5.0 Support for vertical vessels t (A)
Tall Cylindrical Process Columns \ t
• Supported on cylindrical or conical shells (ski • The support skirts are directly welded to the v • The skirt base is stiffened by a continuous sti bottom annulus plates with intermediate verti bending stresses. • They are designed as cantilever beams
(B)
s) ssel bottoms head or shell ening ring, which consists of top and al stiffeners, to reduce localized
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"
, , ~--7
Number of legs: N = 3 for D < 3' 6" N = 4 (or more) for D > 3' 6"
'D
,
H
, ,
Maximum operating temperature = 6500p
1
5.1 Leg Supports • 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.
I
I
I
I
I
I
I
'-'-'-'-r-'-'-'-'
I
.- ._.- .-r·- .-. - .- .
I
I
I
I
I
I
I I I I
'-'-'-'-i'-'-'-'-' I I I
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 round pipe may be used as leg column, which has equal strength in all direction and has a high bending rigidity.
2
Loads on the vessel 1. The wind load (Pw) is horizontal and acts at t e centroid of the projected exposed surface 2. The earthquake load (Pe) acts horizontally on he center of gravity of the vessel 3. 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
Vertical reaction (R) due to turning moment = Mg/D,
(Db = Base diameter)
J~(
R
~
1 ~...::..b __
-7)1
In the case of 3-leg support R '"",,---'-,
3 MD =R(-+-sm30)=-RDb 224 Db
Db'
""
,
0
, 120
.--.-~--
/i I I I
=>
I
General expression:
3
Maximum load on the leeward side (compreSSioj side) is:
Co
=
Wo + 4Mb N
(Operating condition)
NDb
C =WT o N
(Test condition, no win loading)
Maximum axial load on the windward side (upli Wo 4Mb T =--+-o N ND b
(operating condi ion)
T =_We + 4Mb o N NDb
(empty)
The eccentric loads P1 and P2 at the column top are:
11 = ~
+~
(operating condition)
11 = WT
(test condition)
N
D2 1-
-__
Wo + 4M N
P2
=-
a
ND
We + 4M a N
(
. operatmg con dition) rtion
(empty)
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
. CompresslOn stress · Ben dmg stress
=
C
= -
ab
11(d/2)·(a/2)
-.
ba~/12
=
11(d/2) 2
ba 16
Compressive stress must be always greater than
e bending stress. d
b
)1
d
1<
1<
a
)1
Weld Size Shear stress = PI 1 (2L1 + h) Bending moment on the weld joint = C (d/2) + F (L/4)
h
LIrI:::-lLII ge;~etry
Size of Anchor bolt
S
a
·Ab
4M
= ( NDbb
-
WJ N
Anchor bolts are designed to resist the uplift force. 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 longitudinal moment arises. In each case the vessel wall is su ject to the extemallongitudinal moment of 'Fd' where 'F' is the maximum react" on at the support and 'd' is the distance from the centerline of the support to the shell outside surface.
The stresses in the shell induced by the Bracket can be found by the local load method.
Vessel Wall
~
I (d
)
Forces and stresses on the bracket
: Top bar
I
~------~I--------~~
T h Vplate
ta
R
Gusset ------
Base plate
F/2
F (
b
(
d
) )
a )
(
F .d 2
=R
. d sin a
R=
F 2sina
6
Maximum compressive stress (Sg) on the gusset (regarded as an eccentrically loaded plated) I I I
s = g
R (bsina).tg
+ 6Re
i
£t.tg
I I
I where the force eccentricity,
e = (d - b)sina
!
2
Force and stress on the Top Bar
The top bar is assumed to be a simply-supported Fd/ha
earn with uniformly distributed load
Maximum moment occurs at the mid-span M
=(Fd).~=Fda max
ha
" Ben d mg stress
8
=
6M
8h
2
(2":::;c:::;8ta)
fa . C
7
6.0
Saddle Supported Cylindrical Vess Is
The code design of saddle-supported horizontal ylindrical vessel follows the work of L.P. Zick (1951, The Welding Journal Research upplement) who used a modified beam and ring analysis so that the mathematical odel agrees with the experimental results he had available. Most recent work has indicated that Zick's appr ch gives reasonable agreement only when a flexible saddle support is employed. Wh n the saddle is rigid the simple Zick's analysis significantly underestimate the peak str s in the vessel by a factor as much as 50%.
6.1
General considerations
(a)
Saddle supports should be located to cause minimum stress in shell and without additional reinforcement
(b)
Most vessels are supported on two saddle supports. The saddles have an embracing angle between 120 and 150 degrees. Any relative settlement of the supports does not change the support reactions, therefore, the stresses in the shell remain quite the same.
(c)
In the case of very long vessel that rested on more than two supports, the support reactions are calculated based on continuous beam theory and increased by 20 to 50% as a safety factor for relative support settlements.
(d)
The support reactions are highly concentrated and they induce highly localized stresses at the support regions. To reduce highly localized stresses, the saddle must be designed to provide flexibility at the support-shell junctions. Extended plates or wear plates may be used to provide a gradual transition of structural rigidity between the support and the vessel's shell.
(e)
One of the saddles should be designed at the base to provide free horizontal movement, thereby avoiding restraint due to thermal expansion.
8
The Mathematical Model
4.: +--
H ~
I I I I I I I I
Rorr
I I I
-
~-j
_._._._._._.-.-._'
.-.-.-.-.-.-.-.-.-.-.-.-.-.-.~.-.-.-.-.
-
,
-.
I
,
, I
<
A
<
)
A
L
~
<,
, -'
I
WR2/4
)
2Hw/3
2Hw/3 w
~
~/8 Q
< <
A
)1
3H/8 ---.
Q
1<
L
A
)WR2/4
~
)
>
Shear force diagram Q
Bending moment diagram
9
Points to note: (1) (2)
(3) (4) (5)
The support reaction is Q - the total weigh is 2Q 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 from end of parallel The total length is = L + 4HJ3 The uniform load has a intensity w = 2Q/( +4H/3) The hydrostatic pressure that acts normal t the dished end creates a couple given by WR2/4
bending stresses in the she I
6.2
Longitudinal
(a)
The bending moment at the mid-span 2 M} =--Hw 3 =
QL
[1
4
'
(3H wR2 wL L -+- L) +---(-)+Q(--A) 82 4 24 + 2(R2 - H2)/ L2 _ 4A] 1+(4HI3L) L
L 2
=K{~L) Bending stress at the mid-span:
The above expression assumes that the full vessel section is available in resisting bending stresses and the cross section remains circular. For very thin vessels it is found that the cross section does not remain circular especially so during filling with liquid. Nevertheless, the expression gives satisfactory design dimensions for vessels with D/t ratio up to 1250.
(b)
The bending moment at the saddle-support
M2
=:": 2
= QA[I-
(3H S+A
)
2 2 wR wA +-4---22
2
1- AI L + (R - H )/(2AL)] 1 + (4H /3L) 10
The top portion of shell above the saddle support would feform under load and is deemed ineffective in resisting longitudinal moment. So the moment of inerti at this cross section is reduced to that of a ring with its top portion removed.
2i1 = 2(8/2 + ~ 1 )
The effective arc is assumed to be:
I
yJ:---
0o ::
t,
NA
.................... .....- ~+
Effective portion
~C_~
The position of the neutral axis, N.A. and the second moment of area I about this axis can be found.
_ rsini1 y= ~
3.
Cl
= r (sini1 ~
I = r t i1 + sm i1cos i1 - 2 [
J
- cos A ,
. 2 D. A] sm ~
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 the allowable 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 the saddle depend a great deal on how the shell is reinforced. ! !
The inner shear force,
2 H)= ~(L-2A) 3 f + 4H 13
V=Q-w(A+
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)
=-:'---~ <
L+4H13
(A)
2Q(A + 2H 13)
or
L> 4A +4H13
L+4H13
Shell is stiffened by a ring at support region
If the shell is made rigid, the whole section is effective in resisting load-induced shear stresses. The shear flow (shear force per unit arc length) is:
qo
v .
d,
= -SIlly
nr
(~ is measured from the top of the cross section)
The maximum stress flow is when ~ =90 degree. =>
Max. shear stress = qo = t
(B)
Q [ Lnrt
2A
L + 4H 13
]
Shell not stiffened by ring
When the shell is free to deform above the saddle, it is considered that the shear stress acts on a reduced cross section. The upper portion of the shell is considered ineffective. The effective portion is assumed to be:
12
2~ = 2(e / 2 + jJ /20) = -19 ( J( 20
--
()) 2
i I
As a result, the shears in the effective portion will be increased by a factor:
r
2
Factor C = qo(unstiffened shell) = sin ¢d¢ = J( , qo(stiffenedshell) fsin2¢d¢ J(-a+sinacosa
The shear flow, qo
Vsin~) nr
= C(
The shear stress, S2"qo =t
= c(Vsin¢) --srt
The maximum shear is now at the tip of the saddle, i.e. ~ = a
(C)
Shell stiffened by heads (A
If the saddle is close to the end closure the shell is stiffened on the side of the head. It is 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 stiffened region. 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 be I found by summing the shears to one side of the saddle. The sum of vertical shear force for the upper portion is equal to the sum of verticrl shear force in the lower portion.
I I ---------------------------------J
-------
I I
I I
, I
I
Shear force near the enr closure
I
~r--r-
------------------------------1
-----~-,
-------
,
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~) nr
The shear stress, S
qo
=-
t
= ~. Q (a - sin o.cosc.Isin e nr n
Q a - smacosa . . tcrt n - a + smacosa
=-
. sin
~
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 without attachment. Ring compression is caused by shear forces. The ring compression in the region a :s; ¢ :s; J[ The shear flow is: q3
v . d.( = -smlf' Jrr
1[ J[ -
a
)
.
+ smacosa
The total shear force at any point on the shell arc above that point.
Total shear force
¢ =
Q(cos¢ - cosa)
=
fq3rd¢
.
J[ -
a
a + smacosa
The contact pressure between the saddle and shell would induce a tangential compression force similar to the above. That is: Tangential compression force due to saddle force
Therefore, the maximum tangential force is =
=
_ J[ -
Q( cos¢ - cos fJ) J[ - fJ + sin j3 cos j3 Q(1 + cos fJ) fJ + sin f3 cos f3
The contact pressure can be deduced from the tangential force:
1 Contact pressure = _.
Q(cos
d. --='---'----'If'__
r
J[ -
cos /3)
~-'----
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 side plus the width of the saddle, i.e. width = b + -.
In a follow-up paper, he suggested width
=
b + 1.16-Jrl
I
The tangential stress can be calculated, S5 = shea~ force / width The stress S5 is important when concrete saddle diameter vessel.
JI
used. It should be checked for large
Recent experimental and theoretical work on sad les welded to the vessel have found that this tangential stress is very small, about 111 of that predicted by Zick's approach. However, for the saddle not welded to he shell, the Zick's approach gives the correct order of stresses. The ring compression may be reduced by attaching a wea plate somewhat larger than the saddle surface area directly over the saddle.
Allowable stress limits The compressive stress S5 should not exceed 1/2 of S, and is not additive to the pressure stress. If wear plate is used, the combined thickness of wear plate and the shell can be used to calculate S5, provided the wear plate extends r/I 0 inch beyond the horn.
Despite the limitations of Zick's approach it does provide a workable design method that has been used extensively over many years. However, the very high circumferential stresses known to exist at the saddle horn region when the vessel is supported on a rigid saddle at not predicted adequately by the analysis. Although these peak stresses do exist, they are very local to the saddle horn and are unlikely to cause plastic collapse of the support. However, their existence does cause concern when the vessel is subject to high cyclic stressing.
16
Local stresses in shell due to loads on attachment I
Types of attachments:
Nozzles, supporting lugs,i lifting brackets, etc.
! Main concerns - High concentrated stresses at tHe attachment due to combined internal pressure and external loads applied through the attachment can be a source of failure if proper reinforcement is not supplied. ! i
Design 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 the opening, where d is the diameter of the opening. • Sharp junctions should be avoided; fillets should be incorporated to reduce the magnitudes of stresses at the junctions. "The best arrangement is the so-called balanced reinforcement, which consists of about 35-40% of the area on the inside and about 60-65% on the outside. On many designs, however, it is difficult to place reinforcement on the inside. Balanced reinforcement is often used at manway and inspection opening where no nozzle is attached"
Area Replacement
for Nozzles
This method formed the basic design method in many design codes. The origin of the area replacement idea is not entirely clear. Simply expressed one replaces the area cut away by the cross section of the hole in the shell and relocates it around the hole close to 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 to the edge of the opening where the stress field is increased. The extent of the reinforcement is preferably equal the die-out distance of the peak stresses at the edge of the opening. That is why in some codes the extent of reinforcement is expressed as a function of Jrl, the characteristic length parameter for the die-out distance of the discontinuity stress. In any case one simply obeys the rules as stipulated and no explanation is given. It should be noted that the distance for reinforcement is generally quite 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
ri-,
-r------
The shear stress in shell due to the torque Tis:
Maximum shear due to VL or Vt is:
The shear stresses rand
t'
v
r'=_t_
Jrrot
T
- 2Jrrot - 2Jrr;t
or
V
-r,=_L_
Jrrj
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 area Cylindrical attachment: ~ = O.875ro / R Rectangular 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
i
For different loadings, the circumferential and tlhe longitudinal stresses are expressed in different parametric forms as follows: I (1) Radial load, P
0. =(;'XcP~;)y 6;.] +
=Cp(Plt2) =
(2)
(outward force)
C~ (P I t2)
(inward force)
Circumferential (tangential) moment, M,
u, {Nep O"ep
= t2 R~ =
(3)
i
(Mt I R2~)'Y + (M{ I R~)
C (M t
6Mep}
t
1(2 R2B )
Longitudinal moment, ML &
Design considerations (a)
lfthe maximum stress at the attachment is too high, the shell must be reinforced by a reinforcing pad or the thickness of the reinforcing pad required for internal pressure must be increased. The width of the pad is such that stresses at the edges 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-out distance, then their influence on each other must be considered.
Note: The analysis presented above for local loads applied on cylindrical shell is too simplistic. More
detailed and accurate analyses for different types of attachments are available in the literature and recommended by design codes, specifically for loads on the nozzle, and openings. For example, the Welding 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 can be conducted then a better, less conservative, a sessment of the design can be made compared to the usual approach of Design by le. The philosophy was originated in the 1960's in the US. The motivation was drive by the sophisticated design work in the nuclear industry. There were many design £ atures that were not covered directly by the existing Design by Rule methods. In the early years, all design by analysis ideas ere developed based on thin shell analysis and in particular the analysis of discon inuity effects including thermal discontinuities. It was suggested that different types of stress h d different degrees of importance and this 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 specified stress limits that are set at different levels for the different stress categories. This methodology was first incorporated in ASME PV code Section III and Section VIII Division 2 in 1968 and later into BS 5500 as Appendix A. Many countries have now adopted the same basic approach.
Multiaxial Stress States In real world, all stresses are three-dimensional. It is the simplifying assumptions that reduce the 3-D stresses into 2-D and I-D. Yielding in the presence of multiaxial stress states is not governed by the individual component but by some combination of all the stress components. The two commonly used yield criteria are the Von-Mises criterion and the Tresca criterion. Von Mises criterion (distortion energy theory) states that yielding will take place when;
Tresca criterion (maximum shear stress theory);
= +0" y
/2
20
Although it is generally agreed that the Mises criterion is better for common pressure steel, ASME chose to use the Tresca criterion as a framework for the Design by Analysis procedure. The reason is that Tresca is the more conservative and it is easier to apply. The later is longer true now since computer can perform complex calculations at ease. In order to avoid the unfamiliar (and unnecessary) operation of dividing both calculated 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: Throughout
S=(J'
Design by Analysis procedure
y
stress intensities are to be used.
Stress Categories Certain types of stresses are more important than others and that these should be assigned to different categories with different levels of importance having different stress 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
(C)
Peak Stress, F
Stress,
Q
Primary stress is a stress developed by the imposed loading that is necessary to satisfy the law of equilibrium between external and internal forces and moments. The basic 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 and support.
21
Secondary stress is stress developed by the self-constraint of a structure. It must satisfy an imposed strain pattern rather than being in equilibrium with an external load. The secondary stress is self-limiting, its may cause local yielding and minor distortion resulting from discontinuity condition or thermal expansion. Peak stress is the highest stress in the region under consideration. The basic characteristic of a peak stress is that it causes no significant distortion and is objectionable mostly as a possible source of fatigue failure.
Failure modes 1. Excessive elastic deformation incl ding elastic instability 2. Excessive plastic deformation 3. Brittle fracture 4. Stress rupture and creep deformati n 5. Plastic instability - incremental co lapse 6. High strain - low cycle fatigue 7. Stress corrosion 8. 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 Yield
Sm
2S 3 y
Local primary membrane, PL
1.5 Sm
Sy
Primary membrane + bending, (Pm + Pb) or (PL + Pb) Primary + secondary (PL + P, + Q) or (Pm + P, + Q) Fatigue, (PL + Pb + Q + F) or (Pm + Pb + Q + F)
1.5 Sm
Sy
3Sm
2Sy
2Sa (allowable fatigue stress range)
-
General primary membrane, Pm
The above limits are not always applicable; there are a number of special cases. In the case of nuclear vessels the service loadings are classified into normal, upset, emergency and faulted conditions. This is formalized in ASME with k-factors applied to 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 .25 Sm(1.5 Sy) i
Some cautionary words are necessary for the u wary. The manner in which the symbolism is used can lead to confusion. For e ample a stress limit on some combination of stress categories denoted as CPL + P, + Q) needs to be clearly understood. It is the stress intensity evaluated om the principal stresses after the stresses for each category have been added tog ther in the appropriate way. It should not be interpreted as the combination of stress ~ntensity from each category.
I In summary: ONLY add stresses, DO NOT add stress intensities.
A trivial example of the wrong way of summing the stresses in given below: Stresses Sx = S] Sy = S2 Sz = S3
Pm 10 10 -2
Q 25 -5 0
Pm+Q 35 5 -2
The maximum stress intensity for Pm = 12 The maximum stress intensity for Q = 30 The 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 the appropriate locations for the identified combination of loads. The approach is to evaluate all the stresses for the different types of loading. These should be assigned to categories as necessary. Then the stresses in the various categories should be summed and finally the stress intensities calculated for the particular combination of categories required.
23
FE Analysis for Pressure Vessel Desig The Design by Analysis is closely rooted in thi shell discontinuity analyses. When FE (finite element) method is used, some diffi ulties in stress categorization occur. The FE gives accurate stress information for c mplex geometries. These stresses may vary nonlinearly through the thickness. For ass ssment purposes it is necessary to linearize the stress distribution and separate membrane and bending effects. In a simple case the procedure would be straightforward and membrane, bending and peak elements of the stress could be identifies. Unfo unately things are not always so simple. Firstly the Linearization procedure is it elf subject to a number of uncertainties. Secondly the bending componen in general may include primary bending as well as secondary bending. In practice it tends to assume the membrane str ss intensity as primary and the bending stress intensity as secondary (which m y not be conservative). In critical situation 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 the categorization problem or at least simplify its interpretation. The Standards allow the design to be based on limit load analysis with a suitable factor where the factor has to be the same as the main shell (i.e. 1.5). Design may proceed directly with a factor on load without detailed consideration of the stresses. The approach seems promising if it can be extended to complex loading situations it could provide a relatively simple alternative 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 bending stresses 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 by which membrane and bending stresses are determined directly - so there is no much confusion in the classification of stresses. With the advent of finite element (FE) techniques, the transition from the stress distribution to the failure mode requires a different path. The results of axisymmetric or 3-D solid FE analysis are not immediately in a form suitable for the extraction of shell type membrane and bending stresses. Difficulties are associated with linearization procedure used to obtain membrane and bending stresses.
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 and secondary. The problems of assessing primary and second ry stress failure modes and their relationship to stress results from axisymmetri and 3D geometries were first addresses by Hechmer and Hollinger in 1986. "3D stress criteria - a weak link in vessel design and a lysis", PVP Vo1.109, A Symposium on ASME Codes and Recent Advances in PVP and Valve Technology including a Survey of Operational 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, Pressure Vessels, and Components, July 1987, ASME, New York, NY.
The study shows that the 3 approaches can give substantially different results. The most complex of the three approaches is stress-on-a-plane. The definition of the plane for 3D geometries is subjective and the resultant stresses and conclusions are merely engineering judgement. Some issues: It should be emphasized that these issues actually arise from the nature of the Code rules, rather from any deficiency in the finite element solution. In 2D axisymmetric analyses, the bending stress can be calculated using component normal and/or shear stresses or principal stresses. The distinction is that for a given set of geometric reference axes, component stress directions remain constant with location and load application whereas principal stress directions vary with location and load application. This distinction is important when considering the pros and cons of using component versus principal stresses. Which stresses are consistent with bending theory? Code implies that bending is applicable only to normal stress components, because the Code links bending to bending moments. Mathematically, one can calculate linearized shear stresses and call it a bending stress. However, it is difficult to conceptualize a bending moment for any shear 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 the use of planes. Two PVRC grants were established to investigate and document the need to update the ASME B&PV and Piping Code criteria and re~uirements for relating 3D stress distributions 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 and Applications 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 Pm is small (for example, the plate structures). Stresses for Pmneed only be evaluated in basic structural elements. Designer should apply his ingenuity to calculate equilibrium stresses, not to extract stresses from a general FE model. • It is appropriate to calculate PL stresses in the vicinity of all discontinuities. There are discontinuities where PL stress exists, but need not be evaluated. Because code rule reinforcing rules ensure that PL limit is met. o Nozzle-shell junctions o Formed heads to shells o Cones to shells o 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 net force is then subtracted from the net force distribution that is used to calculate the bending 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 the transition elements). The reason is that plastic collapse and gross strain concentration will not occur in the stiff transition elements. They will occur in the more flexible shell element. If a fatigue analysis is to be performed within a transition elements due to high stress concentration, it may be appropriate to consider the (P + Q) in the adjacent structural elements.
• For assessing the membrane stress limits (Pm + Pb), all the stress components (3 normal + 3 shear components) should be included. The average principal stresses must be computed from the average stress components through the thickness and NOT from average principal stresses. That is: compute the average stress
components first then compute the principal stresses. 26
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44
Conventional bolts are usually made to the specific project requirements by steel fabricators or they may be purchased in standard sizes (diameters and lengths) from steel suppliers. The availability and cost of conventional bolts are generally based on demand and fabrication requirements. The types of conventional anchor 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 their wide availability and relatively low cost (see Figure 1). Higher strength bolts, such as ASTM A 325 bolts, are available and can be used, but are more expensive. A washer placed against the bolt head is often used with the intention of increasing the bearing area and thus increasing the anchor strength. However, the actual strength increase obtained by adding a washer is small, if any, and under certain conditions (small edge distances), may actually decrease the tensile strength.
(::]) A) HEX-HEAD
Headed Bolts FIG. 1
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 (see Fig. 3) is available since there currently is no standard governing the geometric properties of bent bar anchors. These anchors are usually made from ASTM A 36 bar stock and are shop-threaded.
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T o v ~
.,-_,,/_~ 1h TO 1% D
~ ,_~
~
~IaI~
0
~
_I
T
A) Bl" BOLT
B) "J" BOLT
"L" and "J" Bent Bar Anchors FIG.2
A) EYE BOLT
S) "U" BOLT
C) ACUTE BEND '~" BOLT
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Other Bent Bar Anchors FIG.3
Plate Anchors. Plate anchors are usually made by welding a square of circular steel plate perpendicular to the axis of a steel bar that is threaded on the opposite end (see Fig. 4). There are no standards governing the dimensions (length, width or diameter) of the plate. The American Institute of Steel Construction does limit the fillet weld size based on the plate thickness (see Table 1). Both the plate and bar are usually made from 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 the masonry and are composed of a threaded rod or bar with a bearing plate located on the surface opposite the attachment (see Fig. 5). In the early 1900's, through bolts were used in loadbearing masonry structures to tie floor and wall systems together. Often decorative cast bearing plates were used since through bolts were visible on the exterior masonry surfaces (see Fig. 6). Today, through bolts are primarily used in industrial 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.
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Through Bolt FIG. 5
Decorative Through Bolt Bearing Plate FIG.6
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Page 60f24
,
-----_._--_.-
._---
---
.._---
-
-
.-
--
* American Institute of Steel Construction
Proprietary Anchor Bolts
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 theories behind the anchors are very similar. For this reason, proprietary anchors can be divided into two generic categories: expansion-type anchors and adhesive or chemical-type anchors. Expansion Anchors. Two different types of expansion anchors are generally recommended by their manufacturers for use in brick masonry: the wedge anchor and the sleeve anchor (see Fig 7). These anchors develop their strength by means of expansion into the base material. Wedge anchors develop their hold by means of a wedge or wedges that are forced into the base material when the bolt is tightened. The wedges create large point bearing stresses within the hole; therefore, this anchor requires a solid base material to develop its full capacity. For this reason, voids formed by brick cores and partially filled mortar joints in some brick masonry may make the construction unsuitable for wedge anchor installation.
A) WEDGE A.NCHOR
B) SLEEVE ANCHOR
Proprietary Expansion Anchors
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FIG. 7
Sleeve anchors develop their strength by the expansion of a cylindrical metal sleeve or shield into the base material as the bolt is tightened. The expansion of the sleeve along the length of the anchor provides a larger bearing surface than the wedge anchor, and is less affected by irregularities and voids in the base material than is the wedge anchor. For this reason, sleeve tnChors are recommended by their manufacturers 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 are typically not recommended by their manufacturers for use i i masonry. The reason for this is due to the embedment and setting characteristics of the two anchors. Both anchors are produced to allow shallow embedment depths and are expanded or set by an impact setting tool. The combination of shallow embedment 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-carrying capacity 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-type anchors 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 expansion anchors. Excessive torque can reduce anchor strength or may lead to failure as excessive torque is applied. 3) Expansion anchors require solid, hard embedment material to develop their maximum capacities. Some brick construction may not provide a good embedment material due to voids formed by brick cores and partially filled mortar joints. Adhesive Anchors. Two basic types of adhesive anchors are currently available. The major difference between the two is that one anchor is manufactured as a pre-mixed, self-contained system, whereas the second type requires measurement and mixing of the epoxy materials at the time of installation. The more popular self-contained types use a double glass vial system (see Fig 9) to contain the epoxy. The outer vial
contains a resin and the inner vial contains a hardener and aggregate The glass vial is placed in a predrilled hole and a threaded rod or bar is driven into the hole with a rotary hammer drill, breaking the vials
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44
In hollow brick construction, the units are laid so that the cells are aligned and provide continuous channels for reinforcing steel placement and for grouting. Depending on the design, every cell or intermittent cells may be reinforced and grouted (see Technical Notes 41 Revised). The anchor embedment detail will depend on the reinforcing pattern used in the construction. Figure 15 shows typical embedment details for conventional anchors embedded between reinforcing cells. The anchor should be solidly surrounded vertically and horizontally by grout for a minimum distance of twice the embedment depth (1b) (Figs. 14 and 15) for full tension cone development. The tension cone theory is discussed in following sections. This may require that some cells be partially grouted. A wire mesh screen can be placed in the bed joint across cells that are to be partially grouted to restrict the grout flow beyond a certain point. Figure 16 shows typical embedment details for conventional anchors embedded in reinforced cells. In this detail, the anchor may be tied with wire to the reinforcing to secure the anchor during the grouting process Again, the anchor should be solidly surrounded by grout to a minimum distance of tWife the actual anchor embedment depth, both vertically and horizontally. I
i f
DMIN.
IF;::::~~·* I;>
P;; "L"
socr
s,
",I" SOLT
CI HEA.OED ;BOLT
O)PlA15 ANCHOR
Conventional Anchors in Reinforced Hollow Brick FIG.15
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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 and concrete block construction are shown in Fig. 17. As shown, anchor bolts may be placed in the collar joint between 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 collar joint. 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 joint dimension is in the 1 in. (25 mm) range, it may become difficult to position anchor bolts in the collar joint and maintain the recommended clear distance between the masonry and the anchor (Fig. 17). The practice of using soaps to accommodate anchors larger than the collar joint is not recommended because the reduction in the brick masonry thickness around the anchor could lead to strength reductions. If the anchor dimensions required are larger than the collar joint, a detail similar to that shown in Fig 17(b) should be considered.
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GROUT STOP
B) ANCHOR IN BLOCK WYTHE
Conventional Anchors in Composite Brick/Block Masonry FIG. 17
Through bolts are typically installed after construction and grouting by drilling through the completed masonry work. When through bolts are to be installed after construction in reinforced brick masonry, care should be taken during installation to avoid cutting or damaging reinforcement while drilling the through bolt holes. Reinforcing bar locations can be identified by specially tooled joints or other marks made during construction.
Proprietary Anchors Proprietary equipment. application, application
expansion and adhesive anchors typically require special installation procedures and The manufacturer should be contacted to determine the appropriate anchor for a particular the correct installation procedure and if any special installation equipment is required. Improper 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 be embedded in head joints when facing or building brick are used. This reduces the possibility of placing anchors 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 intersect grouted cells, or should be placed in holes drilled through the faces of the units into the grouted cells. As with conventional anchors, proprietary anchors should be solidly surrounded vertically and horizontally by grout for a minimum distance of twice their embedment depth.
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Page 16 of24
n
]
"/
u '~It-::ll>
r
11
l
f'(
u
J
'V
IJ
A) GROUTED OOLL~R ,JOINT CONSTRUCnON
'b
J
~
! l ~
.,
~;;J
II
[
~
II
-.
'v
II
Typical Proprietary Anchor Details FIG.18
ANCHOR BOLT DESIGN
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 a combination 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 insure safety, anchor bolt details should receive the same design considerations as would any other structural connection. However, due to a lack of available research and design guides, anchor bolt designs are based largely 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 masonry has been completed. Reports have been issued that evaluate anchor performance and suggest equations to predict ultimate anchor strengths. By combining the research findings with design practices currently used in concrete design, equations for allowable tension, shear and combined tension/shear loads for plate anchors, headed bolts and bent bar anchors are under consideration for adoption in the proposed "Building Code 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 the anchor material. For example, if the embedded depth of an anchor is small relative to its diameter, a tension cone failure of the masonry is likely to occur. However, if the embedded depth of the anchor is large relative to its diameter, failure of the anchor material is likely. For these reasons, the allowable tensile load is based
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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
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.
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---------------aThe manufacturer
should always
~
be consulted
when adhesive
anchors
are to be used in areas where contact
with chemicals
is likely.
''']
W
.J
~ V)
Z W
~
50
'I
'1"1 ""t4_._-r--r-.-rl
" ~rl
50
T,'1-'--'---' -rf "--.",,,,,
100
,..r"Tf""f·T'"f~r"1'I,I_._i
150 TEMPERATURE,
"
T,'I"-'-"--'--'-'-TI 200
"'"""! 25G
'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 are shown in Fig_ 12_The conventional embedded anchors (headed bolts, bent bar and plate anchors) are usually placed at the intersection of a head joint and bed joint. By using this location, the brick units adjacent to the anchor can be chipped or cut to accept the anchor without altering the joint thickness.
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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
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MIX
RESIN
HARD!ENER
PLACE
Site-Mixed Adhesive Anchor FIG. '10
There are special requirements and limitations. that should be considered when contemplating the use of adhesive anchors in brick masonry. They are: 1) Specially designed mixing and/or setting equipment may be required 2) Dust and debris must be removed from the pre-drilled holes to insure proper bond between the adhesive and base material. 3) The adhesive mixture tends to fill small voids and irregularities in the base material. 4) Large voids (due to brick cores, intentional air spaces and partially filled joints) may cause reductions in anchor capacities. This is especially true with the self-contained adhesive anchors since a limited volume of epoxy is available to fill the voids and provide a bond to the anchor. 5) The adhesive bond strength is reduced at elevated temperatures and may also be adversely affected by some chemicals (see Table 2 and Fig. 11).
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44
and mixing the adhesive components. The other type of adhesive anchor requires that the epoxy components be hand-measured and mixed before the epoxy is placed into a pre-drilled hole. A threaded rod or bar is then set into the epoxy mixture, as shown in Fig. 10. Adhesive epoxies usually vary slightly between 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
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TABLE 4 Allowable
Shear
on Anchor
use
Bolts - l-rcrn
1985 Edition"
(a) ,A.LLOV\,I,ABLE SHE.A,R C)~\J.A.f",JCHOF.'BOLTS1 COf\JCF.'ETE t"llASOf\JRY
FOR CLAY .A.f",JD
Tot
Allowable Shear3
Elllbedmenf (inches)
fibs}
:3/8
4 4
1(2
4
270 41D 550
5/B
4,
3/4
~I
7/8
f3
(inches)
1/4
-
~~I
r-
1100 1500 18504 22504
"7 I
1-1 fa
0
u
'P',n anchor bo~ is eIbolt that ,f!.,
h;",,, eIright
elngle extension of elt leelst three dierneters.
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\JED MA,Sm',JF~Y EXCEF'T Ur'"JBURf\JED cu~.,,( U~',JlTS Embedment'
Diameter Bolt (inches)
(inches)
112
4
5/8
4
3/4
5
na
f3
7
e
1-1/8
Solid Masonry (Shear in Pounds!
Grouted Masonry (Shear in Pounds)
350 500 750 1000 1250 1500
550 750
1100 1500 113502 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, The
International Conference of Building Officials."
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on the smaller of the two loads calculated for the masonry and anchor material. Thus, the allowable load in tension 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 (In composite construction, when the masonry cone intersects different materials, fm should be based on 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 to radiate at an angle of 45° (see Fig. 19) from the anchor base. When an anchor is embedded close to a free edge, as shown in Fig 20, a full masonry cone cannot be developed and the area Ap must be reduced so as 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.
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Full Masonry Tension Cone FIG. 19
.
'~"
G',
~..----"~~
A) PROJECTED
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CONE
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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 the surface of the masonry to the plate or head for plate anchors or headed bolts. The effective embedded length of bent bar bolts (1b) is the length of embedment measured perpendicular from the surface of the masonry to the bearing surface of the bent end minus one bolt diameter. Where the projected areas of adjacent anchors overlap, Ap of each bolt is reduced by one-half of the overlap area. Also, any portion of the projected cone falling across an opening in the masonry (i.e., holes for pipes or conduits) should be deducted 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 anchor capacity is governed by either the masonry strength or the anchor material strength. The distance between an anchor and a free masonry edge has an effect on the masonry shear capacity. Calculations have shown that for edge distances less than twelve times the anchor diameter, the masonry shear strength controls the anchor 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 shear lOad is the lesser of: ..-
'i 'i A.
-.",rJ41~f' - ",01
~J rn' A '8
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(Eq 5)
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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 is determined by linear interpolation from a value of VA obtained in Eq. 5 at an edge distance of 12 anchor diameters to an assumed value of zero at an edge distance 0 1 in. (25 mm). This takes into consideration the 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 the allowable pure tension and pure shear loads calculated in Eqs. 1, 2, 5 and 6. Anchors subjected to combinations 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 bolts and 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 proprietary anchors. The combined tension/shear interaction equation produced an average safety factor of 7.0 when compared to test results on proprietary anchors. Therefore, based on comparison to test results, the allowable load equations proposed in this Technical Notes are suggested for use in the design of proprietary anchors in brick masonry. The embedment depth used to calculate the allowable load values should 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 large enough to cause cracking or spelling of the brick when edge distances become small. To date, no research has been conducted in this area. Therefore, due to the lack of information, it is suggested that a minimum edge distance of 12 in. (300 mm) be maintained when expansion anchors are installed in brick masonry.
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Page 21 0[24
Through Bolts There are no known published reports available addressing the strength characteristics of through bolts in brick masonry. However, based on the conservatism in the allowables for bent bar anchors and proprietary anchors, the allowable load equations should provide acceptable allowable load values for through bolts used in brick masonry. The embedment depth used to calculate the allowable load values should be taken as equal to the actual thickness of the masonry.
Current Codes and Standards
i
At the present time, one model code and one design standard contain provisions for anchor bolt design in brick masonry. The BIA Standard, Building Code Requirements for Engineered Brick Masonry, and the Uniform Building Code cover design allowables and embedment depths for anchors loaded in shear. There are no provisions for axial tensile loads or combined tension/shear loads in these documents. Tables 3 and 4 show the allowable shear loads and minimum embedment depths from the two documents. The values in Table 4(a) are based on rational analysis and in Table 4(b) on empirical analysis. As can be seen, the tables are very similar and are generally more conservative than the allowable shear loads obtained from Eqs. 5 and 6 for the same embedment depths (Table 5).
!xl
! L:J
• From Building Code Requirements for Engineered Brick Masonry, Brick Institute of America, August 1969.
'In determining the stresses on brick masonry, the eccentricity due to loaded bolts 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 competent representative.
http://www.hia.org/BINtechnotes/t44.htm
3/15/2008
44
Page 23 of24
'American Concrete Institute/American Society Of Civil Engineers Committee 530 on Masonry 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 anchor bolt types, detailing and allowable loads for anchor bolts in brick masonry. Other Technical Notes in this series will address brick masonry fasteners and ties.
The information and suggestions contained in this Technical Notes are based on the available data and the experience of the technical staff of the Brick Institute of America. The information and recommendations contained herein should be used along with good technical judgment and an understanding of the properties of brick masonry. Final decisions on the use of the information discussed in this Technical Notes are not within the purview of the Brick Institute of America and must rest with the project designer, owner or both
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, PRF7806095, "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, NSF Award No. CEE-8217638, "Static and Cyclic Behavior of Masonry Retrofit Embedments (Earthquake Engineering)", Report No.1, April 1985. 4. Hatzinikolas, M.; Lee, R.; Longworth, J. and Warwaruk, J., "Drilled-In Inserts in Masonry Construction", Alberta Masonry Institute, Edmonton, Alberta, Canada, October 1983.
5. Building Code Requirements for Engineered Brick Masonry, Brick Institute of America, Virginia,
August
McLean,
1969.
6. Uniform Building Code, International
Conference
of Building
Officials,
7. Technical Notes on Brick Construction 17 Revised, "Reinforced Institute of America,
McLean,
Virginia,
October
Whittier,
Brick Masonry,
California,
1985.
Part I of IV", Brick
1981.
8. Technical Notes on Brick Construction 41 Revised, "Hollow Brick Masonry-Introduction", Institute of America,
McLean,
Virginia,
Brick
1983.
9. Specification for the Design and Construction of Load-Bearing Concrete Masonry, National Concrete
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 Alabama,
Building
Code Congress,
International,
12. Technical Notes on Brick Construction 7A Revised, "Water Resistance Materials,
Inc .. Birmingham,
1985.
Part 1\ of III", Brick Institute of America,
http://www.bia.orgIBIA/technotes/t44.htm
Reston, Virginia,
of Brick Masonry-
1985.
311512008
ATS-AB anchor bolts are pre-assembled anchor bolts that have been designed for use with the ATS system. They are available in 18", 24" and 36" lengths and match the strength and material grade of the corresponding Strong-Rod connecting rods. The heavy hex nuts are pressed onto the bolt to keep them in place. Material: Standard (Model ABJ - ASTM A307, Grade A High strength (Model AB_H) - ASTM A449 or ASTM A193, Grade B7 Higher strength (Model AB- H150) - ASTM A434, Class BD or ASTM A354, Claks BD i Finish: None
i
Naming Scheme:
ATS-AB5Hx24 ATS
=::J T
Anchor
Bolt
*
L
Length
Diameter and Grade
Units in Va" Increments (Ex: 9 = 'fa" or 1%")
Anchor Bolt Bolt Diameter Model No. (in)
Plate Washer Size (in)
ATS-AB5
S/s
3fsxl'hxlY,
ATS-AB7
1's
3fs
ATS-AB9
1%
3/8 x 23iA
ATS-AB5H
s/s
3/S
ATS-AB7H
'Is
3,4, X 2'.4 x 2V.
1, (in)
Component Color Cod'e
1v.
Blue
x 2v.. x2.'A
.....
x 2%
xl 'h x 1'/2
ATS-AS9H
11/8
3/ax2:v..x2%
ATS-AB9H150
1%
'hx3x3
ATS-AB10H150
1%
1. 2. 3. 4.
i.
.
r
3reen·,>
PiA
Orange
1'.4
Blue
'.'
ATS-AB Anchor Boll
Green
1'h
.
...
j3iA
.•...Orange
1'/8
Orange
2'h
Purple
....
1 x3'hx3'h
Anchor rods are available in 18",24" and 36" lengths. Standard Anchor bolts are based on minimum Fu = 60,000 psi and Fy = 43,000 psi. High strenqtn anchor bolts are based on minimum Fu = 120,000 psi and Fy = 92,000 psi. H150 anchor bolts are based on minimum Fu = 150,000 psi and Fy = 130,000 psi.
ANCHOR BOLT LOCATIONS Anrnor bolts shall be specified by the Designer. 1-2x4or1-2x6=4'fl 1 - 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;
1 - 2x4 or 1 - 2x6 = 4 Vi 1 - 3x4 or 1 - 3x6 = 5 W 2 - 2x4 or 2 - 2x6 = 6' 2 - 3x4 or 2 ' 3x6 = 8' 1 - 4x6 or 1 - 6x6 = 8Y, 1 - 4x8 or 1 - 6x8 = lOW 1 - 4x10 or 1 . 6x10 = 12'1.
Perpendicular-To-Wall Installation -------
4Yi' 5W 6" 8" ---- 8Yi _10W
-----__
16
1 - 2x4 or 1 - 2x6 1 - 3x4 or 1 - 3x6
2 - 2x4 or 2 . 2x6 2 - 3x4 or 2 . 3x6 1 - 4x6 or 1 - 6x6 1-4x8orl-6x8 1 - 4xl0 or 1 - 6x10
12Y4_
Corner Installation
~ /
~
Mid-Wall Installation
COMPRESSION MEMBERS
COMPRESSION MEMBERS
ATS: Anchor Bolts
Page 4 of 4
Anchor Rod f.'odel No.
ATS·A85
ATS·,l"B7H
ATS-,4B9H ATS·ABBH15G
1. 2. 3.
4.
5. 6. 7.
IBC calculations are based on ACI 3113,Appendix D 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. 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 ductile requirements of ACI 318, Section D.3.3.4. For UBC design anchor design for 2500 psi minimum concrete assumes no special inspection and a multiplier of 2.0 on the concrete per section 1923.3.2. For 3000 psi and 4500 psi concrete, special inspection is assumed and a multiplier of 1.3 is applied. Plate washers have been designed for plate bending. Alternate anchor bolt solutions may be provided by the Designer. Foundation dimensions are for anchorage only. The Designer is responsible for the foundation size and reinforcement for all load conditions.
ReLated Catalog Pages (PDFs): C-ATS07
(AnchorTiedQwn$ysiem),page
"" top 16 (173k)
Printed March 15, 2008 from http://www.strongtie.com/products/ats/connectors/anchor-bolts.html©
http://www.strongtie.com/products/ats/connectors/anchor-bolts.html
Order frE?",_cat§Log~ by mail
2008 Simpson Strong-Tie®
3/15/2008
ATS: Anchor Bolts
Page 3 of 4
Anchor Rod Model Nu.
•• See footnotes below
Wind and Seismic Design 97 USC without
Supplementary
Reinforcing:
Anchor Rod Model 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 Rod IWodel 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
15
Vessel On Beams Vessel
Dimensions
25
130.000 80.000 26.500 42.000 44.500 0.750
26
0.250
20
21 22 23
24
13.500 16.000 12,300 353.9
27 28 29 30
46 47
1.300 <- Foundation
Site Specific
35 44 45
www.pveng.com
(Inch and Lbs): <- H, height <- L, center of gravity <- Is, leg free length <- Do, shell outside diameter <- ds, leg pitch diameter <- t, shell corroded thickness <- ws - leg weld size <- Iw - length of leg to shell weld <- Iwf - length of weld on foot <- W, Weight Ibs <- Pr, Pressure
Seismic Information per NBC-95: 1.000 <- I, occupation importance factor 0.400 <- v, zonal velocity ratio 6.000 <- Za, acceleration-related seismic zone 5.000 <- Zv, velocity-related seismic zone
32
Page 22 of 25
27-Apr-07
Sample Vessel 8 <- Vessel
17
19
Ver2.24
NBC-95
Fv
~ __
+_
\iiiJ
Fh
Fh c.g.
Fv
f3
Factor (F)
v ds
61
Leg Supports: Angles 4" x 5/8" <- Structural Description 4 <- n, number of legs 6.660 <- lx, for one leg 6.660 <- Iy, for one leg 1.200 <- fFactor, Least radius of Gyration 4.610 <- A, Leg Cross Sectional Area 4.000 <- 2cx, Beam Depth 4.000 <- 2cy, Beam Width 0.800 <- K1, Leg Anchor Factor
63
Material
52 53 54 55 56 57 58 59 60
69
71 72 73 74 75
Attachment
Base Shear
103
U = 0.6
104
R = r-=4__
106 107
x
ATTACHMENT
Static Deflection E = 30,000,000 bc = 12.0 leg boundary condition based on fixed or loose leg y = (2*W*ls"3)/(bc*n*E*(lx + Iy» y = 0.024 (2*12300*26.5"3)/(12*4*30000000*(6.66 + 6.66»
84
105
-
Dimensions:
79
78
RECTANGULAR
...
CL
5.657 <- 2C1, Width of rectangular loading 13.500 <- 2C2, Length of rectangular loading
Period of Vibration g = 386 T = 2*pi*sqrt(y/g)
77
-:\.
---------
17,100 <- maximum leg bending stress (Sb) 16,200 <- maximum shell stress (Sa)
65
68
---.u AI..
Properties:
64
67
o
=2 * 3.14 * sqrt(0.02/386)
-:-::--::-;:-, 4.2001<- Seismic Response Factor (S) Ve = v*S*I*F*W = 0.4*4.2*1*1.3*12300/ V = (Ve/R)*U = (26863.2/4)*0.6 .
T = 0.049
I
Ve = 26863 V = 14029
120
Sample Vessel 8 Vessel On Beams Horizontal Seismic Force at Top of Vessel Ftmax = 0.25*V = 0.25 * 4029 Ftp = 0.07 * T * V = 0.07 * 0.049 * 4029 Ft = if (T < 0.7, 0, min(O.07*T*V, Ftmax))
122
Horizontal
115 117 118 119
123
Seismic Force at cg Fh = V - Ft
126
Vertical force at cg Fv = W
128
Overturning
125
129
131
Overturning
132 "0
Moment at Base Mb=L*Fh+H*Ft
Maximum eccentric load f1 = Fv/n + 4*Mto/(n*Do)
137
Axial Load f2 = Fv/n + 4*Mb/(n*ds)
138
140
f3x = 0.5*V*lx/(lx+ly)
142
f3y = 0.5*V*ly/(lx+ly)
145 146 147
149 150 151 152
154 155 ''156 ~57
159 160 161
163 164 165 166 167
= 4029 - 0
Fh = 14,029 Fv = 112,300 Mb = 322,358
= 80 * 4029 + 130 * 0
Mt = 215,577
= 12300/4 + 4*215577/(4
* 42)
f1 = 8,208
= 12300/4 + 4*322358/(4
* 44.5)
f2 = 10,319
Leg Loads
141
144
Page 23 of 25 Ftmax = 1007 Ftp = 13.94 Ft= 1L...:...9 __
Moment at Bottom Tangent Line Mt = (L-ls)*Fh + (H-ls)*Ft = (80 - 26.5) * 4029 + (130 - 26.5) * 0
135
134
27-Apr-07
Leg Bending Moments e= (ds-Do)/2 Mx = f1 *e + f3x*ls My = t1 "e + t3y~ls Leg Bending Stress Sbmax = Sb * 1.25 fx = Mx*cx/lx fy = My*cy/ly
=0.5* 4029*6.66 I( 6.66+6.66) =0.5* 4029*6.66 /( 6.66+6.66)
f3x = 1,007 f3y = 1,007
=(44.5-42)/2 =8208*1.25 + 1007*26.5 =tl2otn.25 + 1UUr20.5
e = 1.25 Mx = 36,955 My = 30,955
=17100 * 1.25 =36955 * 2 f 6.66 =36955 * 2 f 6.66
Leg axial stress K1*Is/r = =0.8 * 26.5 I 1.2 Fa max = AISC code lookup based on K1 *Islr fa = f2/A =10319 14.61
Sbmaxfx
fy = ~'"'--"---'
Acceptable
K1 *Islr = 17.667 Fa max = 25,675 fa = 12,.23$
Maximum Euler Stress Fe = 12*piA2*E/(23*(K1 *L1r)"2) = 12*piA2*30000000/(23*17.667A2) Combined Stress Fc1 = fa/Famax + O.85*fxl«1-fafFe)*Sbmax) 2238/25675 + 0.85*11098/«1-2238/494954)*21375) Fe2 = fa/Famax + 0.85*fy/«1-fa/Fe)*Sbmax) 2238/25675 + 0.85*11098/«1-2238/494954)*21375)
== ,...,.,....,:-,.-........,......,
Acceptable Acceptable
Fe = 494,954
Acceptable Fe1 = 10.53 Acceptable Fc2 = 10.53
171
173
175
176 177
Sample Vessel 8 Vessel On Beams
27-Apr-07
Page 24 of 25 Mz
Beam to Shell Attachment Stresses Beam Dimensions ex = 2cx/2 ey = 2cy/2
ws:
ex = 2.000 cy = 2.000
: IV1x
178
-r I
179
181 182 183 184 185
187 188 189 190
192 193 194 195 196
198 199 200 201 202
204 205 200 207 208
210 211
213 214 215 216
C dimensions weld area wex wez wey
for weld stress = ws*lw wa = 3.375 = Iw/2 wex = 6.750 = ey + ws wez = 2.250 = sqrt(wex"2 + wey"2)
Shear Force Distribution Vx = (V*lx)/( (n/2) "(Ix+ly) Vy= (V*ly)/( (n/2)*(lx+ly» Vg = Win
gravity
= sqrt(6.
5"2 + 7.115"2)
= (4029. 8*6.66)/«4/2)*(6 = (4029 .. 8*6.66)/«4/2)*(6 = 12300/4
Weld Moments of Inertias Iwx = (ws*lw"3/12)*2 = (0.25*13.5"3/12)*2 Iwz = (lw*ws"3/12 + wa*(cy+ws/2)"2)*2 (13.5*0.251\3/12 + 3.375*(2+0.25/2)1\2)*2 Iwy = Iwx + Iwz = 103 + 31 Weld Moments Mx = Vx*(ls+lw/2) + Vg*(ds-Do)/2 1007*(26.5+13.5/2) + 3075*(44.5-42)/2 My1 = Vy*(ls+lw/2) Mz= Vy*(ds-Do)/2
= 1007*(26.5+13.5/2) = 1007*(44.5-42)/2
Weld Stresses Sx Sy Sz Sg
= = = =
= = = =
Mx*wcx/lwx My1 *wcy/lwy Mz*wcz/lwz Vg/(wa*2)
Bending Twisting Torision Gravity
Stress Limits and Ratios Slim = min(Sb,Sa)*OA9 SxR SyR SzR SgR
= = = =
36955*6.75/102.5 33495*7.115/133 1259*2.25/30.5 3075/(3.375*2)
= = = =
2433/7938 1791/7938 93/7938 456/7938 Acceptable
217 218 219 220
222 223
225 226 227
Foot Plate Attachment Stresses waf = ws*lwf Vv= V/n Sv = Vv/waf Sgf = Vg/waf SvRf= SgRf=
Sv/Slim Sgf/Slim
weld area in foot
Vx = 1,007 Vy= 1,007 Vg = 3,075 Iwx = 102.5 Iwz= 30.5 Iwy=
133.0
Mx = 37,339
= min(17100,16200)*0.49
Sx/Slim Sy/Slim SzlSlim Sg/Slim
7.115
wey=
My1 = 33,495 Mz = 1,259 Sx Sy Sz S9
= = = =
2,433 1,791 93 456
Slim = 7,938 SxR = SyR = SzR= SgR = total «1)
0.307 0.226 0.012 0.057 0.601
= 0.25*16 = 4029/4
waf = 4.000 Vv = 1,007
= 1007/4 = 3075/4
Sv = 252 Sgf = 769
= 252/7938 = 768.75/7938 Acceptable
SvRf = 0.032 SgRf = 0.097 total «1) O~129
27-Apr-07
Page 25 of 25
WRC 107 - shell local stress at support TO>
'" Loads (psi and Ib) 1,007,4 10,319.0
<<-
P, Axial Load (=vx) VL, Lon9itudinalload(=f2)
0.0 <- Ve, Circumferential
'"
36,955.0
<- ML, Moment
load
(=My)
0.0 <- Me, Moment 0.0 -c, MT, T orisional 242
Parameters MaxSPm;:;; MaxSPmb::
=
MaxSPmbQ Ri Rm
=
= = Beta 1 = Beta2 = SL = Sc = r
concentration
Sa
for Pm stresses
1.5*Sa
for Pm + Pb stresses
Pm - primary membrance stress Pb - primary bending stress
3·Sa (00-2'T)/2 (00-T)/2
for Pm
o - secondary
No/(P/Rm) see
,
stress
= 5.65712120.625 = 13.512/20.625 = (20.25-0.4'0.75)'353.898/(2'0.75)
2C2/2/Rm (Ri-OA'T)'Pr/(2'T) (Ri+0.6'T)'PrlT
Stress
A
a stresses
= 20.625/0.75
Shell Combined Stresses' Stress
+
RmlT
""
Lookup
Pb
2C 1/2/Rm
283
Pressure
+
factors
Kb Equation
Cat
Au
AL
Bu
BL
Cu
CL
Du
DL
3C 1.88561 0,04871
SC Kn'A'P/(Rm'T) Kb'A'6'PfTA2
Pm Pm Pb
9768 -244 -523
9768 -244
9768 -244
9768 -244
9768 -123
9768 -123
9768 -123
9768 -123
Kn'A'Mc/(RmA2'beta'T)
Pm
Kb' A '6'McI(Rm'beta'TA2) Kn'A'MU(RmA2'beta-T)
Q Pm Q
o
0
0
0
1.05302 0.08268
No/(MLJ(Rm"2'beta» Mo/(MU(Rm"beta»
18 or 18-1
2.75635 0.01754
Kb·A*S*MU(Rm·beta'"TA2)
Pm Pm+Pb Pm+Pb+Q
Nx/(MLI(Rm"2'beta)) Mx/(MLI(Rm'beta )
VIII-1 Code 3C or 4C 1C-1 or 2C 4A 2A
3.74796 0.03139
4B 28 or 28-1
1.88561 0.05870 1.71344 0.03430 1.12882 0.03569
SL Kn'A'P/(Rm'T) Kb'A'6-PfT'2 Kn' A 'McI(RmA2'beta'T) Kb'A-6'McI(Rm'beta'TA2) Kn'A-MU(RmA2'beta'T) Kb'A'6'MU(Rm'beta'P2) Pm Pm+Pb Pm+Pb+Q
Shear vc Tolal
Shear
S1m S2m
Sum
,
S2m+b
S12 S23
, ,
5mb<=
S2m+b+Q
S12 S23
.
S31 Smbce
MaxSPmbQ
,2556 4050 3419 864
Txo
0 0
9645
9645
9645
11351 10265
8776 8776
10514 10514
8776 8776
10514 10514
4707 -123
4707 -123
4707 -123
4707 -244
4707 ,244
4707 -244
4707 -244
•.... -534
0
0
534
0 .
0
534
2556 4050 4681 7236
2556 5118 4487 7043
-2556 5118 5749 3193
4463 4125 4125
4463 4800 4800
4463 4125 4125
4463 4800 4800
-1002
f",'.ne
0 0
0 0
0 0 10,827
,1002
1002
1002
-1002
·1002
1002
1002
9,832 4,276
9,832 4,276
9.832 4,276
9.832 4,276
5,556 4,276 9,832
5,556
5,556 4,276 9,832
9,832
4.276 9,832 9,832
5,556 4,276 9,832 9,832
8.983
10.685
8,983
10,685
3.919 5,064 3,919 8,983 8,983
4.629 6,055 4,629 10,685 10,685
3,919 5,064 3.919 8,983
8,983
4,629 6,055 4.629 10,685 10,685
8,983 3,919 5,064
10,685 4,629 6,055
3,919 8,983
4,629 10,685 10,685
)~.£%
8,220
10,827
4.050
4,050
5,118
5,118
4,170 4.050 8.220 8,220
4,170 4,050 8,220
5,709
5,709
5.118 10,827 10,827
5.118 10.827 10,827
7,696 3.419
8,743
10,304 4.487
11,351 5,749
5,817 4,487 10,304 10,304
5,602 5,749 11,351 11,351 10,265 3,193 7,071
8,983 3,919 5,064
10,685 4,629 6,055
3,193 10,265 10,265
3,919 8,983 8,983
4,629 10,685 10,685
abstS'lmS2m) abs(S2m-O) abs(0-S1m) max(S12,S23,S31)
Acceptable
rxc-z)
( Sx+So)/2)+SQRT«(Sx-So)/2)A2+ (Sx+So)/2)-SQRT« Sx-So)/2 abs(S1 m, S2m) abs S2m-0) abs(0-S1m) max(S12,S23,S31
MaxSPmb
S1m-rb-rQ
0 Sx Sx Sx
9645
10304 11390
8,220
S31
a
-534
1304 -1086 10827
T xc-z)
MaxSPmb
,
4707 -123 -631
1304 1086 10827
Txo'2)
831
S1m+b
Pm Pm Pb Pm Q Pm
,1304 1086 8220 8743 9829
«Sx+So)/2),SQRT«(Sx-So)/2)'2+
823
Sm<=
of shears
-1304 -1086 8220 7696 6610
( Sx+So)/2)+SQRT«(Sx-So)/2)A2+
S12
,
So So So
VU(Pi'sqrt(cl'c2)"T) VC/(Pi'sqrt(c1'c2)'T)
Shear VL
,
1
A Value
38
Mx/P
Sc = Kn =
= 1
4C 3,74796 0,08088
3A 1A
Mx'(Mc/(Rm~beta)
= =
A Value
Mo/(MC/(Rm"'bela»
Nx/(McJ(Rm"2°bela»
Bela2 SL
-1
Curve
1C or 2C-1
Stress
r-~ t =
16,200 24,300 0" '--1-"'--4>f-=': 48,600 0L 20.25 -~ 20.625 -----"-,... r?7
VIII-1 Code 3C or 4C
Mo/P
Nxf(P/Rm)
Beta
= (20.25+0.6'0.75)'353.898/0,75
No/(Mc/(Rm'2'beta»
Pressure
MaxSPm = MaxSPmb = MaxSPmbO = Ri = Rm =
A2+ Txo'2)
4.277 3,419 7,696 )
Acceptable
«sx-se
/2 +SQRT(((Sx-So)/2 '2+ Txo'2 «Sx+So)/2 -SQRT«(Sx-So)/2)A2+ rxc-z) abs(S1m, S2m) abs(S2m-O) abs O,S1m) max(S12,S23,S31)
Acceptable
8,220 4,681 4.062 4,681 8,743
7,696
8,743
6,610 864 5,746
9,829 7,236 2.593
11,390 7,043 4,347
864 6,610 6,610
7,236 9,829 9,829
7,043 11,390 11,390
8,983
9,832
