COMPUTERS AND STRUCTURES, INC., FEBRUARY 07 CONCRETE SHELL REINFORCEMENT DESIGN Technical Note Backgroun The esign of reinforcement for concrete shells in accorance with a preetermine fiel of moments, as implemente in the software, is base on the provisions in DD ENV 99-- 99 Eurocoe : Design of Concrete Structures. Generally, slab elements are subjecte to eight stress resultants. Using the software terminology, those resultants are the three membrane force components f, f an f ; the two flexural moment components m an m an the twisting moment m ; an the two transverse shear force components V 3 an V 3. For the purpose of esign, the slab is conceive as comprising two outer layers centere on the mi-planes of the outer reinforcement layers an an uncracke core this is sometimes calle a "sanwich moel." The covers of the sanwich moel (i.e., the outer layers) are assume to carry moments an membrane forces, while the transverse shear forces are assigne to the core, as shown in Figure. The esign implementation in the software assumes there are no iagonal cracks in the core. In such a case, a state of pure shear evelops within the core, an hence the transverse shear force at a section has no effect on the in-plane forces in the sanwich covers. Thus, no transverse reinforcement nees to be provie, an the in-plane reinforcement is not enhance to account for transverse shear. The following items summarize the proceure for concrete shell esign, as implemente in the software:. As shown in Figure, the slab is conceive as comprising two outer layers centere on the mi-planes of the outer reinforcement layers.. The thickness of each layer is taken as equal to the lesser of the following: Twice the cover measure to the center of the outer reinforcement. Backgroun Page of 0
Twice the istance from the center of the slab to the center of outer reinforcement. Figure : Statics of a Slab Element Sanwich Moel Backgroun Page of 0
3. The six resultants, f, f, f, m, m, an m, are resolve into pure membrane forces N, N an N, calculate as acting respectively within the central plane of the top an bottom reinforcement layers. In transforming the moments into forces, the lever arm is taken as the istance between the outer reinforcement layers. 4. For each layer, the reinforcement forces NDes, NDes, concrete principal compressive forces Fc, Fc, an concrete principal compressive stresses Sc an Sc, are calculate in accorance with the rules set forth in Eurocoe 99. 5. Reinforcement forces are converte to reinforcement areas per unit with Ast an Ast (i.e., reinforcement intensities) using appropriate steel stress an stress reuction factors. Basic Equations for Transforming Stress Resultants into Equivalent Membrane Forces For a given concrete shell element, the variables h, Ct, Ct, Cb, an Cb, are constant an are expecte to be efine by the user in the area section properties. If those parameters are foun to be zero, a efault value equal to 0 percent of the thickness, h, of the concrete shell is use for each of the variables. The following computations apply: t h Ct t h Ct b h Cb h Cb ; ; ; b Cb h Ct ; h Ct Cb ; min = Minimum of an b min = Minimum of b an b t min = Minimum of t an t Page 3 of 0
The six stress resultants obtaine from the analysis are transforme into equivalent membrane forces using the following transformation equations: N N N top m f b ; m f b N bot m f t top ; N bot m f b m f t min min top ; N bot min m f t min Equations for Design Forces an Corresponing Reinforcement Intensities For each layer, the esign forces in the two irections are obtaine from the equivalent membrane forces using the following equations, in accorance with Eurocoe -99. In the equations below, F, F, F, NDes, an NDes are temporary variables. For reinforcement in top layer - If N top N (top), then F N(top); F N(top); F N(top) If N top > (top), If F F, then N then F N(top); F N(top); F N(top) NDes F NDes F F F Fc top F Equations for Design Forces an Corresponing Reinforcement Intensities Page 4 of 0
If F < F, then NDes 0 NDes Fc F F F F top F F If N top N(top), then NDes top NDes ; NDes top If N top N(top), then NDes top NDes ; NDes top NDes NDes For reinforcement in bottom layer - If N bot N (bot), then F N(bot); F N(top); F N(bot) If N bot > (bot) N then F N(bot); F N(bot); F N(bot) If F F, then NDes NDes Fc F F F F bot F If F < F, then NDes 0 NDes F F F Equations for Design Forces an Corresponing Reinforcement Intensities Page 5 of 0
Fcbot F F F If N bot N(bot), then NDes bot NDes ; NDes bot If N bot N(bot), then NDes bot NDes ; NDes bot NDes NDes Following restrictions apply if NDes or NDes is less than zero: If NDes top 0, then NDes top 0 NDes top 0 If NDes top 0, then If NDes bot 0, then NDes bot 0 NDes bot 0 If NDes bot 0, then The esign forces calculate using the preceing equations are converte into reinforcement intensities (i.e., rebar area per unit with) using appropriate steel stress from the concrete material property assigne to the shell element an the stress reuction factor, s. The stress reuction factor is assume to always be equal to 0.9. The following equations are use: Ast Ast top top NDes top ; Ast bot 0.9( f ) y NDes top ; Ast bot 0.9( f ) y NDes 0.9( f ) bot y NDes bot 0.9( f ) y Equations for Design Forces an Corresponing Reinforcement Intensities Page 6 of 0
Principal Compressive Forces an Stresses in Shell Elements The principal concrete compressive forces obtaine earlier are use to compute the principal compressive stresses to top an bottom layers as follows: Sc top Fc top ; Minimum( Ct, Ct ) Sc bot Fc bot Minimum( Cb, Cb ) Principal Compressive Forces an Stresses in Shell Elements Page 7 of 0
Notations The algorithms use in the esign of reinforcement for concrete shells are expresse using the following variables: Ast (bot) Ast (top) Ast (bot) Ast (top) Cb Cb Ct Ct Reinforcement intensity require in the bottom layer in local irection Reinforcement intensity require in the top layer in local irection Reinforcement intensity require in the bottom layer in local irection Reinforcement intensity require in the top layer in local irection Distance from the bottom of section to the centroi of the bottom steel parallel to irection Distance from the bottom of the section to the centroi of the bottom steel parallel to irection Distance from the top of the section to the centroi of the top steel parallel to irection Distance from the top of the section to the centroi of the top steel parallel to irection Lever arm for forces in irection Lever arm for forces in irection b b Distance from the centroi of the bottom steel parallel to irection to the mile surface of the section Distance from the centroi of the bottom steel parallel to irection to the mile surface of the section b min Minimum of b an b Notations Page 8 of 0
min t t Minimum of an Distance from the centroi of the top steel parallel to irection to the mile surface of the section Distance from the centroi of the top steel parallel to irection to the mile surface of the section t min Minimum of t an t f Membrane irect force in local irection f Membrane in-plane shear forces f Membrane irect force in local irection Fc(bot) Fc(top) f y h Principal compressive force in the bottom layer Principal compressive force in the top layer Yiel stress for the reinforcement Thickness of the concrete shell element m Plate bening moment in local irection m Plate twisting moment m Plate bening moment in local irection N (bot) Equivalent membrane force in the bottom layer in local irection N (top) Equivalent membrane force in the top layer in local irection N (bot) N (top) N (bot) Equivalent in-plane shear in the bottom layer Equivalent in-plane shear in the top layer Equivalent membrane force in the bottom layer in local irection N (top) Equivalent membrane force in the top layer in local irection Notations Page 9 of 0
NDes (top) Design force in the top layer in local irection NDes (top) Design force in the top layer in local irection NDes (bot) Design force in the bottom layer in local irection NDes (bot) Design force in the bottom layer in local irection Sc(bot) Sc(top) s Principal compressive stress in the bottom layer Principal compressive stress in the top layer Stress reuction factor References DD ENV 99--: 99 Eurocoe : Design of Concrete Structures, Part. General rules an rules for builings References Page 0 of 0