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Shell StructuresDesign 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.
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 PlatesFolded 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.
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 Analysis1. Deflection under load are small.
2. Points on the normal to the middle surface of deformation
will remain on the normal after deformation
3. Shear stresses normal to the middle surface can be neglected
Spherical Shells
ra
a
H
N
BD
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
ra
a
H
N
BD
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’
ra
a
H
N
BD
A
C
d
E
F
N
N+dN
N+dN
N
D
Bd
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
d
N ad
N
N
cos
causes hoop tension a cos sin
The horizontal component of +d is +d cos
+d causes hoop tension +d cos
similarly
cos
N
N N
N N N N d
N N N N d a d
ra
a
H
N
BD
A
C
d
E
F
N
N+dN
N+dN
N
D
Bd
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
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
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 be 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.
Edge Forces
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.
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
Example: Design of a spherical dome
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
Example: Design of a spherical dome
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
Example: Design of a spherical dome
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 aN
W aat N t m
at foot
N t m
cm
Example: Design of a spherical dome
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
Example: Design of a spherical dome
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