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. 2.2. 2.3. 2.4. 2.5. General discussion...................................................................................................................... 6 Coordinates ................................................................................................................................. 6 Equations of equilibrium............................................................................................................ 7 Stresses under some special conditions ..................................................................................... 8 Discussion on the stress results of the table 2-1 ...................................................................... 10 3. Bending theory .................................................................................10 3.1. 3.2. 3.3. 3.4. a.) b.) 3.5. General discussion.................................................................................................................... 10 General bending theory ............................................................................................................ 11 Stresses and displacements from the loaded membrane theory of circular shell roofs .......... 14 Schorer’s bending theory for circular shell roof ..................................................................... 15 Single shell without edge beams ........................................................................................... 20 Single shell with deep edge beams (Fig. 3.3)........................................................................ 20 Other results, discussion, and result of Donnell’s theory ....................................................... 21 4. Design of circular shell roofs...........................................................26 4.1. 4.2. 4.3. Selection of shell ....................................................................................................................... 26 Design steps .............................................................................................................................. 26 Determine reinforcement ......................................................................................................... 27 5. Conclusion.........................................................................................27 The design steps will show how to use these results. Different results of different authors who make different assumptions are given and a discussion on using the results is also made.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. Page 2 of 28 . Membrane theory and bending theory of cylindrical shell roofs are then introduced in the form of governing differential equations to exactly solve the problem. The appendix is devoted for a numerical example to make the calculation of shell roof structures approachable. An analogy between mathematics and structures in solving the problem is next discussed. a cycloid.1: Geometry of cylindrical shell (Courtesy of G. P. a catenary. (Fig.2. Billington) Page 3 of 28 . Introduction to cylindrical shell roof structure A cylindrical shell roof structure is a particular type of cylindrical shell structure including water tanks. 1. an ellipse.3).CEE515 Term Report 1. or a parabola. The traverses can be trusses. 1. 1. S. For roofs. 1. 1. the directrix can be a chord of a circler.2: Single shell supported by ribs as traverses (Courtesy of D. the shell is a singly curved surface. reinforced concrete diaphragms or ribs.1). The roof is usually composed of a single shell or of multiple shells supported by traverses and/or edge beams. domes being an example of doubly curved shell. generated by a straight line generator running along a cylindrical directrix (Fig. Fig. Geometrically. Ramaswamy) Fig. 3: Multiple shells supported by edge beams and traverses on columns (Courtesy of D. a negligible stress normal to the middle surface as reference surface of the shell. this assumption implies that the ratio d/R can be neglected compared to unity in derivation of the thin elastic shell theory [7]. P. this assumption infers a small defection of shells. There are four assumptions of the theory. A shell is considered to be thin if d/R ≤ 1/20 [1]. It is noted that the designation of the middle Page 4 of 28 . Mathematically.CEE515 Term Report Fig. 1. The first assumes that the shell must be thin. Billington) The shell roofs discussed in the report are within the range of thin elastic shell theory. and a preservation of normals to the middle surface of the shell. where d and R are respectively the thickness and the radius of the shell. Structurally. The third assumption is an extension the Bernoulli hypothesis in beam theory. The second inference is not correct in the area of the shell subjected to concentrated loads. which states that plane sections remain plane after deformation. S. a. All four assumptions are often referred to as the Love’s hypotheses.) Page 5 of 28 . Ramaswamy) b. Thus. shear stress Nxφ (membrane theory.CEE515 Term Report surface as the reference surface assumes a homogeneous material of the shell. 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. Therefore.4: Stresses in membrane theory (a) and bending theory (b. Nφ . 1. Qφ and couples Mx .) Fig. Mφ . Under these assumptions. this theory is applicable to uniform loads. Mxφ (bending theory. the stresses in a shell element (Fig. Article 3).) c. c) (Courtesy of G. which are dead load and live load in the report.4) are only normal stresses Nx . Article 2) or might include transverse shear stresses Qx . 1. General discussion In membrane theory. which lay in the shell middle surface. and discussed in this report. This particular kind of shells is often referred to as simply-supported shells. there are the normal stresses only. not straight like slabs. That is why beam method is also valid for solving long shell problem.1). or long depending on the ratio ℓ/R.4a). In general. the y. 2. It is well known that circular pipes or vessels subjected to internal pressure are examples of shells subjected to direct stresses only. of course). while there is the longitudinal tensile stress Nx (σ2 in Fig. which carry loads by bending moments. which requires bending theory in Article 3. If direct stresses are compressive. the load will be carried by both the traverses and edge beams. The only stress in the pipes is the ring tensile stress Nφ (σ1 in Fig. the loads are considered to be carried only by in-plane direct stresses Nx . i. They are subdivided into short. which is discussed in more detail in Article 3. This is quite reasonable because it has a small rigidity (thin shell) and is curved. Therefore. the bending moments develop.1.1). Membrane theory 2. Fig. Nφ . 2.CEE515 Term Report If the edges beams are supported on a continuous foundation. However. the short shell behaves almost like a shell and the long like a beam and the immediate somewhere in between.axes change their directions depending on the position of the considered point.e. 2. P. none of them is the case of simply-supported shell roofs. immediate.e.1: Idealized shell (Courtesy of J.2. the loads Y. At the free edges. 2. and Nxφ (Fig. the shell behaves like an arch. 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. In addition. 2. Z in Page 6 of 28 .1. to exist.2. i. If the edges are free. in-plane direct stresses.and z. It is noted that in order for such kind of the stress state. 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. the shell must be closed (case of circular. M. they can cause a buckling effect that is similar to a column under axial forces. which have free edges not satisfying the condition specified above.Coordinates The coordinate system used in membrane theory is shown in Fig. Timoshenko) 2. Gere and S. Equations of equilibrium The shell problem in the membrane theory is straightforward in that it has three unknowns. Ramaswamy) 2. Page 7 of 28 .and z.axes respectively. Fig.2 Coordinate system and stresses in membrane (Courtesy of G. two normal stresses Nx . S. and transverse shear stress Nxφ and three equations of equilibrium are enough for solving the problem. c) is in the same direction of the y.3.CEE515 Term Report the equilibrium Equations (2-1b. 2. Nφ . 1 for cycloid. which represent most common problems. the stresses are as follows: Nφ = − gRo cos n +1 θ N xφ = −(n + 2) gx sin θ N x= − n+2 2 ⎛ l2 1 2⎞ ⎟ − g⎜ x ⎜4 ⎟ R cos n −1 θ ⎝ ⎠ o for dead load g (2-4a.1). Nxφ are the functions of the angle θ and the coordinate x. by traverses and the load X is null. for instance. c) By replacing appropriate value of n. c) It is noted in Equations (2-1) that the radius R.Stresses under some special conditions The general equation of radius of a cylindrical shell is as follows: R = Ro cosn +1 θ (2-2) where: Ro is a constant radius n = 0 for a circular shape. Z and stresses Nx . Z = p o cos θ 2 If the shell is simply-supported. Y. c. b. d) Y = p o sin θ cos θ . b. loads X.b. 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: Y = g sin θ .CEE515 Term Report ∂N x 1 ∂N xθ + +X =0 ∂x R ∂θ ∂N xθ 1 ∂N θ + +Y = 0 R ∂θ ∂x N θ + ZR = 0 (2-1a. -2 for catenary. c) N o = − p o Ro cos n + 2 θ N xφ = −(n + 3) p o x sin θ cos θ Nx = − ⎞ cos 2 θ − sin 2 θ n + 3 ⎛ l2 ⎜ − x2 ⎟ ⎟ 2 Ro ⎜ cos 2 θ ⎝ 4 ⎠ for snow load po (2-5a.4. we can get the stresses for different shapes of shells (Table 2. Z = g cos θ (2-3a. b. 2. Page 8 of 28 . Nφ . S.1: Membrane stresses (Courtesy of G. Ramaswamy) .CEE515 Term Report Page 9 of 28 Table 2. If the shells are terminated at the edges.4). If these shells are idealized. The shell behaves like an arch.e. the edges have to carry all the compressive stress to the support by behaving like the beams. 2.3). Fig. this result is reasonable.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. there exist the stresses Nx and Nxφ . the membrane theory cannot be solely used to solve the simply-supported shell roof problem.3: Behavior of edge beams In almost other cases. Billington) 3. 2.1. 2. and the edges beam are usually introduced to carry this stress (Fig. This is because the theory shows that the stresses Nφ and Nxφ exist everywhere in the shell whereas the free edges without edge Page 10 of 28 .CEE515 Term Report 2.5. Fig.General discussion As discussed in the article 2. If the shell is terminated at the edge. the only stress is Nφ and compressive. P. which dictate the beam action of the shell. i.4. The stresses Nx and Nxφ of a closed circular shell are exactly the same as those obtained of a hollow circular beam from the bending theory [2] (Fig. the edges are subjected to tensile force while the shell near the crown subjected to compressive force. 2. Bending theory 3.4: Shear stress N’φx (or Nxθ) and normal stress Nx diagram (Courtesy of D.1 and 2. expanding into infinity. A stiffness ratio between the shell and the edge beam may be introduced to clarify this situation. which are usually referred to as corrective line load. In addition to these stresses as unknowns. however. that if the free edges are associated with the stiff edge beams.1). and three couple stresses Mx . which is not the scope of the membrane theory. it is necessary to introduce fictive stresses. The result of the two theories.2). as being described later in the report. there are also the three displacements u .2. Z. Mφ .e. 1. the bending theory introduces the shear stresses Qx . Qφ .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. Therefore. The above process. y . It should be noted immediately that the fictive stresses in the bending theory are lately treated as a boundary condition rather than loads. 3.1: (a) Stresses from membrane theory (b) Fictive stresses to be realized in bending theory 3. it is necessary to introduce the bending theory under the effect of the fictive stresses. reflects the steps in solving differential equations of the general bending theory in Article 3. However. no substantial displacements. The fictive stresses are the value from the membrane theory with the angle φC at the edge. along edge to realize these particular boundary conditions (Fig. the membrane theory under actual loads.4b. Nφ . w corresponding to the three axes x . 3. 3. is superimposed to get the final result of the shell roof problem. Mxφ (Fig. referred to as loaded membrane theory in this report. i. Therefore. v .CEE515 Term Report beams dictate that the stresses must be null (free + no edge beams)*.General bending theory Instead of considering the loads to be carried by the three normal stresses Nx . and the bending theory under fictive stresses only referred to as unloaded bending theory. this introduction will cause the shell to bend [1]. z (Fig. they can readily give reaction forces to the shell. These eight stresses are supposed to carry the load X. The reporter think. Page 11 of 28 . c). Nxφ as in the membrane theory. Y. -Nxφ -Nφ (a) (b) Fig. which may then behaves almost like an arch. the relations between the couple stresses and displacement can be written as follows: ⎡ ∂ 2 w υ ⎛ ∂v ∂ 2 w ⎞⎤ ⎟ M x = −D⎢ 2 + 2 ⎜ ⎜ ∂φ + ∂φ 2 ⎟⎥ ∂ x a ⎢ ⎝ ⎠⎥ ⎣ ⎦ ⎡ ∂v ∂2w ∂2w⎤ M x = −D⎢ 2 + 2 2 + υ 2 ⎥ ∂x ⎦ ⎣ a ∂φ a ∂φ ⎛ ∂2w 1 ∂v ⎞ ⎟ M xφ = D(1 − υ )⎜ + ⎜ a∂x∂φ 2a ∂x ⎟ ⎝ ⎠ (3-4) Page 12 of 28 . 3.2. Three equations of stress strain relations (3-1a. − a ∂φ a ∂φ ∂x ∂x 3. + + − +Y = 0 ∂x a ∂φ a ∂φ a ∂x ∂Q x 1 ∂Qφ N φ + +Z =0 + a ∂φ a ∂x ∂M xφ 1 ∂M φ ∂M x 1 ∂M φ x − Qx = 0 + + Qφ = 0 .3. ( ) (3-3) 3.2.1.2.2. c) K = Ed / 1 − υ 2 : extensional rigidity υ: Poison ratio. b. the relations between the normal stresses and displacement can be written as follows: ⎡ ∂u ⎛ ∂v w ⎞⎤ N x = K ⎢ + υ⎜ ⎟⎥ ⎜ a∂φ − a ⎟ ⎠⎦ ⎝ ⎣ ∂x ⎡ ∂v w ∂u ⎤ − +υ ⎥ Nφ = K ⎢ ∂x ⎦ ⎣ a∂φ a ⎛ ∂u ∂v ⎞ N xφ = K (1 − υ )⎜ ⎟ ⎜ a∂φ + ∂x ⎟ ⎠ ⎝ where (3-2a. c. b.2: Coordinate system in bending theory 3. d. Three equations of moment curvature relations From the three equations of moment curvature relation above. Five equations of equilibrium The equilibrium equations can be written as follows: ∂N xφ 1 ∂N φ Qφ ∂N x 1 ∂N φ x +X =0 . e) From the three equations of stress strain relations.CEE515 Term Report Fig. The combination of the equations leads to a partial differential equation in term of one of any of eleven unknowns above.. * Page 13 of 28 .. Z ) . This means that the shell is flat enough (or shallow) so that the curvature changes is due mainly to the displacement w and that Qφ plays a little role in carrying the load. m2. C2. The reporter does not know who is the first person deriving the differential equation (3-6) for the shallow shell theory. Y .2.4. C8.Y . C8 are non-dimension constants depending on the boundary conditions and m1. c) ∇ w=⎜ ⎜ ∂x 2 + a 2∂φ 2 ⎟ ⎟ w ⎝ ⎠ ν ∂ 3 X 2 + ν ∂ 3Y 1⎛ 4 1 ∂3 X 1 ∂ 3Y ⎞ ⎟ f "( X .5.. C2. …. C1.. In term of mathematics. m8) (3-7) where x. the particular solution is used to find the non-dimension coefficient. b. The displacement together with that from the particular solution (integral) will be the solution to the equation (3-6). . ∇8 w + 8 Ed ∂ 4 w = f " ( X . where Da 2 ∂x 4 4 ⎛ ∂2 ∂2 ⎞ (3-6a. φ are coordinates as described previously to dictate the displacements at different points on the shell. The solution of the equation will then be used to calculate other unknowns. 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 Qφ is then negligible [2]. (3-2) and (3-4) form the basis of the general bending shell theory. φ.CEE515 Term Report where D = Ed 3 / 12(1 − υ 2 ) : flexural rigidity 3. Governing differential equation (3-5) The eleven equations (3-1). 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.. C1.2. m8 are the eight roots from a characteristic equation to the eighth order depending on the geometry of the shell.. Billington* derived the differential equation of the shallow shell theory as below. It is noted that in deriving the equation (3-6). …. [2]. Z ) = ⎜ Z ∇ − − + − D⎜ a ∂x 3 a 2 ∂x 2∂φ a 3 ∂x∂φ 2 a 4 ∂φ 3 ⎟ ⎝ ⎠ 3. . The equation (3-6) is adopted from Ref. m1. m2. c) Page 14 of 28 . b. Therefore. b. This homogeneous equation can be also obtained by using the equilibrium Equations (3-1) without the loads X.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). the homogeneous equation represents unloaded bending theory. If the first term is considered only. 3. Z. c) cos(φc − φ ) sin(φ c − φ ) Y =− π g cos 4 πx l cos(φ c − φ ) l πx 8l N xφ = 2 g cos sin(φ c − φ ) l π 2 8 gl πx Nx = − sin cos(φ c − φ ) 3 l aπ Nφ = − π ag cos πx (3-9a. 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 dead load can be convenient to be expressed in the Fourier form. 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. with the Poison ratio υ = 0. c) u= l4 πx 2l 2 8g sin(φ c − φ ) cos ( 2 + 2 4 ) l π πEd a π 2 l4 πx 2l 8g v=− sin(φ c − φ ) cos ( 2 + 2 4 ) ) l π πEd a π 2 πx 2l l4 8g w= cos(φ c − φ ) cos ( 2 + 2 4 ) l π πEd a π (3-10a. stresses and displacements are as follows: g= Z= 4 g ∑ (−1) n +1 1 n π 4 π g cos 4 πx l 1 nπx cos n l (3-8 a. Y or Z in the homogeneous part of (3-6). b. In addition. Z. The membrane solution is then used to find the non-dimension constants. the loads Y. Y.CEE515 Term Report It is noted that there is no external forces X. Schorer’s bending theory for circular shell roof This part introduces the Schorer’s theory which is the fundamental to explain the result of others.4. Schorer derives the following differential equation: D :::: 1 1 w + Edw' ' ' ' = 2 ( Z :: − Y . small tangential and shear strain in compared to longitudinal strain and υ = 0. Z.. 3. ) + X '. = ∂x ∂φ The homogeneous equation corresponding to the equation (3-14) w:::: + a4 w' ' ' ' = 0 k where k= d2 12a 2 (3-14) (3-15) 3.4. the loads Y. stresses and displacements in case of snow load are as follows: po = (−1) n +1 cos ∑ π n 1 4 n 1 πx l (3-11) Nφ = − N xφ cos 2 (φ c − φ ) l 6 po l πx sin sin 2(φ c − φ ) = 2 l π 4 po a π cos πx (3-12) N xφ = − 12 p o l aπ 3 2 cos πx l cos 2(φ c − φ ) u= 12 p o l 3 Edπ a 4 cos 2(φ c − φ ) sin 4⎡ πx l (3-13) 12 p o a 2 ⎛ l ⎞ v=− ⎜ ⎟ Edπ 5 ⎝ a ⎠ 2 ⎤ πx ⎛ πa ⎞ ⎢⎜ ⎟ + 2⎥ sin 2(φ c − φ ) cos l ⎢⎝ l ⎠ ⎥ ⎣ ⎦ 4 2 ⎤ 24 p o a 2 ⎛ l ⎞ ⎡⎛ πa ⎞ πx + 2 w=− ⎢ ⎥ cos 2(φ c − φ ) cos ⎜ ⎟ ⎜ ⎟ 5 l Edπ ⎝ a ⎠ ⎣ ⎢⎝ l ⎠ ⎥ ⎦ 3.2. Governing differential equation By assuming Mx = Mxφ = Qx= 0 .. ∂( ) where ' = . 6 a a a ∂( ) .CEE515 Term Report Similarly.. Solution to the homogeneous equation Page 15 of 28 .4.1. 924 ρ (2 1/ 2 ρ 81 / 4 − 1) 1/ 2 ≈ 0.384 ρ .CEE515 Term Report The general solution to (3-15) has the form: w = He mφ cos λn x a (3-16) where λn = nπa/ℓ C. m 2 = α 1 − iβ 1 . m 2 = α 2 − iβ 2 m5 = −m1 . m8 = −m 4 where (3-18) α1 = β 2 = α1 = β 2 = ρ 8 1/ 4 (21 / 2 + 1)1 / 2 ≈ 0. we have the following characteristic equation: m + 8 λ4 n k =0 (3-17) The roots of (3-17) are formed in complex conjugate pairs as follows: m1 = α 1 + iβ 1 . the displacement w. m7 = −m3 . ρ = /2 λ1 n (3-19) k 1/ 8 With these roots. m are constant as explained in (3-7) By substituting (3-16) into (3-17). m3 = α 2 + iβ 2 . m6 = −m 2 . stresses and other displacements are as the equations (3-20) Page 16 of 28 . e. d. i) Where ϑ is the rotation of the tangent to the shell directrix at the considered point. Page 17 of 28 . f. b. h. c.CEE515 Term Report (3-20a. g. 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. Z have the form as in the equations (3-8b. Dn . c). S. c) in (3-14). ∂( ) 1 ⎡ 8g πx ⎤ D :::: . Substituting (3-8b.… Dni which are described in Table (3-1).… Bni . Table 3.… Ani . Anc.CEE515 Term Report The expressions (3-20) contain the constants An. we get the following differential equation due to the dead load g: ∂( ) . Bnc. It is noted from Table (3-1) that all the constant above are expressed in term of the four constants An.… Cni . Cn. = w + Edaw' ' ' ' = ⎢ cos(φ c − φ ) cos ⎥ where ' = 5 ∂x l⎦ ∂φ a⎣π a (3-21) Page 18 of 28 . Dnc. Bn. Cnc. Dn.3. Cn. Ramaswamy) 3.4. The only thing to do next is to determine these four constants from the boundary conditions. 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.1 (Courtesy of G. Bn. Dn as in the Schorer’s theory.4. there are four non-dimension constants as in (3-7). the number of boundary conditions is four corresponding to the four constants An. Bn. Dn In a general shell problem. b.CEE515 Term Report A particular integral of (3-21) has the following form: w = C cos(φ c − φ ) cos πx l (3-22) Substitute (3-22) in (3-21). Boundary conditions for determination of the non-dimension constants An. 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. we get the following value of C C= 8g 4 ⎡ Ed 3 ⎤ ⎛π ⎞ πa ⎢ Eda⎜ ⎟ + 5⎥ l 12 a ⎝ ⎠ ⎥ ⎢ ⎦ ⎣ (3-23) The eleven equations (3-1). Bn.4. Cn. Cn. These boundary conditions depend on the edge configuration of the shell. in term of loads and geometry. Therefore. In a symmetrical shell problem. The following case is considered: Page 19 of 28 . 3. 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]. as mentioned in article 3.4. c) 8 gl π sin(φc − φ ) sin πx πx 8 gl 2 Nx = − cos(φ c − φ ) cos 3 l aπ 8 gl 3 πx u=− 4 cos(φ c − φ ) sin l π Eda πx v = −C sin(φ c − φ ) cos with C as in (3 − 23) l w as in (3 − 22) (3-25a. it can be concluded. (3-2) and (3-4) with Schorer’s assumptions will then be used to derive the following result of displacement and stresses: M φ . Qφ is negligible Nφ = − N xφ = 4 ga π 2 cos(φc − φ ) cos l πx l (3-24a. ) Single shell without edge beams At φ = 0: M φ = (M φ ) b + (M φ ) m = 0 Qφ = (Qφ ) b + (Qφ ) m = 0 Nφ = ( Nφ ) b + ( Nφ ) m = 0 N xφ = ( N xφ ) b + ( N xφ ) m = 0 (3-26a. they are designated to resist vertical forces form the shell rather horizontal forces and torsion. b. It is note that the equations (3-26) must hold true for all values x along the shell. The other two are formulated by the compatibility of vertical and horizontal deflection of the shell edges and edge beams at their junction. b. their deflections being equal. This assumption formulates the first two boundary conditions.CEE515 Term Report a. All four boundary conditions are as follows: Page 20 of 28 . m in (3-26) denotes the stresses results from the unloaded bending theory and the loaded membrane theory respectively. d) The subscript b. 3.) Single shell with deep edge beams (Fig. c. The compatibility must hold true for all value x along the shell.3) When the deep edge beams are provided at the shell edges. 3 Boundary conditions of single shell with deep edge beams (Courtesy of G. Jenkins. Ramaswamy) At φ = 0: M φ = (M φ ) b + (M φ ) m = 0 (u ) b + (u ) m = δ h φ b H = [( N φ ) b + ( N φ ) m ]cos φ c + [(Qφ ) b + (Qφ ) m ]sin φ c = 0 ) + (vφ ) m ]sin φ c + [( wφ ) b + ( wφ ) m ]cos φ c = δ v (3-27a. 3. c. Thus.2” gives the result the same as that of Donell’s theory [1]. and result of Donnell’s theory We have so far examined Schorer bending theory.CEE515 Term Report Fig. discussion. Lundgren’s. and Vlasov also have the same theory as Donell’s. Holand’s.5. Aas-Jacoben’s. Kármán. there are also other literatures in which different assumptions are made.2”.28a.1” and “Approximation No. W: the weight per unit length of the beam 3 4 4 3 [ ] [ ] [ ] (3 . Donell’s theory is sometimes known as D-K-J’s theory [1]. “Approximation No. b. In addition. d) [(v Where: δh: horizontal displacement of the edge beams δv: vertical displacement of the edge beams H: horizontal force from the shell δh and δv have the following form: ⎛l⎞ δh = ⎜ ⎟ ⎝π ⎠ 2 2 ⎞ ⎛ 1 a1 ⎟ ( N xφ ) b + ( N xφ ) m ⎜ + ⎜ AE I ⎟ ⎠ ⎝ [ ] [ ] 3 ⎛l⎞ a ⎛l⎞ a + ⎜ ⎟ 1 { ( N φ ) b + ( N φ ) m sin φc − (Qφ ) b + (Qφ ) m cos φc }− ⎜ ⎟ 1 W ' ⎝π ⎠ I ⎝π ⎠ I ⎛l⎞ 1 {( Nφ ) b + ( Nφ ) m sin φc − (Qφ ) b + (Qφ ) m cos φc } δv = ⎜ ⎟ ⎝ π ⎠ EI ⎛l⎞ 1 ⎛l⎞ a W' + ⎜ ⎟ 1 N xφ − ⎜ ⎟ ⎝ π ⎠ EI ⎝ π ⎠ EI 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/π . The ASCE Manual also provides two methods known as “Approximation No. Page 21 of 28 .Other results. Basically. b) 3. S. and Finsterwalder’s. Dischinger’s. They include Flügge’s theory. Donell’s. Theory Assumptions Membrane theory as a particular integral NO‡ YES NO NO 1. ASCE Manual. Finsterwalder 9. Flügge 2. 2 Donell. Aas-Jacoben 4. In the table 3. k= l 12a 2 where λ n = ‡ Flügge does not use the membrane solution a particular integral.1 Ramaswamy [1] assumes the Poisson ratio υ to be zero and uses the characteristic equations to compare different theories (Table 3. 1 6. Vlasov 8. Lundgren 5.4. b.CEE515 Term Report Table 3. Page 22 of 28 .4 Mx = Mxφ = Qx= 0 Mx = Mxφ = Qx= 0 εφ = γxy = 0 Table 3.2). Schorer Love’s hypotheses Love’s hypotheses see 3. ASCE Manual.2.1 summarizes the assumptions made by different theories.2.3. but compare the two numerically and has the same conclusion as in article 3. Holand 7. the following notations are introduced: m = m/ ρ ρ= /2 λ1 n k 1/ 8 ⎞ ⎟ ⎟ ⎠ 2 ⎛ λn κ =⎜ ⎜ ρ ⎝ (3-29a. Dischinger 3. c) nπa d2 . 2 Page 23 of 28 .CEE515 Term Report Table 3. 67 Immediate shells 0. Table 3. No. No. the stresses and displacements for Donell’s theory will be given for completeness.3 Because Ramaswamy’s suggestion will be used to treat the examples in the Appendix A.67 Suggested Donell’s method theory π ≤ ℓ/a Schorer’s theory or beam method Table 3. Short shells Gibson ASCE Manual Ramswamy Range Range ℓ/a ≤ 0.2” in the ASCE Manual and Schorer’s theory to distinguish among short.2 Range ℓ/a ≤ 1. Ramaswamy bases on the “Approximation No.5 ≤ ℓ/a 1. Page 24 of 28 . immediate and long shells of Gibson [3].67 < ℓ/a <5 Appr. immediate and long shells and then suggests the method used.1 π ≤ ℓ/a Schorer’s theory Very long shells 5 < ℓ/a Method Appr. and Ramaswamy [1].67 < ℓ/a <π Any theory from 1-7 in the table 3.5 < ℓ/a <2.5 1.3 shows the classification of short.CEE515 Term Report It is now important to discuss which theory should be used in designing circular shell roofs.2 Long shells 2. ASCE Manual [8].5 ℓ/a ≤ 1. Bn. Dnd are expressed in term of An. Cnc . Dn as in Table 3. f. b. c).2α2β2Dn Dnd = 2α2β2Cn . e.29b.1. Bnd. k) The constants ρ. i. Cn. The constants Anc .2α1β1Bn Bnd = 2α1β1An . The constants And . Cnd. κ have the form as in (3. Dn as follows: And = (α12-β12)An .(α12-β12)Bn Cnd = (α22-β22)Cn . g.(α22-β22)Dn Page 25 of 28 . Cn.CEE515 Term Report (3-30a. h. Bnc. Bn. Dnc are expressed in term of An. j. β2 have the following form. b.1.7cm to 8cm recommended 5.Design steps As discussed to some extent in article 3. radius a.Selection of shell The table 4.2.CEE515 Term Report The constants α1. Thickness .practical span of long shells is 100ft. for example (3-9). Radius . d) 1−κ 2 81 / 4 ⎢ ⎣ ρ ⎡ ( 2 ⎤ +1 + 1−κ 2 ⎥ ⎦ ( ) 1/ 2 4. dead load g and snow load po . the shell calculation usually has the following steps a.for short shells: shell depth / chord width > 1/10 .for long shell: 1/12 < shell depth / span < 1/6 6. traverses should be provided .minimum of 4 cm to 5 cm (2inches) .1: Parameter of shell roofs 4. α2. Parameter 1.from 30° to 45° 4. b.4. (3-10). edge beams should be provided. α1 = α2 = β1 = β2 = 81 / 4 ⎢ ⎣ ρ ⎡ ρ ⎡ (1 + κ 2 ) ( ( ) 2 ⎤ +1 + 1+ κ 2 ⎥ ⎦ ( ) 1/ 2 1−κ 2 81 / 4 ⎢ ⎣ 1+ κ 2 81 / 4 ⎢ ⎣ ) ) 2 ⎤ +1 − 1−κ 2 ⎥ ⎦ ⎤ +1 − 1+ κ 2 ⎥ ⎦ ( ) 1/ 2 ρ ⎡ 2 ( ) 1/ 2 (3-31a. semi-central angle φc.1 summarizes the suggestions in [1] of shell parameters. β1. Shell depth . Semi-central angle .) Compute the stresses and displacements from the loaded membrane theory using equations. short shell Suggestion . 2. prestressing may be considered. Edge beam . thickness d. Page 26 of 28 .acoustic consideration: center of curvature should not lie at the working level 3. Design of circular shell roofs 4.short shells for covering large area such as hangars. Long shell vs. chord width of 400ft is possible.) Prepare preliminary data including span ℓ.minimum of 6 inches for practical purposes Table 4. c. Cnc. Billington [1] suggests first to calculate the principal stresses and then to distribute the reinforcement such that it can carry the principal stresses. constants α1. κ using equations (3-29b. This report is restricted in many dimensions. the calculation is lengthy and may not be clear without numerical examples. Dn are next determined using boundary conditions as in (3-26) or (3-27). but cause difficulty for construction. The arch analysis is then done with the shear stresse Nxφ as the loads in addition to dead load and live load in order to determine the stresses Mφ. This step requires the calculation of constants ρ. Bn. Bn.1.) and c.3. Due to time constraint. β2 using. the latter is more popular. Qφ. Although the concept is simple. However. α2. e. Similarly. c). Dn . for example Table 3. It then shows design how to use them by introducing design steps. the reinforcement can be calculated without too much difficulty. Conclusion This report deals mostly with the calculation of simply-supported shell roofs. Anc. The reinforcement can be placed along the principal stress trajectories or in the rectangular mesh. Cn. The constants An.… Ani .) to get the final stresses and displacements. 5.) Compute the stresses and displacements from the unloaded bending theory using equations.) Superimpose the stresses determined from b.) See Appendix A for the numerical examples 4. the reporter hopes that this report provide the basis for any further study on reinforced concrete cylindrical shell roofs. Although the former one can save more reinforcement.) Compute reinforcement using the stresses and displacements from d. Cn. Dnc. for example (3-20). The stresses and displacements are then determined in term of non-dimension constants An. The beam method employs the well known beam theory for a curved cross section to determine the axial stresse Nx and shear stresses Nxφ using the formulas Mc/I and VQ/Ib as seen in beam theory. d.CEE515 Term Report c.… Cni . Bnc. and Nφ. Dn. the calculation of edge beams traverses such as ribs employs no more than simple beam or arch analysis. for example (3-19) and then constants An. It explains the membrane theory and the bending theory and their relation in the shell roof problems.Determine reinforcement After determining all the stresses. but need detailed explanation so that reinforcement can be distributed properly.… Dni using. It should cover parts such as beam method commonly used for long shell roof design. edge beam and traverse design. Bn. Page 27 of 28 . β1. these topics have not been in this report. Cn.… Bni . 1947. Chinn. May 1959. chapter 1. 55. “The design of Shell Roofs. and 6. 1967. J. Page 28 of 28 . Kraus. vol..S. 31. “Design and Construction of Concrete Shell Roofs. 1968... 1982. 5... “Cylindrical Shell Analysis Simplified by Beam Method. 2. S.” McGrawHill Book Company. 1930. H..” Manual of Engineering Practice. Springer. 1968. Theory and Design of Cylindrical Shell Structures. N.” ACI Journal of Structural Engineering. R. 6. “The column Analogy. Spon Ltd. G..” John Wiley & Sons. & F. Gibson. 4. 8. ASCE.” McGraw-Hill Book Company.” E. “Design of Cylindrical Concrete Shell Roofs. J. David P. London W1. Billington. 5. 4. Inc. E. vol. 1952.CEE515 Term Report Reference: 1. Colquhoun House. New York.” University of Illinois Engineering Experiment Station Bulletin 215. Jenkins. H. The O N Arup Group of Consulting Engineers. Stresses in shells. 1962 9. “Thin Shell Concrete Structures. “Thin Elastic Shell.. Cross.. Berlin. W. 3. Ramaswamy. Fl ügge. 7.