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o_12_rc_th_sh_st

# o_12_rc_th_sh_st - 1.054/1.541 Mechanics and Design of...

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1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk 1 / 11 Massachusetts Institute of Technology 1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 12 Reinforced Concrete Thin Shell Structures ± Thin shell o Definition – A thin shell is a curved slab whose thickness h is small compared with its other dimensions and compared with its principal radius of curvature. o Middle surface The surface that bisects the shell is called the middle surface. It specifies the form of this surface and the thickness h at every point. o Analysis of thin shells consists the following steps: ² Establish equilibrium of a differential element cut from the shell ² Achieve strain compatibility so that each element remains continuous with each adjacent element after deformation. o Stress resultants and stress couples ± Shell theories o The Kirchhoff-Love theory – The first-approximation of shells ² Assumptions: (1) The shell thickness is negligibly small in comparison with the least radius of curvature of the shell middle surface. (2) Strains and displacements that arise within the shells are small. (3) Straight lines that are normal to the middle surface prior to deformation remain straight and normal to the middle surface during deformation, and experience no change in length.

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1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk Outline 12 ( Analogous to Navier’s hypothesis for beams – Bernoulli-Euler theory for beams ) (4) The direct stress acting in the direction normal to the shell middle surface is negligible. ± Results of the assumptions: Normal directions to the reference surface remain straight and normal to the deformed reference surface. The hypothesis precludes any transverse-shear strain, i.e., no change in the right angle between the normal and any line in the surface. It is strictly applicable to thin shells. It is not descriptive of the behavior near localized loads or junctions. (A ssumption (4) is not valid in the vicinity of concentrated transverse loads. ) o The Flügge-Byrne theory – The second-approximation of shells ± Assumptions: It adopts only assumption (2). It is referred to as “higher-order approximations” of the Kirchhoff-Love assumptions ² Classification of shells: o Classified by governing equation of geometry: ± Paraboloid of revolution ± Hyperboloid of revolution ± Circular cylinder ± Elliptic paraboloid ± Hyperbolic paraboloid ± Circular cone 2 / 11
1.054/1.541 Mechanics and Design of Concrete Structures Spring 2004 Prof. Oral Buyukozturk Outline 12 ± Geometrical analysis of shells: o Orthogonal curvilinear coordinates Consider the position vector () , r αβ = ( 1 , f ) u + ( ) 2 , f v + ( ) 3 , f w where 1 , f , ( ) 2 , f , and ( ) 3 , f are continuous, single-valued functions. The surface is determined by α and β uniquely. and β are called curvilinear coordinates. u , v , and w are unit vectors in the Cartesian coordinate system.

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o_12_rc_th_sh_st - 1.054/1.541 Mechanics and Design of...

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