spherical dome 2013 m

8/18/2019 Spherical Dome 2013 m

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Design of Spherical Shells (Domes)

The Islamic University of Gaza

Department of Civil EngineeringENGC 6353


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Shell Structure

A thin shell is defined as a shell with a relatively small

thickness, compared with its other dimensions.


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Shell Structure


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Four commonly occurring Shell Types:

Hyperbolic Paraboloid (Hypar)

Dome

Barrel Vault

Folded Plate


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To answer this question, we have to investigate some important notionsof structural design.

What is a shell structure?


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Two-dimensional structures: beams and arches

A beam responds to loading by bending

the top elements of the beam are

compressed and the bottom is extended:the development of internal tension andcompression is necessary to resist theapplied vertical loading.

An arch responds to loading bycompressing.

The elements through the thickness ofthe arch are being compressedapproximately equally. Note that thereis some bending also present.


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Plate Bending

A plate responds to transverse loads by bending

This is a fundamentally inefficient use of material, by analogy tothe beam. Moreover, bending introduces tension into theconvex side of the bent plate.


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Plate bending vs. membrane stresses

This slide shows a concrete plate of 6”thickness, spanning 100 feet, resistingits own weight by plate bending

If the plate is shaped into a box,then each of the sides of the boxresists bending by the developmentof membrane stresses. The box

structure is much stronger andstiffer!

Note: this is an experiment you can try yourself by folding a sheetof paper into a box.


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A shell is shaped so that it will develop membrane

stresses in response to loads

The half-dome shell responds to transverse loads by development ofmembrane forces. Note that lines on the shell retain approximately theiroriginal shape.

Domes


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Domes

The primary response of a dome to loading is development of membranecompressive stresses along the meridians, by analogy to the arch.

The dome also develops compressive or tensile membrane stresses alonglines of latitude. These are known as ‘hoop stresses’ and are tensile at the

base and compressive higher up in the dome.

Meridional Compressive Stress

Circumferential Hoop Stress(comp.)

Circumferential Hoop Stress (tens.)


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The half-dome shell does develop membrane tensile stresses, below about50 ‘north latitude.’ These are also known as ‘hoop stresses’

In this figure, the blue colorrepresents zones ofcompressive stress only.The colors beyond bluerepresent circumferential

tensile stresses, intensifyingas the colors move towardsthe red.

A dome that is a segment of

a sphere not includinglatitudes less than 50 doesnot develop significant hooptension.


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Thin Shell Structures

Two type of stresses are produced:1. Meridional stresses along the direction of the meridians

2. Hoop stresses along the latitudes

Bending stresses are negligible, but become significant when the rise

of the dome is very small

(if the rise is less than the about1/8 the base diameter the shell is

considered as a shallow shell)


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Thin Shell Structures

Assumption of Analysis1. Deflection under load are small.

2. Points on the normal to the middle surface deformationwill remain on the normal after deformation

3. Shear stresses normal to the middle surface can be neglected


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Spherical Shells

r a

a

H

N

B D

A

C

d

E

F

N

N +dN

Internal Forces due to dead load w/m3

Consider the equilibrium of a ring enclosed between two

Horizontal section AB and CD

The weight of the ring ABCD itself acting vertically downward

The meridional thrust N per unit length acting tangentially at B The reaction thrust N +d N per unit unit length at point D


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r a

a

H

N

B

D

A

C

d

E

F

N

N +dN

2

2

2

Surface area of shell AEB

21 cos

2

2 (1 cos )

(2 )sin 2 (1 cos )

sin

(1 cos ) (1 cos )

sin (1 cos )(1 cos ) 1 cos

Meridional Force

D D

D

D

D D D

A a EF EF a

W w A a EF

W w a

N r w a

r a

w a w a w a N

N

+ve compression

-ve tension


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Spherical Shells

r a

a

H

N

B

D

A

C

d

E

F

N

N +dN

N +dN

N

D

B

d

W

The difference between the and which respectively acts at

angles and with the horizontal give rise to the hoop force.

Hoope force =

The horizontal component of is

Hoop Force

N N dN

N ad

N N

N

cos

causes hoop tension a cos sin

The horizontal component of +d is +d cos

+d causes hoop tension

similar

+d cos c

l

s

y

o

N N

N N N N d

N N N N d a d


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Spherical Shells

r a

a

H

N

B

D

A

C

d

E

F

N

N +dN

N +dN

N

D

B

d

W

2

2

When increasein issmall d tends to be zeroa cos sin

(1 cos )

sin

1 cos cos 1 cos -

1 cos 1 cos

D

D D

N N ad d N

where

w a N

N w a w a


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Spherical Shells

'

o '

o '

1 cos -

1 cos

0 ( )2

90 (when

)0 51 49

51 49 willbecompressive

51 49 will be tensile

HoopForce

D

o

N w a

wa At crown N

At base N wa N

compress

for N

fo

ion

tensio

r N

n

N


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Spherical Shells


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Spherical Shells

o '

max

coscos

1 cos

at 51 49 0 & is maximum

0.

Ri

382

ngForce H

D

D

H N w a

N H

H w a


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Spherical Shells

2 2 2

2 2

o

max

MeridionalForceT

Hoop Forc

in

(1 cos )sin

2 sin sin in

2

cos 2

2

coscos

2

at 45 0 & H is maximum 0.35

e

35

N

L L

L

L

L

L

L

W w r w a s

y ar a

N a w a s

w a

N

w a N

Ring Tension

H N w a

N H w a

Internal forces due to Live load (w L/m 2 )horizontal


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Spherical Shells


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In conical shells and flat spherical dome, bending moments

will be developed due to the big difference between the high

tensile stress in the foot ring and compressive stresses or lowtensile stress in the adjacent zones of the shell


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Ring beam design

max(see the tables of circu

0.9

Vertical Uniform load ( ) sin .

2

# of supports

Horizontal Load

Vertical L

2

1

o d

a

Design of the Circular Beam

Ultimate Load

s

y

V

V

ve

T A f

T H r

w N o w

r Span length l

P r w

M C P r

max

lar beams

(see the tables of circular beams)

2

)

ve M C P r


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Edge Forces

In flat spherical domes, bending moments will be developed due to

the big difference between the high tensile stress in the foot ring and

compressive stresses in the adjacent zones

It is recommended to use transition curves at the edge and to

increase the thickness of the shell at the transition curve.

Bending moments can avoided if the shape of meridian is changed

in a convenient manner. This change can be done by a transitioncurve, which when well chosen gives a relief to the stress at the foot

ring.

In order to decrease the stress due to the forces at the foot ring, it isrecommended to increase the thickness of the shell in the region of

the transition curve.


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Edge Forces

In flat spherical domes, bending moments will be developed due to

the big difference between the high tensile stress in the foot ring and

compressive stresses in the adjacent zones

It is recommended to use transition curves at the edge and to

increase the thickness of the shell at the transition curve.


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Ring Beam

At the free edge of the dome, meridian stresses have a large

horizontal component which is taken care of by providing a ring

beam there. This ring beam is subjected to hoop tension.

In case of hemispherical domes, no ring beam are required since

the meridional thrust is vertical at free end


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Reinforcement

Steel is generally placed at the center of the thickness of the

dome along the meridians and latitudes. If all the meridional

lines are led to the crown, there will be a lot of congestion of bars

and their proper anchorage may be difficult.

To overcome this problem, small circle is left at the crown and

all the meridional steel bars are stopped at this circle. Area

enclosed by this small circle at the top is reinforced by a separatemesh.


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Example: Design of a spherical dome

Design a spherical shell roof for a circular tank 12m indiameter as shown in the figure. Assume the followingloading: Covering material = 50 kg/m2 and LL= 100 kg/m2

Use ' 2 2300 / 4200 /c ykg cm and f kg cm

r=6m

y=1.4m

a


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22 2

2 2 2 2

2 2 2 2

2

6 1.4Radius of the Shell 13.562 2 1.4

6sin = 0.442

13.56

26.23 cos 0.896 tan 0.493

a r a y

a r a y ay

r ya m y

r=6m

y=1.4m

a

Loading on roof

Assume shell thickness = 10 cm

Own weight = 0.1(2.5)= 0.25 t/m2

Covering materials = 0.05 t/m2

LL= 0.1 t/m2

Note: the live load is considered as loading per surface area


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Design of Ring Beam:Wu= 1.2(0.2+0.05)+1.6(0.1)=0.52 t/m2

Total load on roof =

2 ayWu=2 (13.56)(1.4)(0.52)=62 ton

Vertical Load per meter of cylindrical wall

=62/(2*6)=1.645 ton/m’

Outward horizontal force =1.645/tan=3.337 t/m’

2

Ring tension in beam

3.337 5 20

20*1000 5.350.9 4200

use 8 10 mm

s

y

T H r tons

T cm f


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Design of the ShellMeridian Force

Meridian force per unit length of circumference

2

s

1 cos

0.52*13.560 3.52 / ' (compression)

2 2

cos 0.896

0.52 13.563.72 / ' (compression)

1 0.896

Use minimum reinf. ratio = 0.0018

A 0.0018(10)(100) 1.8

use 5 8 mm/m

u

u

W a

N

W aat N t m

at foot

N t m

cm


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Ring (Hoop) Force

2

s

1 cos -

1 cos

0.52(13.56)0 3.52 / ' (compression)

2 2

cos 0.896 2.59 / ' (compression)

A 0.0018(10)(100) 1.8use minmum reinf. 5 8 mm/m

u N w r

wr At crown N t m

At foot N t m

cm


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22

6

min2

0.6 0.6 13.56 0.15 0.85

0.52 0.85Fixing moment 0.188 /

2 2

15 3 12

0.85 300 1 2.61 10 0.1881 0.00034200 100 12 300

use minmum reinf. 5 8 mm/m

u

x at cm

W x M t m

d cm

Bending Moment

Assume that the thickness at the foot = 15 cm


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Example: Design of a spherical Dome

Reinforcement details

h l h ll d l d


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Spherical Shells under General Loading

Internal Forces Due to Others Loading


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spherical dome 2013 m

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