:: 1.5*Sa for Pm + Pb stresses

MaxSPmbQ = 3·Sa for Pm + Pb + a stressesRi = (00-2'T)/2

Rm = (00-T)/2r = RmlT

Beta 1 = 2C 1/2/RmBeta2 = 2C2/2/Rm

SL = (Ri-OA'T)'Pr/(2'T)Sc = (Ri+0.6'T)'PrlT

"" Stress concentration factors283 Shell Combined Stresses'

Pm - primary membrance stressPb - primary bending stresso - secondary stress

MaxSPm = 16,200MaxSPmb = 24,300 0" '--1-"'--4>f-=':

MaxSPmbO = 48,600 0LRi = 20.25 -~

Rm = 20.625 ---- --"-,...- r?7<n'I A. -1 !llIl "

r-~ Al Cull t

Beta t = 0137--------- --- .•.•--Bela2 = 0:327. C L :

SL = 4,707 "tCT4HGULAR ATTACHMt •• rSc = 9,768Kn = 1

= 20.625/0.75

= 5.65712120.625= 13.512/20.625= (20.25-0.4'0.75)'353.898/(2'0.75)= (20.25+0.6'0.75)'353.898/0,75

Kb = 1

see

Lookup A Curve A Value A Value Equation Cat Au AL Bu BL Cu CL Du DLPressure Stress VIII-1 Code 4C 3C SC Pm 9768 9768 9768 9768 9768 9768 9768 9768

No/(P/Rm) 3C or 4C 3,74796 1.88561 Kn'A'P/(Rm'T) Pm -244 -244 -244 -244 -123 -123 -123 -123Mo/P 1C or 2C-1 0,08088 0,04871 Kb'A'6'PfTA2 Pb -523

-No/(Mc/(Rm'2'beta» 3A 1.05302 Kn'A'Mc/(RmA2'beta'T) PmMo/(MC/(Rm"'bela» 1A 0.08268 Kb' A '6'McI(Rm'beta'TA2) Q o 0 0 0

No/(MLJ(Rm"2'beta» 38 2.75635 Kn'A'MU(RmA2'beta-T) Pm -1304 ,1304 1304 1304, Mo/(MU(Rm"beta» 18 or 18-1 0.01754 Kb·A*S*MU(Rm·beta'"TA2) Q -1086 1086 1086 -1086

Pm So 8220 8220 10827 10827 9645 9645 9645 9645Pm+Pb So 7696 8743 10304 11351 8776 10514 8776 10514

Pm+Pb+Q So 6610 9829 11390 10265 8776 10514 8776 10514Pressure Stress VIII-1 Code SL Pm 4707 4707 4707 4707 4707 4707 4707 4707

Nxf(P/Rm) 3C or 4C 3.74796 1.88561 Kn'A'P/(Rm'T) Pm -123 -123 -123 -123 -244 ,244 -244 -244Mx/P 1C-1 or 2C 0.03139 0.05870 Kb'A'6-PfT'2 Pb -631

-Nx/(McJ(Rm"2°bela» 4A 1.71344 Kn' A 'McI(RmA2'beta'T) PmMx'(Mc/(Rm~beta) 2A 0.03430 Kb'A-6'McI(Rm'beta'TA2) Q •.... 0 0 0 0

Nx/(MLI(Rm"2'beta)) 4B 1.12882 Kn'A-MU(RmA2'beta'T) Pm -534 -534 534 534 .Mx/(MLI(Rm'beta ) 28 or 28-1 0.03569 Kb'A'6'MU(Rm'beta'P2) 0 ,2556 2556 2556 -2556

Pm Sx 4050 4050 5118 5118 4463 4463 4463 4463Pm+Pb Sx 3419 4681 4487 5749 4125 4800 4125 4800

Pm+Pb+Q Sx 864 7236 7043 3193 4125 4800 4125 4800Shear VL VU(Pi'sqrt(cl'c2)"T) f",'.ne -1002 ,1002 1002 1002Shear vc VC/(Pi'sqrt(c1'c2)'T) 0 0 0 0 )~.£%

, Tolal Shear Sum of shears Txo 0 0 0 0 -1002 ·1002 1002 1002S1m ( Sx+So)/2)+SQRT«(Sx-So)/2)A2+ T xc-z) 8,220 8,220 10,827 10,827 9,832 9,832 9.832 9.832S2m «Sx+So)/2),SQRT«(Sx-So)/2)'2+ Txo'2) 4.050 4,050 5,118 5,118 4,276 4,276 4,276 4,276S12 abstS'lm- S2m) 4,170 4,170 5,709 5,709 5,556 5,556 5,556 5,556823 abs(S2m-O) 4.050 4,050 5.118 5.118 4,276 4.276 4,276 4,276, 831 abs(0-S1m) 8.220 8,220 10,827 10.827 9,832 9,832 9,832 9,832

, Sm<= MaxSPmb max(S12,S23,S31) Acceptable 8,220 8,220 10,827 10,827 9,832 9,832 9,832 9,832S1m+b ( Sx+So)/2)+SQRT«(Sx-So)/2)A2+ rxc-z) 7,696 8,743 10,304 11,351 8.983 10.685 8,983 10,685S2m+b (Sx+So)/2)-SQRT« Sx-So)/2 A2+Txo'2) 3.419 4,681 4.487 5,749 3.919 4.629 3,919 4,629

S12 abs(S1 m, S2m) 4.277 4.062 5,817 5,602 5,064 6,055 5,064 6,055, S23 abs S2m-0) 3,419 4,681 4,487 5,749 3,919 4,629 3.919 4.629a S31 abs(0-S1m) 7,696 8,743 10,304 11,351 8,983 10,685 8,983 10,685, 5mb<= MaxSPmb max(S12,S23,S31 ) Acceptable 7,696 8,743 10,304 11,351 8,983 10,685 8,983 10,685

S1m-rb-rQ «sx-se /2 +SQRT(((Sx-So)/2 '2+ Txo'2 6,610 9,829 11,390 10,265 8,983 10,685 8,983 10,685, S2m+b+Q «Sx+So)/2 -SQRT«(Sx-So)/2)A2+ rxc-z) 864 7,236 7,043 3,193 3,919 4,629 3,919 4,629

S12 abs(S1m, S2m) 5,746 2.593 4,347 7,071 5,064 6,055 5,064 6,055S23 abs(S2m-O) 864 7,236 7,043 3,193 3,919 4,629 3,919 4,629. S31 abs O,S1m) 6,610 9,829 11,390 10,265 8,983 10,685 8,983 10,685

Smbce MaxSPmbQ max(S12,S23,S31) Acceptable 6,610 9,829 11,390 10,265 8,983 10,685 8,983 10,685