The Islamic University of Gaza Department of Civil Engineering ENGC 6353
Design of Spherical Shells (Domes)
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:
Barrel Vault
Dome
Folded Plate Hyperbolic Paraboloid (Hypar)
What is a shell structure? To answer this question, we have to investigate some important notions of structural design.
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 Note: this is an experiment you can try yourself by folding a sheet of paper into a box.
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!
Domes
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 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.)
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. The half-dome shell does develop membrane tensile stresses, below about 50° ‘north latitude.’ These are also known as ‘hoop stresses’
Barrel Vaults A barrel vault functions two ways compression Arch (compression) In the transverse direction, it is an arch developing compressive membrane forces that are transferred to the base of the arch
tension
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.
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 top and compression at the bottom, 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. Click on the picture, if you need to see the action of the shell under load, or click on the answer below.
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)
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
Thin Shell Structures Membrane theory of surfaces of revolution
r1 = radius of curvature r2 = cross radius of curvature Nϕ = Resultant meridian force per unit length Nθ = Resultant ring force per unit length H = horizontal thrust of shell per unit length Wϕ = Sum of vertical forces above level Z
∑F
y
=0
Wϕ / 2π r = Nϕ sin ϕ ⇒ Nϕ = Wϕ / 2π r sin ϕ but r = r2 sin ϕ
then
Nϕ = Wϕ / 2π r2 sin 2 ϕ H = Wϕ / 2π r tan ϕ = Nϕ cos ϕ
Nθ ds dθ = ( H + dH )(r + dr )dθ − H r dθ Nθ = d ( H r ) / ds If the dome were simply supported, the maximum ring force at the lowest strip would be max . Nθ = H rmax
For the element → ds1 × ds 2 The external forces on the element = Pr ⋅ ds1 ⋅ ds 2 From equilibrium Nϕ
Nθ Pr + = r1 r2
r= a For a shperical surface r= 1 2 Nϕ +Nθ = Pr a For a conical surface N θ = Pr r2
Spherical Shells 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 Nϕ per unit length acting tangentially at B The reaction thrust Nϕ +d Nϕ per unit unit length at point D Nϕ
E F
A
θ
C
B
D Nϕ+dNϕ
ϕ
H Nϕ
a
r a
ϕ
dϕ
Spherical Shells Nϕ
E F
A C
Meridional Force N ϕ Surface area of shell AEB A 2π a ⋅ EF = EF = a (1 − cos ϕ )
θ
ϕ
H
W = w D ⋅ A = ωD ⋅ 2 π a ⋅ EF
Nϕ
a
r a
W = w D ⋅ 2 π a 2 (1 − cos ϕ ) N ϕ (2π r ) sin ϕ = w D ⋅ 2 π a 2 (1 − cos ϕ ) r = a sin ϕ w D ⋅ a (1 − cos ϕ ) w D ⋅ a (1 − cos ϕ ) w Da Nϕ = = sin 2 ϕ (1 − cos ϕ )(1 + cos ϕ ) 1 + cos ϕ
ϕ
dϕ
B
D Nϕ
Spherical Shells Internal forces due to Dead load
wD / m 2 surface
Hoop Force N θ 1 N θ = w D r cosϕ ϕ 1 cos + wr At crow n ϕ 0= Nθ = 2 At base ϕ = 90 N θ = −wr
θ 51o 49' N θ 0= when= for ϕ < 51o 49'
N θ will be compressive
for ϕ > 51o 49'
N θ will be tensile
Spherical Shells Internal forces due to Dead load
wD / m 2 surface
Ring Force H H
N= wD ϕ cos ϕ
cos ϕ a 1 + cos ϕ
at θ= 51o 49' ⇒ N θ= 0 & F is maximum Fmax = 0.382 w D a
Spherical Shells
Spherical Shells Internal forces due to Live load (wL/m2)horizontal Meridional Force T W = w L ⋅π r 2 = w L ⋅ π a 2 sin 2θ = y a (1 − cos θ ) r = a sin θ
θ w L ⋅ π a 2 sin 2θ N ϕ ( 2π a sin θ ) sin= w La 2 Hoop Force Nθ Nϕ =
wL a ( cos 2ϕ ) 2 Ring Tension Nθ =
cos ϕ 2 at ϕ = 45o ⇒ N θ = 0 & H is maximum H max = 0.3535 w L a
= H N= wL a ϕ cos ϕ
Spherical Shells
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 low tensile stress in the adjacent zones of the shell
Ring beam design Design of Horizontal Beam As =
(T )Ultimate Load 0.9f y
T= H × r Design of Vertical Beam Vertical Uniform load (w V ) N φ sin φ + o .w = 2π r Span length ( l ) = # of supports P = 2 π r wV M max −ve= C 1× P r M max += C 2× P r ve
see the tables of circular beams see the tables of circular beams
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 ' f c 300 = kg / cm 2 and f y 4200 kg / cm 2
y=1.4m r=6m
Spherical Shells under General Loading
Internal Forces Due to Others Loading
