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University of Michigan Department of Civil & Environmental Engineering
---oOo---
ANALYSIS and DESIGN of REINFORCED CONCRETE
CYLINDRICAL SHELL ROOF STRUCTURE
CEE515 Term Report by Hai Dinh
-December 17, 2004-
TABLE OF CONTENT
1. Introduction to cylindrical shell roof structure ..............................3
2. Membrane theory...............................................................................6 2.1. General discussion...................................................................................................................... 6 2.2. Coordinates ................................................................................................................................. 6 2.3. Equations of equilibrium............................................................................................................ 7 2.4. Stresses under some special conditions ..................................................................................... 8 2.5. Discussion on the stress results of the table 2-1 ...................................................................... 10
3. Bending theory .................................................................................10 3.1. General discussion.................................................................................................................... 10 3.2. General bending theory ............................................................................................................ 11 3.3. Stresses and displacements from the loaded membrane theory of circular shell roofs.......... 14 3.4. Schorers bending theory for circular shell roof ..................................................................... 15
a.) Single shell without edge beams ........................................................................................... 20 b.) Single shell with deep edge beams (Fig. 3.3)........................................................................ 20
3.5. Other results, discussion, and result of Donnells theory ....................................................... 21
4. Design of circular shell roofs...........................................................26 4.1. Selection of shell....................................................................................................................... 26 4.2. Design steps .............................................................................................................................. 26 4.3. Determine reinforcement ......................................................................................................... 27
5. Conclusion.........................................................................................27
CEE515 Term Report
Analysis and Design of Reinforced Concrete Cylindrical Shell Roof Structure
Abstract: Shell structure is a classical and broad topic in reinforced concrete structures. This report deals with reinforced concrete cylindrical shell roof structures. A general discussion on shell roof problems will be provided. Membrane theory and bending theory of cylindrical shell roofs are then introduced in the form of governing differential equations to exactly solve the problem. An analogy between mathematics and structures in solving the problem is next discussed. Different results of different authors who make different assumptions are given and a discussion on using the results is also made. The design steps will show how to use these results. The appendix is devoted for a numerical example to make the calculation of shell roof structures approachable.
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1. Introduction to cylindrical shell roof structure
A cylindrical shell roof structure is a particular type of cylindrical shell structure including water tanks. Geometrically, the shell is a singly curved surface, domes being an example of doubly curved shell, generated by a straight line generator running along a cylindrical directrix (Fig. 1.1). For roofs, the directrix can be a chord of a circler, an ellipse, a cycloid, a catenary, or a parabola. The roof is usually composed of a single shell or of multiple shells supported by traverses and/or edge beams. (Fig. 1.2, 1.3). The traverses can be trusses, reinforced concrete diaphragms or ribs.
Fig. 1.1: Geometry of cylindrical shell (Courtesy of G. S. Ramaswamy)
Fig. 1.2: Single shell supported by ribs as traverses (Courtesy of D. P. Billington)
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Fig. 1.3: Multiple shells supported by edge beams and traverses on columns (Courtesy of D. P. Billington)
The shell roofs discussed in the report are within the range of thin elastic shell theory. There are four assumptions of the theory. The first assumes that the shell must be thin. A shell is considered to be thin if d/R 1/20 [1], where d and R are respectively the thickness and the radius of the shell. Mathematically, this assumption implies that the ratio d/R can be neglected compared to unity in derivation of the thin elastic shell theory [7]. Structurally, this assumption infers a small defection of shells, a negligible stress normal to the middle surface as reference surface of the shell, and a preservation of normals to the middle surface of the shell. It is noted that the designation of the middle
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surface as the reference surface assumes a homogeneous material of the shell. The second inference is not correct in the area of the shell subjected to concentrated loads. Therefore, this theory is applicable to uniform loads, which are dead load and live load in the report. The third assumption is an extension the Bernoulli hypothesis in beam theory, which states that plane sections remain plane after deformation. Thus, it assumes that there is no strain in the direction of the normals and that the strain is linearly distributed across the thickness of the shell. All four assumptions are often referred to as the Loves hypotheses.
Under these assumptions, the stresses in a shell element (Fig. 1.4) are only normal stresses Nx , N , shear stress Nx (membrane theory, Article 2) or might include transverse shear stresses Qx , Q and couples Mx , M , Mx (bending theory, Article 3).
Fig. 1.4: Stresses in membrane theory (a)
a.) b.)
c.)
and bending theory (b, c) (Courtesy of G. S. Ramaswamy)
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If the edges beams are supported on a continuous foundation, the shell behaves like an arch, i.e. there are the normal stresses only. If the edges are free, the load will be carried by both the traverses and edge beams. This particular kind of shells is often referred to as simply-supported shells, and discussed in this report. They are subdivided into short, immediate, or long depending on the ratio /R, which is discussed in more detail in Article 3. In general, the short shell behaves almost like a shell and the long like a beam and the immediate somewhere in between. That is why beam method is also valid for solving long shell problem.
2. Membrane theory
2.1.General discussion
In membrane theory, the loads are considered to be carried only by in-plane direct stresses Nx , N , and Nx (Fig.1.4a), which lay in the shell middle surface. This is quite reasonable because it has a small rigidity (thin shell) and is curved, not straight like slabs, which carry loads by bending moments. If direct stresses are compressive, they can cause a buckling effect that is similar to a column under axial forces.
It is noted that in order for such kind of the stress state, i.e. in-plane direct stresses, to exist, the shell must be closed (case of circular, elliptical or cycloidal shells) or extend to infinity (case of parabolic or catenary shells) so that the stresses can be self-balanced (with the load, of course). It is well known that circular pipes or vessels subjected to internal pressure are examples of shells subjected to direct stresses only. The only stress in the pipes is the ring tensile stress N (1 in Fig. 2.1), while there is the longitudinal tensile stress Nx (2 in Fig. 2.1). However, none of them is the case of simply-supported shell roofs, which have free edges not satisfying the condition specified above. At the free edges, the bending moments develop, which requires bending theory in Article 3.
Fig. 2.1: Idealized shell (Courtesy of J. M. Gere and S. P. Timoshenko)
2.2.Coordinates
The coordinate system used in membrane theory is shown in Fig. 2.2. It is noted that the y-axis is tangent to the plane directrix at the considered point and that the z-axis coincides with the radius at the considered point. Therefore, the y- and z- axes change their directions depending on the position of the considered point. In addition, the loads Y, Z in
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the equilibrium Equations (2-1b, c) is in the same direction of the y- and z- axes respectively.
Fig. 2.2 Coordinate system and stresses in membrane (Courtesy of G. S. Ramaswamy)
2.3.Equations of equilibrium
The shell problem in the membrane theory is straightforward in that it has three unknowns, two normal stresses Nx , N , and transverse shear stress Nx and three equations of equilibrium are enough for solving the problem.
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0
01
01
=+=+
+
=++
ZRN
YN
RxN
XN
RxN
x
xx
(2-1a, b, c)
It is noted in Equations (2-1) that the radius R, loads X, Y, Z and stresses Nx , N , Nx are the functions of the angle and the coordinate x.
2.4.Stresses under some special conditions
The general equation of radius of a cylindrical shell is as follows:
1cos += noRR (2-2)
where: Ro is a constant radius n = 0 for a circular shape, 1 for cycloid, -2 for catenary, and -3 for parabola
The loads to be considered are dead load g on the shell surface and snow load po on the horizontal surface:
2cos;cossin
cos;sin
oo pZpY
gZgY
====
(2-3a, b, c, d)
If the shell is simply-supported, for instance, by traverses and the load X is null, which represent most common problems, the stresses are as follows:
12
2
1
cos1
422
sin)2(
cos
+
+=
+==
no
x
x
no
RxgnN
gxnNgRN
l for dead load g (2-4a, b, c)
2
222
2
2
cossincos
423
cossin)3(cos
+=
+== +
xR
nN
xpnNRpN
ox
ox
nooo
l for snow load po (2-5a,b, c)
By replacing appropriate value of n, we can get the stresses for different shapes of shells (Table 2.1).
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Tab
le 2
.1: M
embr
ane
stre
sses
(Cou
rtes
y of
G. S
. Ram
asw
amy)
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2.5.Discussion on the stress results of the table 2-1
In the case of catenary shells under dead loads or parabolic shells under snow loads, the only stress is N and compressive. The shell behaves like an arch. If these shells are idealized, i.e. expanding into infinity, this result is reasonable. If the shells are terminated at the edges, the edges have to carry all the compressive stress to the support by behaving like the beams; and the edges beam are usually introduced to carry this stress (Fig. 2.3).
Fig. 2.3: Behavior of edge beams
In almost other cases, there exist the stresses Nx and Nx , which dictate the beam action of the shell. The stresses Nx and Nx of a closed circular shell are exactly the same as those obtained of a hollow circular beam from the bending theory [2] (Fig. 2.4). If the shell is terminated at the edge, the edges are subjected to tensile force while the shell near the crown subjected to compressive force.
Fig. 2.4: Shear stress Nx (or Nx) and normal stress Nx diagram (Courtesy of D. P. Billington)
3. Bending theory
3.1.General discussion
As discussed in the article 2.1 and 2.4, the membrane theory cannot be solely used to solve the simply-supported shell roof problem. This is because the theory shows that the stresses N and Nx exist everywhere in the shell whereas the free edges without edge
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beams dictate that the stresses must be null (free + no edge beams)*. Therefore, it is necessary to introduce fictive stresses, which are usually referred to as corrective line load, along edge to realize these particular boundary conditions (Fig. 3.1). The fictive stresses are the value from the membrane theory with the angle C at the edge. However, this introduction will cause the shell to bend [1], which is not the scope of the membrane theory. Therefore, it is necessary to introduce the bending theory under the effect of the fictive stresses. The result of the two theories, i.e. the membrane theory under actual loads, referred to as loaded membrane theory in this report, and the bending theory under fictive stresses only referred to as unloaded bending theory, is superimposed to get the final result of the shell roof problem. It should be noted immediately that the fictive stresses in the bending theory are lately treated as a boundary condition rather than loads. The above process, as being described later in the report, reflects the steps in solving differential equations of the general bending theory in Article 3.2.
-N
-Nx
(a) (b)
Fig. 3.1: (a) Stresses from membrane theory (b) Fictive stresses to be realized in bending theory
3.2.General bending theory
Instead of considering the loads to be carried by the three normal stresses Nx , N , Nx as in the membrane theory, the bending theory introduces the shear stresses Qx , Q , and three couple stresses Mx , M , Mx (Fig. 1.4b, c). These eight stresses are supposed to carry the load X, Y, Z. In addition to these stresses as unknowns, there are also the three displacements u , v , w corresponding to the three axes x , y , z (Fig. 3.2). The total eleven unknowns may be got from a system of eleven equations described as follows: * This can be understood that there is no reaction at the free edge. The reporter think, however, that if the free edges are associated with the stiff edge beams, no substantial displacements, they can readily give reaction forces to the shell, which may then behaves almost like an arch. A stiffness ratio between the shell and the edge beam may be introduced to clarify this situation.
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Fig. 3.2: Coordinate system in bending theory 3.2.1. Five equations of equilibrium The equilibrium equations can be written as follows:
01;01
01
01;01
=+
=+
=++
+
=++
=++
xxxx
x
xxx
QM
axM
QM
axM
Za
NQax
Q
Ya
QNax
NX
Nax
N
(3-1a, b, c, d, e)
3.2.2. Three equations of stress strain relations
From the three equations of stress strain relations, the relations between the normal stresses and displacement can be written as follows:
( )
+
=
+
=
+
=
xv
auKN
xu
aw
avKN
aw
av
xuKN
x
x
1
(3-2a, b, c)
where ( )21/ = EdK : extensional rigidity (3-3) : Poison ratio.
3.2.3. Three equations of moment curvature relations
From the three equations of moment curvature relation above, the relations between the couple stresses and displacement can be written as follows:
( )
+
=
+
+=
+
+=
xv
axawDM
xw
aw
avDM
wvax
wDM
x
x
x
211
2
2
2
22
2
2
2
2
22
2
(3-4)
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where : flexural rigidity (3-5) )1(12/ 23 = EdD
3.2.4. Governing differential equation
The eleven equations (3-1), (3-2) and (3-4) form the basis of the general bending shell theory. The combination of the equations leads to a partial differential equation in term of one of any of eleven unknowns above. The solution of the equation will then be used to calculate other unknowns.
Billington* derived the differential equation of the shallow shell theory as below. It is noted that in deriving the equation (3-6), there are two assumptions to be made: (1) the in-plane displacement v is negligible in the expression of curvature changes and (2) the shear stress Q is then negligible [2]. This means that the shell is flat enough (or shallow) so that the curvature changes is due mainly to the displacement w and that Q plays a little role in carrying the load.
++
=
+
=
=+
3
3
42
3
32
3
23
34
4
22
2
2
28
4
4
28
1121),,("
where;),,("
Y
axX
axY
axX
aZ
DZYXf
wax
w
ZYXfxw
DaEdw
(3-6a, b, c)
3.2.5. Solution to the governing differential equation
It is well known in solving differential equations like (3-6) that the solution to the equation (3-6) is the sum of a particular solution (or integral) to the equation (3-6)and the general solution to the homogeneous equation of (3-6) corresponding to that particular integral. The general solution w to the homogeneous equation of (3-6) has the form of:
w = f(x, , C1, C2, ..., C8, m1, m2, , m8) (3-7)
where x, are coordinates as described previously to dictate the displacements at different points on the shell, C1, C2, ..., C8 are non-dimension constants depending on the boundary conditions and m1, m2, , m8 are the eight roots from a characteristic equation to the eighth order depending on the geometry of the shell. In term of mathematics, the particular solution is used to find the non-dimension coefficient. The displacement together with that from the particular solution (integral) will be the solution to the equation (3-6).
* The reporter does not know who is the first person deriving the differential equation (3-6) for the shallow shell theory. The equation (3-6) is adopted from Ref. [2].
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It is noted that there is no external forces X, Y or Z in the homogeneous part of (3-6). Therefore, the homogeneous equation represents unloaded bending theory. This homogeneous equation can be also obtained by using the equilibrium Equations (3-1) without the loads X, Y, Z. In addition, it will be shown later that the solution to the membrane theory is the particular solution to the equation (3-6) to some extent [1]. The membrane solution is then used to find the non-dimension constants. This shows that the combination of the loaded membrane solution and the unloaded bending solution which employs the loaded membrane solution as the boundary condition reflects the solving of a differential equation like (3-6) of a general bending theory.
3.3.Stresses and displacements from the loaded membrane theory of circular shell
roofs
This part will report the stresses and displacements from the loaded membrane theory in the coordinate system of bending theory (so that they can be readily used), with the Poison ratio = 0.
The dead load can be convenient to be expressed in the Fourier form. If the first term is considered only, the loads Y, Z, stresses and displacements are as follows:
)sin(cos4
)cos(cos4
cos1)1(4
1
1
=
=
= +
c
c
nn
xgY
xgZ
xnn
gg
l
l
l
(3-8 a, b, c)
)cos(sin8
)sin(cos8
)cos(cos4
3
2
2
=
=
=
cx
cx
c
xagN
xgN
xagN
ll
ll
l (3-9a, b, c)
)2(cos)cos(8
)2(cos)sin(8
)2(cos)sin(8
42
4
2
2
42
4
2
2
42
4
2
2
ax
Edgw
ax
Edgv
ax
Edgu
c
c
c
lll
lll
lll
+=
+=
+=
) (3-10a, b, c)
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Similarly, the loads Y, Z, stresses and displacements in case of snow load are as follows:
+= n no lxnp 1 1 cos1)1(4 (3-11)
)(2coscos12
)(2sinsin6
)(coscos4
3
2
2
2
=
=
=
co
x
co
x
co
xap
N
xpN
xapN
ll
ll
l
(3-12)
lll
lll
ll
xaaEd
apw
xaaEd
apv
xaEd
pu
co
co
co
cos)(2cos224
cos)(2sin212
sin)(2cos12
24
5
2
24
5
2
4
3
+
=
+
=
=
(3-13)
3.4.Schorers bending theory for circular shell roof
This part introduces the Schorers theory which is the fundamental to explain the result of others.
3.4.1. Governing differential equation
By assuming Mx = Mx = Qx= 0 , small tangential and shear strain in compared to longitudinal strain and = 0, Schorer derives the following differential equation:
=
=
+=+)(;)('
'1)(1''''
.
.....::2
::::6
xwhere
Xa
YZa
EdwwaD
(3-14)
The homogeneous equation corresponding to the equation (3-14)
2
24::::
120''''
adkwherew
kaw ==+ (3-15)
3.4.2. Solution to the homogeneous equation
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The general solution to (3-15) has the form:
a
xHew nm
cos= (3-16) where n = na/ C, m are constant as explained in (3-7) By substituting (3-16) into (3-17), we have the following characteristic equation:
04
8 =+k
m n
(3-17)
The roots of (3-17) are formed in complex conjugate pairs as follows:
48372615
222223112111
;;;;;;
mmmmmmmmimimimim
=====+==+=
(3-18)
where
8/1
2/12/12/1
4/121
2/12/14/121
;384.0)12(8
924.0)12(8
kn
===
+== (3-19)
With these roots, the displacement w, stresses and other displacements are as the equations (3-20)
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(3-20a, b, c, d, e, f, g, h, i)
Where is the rotation of the tangent to the shell directrix at the considered point.
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The expressions (3-20) contain the constants An, Anc, Ani ; Bn, Bnc, Bni ; Cn, Cnc, Cni ; Dn, Dnc, Dni which are described in Table (3-1). It is noted from Table (3-1) that all the constant above are expressed in term of the four constants An, Bn, Cn, Dn . The only thing to do next is to determine these four constants from the boundary conditions.
Table 3.1 (Courtesy of G. S. Ramaswamy)
3.4.3. Particular integral due to dead load g This part will partially show that the solution from the loaded membrane theory is a particular integral of the unloaded bending theory. The dead load g expressed in Fourier series and considered the first term has the form as in the equation (3-8a) and its components Y, Z have the form as in the equations (3-8b, c). Substituting (3-8b, c) in (3-14), we get the following differential equation due to the dead load g:
==
=+ )(;)('cos)cos(81'''' .::::5 xwhere
xga
EdawwaD
c l (3-21)
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A particular integral of (3-21) has the following form:
lxCw c
cos)cos( = (3-22) Substitute (3-22) in (3-21), we get the following value of C
+
=5
34
12
8
aEd
lEdaa
gC (3-23)
The eleven equations (3-1), (3-2) and (3-4) with Schorers assumptions will then be used to derive the following result of displacement and stresses:
ll
ll
l
xagN
xgN
xgaN
negligibleisQM
cx
cx
c
cos)cos(8
sin)sin(8
cos)cos(4
,
3
2
2
=
=
=
(3-24a, b, c)
)223(
)233(cos)sin(
sin)cos(843
=
=
inasw
inasCwithxCv
xEdagu
c
c
l
ll
(3-25a, b)
It can be seen that the stresses in (3-24) and displacement in (3-25) are similar to those in (3-9) and (3-10) from the loaded membrane theory. Therefore, it can be concluded, as mentioned in article 3.4, that the solution to the loaded membrane theory can be used as a particular integral without serious error if the load on shell is uniform [1].
3.4.4. Boundary conditions for determination of the non-dimension constants An, Bn,
Cn, Dn
In a general shell problem, there are four non-dimension constants as in (3-7). In a symmetrical shell problem, in term of loads and geometry, the number of boundary conditions is four corresponding to the four constants An, Bn, Cn, Dn as in the Schorers theory. These boundary conditions depend on the edge configuration of the shell. The following case is considered:
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a.) Single shell without edge beams At = 0:
0)()(
0)()(
0)()(
0)()(
=+==+=
=+==+=
mxbxx
mb
mb
mb
NNNNNN
QQQMMM
(3-26a, b, c, d)
The subscript b, m in (3-26) denotes the stresses results from the unloaded bending theory and the loaded membrane theory respectively. It is note that the equations (3-26) must hold true for all values x along the shell. b.) Single shell with deep edge beams (Fig. 3.3) When the deep edge beams are provided at the shell edges, they are designated to resist vertical forces form the shell rather horizontal forces and torsion. This assumption formulates the first two boundary conditions. The other two are formulated by the compatibility of vertical and horizontal deflection of the shell edges and edge beams at their junction, their deflections being equal. The compatibility must hold true for all value x along the shell. All four boundary conditions are as follows:
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Fig. 3.3 Boundary conditions of single shell with deep edge beams (Courtesy of G. S. Ramaswamy) At = 0:
(3-27a, b, c, d) [ ] [ ]
[ ] [ ] vcmbcmb hmbcmbcmb
mb
wwvvuu
QQNNHMMM
=+++=+
=+++==+=
cos)()(sin)()()()(
0sin)()(cos)()(0)()(
Where: H: horizontal force from the shell h: horizontal displacement of the edge beams v: vertical displacement of the edge beams h and v have the following form:
[ ][ ] [ ]{ }
[ ] [ ]{ }b) 28a,-(3'1
cos)()(sin)()(1
'cos)()(sin)()(
)()(1
41
3
4
13
13
21
2
WEI
NEIa
QQNNEI
WIaQQNN
Ia
NNI
aAE
x
cmbcmbv
cmbcmb
mxbxh
+
++
=
++
+
+
+
=
ll
l
ll
l
where A: area of the cross section of the edge beams I: moment of inertia of the cross section of the edge beams 2a1: height of the edge beams W = 4W/ , W: the weight per unit length of the beam
3.5.Other results, discussion, and result of Donnells theory
We have so far examined Schorer bending theory. In addition, there are also other literatures in which different assumptions are made. They include Flgges theory, Dischingers, Aas-Jacobens, Lundgrens, Holands, Donells, and Finsterwalders. Krmn, Jenkins, and Vlasov also have the same theory as Donells. Thus, Donells theory is sometimes known as D-K-Js theory [1]. The ASCE Manual also provides two methods known as Approximation No.1 and Approximation No.2. Basically, Approximation No.2 gives the result the same as that of Donells theory [1].
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Table 3.1 summarizes the assumptions made by different theories.
Theory Assumptions Membrane
theory as a particular integral
1. Flgge Loves hypotheses
NO
2. Dischinger - - 3. Aas-Jacoben - - 4. Lundgren - - 5. ASCE Manual, 1
Loves hypotheses
YES
6. Holand - NO 7. ASCE Manual, 2 Donell, Vlasov
see 3.2.4 -
8. Finsterwalder Mx = Mx = Qx= 0
-
9. Schorer Mx = Mx = Qx= 0 = xy = 0
NO
Table 3.1
Ramaswamy [1] assumes the Poisson ratio to be zero and uses the characteristic equations to compare different theories (Table 3.2). In the table 3.2, the following notations are introduced:
2
2
2
8/1
2/1
12;
/
adkanwhere
k
mm
n
n
n
==
=
==
l
(3-29a, b, c)
Flgge does not use the membrane solution a particular integral, but compare the two numerically and has the same conclusion as in article 3.4.3.
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Table 3.2
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It is now important to discuss which theory should be used in designing circular shell roofs. Ramaswamy bases on the Approximation No.2 in the ASCE Manual and Schorers theory to distinguish among short, immediate and long shells and then suggests the method used. Table 3.3 shows the classification of short, immediate and long shells of Gibson [3], ASCE Manual [8], and Ramaswamy [1].
Short shells Immediate
shells Long shells Very
long shells
Gibson Range /a 0.5 0.5 < /a
CEE515 Term Report
(3-30a, b, e, f, g, h, i, j, k) The constants , have the form as in (3.29b, c). The constants Anc , Bnc, Cnc , Dnc are expressed in term of An, Bn, Cn, Dn as in Table 3.1. The constants And , Bnd, Cnd, Dnd are expressed in term of An, Bn, Cn, Dn as follows: And = (12-12)An - 211Bn Bnd = 211An - (12-12)Bn Cnd = (22-22)Cn - 222Dn Dnd = 222Cn - (22-22)Dn Page 25 of 28
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The constants 1, 2, 1, 2 have the following form.
( ) ( )( ) ( )( ) ( )( ) ( ) 2/124/12
2/12
4/11
2/12
4/12
2/12
4/11
211218
211218
211218
211218
++=
+++=
+=
++++=
(3-31a, b, c, d)
4. Design of circular shell roofs
4.1.Selection of shell
The table 4.1 summarizes the suggestions in [1] of shell parameters.
Parameter Suggestion 1. Long shell vs. short shell
- short shells for covering large area such as hangars; chord width of 400ft is possible; traverses should be provided - practical span of long shells is 100ft; prestressing may be considered; edge beams should be provided.
2. Radius - acoustic consideration: center of curvature should not lie at the working level
3. Semi-central angle - from 30 to 45 4. Thickness - minimum of 4 cm to 5 cm (2inches)
- 7cm to 8cm recommended 5. Shell depth - for short shells: shell depth / chord width > 1/10
- for long shell: 1/12 < shell depth / span < 1/6 6. Edge beam - minimum of 6 inches for practical purposes
Table 4.1: Parameter of shell roofs
4.2.Design steps
As discussed to some extent in article 3.4, the shell calculation usually has the following steps a.) Prepare preliminary data including span , radius a, thickness d, semi-central
angle c, dead load g and snow load po . b.) Compute the stresses and displacements from the loaded membrane theory using
equations, for example (3-9), (3-10).
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c.) Compute the stresses and displacements from the unloaded bending theory using equations, for example (3-20). This step requires the calculation of constants , using equations (3-29b, c), constants 1, 2, 1, 2 using, for example (3-19) and then constants An, Anc, Ani ; Bn, Bnc, Bni ; Cn, Cnc, Cni ; Dn, Dnc, Dni using, for example Table 3.1. The stresses and displacements are then determined in term of non-dimension constants An, Bn, Cn, Dn . The constants An, Bn, Cn, Dn are next determined using boundary conditions as in (3-26) or (3-27).
d.) Superimpose the stresses determined from b.) and c.) to get the final stresses and displacements.
e.) Compute reinforcement using the stresses and displacements from d.)
See Appendix A for the numerical examples
4.3.Determine reinforcement After determining all the stresses, the reinforcement can be calculated without too much difficulty. Billington [1] suggests first to calculate the principal stresses and then to distribute the reinforcement such that it can carry the principal stresses. The reinforcement can be placed along the principal stress trajectories or in the rectangular mesh. Although the former one can save more reinforcement, but cause difficulty for construction, the latter is more popular. 5. Conclusion
This report deals mostly with the calculation of simply-supported shell roofs. It explains the membrane theory and the bending theory and their relation in the shell roof problems. It then shows design how to use them by introducing design steps. This report is restricted in many dimensions. It should cover parts such as beam method commonly used for long shell roof design, edge beam and traverse design. The beam method employs the well known beam theory for a curved cross section to determine the axial stresse Nx and shear stresses Nx using the formulas Mc/I and VQ/Ib as seen in beam theory. The arch analysis is then done with the shear stresse Nx as the loads in addition to dead load and live load in order to determine the stresses M, Q, and N. Although the concept is simple, the calculation is lengthy and may not be clear without numerical examples. Similarly, the calculation of edge beams traverses such as ribs employs no more than simple beam or arch analysis, but need detailed explanation so that reinforcement can be distributed properly. Due to time constraint, these topics have not been in this report. However, the reporter hopes that this report provide the basis for any further study on reinforced concrete cylindrical shell roofs.
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Reference: 1. Ramaswamy, G. S., Design and Construction of Concrete Shell Roofs, McGraw-
Hill Book Company, 1968. 2. Billington, David P., Thin Shell Concrete Structures, McGraw-Hill Book
Company, 1982, chapter 1, 4, 5, and 6. 3. Gibson, J. E., The design of Shell Roofs, E. & F. N. Spon Ltd, 1968. 4. Design of Cylindrical Concrete Shell Roofs, Manual of Engineering Practice,
ASCE, vol. 31, New York, 1952. 5. Chinn, J., Cylindrical Shell Analysis Simplified by Beam Method, ACI Journal of
Structural Engineering, vol. 55, May 1959. 6. Cross, H., The column Analogy, University of Illinois Engineering Experiment
Station Bulletin 215, 1930. 7. Kraus, H., Thin Elastic Shell, John Wiley & Sons, Inc., 1967. 8. Fl gge, W., Stresses in shells, Springer, Berlin, 1962 9. Jenkins, R.S., Theory and Design of Cylindrical Shell Structures, The O N Arup
Group of Consulting Engineers, Colquhoun House, London W1, 1947.
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