Design of Spherical Shells (Domes)

The Islamic University of Gaza

Department of Civil Engineering

ENGC 6353

Shell Structure

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

thickness, compared with its other dimensions.

Shell Structure

Four commonly occurring Shell Types:

Hyperbolic Paraboloid (Hypar)

Dome

Barrel Vault

Folded Plate

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

What is a shell structure?

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 and compression is necessary to resist the applied vertical loading.

An arch responds to loading by compressing.

The elements through the thickness of the arch are being compressed approximately equally. Note that there is some bending also present.

Plate Bending

A plate responds to transverse loads by bending

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

Plate bending vs. membrane stresses

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

If the plate is shaped into a box, then each of the sides of the box resists bending by the development of membrane stresses. The box structure is much stronger and stiffer!

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

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 of membrane forces. Note that lines on the shell retain approximately their original shape.

Domes

Domes

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

The dome also develops compressive or tensile membrane stresses along lines 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.)

The half-dome shell does develop membrane tensile stresses, below about 50 ‘north latitude.’ These are also known as ‘hoop stresses’

In this figure, the blue color represents zones of compressive stress only. The colors beyond blue represent circumferential tensile stresses, intensifying as the colors move towards the red.

A dome that is a segment of a sphere not including latitudes less than 50 does not develop significant hoop tension.

A barrel vault functions two ways

In the transverse direction, it is an arch developing compressive membrane forces that are transferred to the base of the arch

When unsupported along its length, it is more like a beam, developing compressive membrane forces near the crown of the arch, and tensile membrane forces at the base.

Arch (compression)

tension

compression

Barrel Vaults

Barrel Vaults

A barrel vault is a simple extension of an arch shape along the width. It can be supported on continuous walls along the length, or at the corners, as in this example. If supported on the corners, it functions as an arch across the width, and as a beam, with compression on the top and tension on the bottom in the long direction. This form is susceptible to distortion.

Barrel Vault, continued

As with any arch, some form of lateral restraint is required--this figure shows the influence of restraining the base of the arch--the structure is still subject to transverse bending stresses resulting from the distortion of the arch.

Folded Plates Folded plate structures were widely favored for their simplicity of forming, and the variety of forms that were available.

Perpendicular to the main span, the shell acts as short span plates in transverse bending

In the main span direction, the shell develops membrane tension at the bottom and compression at the top, in analogy to a beam in bending

What’s wrong with this Folded Plate Structure?

Compare to the discussion of barrel vaults, and see if you can tell what key element is missing from the folded plate shown.

It is missing transverse diaphragms, especially at the ends.

This animation shows the effect of adding a diaphragm at the two ends and at midspan. The folded plate shell distorts much less.

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)

Thin Shell Structures

Assumption of Analysis 1. Deflection under load are small.

2. Points on the normal to the middle surface deformation

will remain on the normal after deformation

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

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

r a

a

H

N

B D

A

C

d

E

F

N

N+dN

2

2

2

Surface area of shell AEB

2

1 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 aN

N

+ve compression

-ve tension

r’

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

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 zero

a cos sin

(1 cos )

sin

1 cos cos 1 cos -

1 cos 1 cos

D

D D

N

N ad d N

where

w aN

N w a w a

Spherical Shells

'

o '

o '

1 cos -

1 cos

0 ( )2

90 (

when

)

0 51 49

51 49 will becompressive

51 49 will be tensile

HoopForce

D

o

N w a

waAt crown N

At base N wa

N

compress

for N

fo

ion

tensio

r N

n

N

Summary

1 cos

1 cos

1 cos

D

D

w aN

N w a

In order to make the

– Negative sign for compression and

+Positive sign for tension for Meridional and Hoop forces

The previous equations can be rewritten as follows:

Spherical Shells

Spherical Shells

o '

max

coscos

1 cos

at 51 49 0 & is maximum

0.

Ri

382

ng Force H

D

D

H N w a

N H

H w a

Spherical Shells

2 2 2

2 2

o

max

Meridional ForceT

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 N

35

L L

L

L

L

L

L

W w r w a s

y a

r a

N a w a s

w aN

w aN

Ring Tension

H N w a

N H w

a

Internal forces due to Live load (wL/m2)horizontal

Spherical Shells

Ring beam design

max 1(see the tables of circu

0.9

Vertical Uniform l

Horizontal Load

Vertical Load

oad ( ) sin .

2

# of supports

2

Design of the Circular Beam

Ultimate Load

s

y

V

V

ve

TA

f

T H r

w N o w

rSpan length l

P r w

M C P r

max 2

lar beams

(see the tables of circular beams)

)

ve

M C P r

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 transition

curve, 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 is

recommended to increase the thickness of the shell in the region of

the transition curve.

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.

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

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 separate

mesh.

Example: Design of a spherical dome

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

Use

' 2 2300 / 4200 /c yf kg cm and f kg cm

r=6m

y=1.4m

a

22 2

2 2 2 2

2 2 2 2

2

6 1.4Radius of the Shell 13.56

2 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

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

Total load on roof =

2ayWu=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*10005.35

0.9 4200

use 8 10 mm

s

y

T H r tons

TA cm

f

Design of the Shell Meridian 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 aN

W aat N t m

at foot

N t m

cm

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) 1.8

use minmum reinf. 5 8 mm/m

uN w r

wrAt crown N t m

At foot N t m

cm

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 0.0003

4200 100 12 300

use minmum reinf. 5 8 mm/m

u

x at cm

W xM t m

d cm

Bending Moment

Assume that the thickness at the foot = 15 cm

Example: Design of a spherical Dome

Reinforcement details

Spherical Shells under General Loading

Internal Forces Due to Others Loading