Tubular Conveyor Gallery

Tubular conveyor galleries M. S. Troitsky

Department of Civil Engineering, Concordia University, Montreal, Canada {Received January 1981; revised October 1981)

This paper discusses the structural analysis and design of tubular conveyor gallaries, used in industry for supporting and housing of conveyors handling bulk materials. The design of such galleries has not so far been treated in detail in the literature.

The loads and forces acting on the gallery are established and the structural behaviour of the gallery has been analysed in detail. The expression for stresses in the gallery under vertical and horizontal loading or their combinations are shown. Also, local and overall buckling stability requirements are established. The design of intermediate stiffening rings, as well as ring girders, considering vertical and saddle-type supports are treated.

Key words: conveyors, gallery, tubular

Introduction

The purpose of this paper is to outline the structural analysis and design of tubular conveyor galleries, which relatively recently have found an application in industry. 1-3 They are used for the support and housing of conveyor handling bulk materials.

Tubular types offer the following advantages:

(a) They act as structural members in carrying the conveyors and workways, and protecting them from adverse weather. (b) Their tubular cross-section has an aerodynamically superior shape against wind action. (c) The tubular section provides an equal resistance to both the horizontal and vertical loads. (d) The circular tube is an ideal section for resistance to torsional stresses imposed by eccentrically located loads.

These conditions are best fulfilled by a welded tubular conveyor gallery, shown in cross-section in Figure 1.

The tubes are approximately 8 ft 6 in to 10 ft in diameter, depending on the size of conveyor that they house.

It should be noted that the range of wall thicknesses encountered in practice in the design of a tubular conveyor galley depends on the following factors: accepted diameter of the tube, span, conveyor installations, weight of the material transported by the belt and type of steel used for the structure. In addition, external loadings such as the intensity of the wind or earthquake forces influence the design of the tube. As experience indicates, the most common thicknesses of tube walls may vary within the range of ~ in to ½ in.

At intervals along the length of a conveyor gallery, supports must be provided to carry the gallery between its junctions with the plant. Two types of gallery supports, both fabricated of steel pipe, are used. The first type of support is the inverted V-bent, (Figure 2a) and is capable of taking both the horizontal and vertical loads. The second type of support is a single pipe post (Figure 2b), taking vertical loads only.

The latest proposal is to house the conveyor in an elliptically-shaped tube of light-weight, high-strength steel, supported by an A-frame, Figure 3.

Convey

\ \ I ,r ~ .g I Walkway

Re]~r'n ~ / , '~ " / ' ~

:' " Cooc,e,e

t ' igure 1

/ Tubular conveyor gallery

sluice w a y

0141/0296/82/02119-10/$03.00 © 1982 Butterworth & Co. (Publishers) Ltd Eng. Struet., 1982, Vol. 4, April 119

Tubular co#veyor galleries: M. S. Troitsky

1 I

Figure 2 Types of supports (a), inverted V bent support, (b), single post support

The advantages of the V-bent and single post support structures for conveyor galleries include the following:

(a) The supports constitute a minimum of obstruction to plant operations. (b) The heavy pipe sections used are less vulnerable to damage by mobile equipment than are the supports with bracing of the conventional type. (c) Less frequent painting maintenance is required for the supports.

Fabrication

Nearly all of the steel is shop-welded and field-bolted, using high strength bolts. Normally, the structures are shop-assembled into the largest pieces which can be eco- nomically shipped and handled in the field. The tubes are fabricated in up to 60 ft sections, which can be transported by rail to the plant site for erection. All the conveyor components are assembled inside the tube before erection.

Structural grade steels, ASTM A-36 are used for most of the work, although ASTM A-242 may be used for corrosion resistance and alloy steels for reduction in dead weight.

Ver t ical loads and forces

Loading conditions vary, depending on the project, but the following may be considered:

Dead loads

Own weight o f tubular structure:

nWst qt = zrDitPs+ lb/ft (1)

l

where: Di, inside diameter of tube, ft; t, thickness of wall, ft; ps, unit weight of steel shell, lb/ft3;n, number of stiffeners in span; Wst, weight of one stiffener, lb and l, span between bents or supports, ft.

Weight o f mechanical components and equipment.

qe = belts + idlers + pulleys + drive + switches + wiring + other, lb/ft (2)

Weight o f walkway:

qp = checkered plate + grating + stringer + posts, lb/ft (3)

qs = weight of concrete sluiceway, shown in Figure 1, lb/ft (4)

Live loads

Live load due to material on belt."

qt.t, lb/ft (5)

No allowance is made for the impact due to the bounc- ing of lumps as they wave along the belt.

Live load on walkway."

ql.w, lb/ft (6)

Normally, it is taken at 100 lb/ft 2 locally, but 25 lb/ft 2 for the span.

Icing and snow {if applicable):

qi+s, lb/ft (7)

Belt pull during starting, stopping or running."

P, lb (longitudinal force) (8)

Longitudinal forces

Sometimes, the head pulley of the conveyor is located within the gallery, usually at one end, the highest of an inclined gallery. As a result, a longitudinal compressive force, originated by the head pulley of the conveyor, has to be resisted by the tube, and transmitted from one end

F . t

Figure 3 Elliptical tube for conveyor gallery

120 Eng. Struct., 1982, Vol. 4, April

- I - - F

2 S i n g l e p o s t I n v e r t e d V- ben t

Figure 4 General arrangement of conveyor gallery

-I

of the gallery to the other, which is held by braced bent against longitudinal displacement. The unsupported length of the tube is 2l, since the single posts are considered as having zero flexural regidity. This compressive longitudinal force at one end of the conveyor gallery is usually eccentric, with respect to the centre of the tube. As a result, we have two moments, one vertical and one horizontal, at this end of the gallery.

Temperature effect

Due to the temperature changes, the inverted V-bents and single posts, being flexible structures, allow the tube to expand freely.

A factor that must be taken into consideration in designing long-span tubular galleries, is the effect of the sun's heat on the tube, one side of which may be in full sun, while the other side is in full shade, with the conse- quent distortion of the tube to a degree where it may affect the conveyor operation.

Structural behaviour

Stresses under vertical loads. Vertical loads generally include the following components:

qv=qt+qc+qp+qs-Fql.l+ql.w+qi+s lb/ft (9)

Vertical loads are the most important, because they prevail in magnitude and frequency of occurrence. They are uniformly distributed along the length of the gallery.

Tubular conveyor galleries." M. S. Troitsky

The tubular gallery spannings over the inverted V-bents and single posts have to resist in flexure, as a continuous beam of hollow circular cross-section over multiple supports. In addition to flexure, the vertical loads usually cause uniformly distributed torsional moments.

The resultant of the vertical loads in a typical cross- section of the gallery does not usually coincide with the centre of the tube in creating a torsional moment, but the tubular section is the ideal section to resist torque.

In the following analysis we consider the general arrangement of the conveyor gallery, as shown in Figure 4.

The statical values, due to the effect of a vertical uniformly distributed load on a continuous structure, are shown in Figure 5.

Normal stresses due to maximum bending moment

My considering a multispan structure, Figure 5, under a uniformly-distributed load qv is:

My 4M~ Max o v - - psi

S l tD , ( t - - c )

where: S, section modulus of tube, in3; Di, internal diameter, in; t, thickness of wall, in and; c, corrosion allowance, in.

(10)

Shearing stress due to internal torque MT. Due to the asymmetric arrangement of the equipment inside the tube, there will originate an internal torsional moment (see Figure 6).

With reference to Figure 6, internal torsional moment is:

MT = Re = ZPe lb in (11)

where: Px and P2 are the loads due to the conveyor installa- tion, material on the belt and sidewalk; e is the eccentricity of the resulting load, in and:

MTDi 2M T "/'max - - 2~p-p - lrO~(t - -c i psi (12)

where: lp, is the polar moment of inertia of the tube, in 4. An internal torsional moment action on the tube may be

transferred only to inverted V-bents spaced at spans of 2l,

No Scheme of continuous structure Mspan

J v

1 IIIIIII1~111111~ .o o7oq/~

/ _1_ / _j I- -] ]

A A A A I / _1 / _L_ / _1 r-- - I - -I - I

,/qv 3 A I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 ~ , A A A *00773ql2

- i - I - _ 7 _

qv

4 IIIIIIIIIIIIIIIl111111~111111111111111111 .0 0789q/e A /', ,", /', A /', I_ / _1_ / I_ l _t_ / _t_ / .I F I - I- -F - [ i

Bending moments and shear forces in continuous gallery Figure 5

Msupport Shear Reaction

-0 124q12 0 625q1 ~ 1 25gl 0 625q/

_Q 100q/2 QS00q/ [ ' ~ 1 lOql 0 600 ql

-O 107q/2 0 536ql ~ 1 143q1 0 607q/

-0 105q/~ 0526q1 ~ 1 151ql 0 605q/

Eng. Struct., 1982, Vol . 4, Apr i l 121

Tubular conveyor galleries." M. S. Troitsky

#4, : Re

P1

\ /

J i

Figure 6 Equipment loads producing torsional moment M T

,ll,~------_--~ V bent . . . . . . . . . . . . -- -----&l|~y bent-- ~ l w

/ / / 1 / / / / / / / / / / / / / / / / / / / / / / / / / i f / / / / / / / /

F i g u r e 7 Wind a c t i o n on t a b u l a r c o n v e y o r

because single posts due to their flexibility cannot provide the necessary rigidity.

Principal stresses due to normal and shear stresses

The stress due to the shearing force is usually only of secondary importance. Its maximum value occurs at the neutral axis where the normal stress due to bending is zero. Therefore, the maximum combined stress usually occurs at the point where normal and shearing stresses due to torque are a maximum. In the case under consideration - at the tel: and bot tom surfaces of the tube the principal stresses due to the combination of normal and shear stresses are :4

O V + I f ~ Oma x . . . . . gN/O v + 4~ -2 (13) ,-)

Substituting in this equation the values from equations (10) and (12), we obtain:

1

Oma x -- (M v +x/ (Mv2+M}) psi (14) 2S

Stresses under horizontal loads

Wind acting on structure. The wind forces should be determined according to the National Building Code.S

The horizontal wind force is:

qw = qCcCgCp lb/ft 2 (15)

where: q, wind pressure, lb/ft2; C~, exposure factor and

Cg, gust effect factor. Total force:

w = qwDo lb/ft (16)

where: Do is the outside diameter of the tube. The intermediate posts, due to their flexibility cannot

provide rigid support in a lateral direction. Therefore, continuous tubular structures have spans of 21 between the inverted V-bents, as shown in Figure 7.

After determining the maximum bending moment due to the wind, Mw, the stress in the tube wall is:

Mw 4Mw Ow - - (17)

S nD~ (t -- c)

Local wind action. By indicating the intensity of the horizontal wind pressure as qw, the distribution of wind pressure around the tube is presented in Figure 8.

Due to the local wind action, the bending moment Mw; and normal force Nwb acting on the tube, are calculated by the formulae developed by Krupka. 6

The bending moment acting on the tube in the interval:

is:

- - 0.225 ¢ sin 0) lb in

The maximum value is at ¢ = n, or:

max Mw; = 0.261 q w R ~

The normal force acting in the wall of the tube in the interval:

is:

Mwt = qwR2m( - 0.101 + 0.707 sinq5 - 1.362 c a s e

(]8)

(]9)

Nwl = qwRm( - 0.700 -- 0.921 cos~ + 0.707 sinq5

-- 0.225 0 sinq~) lb (20)

At ¢ = n, the value of this force is:

Nwl = 0.212 qwRm lb (21)

The maximum stress in the wall of the tube under local wind action is:

W~nd

Figure 8

qw

© • qw ~

/ 0 3q w

Distribution of horizontal wind pressure


1.566 qwR2m 0.212qwRm max O w l - - - - + psi

t 2 t (22)

where: Rm, is the middle radius of tube, in, and t, thickness of tube wall, in.

Earthquake forces. Tubular galleries in earthquake areas should be designed to withstand earthquake forces. The seismic action should be considered according to the National Building Code r (Figure 9).

The lateral earthquake force acting on the top of the tube is:

Ft = 0.004 V ( ~ ) kips (23)

or:

F t = 0.15 V kips

whichever value is greater. The minimum lateral seismic force, V, is:

V = ASKIFW kips

where:

h t = height of force Ft above base, ft D = diameter of tube, ft A = acceleration ratio, or ratio of specific ground

acceleration to acceleration due to gravity

0.5 S 3x/'T seismic response factor

T - m

(24)

Figure 9

(25)

0.05 h t - fundamental period of vibration of structure

x / ~ in seconds

K = structural coefficient that reflects material and type of structure, damping, ductility and/or energy-absorption capacity of structure

I = importance factor of structure F = foundation factor depending on type and depth of soil

measured from fundament W = operating weight of structure of combined dead load

of tube (including conveyor installations in tube and material on belt) and weight of supports, kips

yS"

h n

Fx

g 1 J

g Earthquake action

Tubular conveyor galleries: M. S. Troitsky

/

Ve H

i f ' l / i / I / i / / / / / / / / / / / / / / / / I i i i i /

I__ 2 ( R * e )

Direction of eerthquake

Acceleration

Figure 10 Ring-girder under earthquake forces

The additional lateral forces acting on the structure are:

( V -- Ft) Wth , F1 = (26)

[Wth, + (W t + Ws)h2 ]

( V - - F t ) (W t + Ws) h2 F2 = (27)

[Wth, + (Wt + Ws) h21

where:

W t = dead weight of tube, including conveyor installations and material on belt, kips

W s = dead weight of supports, kips h 1 = height of force F l above base, ft h2 = height of force F 2 above base, ft

The tubular gallery is usually stiffened above its supports by the ring-girder. According to Foster, 8 the ring-girder is subjected to the bending moment and normal force, see Figure ! O.

The expression of the bending moment is:

- - - 1 + - ( 2 8 ) Ma 4rr R + e

and the value of the normal force is:

Na_ Q e [ L 1 H1[ ]

n LR 4 4 ( R + e )

where:

(29)

Qe = total horizontal reaction of tube support, trans- mitting earthquake acceleration to tube

R = external radius of tube, in r = internal radius of tube, in H = height of support, in e = eccentricity of vertical reaction, in

Head pulley inside gallery

Under the action of the head pulley inside the gallery, the tube is under the action of axial compression and biaxial flexure due to internal moments acting on the vertical and horizontal planes, see Figure 11.

Eng. Struct., 1982, Vol. 4, April 123

Tubular conveyor galleries: 114. S. Troitsky

Pul P

e V

_ _ _ I

F i g u r e 1 1 Biaxial moments

The longitudinal compressive force, due to pull, acts as an axial force along the continuous gallery. Compressive action is used for 2l spans, between the inverted V-bents.

The moment My = Per acts on the continuous tube, having multiple spans l, and a horizontal moment MH = Pert acts upon the continuous tube having multiple spans 21, see Figure 12.

Buckling s tab i l i ty r e q u i r e m e n t s

General data A thin-walled tube subjected to compression in the

direction of its longitudinal axis may fail, either by the instability of the tube as a whole, involving the bending of the axis, or by the local instability of the wall of the tube, which may not involve the lateral distortion of the axis, at all.

We can classify a tube under axial load into three categories: 9

Very short tube."

1 < 1.72x/Rt, ~2E(t3/12)

%r- (3o) (1 - -u 2) l

Intermediate length tube:

172 X / ~ ~< 1 ~< 2 . 8 5 x / ~ t

E t 2 E' t

Ncr--Rx/~(I__v2 ) ' ° c r - R ~ / 3 ( l _ v = )

Long tube:

L~t ( TrRI2 l > 2.85x/R~, N~. = ~ t T - /

or

zr2EI Per = 21rRNer - 12

which is the Euler formula.

(31)

(32)

(33)

Local buckfing or wrinkling Local buckling is the governing consideration in tire

design of tubular galleries of moderate length. Failure of this type is due to the formation of characteristic wrinkles or bulges, circular or lobed in shape. Wrinkling is local in nature and depends on the combined compressive stress which occurs at a point of the dangerously loaded cross- section and has to be smaller than the allowable buckling stress.

In designing thin-walled tubular structures, two con- siderations are of importance. First, the local buckling should be prevented at stresses below yield strength; secondly, a more severe restriction is that the tendency to buckle locally should not reduce the overall buckling load of the whole structure.

The allowable buckling stresses which will be presented below, are for elastic buckling and tubes with simply supported edges. The edge of a tube is assumed to be simply supported if at the edge the radial and circumferential displacements are zero and there is no restraint against translation or rotation in the axial direction.

Ax&l compression The critical buckling stress under axial compression given

by Plantema 1° is:

662 Ocr = - - + 0.399 Fy ksi (34)

D/t

where: by = 36 ksi, the yield point for steel and tile ratio D/t is valid for:

3300 D 13 000 - - - - < - - < - - (35)

@ t Fy

Wilson and Newmark give: al

8000 O e r - - ksi (36)

D/t

or, by assuming a factor of safety of 1.5:1=

5333 Oer - ksi (37)

D/t

a

~-t I [ t l i l I I I I I I I [ [ I I [ I I I I I I I ] I I ] I I [ [ I ILI [ I I I I ]~I~E[? A

~_ 2, j ~ . 2, -4

b

U U / / 4 - " M v : P e v

c

/ 2 / ~]

M , , : P e ,

F i g u r e 1 2 Axial force and resulting bending moments. (a), longitudinal force; (b), vertical bending moment; (c), horizontal bending moment


lO O9

0.8 >-

-0.7 ~ 0 . 6 ~ o 5 ~_o4 ~ o 3 ~ o 2

L) Ol

- L

I I I I l l i l l I I I I L I I I I t I [ I I I I I C 2 3 4 ~ 6 7 B 9 2 ? 4 [ 6 7 8 8 2 3 4 t 6789

10 2 10 3 10 4 R/ t

Figure 13 Correlation factor for unstiffened circular cylinders subjected to axial compression

30

25

2C c_

[_ ~ 15

[ ] lO

0

Figure 14

Ocr = 6 6 2 _ O 399F~ ( R e c o m m e n d e d D/t ~ AISi f o r m u l a )

I ~ \ / O c r = 8 0 0 0 ( W , l s o n - N e w m o r k )

iX \ o / ,

t \ ~ \ o - . = 5 3 3 3 ( R e c o m m e n d e d )

I I I I I I t I [ I 2 0 0 4 0 0 6 0 0 8OO 1 0 0 0 1 2 0 0 1 4 0 0 1 6 0 0 18OO 2 0 0 0

92 361 D / t Recommended allowable buckling stresses

Finally, according to Baker: 13

Oct = 0.6 7E( t /R) ksi

is valid for moderately long tubes, where:

and

lr 2

7Z > - -

l 2

Z = - - X/1 -- p2 R t

7 = correlation factor obtained from Figure 13 E = modulus of elasticity, ksi l = span of tube, ft t -- thickness of tube wall, ft

The recommended allowable buckling stresses in the

(38)

(39)

function o f D/ t are shown in Figure 14.

(40)

Bending

Buckling tests on tubes similar to those tested in axial compression indicate that buckling occurred over the compression side of the tube in the same wave form, with


approximately the same wave-lengths, as those in the axially-loaded tubes. A comparison of the axial compression and the pure-bending test results show exactly the same decrease of axial load P with an increase of the ratio R/ t , as shown by the axially-loaded tubes.

The influence of the initial imperfections in a tube, considering bending, should not be as great as the axial compression. However, during the bending of the tubes a certain flattening of the cross-section occurs, which leads to an increase of maximum stress.

The design-allowable buckling stress for a thin-walled circular tube subjected to bending, as given by Baker, is:

Ocr = 0 . 6 7 , E ( t / R ) psi (41)

where: T1 = correlation factor obtained from Figure 15. For elastic stresses, the allowable moment is:

Mcr = 7rR2tOcr = 0.6 %ETrRt 2 (42)

Ovalling ef fect under bending

During the bending of the tube a certain flattening of the cross-section may occur which leads to an ovalling of the cross-section and consequently, to an increase of the stress. This problem was investigated by Brazier and is known as the 'Brazier effect' 14 (see Figure 16).

Brazier proved that the ovalling of the tube results in the substantial reduction of the tube stiffness and consequently leads to a loss of stability. According to his theory, the critical bending moment is:

2X/~ E~rR m t 2 Mcr - , - - (43) 9 ~ Ib in

09

o 8

oo6

t

O c r - - Y C b L ~ - C ' b = ~ - 3 ( 1 ~ 0

I I 1 [ I I I I I I I l I I I l l ] I I 10 102 103

R/ t Figure 15 Correlation factor for unstiffened circular tube subjected to bending

Figure 16

l / l l l l l l l l l / I / / l l l l l l l l l l l l l l / l l / l ' 1 1 1 / / / / / / /

( Brazier effect

Eng. Struct., 1982, Vo l . 4, Apr i l 125

Tubular conveyor galleries: 114, S. Troitsky

T: h2.- \

i (, 5,q i

~ f) 52i ~ v'ohd for

,7 > 1C:0 for simply supported ' ,%S. ~ edges

Z > ~'OC' for dcmped edges

4{ ,

' ; 4 4 I I I !(,() ~ 2(O6) 300c; 4 0 0 0

R/t

Figure 17 Buck l ing s t r e s s c o e f f i c i e n t C s f o r c i rcu la r tubes subjected to to rs ion

and the critical buckling stress:

oe~ = 0.33 E ( t / R ~ ) psi (44)

where: E, modulus of elasticity, psi; t, thickness of wall, in; and R m , middle radius of tube, in.

Therefore, considering the ovalling deformation of the tube, the following condition should be satisfied:

Mcr<~ 0.33 E ( t / R m ) psi (45) S

It is important to know the ratio R / t above which the cross-section may be distorted, considering the ovalling effect under bending. According to the Brazier effect, the loss of stability or distortion of the tube may occur if:

R 0.33 ES ~> (46)

t Mc~

Buckl ing shear stress due to torsion

The design-allowable buckling stress for tubes of moderate length 100 -<. Z < 78(R/ t )2 ( l -- u 2) subjected to torsion is given by: is

Et r,.~ = C s R Z 1 m (47)

For long tubes, Z > 78(R/ t )a ( l -- u2), the design- allowable buckling stress is:

rot (1 -- u2) s/4 (48)

where:

12 e = - - X/i -- u z (49)

R t

and the coefficient Cs is given in Figure 1 Z

Intermediate stiffening rings To prevent possible ovaUing of the tube along the length of a tubular gallery at equal intervals, intermediate circumferential stiffening rings are installed. They consist of flat plates, angles or T-sections continuously welded onto the inside surface of the tube. These rings are usually

located inside the tube and serve a dual structural purpose, namely: (1) To maintain the cross-section of the tube circular, and thus to allow the tube to act as a beam. (2) To receive the posts and bangers, which support at equal intervals the idlers of the conveyor and stringers of the walkway.

According to Rumman, 16 the bending moment of the ring stiffener is:

M R -- (50) 4Err 2R ~ t

and the normal (axial) force in the stiffener is:

M 2 AIR -- kips (51 )

- 2 4 Err Rmt

where: M, bending moment, kip-R; s, spacing of stiffeners, ft; R m , middle radius of tube, ft and t, thickness of walt of tube, ft.

Ring girders at supports

Rigid ring girders provide an effective support for the tubular structure. These girders prevent the distortion of the tube at the supports and thus maintain its ability to act as a beam.

Usually ring girders are located outside of the tube and consist of one or two stiffening rings continuously welded on to the tube. A design for the ring girder construction, based on elastic theory, was developed by Krupka. a7

Vertical supports

The tube transmits to the ring a concentrated vertical road Q, which is the reaction of a tube acting generally as a continuous beam. In the case of two vertical supports, heaving reactions Q/2 and acting on the ring girder with the moments and normal (axial) forces along the circumference of the ring girder, see Figure 18.

i i

Figure 18 Ring g i rder having t w o vert ical suppor ts


We consider two cases, namely:

Bending moment and normal force for any section below support in interval 0 <~ 4) <<- O:

M 1 - Q R o s 0 - - - ( r r - - 0 ) + + cosy - 21r R R-2

+ siny] (52)

Q [ y s i n y - (1.5 ~ ) cosy] (53)

Bending moment and normal force for any section above support for 0 <~ 0 <- rr:

M2 QR [o a [1 + a2~ = - + cos0 + cosy 2rr I_ R \ 2 R-2]

(y --rr) siny] (54) +

2°- 0.' cos0] where: Q is the reaction at support due to total operating weight of tube, which consists of dead load of tube, conveyor installations in tube and material on belt, kips.

The combination for the simultaneous action of the maximum moment , Mmax, and the corresponding normal force, N, and also the maximum value of the normal force, Nmax and the corresponding moment , M, for the typical values of the angle 0, are shown in Table 1.

Saddle support In a case when the tube is supported by the saddle, see

Figure 19, the bending moments and the normal forces in the ring girder are calculated, using the following formulae developed by Krupka: 17

Bending moment and normal force for any section below support in interval 0 <~ 0 <~ O:

Ma = Q R {(2 sin0 - ½ sin 20 - 0) 27rA1

i i

Figure 19 Saddle suppo r t

Table I Values o f m o m e n t s and no rma l forces

0 ° q5 ° M m a x N q~o Nrna x M

4 5 ° 90 ° - - 0 . 0 4 9 Q R - - 0 . 2 5 0 Q 50 ° - - 0 . 3 7 8 Q 0 . 0 2 6 Q R 90 ° 70 ° 0 . 0 1 4 Q R 0 . 1 5 5 Q 90 ° - + 0 . 2 5 0 Q 0


Table 2 Values o f m o m e n t s and no rma l forces

0o ~o eo M m a x N N m a x M

4 5 ° 80 ° 0 . 0 7 9 Q R - 0 . 3 0 9 Q 45 ° - 0 . 4 1 2 Q 0 . 0 1 2 Q R 9 0 ° 90 ° - - 0 . 0 4 7 QR - 0 . 2 5 0 Q 0 ° - 0 . 4 7 7 Q 0 .043

and

+ [4 t sin20 --½ 0 + (0 --rr) sin20] cosy

+ (0 -- Y -- ½ sin 20) r r s i n y + A l [ l + ½ c o s y

+ (y -- 7r) sin y]} (56)

N3 = ~ {[¼ s i n 2 0 - 3 0 + (0 --rr) sin20] cosy 27rA1

+ (0 -- Y + sin 20) 7r sin Y + A 1 [ ( y - - 7/') sin y

_ =a cos Yl} (57)

Bending moment and nothia'i force for any section above support for 0 <<. y <~ rr:

M4 = O R {2 sin0 --0 --1 sin 20 + (0 sin20 2rrA1

0 + ~ sin 20) cos y + A 1 [1 + ~ cos y

+ (y -- rr) sin y]} (58)

also see Table 2.

QR N4 = 2rrA~ {(0 sin 2 0 + 41- sin 20 -- ½) cos y

+ A1 [(Y -- rr) sin y -- 3 cos Y]} (59)

where:

A1 = 0 + sin0 cos0 (60)

The combinations of the simultaneous actions of the moments and normal forces are shown in Table 2.

For both types of supports, either vertical or in the shape of the saddle, the maximum stress in the ring girder should be calculated using the formula:

M N f - - - - - + - - (61)

W A

where: W is the section modulus of cross-section of ring girder and A is the cross-sectional area of ring girder.

References

1 Mylar, D. T. 'Belt conveyor structures', paper presented at SME Meeting, Seattle, Wash., Preprint No. 71-B-302, September 1971

2 Troitsky, M. S. 'Design guidelines for steel tubular thin-walled structure', 4th Progress Report, CSICC Project No. 727, January 1974

3 Troitsky, M. S. 'On the analysis of tubular conveyor galleries', Proc. Canadian Soc. for Or. Eng. Conf., 7-8 June, 1979, Montreal, pp. 335-340

4 Timoshenko, S. 'Strength of materials', Part I, D. Van Nostrand Company, Inc., New York, 1957, p. 264

5 National Building Code of Canada. Issued by the National Research Council of Canada, Ottawa, 1980, p. 145

6 Krupka, V. 'Analysis of cylindrical thin-waUed metal pipeline structures', Prague, Edition SNTL, 1967 (in Czechoslovakian), p. 32

Eng. Struct., 1982, Vol. 4, April 127


7 National Building Code of Canada. Issued by the National Research Council of Canada, Ottawa, 1980, p. 147

8 Foster, H. A. 'Formulas indicate earthquake forces in design of ring-girder-supported pipes', Cir. Eng., 1949,19, 45

9 McGuire, W. 'Steel structures', Prentice-Hall, Englewood Cliffs, N.J. 1968, pp. 415-418

10 Plantema, F. J. 'Collapsing stresses of circular cylinders and round tubes', Rep. S.280, Nat. Luchtvaart-Laboratorium, Amsterdam, Netherlands, 1946

11 Wilson, W. M. and Newmark, N. M. 'The strength of thin cylindrical shells as columns', Bull. 255, University of Illinois, February 1933

12 Troitsky, M. S. 'On the local and overall stability of thin- walled large diameter tubular structures', Proc. Canadian Struct. Eng. Con f , Montreal, February 22-24, 1976, pp. 1-32

13 Baker, E. H. et al., 'Structural analysis of shells', McGraw-Hill, New York, 1972, p. 230

14 Brazier, E. G. 'On the flexure of thin cylindrical shells and other "thin" sections', Proc. Roy. Soc. London. Ser. A., 1927, 114,104

15 Baker, E. It. et al., 'Structural analysis of shells', McGraw-Hill, New York, 1972, p. 232

16 Rumman, W. S. 'Stresses in ring stiffeners in cylinders', Proc. ASCE, J. Struct. Div. Proc., December, 1961,87,161

17 Krupka, V. 'Analysis of cylindrical thin-walled metal pipeline structures', Prague, Edition SNTL, 1967 (in Czechoslovakian), pp. 39-40, 47-48

Nomenclature

Di

Do R Rm C C

ht, hl , hz l r

s

t A

S W

&,F,,& P

qc

qs

internal diameter of tube external diameter of tube external radius of tube middle radius of tube corrosion allowance eccentricity heights of earthquake forces above base span between supports internal radius of tube spacing of stiffeners thickness of tube wall cross-sectional area of ring-girder polar moment of inertia of cross-sectional area of tube section modulus of cross-sectional area of tube section modulus of cross-sectional area of ring girder horizontal earthquake forces longitudinal force weight of mechanical components and equipment weight of concrete sluiceway

qi ~s qll

qlw

qt q; G w,

V

Ps

Mcr

My g~

Mw

Mwt

M~,M2

M3, M4

G X~r X~

N3, N4

Ocr O'ma x

0-7)

OW

Owl

f "refit

Trnax E

G P

7 ")'1

load due to ice and snow weight of material on belt live load on walkway own weight of tube weight of walkway weight of support of tube weight of tube, including installations and material on belt weight of stiffener base shear unit weight of steel bending moment acting on ring-girder bending moment due to Brazier effect bending moment on ring stiffener torsional moment bending moment due to vertical load acting o11 continuous tubes bending moment due to wind load, acting on continuous tubes bending moment acting on tube under local wind bending moments, acting on ring girder, having vertical supports bending moments, acting on ring girder, having saddle support ring-girder normal force critical axial buckling force normal force in ring stiffener normal force in tube wall under the local wind action normal forces in ring stiffener, having vertical supports normal forces in ring stiffener, having saddle support critical buckling stress under axial load principal stress due to combination of normal and shear stresses bending stress in tube under vertical load bending stress in tube under wind load stress in tube under local action of wind load maximum bending stress in ring-girder critical buckling stress under torsion maximum shear stress due to torsional moment modulus of elasticity of steel coefficient in formula for torsion Poisson's ratio for steel correlation factor for axial stress correlation factor for bending stress


Tubular Conveyor Gallery

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