Criteria for design of reinforced concrete shell structures and folded plates

IS 2210: Scope & Key Specifications for Circular Cylindrical Shells

Scope (Clause 8.1.1.3 & 8.1.2.2)

• Provides simplified analysis of circular cylindrical shells using precompiled tables (Appendix B).
• Tables assist in calculating stresses, moments, and buckling parameters efficiently.
• Appendix B includes:
  • B-1: Tables for shell analysis.
  • B-2: Beam method for cylindrical shells.

Important Notations (Clause 3.1)

Key Formulas (from IS 2210 & general shell theory)

• Flexural rigidity: [ D = \frac{E_c d^3}{12(1 - v^2)} ]
• Buckling stress (critical): [ \sigma_{cr} = \frac{k \pi^2 D}{(b)^2} ] where k = buckling coefficient, b = effective width.

Summary

• Use Appendix B tables for quick design checks.
• Refer to notations for consistent parameter use.
• Shell thickness, curvature, and material properties govern design.

flowchart TD
    A[Start: Shell Geometry & Material] --> B[Select Shell Type]
    B --> C[Use Appendix B Tables]
    C --> D[Calculate Moments & Forces]
    D --> E[Check Buckling & Stresses]
    E --> F[Design Verification]

For detailed design, always refer to Appendix B and relevant clauses.

IS 2210: Definitions & Classification of Shells

Key Notations (Clause 3.1)

• a, b: Semi-major & semi-minor axes of elliptical shells
• d: Thickness of shell
• Ec, Es: Modulus of elasticity of concrete & steel
• fck: Characteristic strength of concrete
• H, h: Total depth & rise of shell
• Mx, My, Mxy: Bending & twisting moments
• Nx, Ny, Nxy: Membrane forces
• Rc, R1, R2: Radii of curvature
• v: Poisson's ratio

Classification of Shells (Appendix A, Clause 4.2)

Reinforcement Spacing (Clause 12.3.2)

• Max spacing ≤ 5 × shell thickness (d)
• Max unreinforced panel area ≤ 15 × d²
• Edge members follow IS 456

Analysis Aids

• Use Appendix B tables for circular cylindrical shells (Clause 8.1.1.3) to simplify calculations.

flowchart TD
    A[Shells] --> B[Singly-Curved (Gauss=0)]
    A --> C[Doubly-Curved (Non-developable)]
    C --> D[Anticlastic (Negative Curvature)]
    C --> E[Other Special Types (Positive Curvature)]
    B --> F[Cylindrical, Conical]
    D --> G[Hyperbolic Paraboloids,

IS 2210 - Notations & Symbols (Clause 3.1) Key Points

• Geometric parameters:

  • a, b: Semi-major and semi-minor axes of elliptical shells
  • B: Chord width
  • d: Thickness of shell
  • h: Rise of shell
  • H: Total depth (crown to edge)
  • Rc, R, R1, R2: Radii of curvature at points on shell

• Material properties:

  • Ec: Modulus of elasticity of concrete (long term)
  • Es: Modulus of elasticity of steel
  • fck: Characteristic strength of concrete
  • v: Poisson's ratio

• Stress & force notations:

  • F: Stress function (in-plane stresses with bending)
  • Tx, Ty: Normal stresses in x and y directions
  • S: Shear stress
  • Mx, My, Mxy: Bending and twisting moments
  • Nx, Ny, Nxy: Membrane forces
  • w: Deflection along z-axis

• Flexural rigidity: [ D = \frac{E_c d^3}{12(1 - v^2)} ]

• Bending analysis equations (for shells of constant curvature): [ D \nabla^4 w + F = Z ] where (Z) is vertical load/unit area.

• Stress resultants from stress function (F) and deflection (w): [ T_x = \frac{\partial^2 F}{\partial y^2}, \quad T_y = \frac{\partial^2 F}{\partial x^2}, \quad S = -\frac{\partial^2 F}{\partial x \partial y} ] [ M_x = D \frac{\partial^2 w}{\partial x^2}, \quad M_y = D \frac{\partial^2 w}{\partial y^2}, \quad M_{xy} = -D(1-v) \frac{\partial^2 w}{\partial x \partial y} ]

• Tables for circular cylindrical shells analysis: See Appendix B for beam method tables

IS 2210: Classification of Shells — Key Points

1. Classification (Appendix A, Clause 4.2)

Shells are classified based on curvature and geometry:

2. Reinforcement Spacing (Clause 12.3.2)

• Max spacing of reinforcement ≤ 5 × shell thickness (t)
• Max unreinforced panel area ≤ 15 × (t²)
• Note: Edge members follow IS 456-1978.

3. Simplifications & Tables (Clause 8.1.1.3)

• Use Appendix B tables for circular cylindrical shells analysis.
• These tables simplify complex shell calculations.

Summary Diagram of Shell Types

graph TD
    A[Shells] --> B[Singly-Curved Developable]
    A --> C[Doubly-Curved Non-developable]
    A --> D[Anticlastic (Negative Curvature)]
    B --> B1[Cylindrical Shells]
    B --> B2[Conical Shells]
    C --> C1[Circular Domes]
    C --> C2[Ellipsoids, Paraboloids]
    D --> D1[Hyperbolic Paraboloids]
    D --> D2[Hyperboloids]

References:

• Appendix A for classification details.
• Appendix B for analysis tables.
• Clause 12.3.2 for reinforcement spacing limits.

IS 2210 - Materials: Key Formulas, Tables & Specifications

1. Notations (Clause 3.1)

• d = Thickness of shell
• Ec, Es = Modulus of elasticity of concrete and steel
• fck = Characteristic compressive strength of concrete
• D = Flexural rigidity = (\frac{E_c d^3}{12(1 - v^2)})
• v = Poisson's ratio
• R, Rc = Radius of curvature (general and at crown)
• Mx, My, Mxy = Bending and twisting moments
• Tx, Ty, S = Normal and shear stresses
• w = Deflection along z-axis

2. Flexural Rigidity Formula

[ D = \frac{E_c d^3}{12(1 - v^2)} ]

3. Governing Equations for Bending Analysis (Clause 2.1)

For shells with constant curvature: [ \begin{cases} D \left(\frac{\partial^4 w}{\partial x^4} + 2 \frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4}\right) + Z = 0 \ \text{(where } Z = \text{vertical load/unit area)} \end{cases} ]

4. Stress Resultants from Stress Function (F) and Deflection (w):

[ \begin{aligned} T_x &= \frac{\partial^2 F}{\partial y^2}, \quad T_y = \frac{\partial^2 F}{\partial x^2}, \quad S = -\frac{\partial^2 F}{\partial x \partial y} \ M_x &= D \left(\frac{\partial^2 w}{\partial x^2} + v \frac{\partial^2 w}{\partial y^2}\right), \quad M_y = D \left(\frac{\partial^2 w}{\partial y^2} + v \frac{\partial^2 w}{\partial x^2}\right) \ M_{xy} &= -D(1 - v) \frac{\partial

IS 2210 - Loads: Key Points and References

• Dead Loads:
  Calculated using unit weights from IS 875 (Part 1)-1987 (Clause 6.2).

• Live, Wind, and Snow Loads:
  Taken as specified in IS 875 (Parts 2 to 4)-1987 (Clause 6.3):

  • Part 2: Live loads
  • Part 3: Wind loads
  • Part 4: Snow loads

• Load Combinations for Design (Clause 6.1):
  Shells and folded plates must be designed for combinations such as:

  • Dead Load + Live Load or Snow Load
  • Dead Load + Live Load + Wind Load
  • Dead Load + Live Load + Seismic Load

• Concentrated Loads:
  Require special analysis and design considerations (Clause 6.5).

Typical Load Combination Format (IS 875 & IS 2210)

Where:

• D = Dead Load
• L = Live Load
• W = Wind Load
• S = Snow Load
• E = Earthquake Load

Reference for Unit Weights (IS 875 Part 1)

flowchart TD
    A[Loads as per IS 875] --> B[Dead Load (IS 875 Part 1)]
    A --> C[Live Load (IS 875 Part 2)]
    A --> D[Wind Load (IS 875 Part 3)]
    A --> E[Snow Load (IS 875 Part 4)]
    F[Design Load Combinations] -->|Use| B
    F -->|Use| C
    F -->

IS 2210: Geometrical Requirements & Dimensions for Shells

Key Parameters (Clause 3.1)

• d = thickness of shell
• H = total depth from crown to edge
• h = rise of shell
• a, b = semi-major & semi-minor axes (elliptical shells)
• Rc, R, R1, R2 = radii of curvature at crown or any point
• B = chord width

Important Stresses & Forces

• Mx, My, Mxy = bending & twisting moments
• Nx, Ny, Nxy = membrane forces
• Tx, Ty, S = normal & shear stresses
• F, @ = stress functions for in-plane and membrane analysis

Flexural Rigidity

[ D = \frac{E_c d^3}{12(1 - v^2)} ]

Where:

• (E_c) = modulus of elasticity of concrete
• (v) = Poisson's ratio
• (d) = shell thickness

Design Considerations (Clause 6.5)

• Concentrated loads require special analysis beyond membrane theory.

Dimension Selection (Clause 8.1.2.2 & Appendix B)

• Use Tables B-1 and B-2 for circular cylindrical shells for simplified analysis and beam methods.
• Refer to Appendix B for tabulated values of buckling stresses, moments, and load factors.

Summary Table: Typical Geometrical Parameters

flowchart TD
    A[Shell Geometry] --> B[Thickness (d)]
    A --> C[Radius of Curvature (R)]
    A --> D[Rise (h)]
    A --> E[Depth (H)]
    B --> F[Flexural Rigidity D]
    F --> G[Design Stresses]
    C --> H[

IS 2210 - Analytical Methods for Shells and Folded Plates (Clause 8 & Appendix B)

Key Points:

• Two-stage analysis for continuous shells (8.1.3.1):

  1. Assume shell simply supported over one span; calculate stress resultants.
  2. Apply continuity corrections and superimpose on stage 1 results.

• Approximate cylindrical shells as folded plates for analysis using beam and folded plate methods (Appendix B).

• Tables (Appendix B, B-1): Provide values for stress resultants and deflections for circular cylindrical shells under various loading and boundary conditions.

• Beam Method (Appendix B, B-2): Treat folded plates as beam elements for simplified analysis.

Analytical Methods:

• Classical methods: Use tables from Appendix B for common shells without complexities.
• Finite Element Method (FEM): Recommended for complex cases (irregular geometry, openings, large spans >30m, variable loads, nonlinear behavior).
• Finite Strip Method (FSM): Efficient for prismatic folded plates and cylindrical shells; discretizes shells into strips or rings.

Typical Formula for Cylindrical Shell Bending Stress (from elasticity theory):

[ \sigma_\theta = \frac{M_x}{Z_x} = \frac{M_x}{t \cdot r^2} ]

Where:

• (\sigma_\theta) = hoop stress
• (M_x) = bending moment per unit length
• (t) = shell thickness
• (r) = radius of curvature

Summary Table (Simplified):

flowchart TD
    A[Shell Analysis] --> B[Classical Methods]
    A --> C[Finite Element Method]
    A --> D[Finite Strip Method]
    B --> E[Use Appendix B Tables]
    B --> F[Beam Method for Folded Plates]
    C

IS 2210 - Permissible Stresses Summary

1. Permissible Stresses Reference

• Clause 9.1.1 states:
  Permissible stresses in steel reinforcement and concrete for shells and folded plates shall follow IS 456-1978.

2. Key Parameters for Shells (Clause 2.1)

• Flexural rigidity:
  [ D = \frac{E_c d^3}{12(1 - \nu^2)} ]
  where:

  • (E_c) = Modulus of elasticity of concrete
  • (d) = Thickness of the shell
  • (\nu) = Poisson's ratio

• Stress functions and bending moments:
  For shells under vertical load (Z), bending stress resultants are:
  [ M_x = D \frac{\partial^2 w}{\partial x^2}, \quad M_y = D \frac{\partial^2 w}{\partial y^2}, \quad M_{xy} = -D(1-\nu) \frac{\partial^2 w}{\partial x \partial y} ]

• (w) = deflection along z-axis

• (F) = stress function for in-plane stresses

3. Permissible Stresses from IS 456-1978 (Typical Values)

4. Notes

• Use IS 456 for exact permissible stresses for concrete and steel.
• Shell design involves solving partial differential equations for stress functions and deflections.
• Tables and beam methods for shell analysis are given in Appendix B of IS 2210.

flowchart LR
    A[Vertical Load (Z)] --> B[Shell Element]
    B --> C[Calculate Deflection (w)]
    C --> D[Calculate Bending Moments (Mx, My, Mxy)]
    D --> E[Determine Stresses using IS 456 permissible limits]

Summary:

Design of Traverses (IS 2210: Clauses 10.2 - 10.4)

• Loads on Traverses (Clause 10.2):
  Traverses carry:

  • Self-weight
  • Shear forces from shell reactions
  • Direct loads on them
    For preliminary design, consider the total load on half the span as a uniformly distributed vertical load on the diaphragm.

• Load Resolution & Analysis (Clause 10.3):

  • Shear forces from shell to end frames are resolved into vertical and horizontal components.

  • The traverse and shell act monolithically; shell participates in bending.

  • Effective shell width acting with traverse:

    Traverse Type	Effective Width (on one side)
    Intermediate Traverse	0.33 × 'Rd' to 0.76 × VRd
    End Traverse	On one side, same range applies

    Where:

    • 'Rd' and 'VRd' = Design resistances (refer to IS 2210 for exact definitions)
    • Higher value for rigid ribs, lower for flexible ribs.

• Connection & Expansion (Clause 10.4):

  • Traverses may be hinged to columns, except when designed integrally (e.g., portal frames).
  • Columns must allow for thermal expansion/contraction of traverses.

Summary Table for Effective Width

Diagram: Load Transfer in Traverses

flowchart LR
    Shell -- Shear Forces --> Traverse
    Traverse -- Shear & Bending --> End Frames
    Traverse -- Expansion/Contraction --> Columns

Note: Refer to IS 2210 for detailed definitions of 'Rd', 'VRd', and load factors.

IS 2210: Design of Edge Beams - Key Points

1. Edge Beam Function (Clause 11.1)

• Edge beams stiffen shell edges and share loads with the shell.
• Vertical beams: for long cylindrical shells (cylindrical action).
• Horizontal beams: for short cylindrical shells (transverse arch action).
• Edge beams should maintain compatibility with shell boundary conditions and consider eccentricities between shell centerline and beam.

2. Dimensions (Clause 11.1.1)

• Width = 2 to 3 × shell thickness
• Minimum width = 15 cm

3. Reinforcement (Clause 11.1.2)

• Edge beams resist longitudinal tensile forces (Nx).
• Provide main reinforcement layers to keep stresses within permissible limits.
• Design for:
  • Self-weight
  • Live load on shell part
  • Wind load
  • Horizontal earthquake forces

4. Design Considerations

• Edge beams may be interior or exterior.
• Consider support conditions and possible eccentricities.
• Ensure compatibility with shell deformation.

Typical Design Formula for Reinforcement Area (As):

[ A_s = \frac{N_x}{f_{sd}} ]

Where:

• (N_x) = Longitudinal tensile force in beam
• (f_{sd}) = Design stress in steel (permissible stress)

Summary Table for Edge Beam Width

graph LR
A[Shell Edge] --> B[Edge Beam]
B --> C[Loads: Nx, Self-weight, Live, Wind, Earthquake]
B --> D[Reinforcement: Longitudinal Tensile Bars]
B --> E[Compatibility with Shell Boundary]

Note: For detailed reinforcement calculation, refer also to IS 456 for concrete and steel design.

Design of Reinforcement as per IS 2210 (with reference to IS 456-1978):

Key Points:

• Minimum Reinforcement: Follow IS 456-1978 for minimum reinforcement limits (Clause 12.3).
• Edge Beams: Must carry longitudinal tensile forces (Nx), self-weight, live load, wind, and seismic forces (Clause 11.1.2). Multiple layers of reinforcement may be needed.
• Stress Limits: Ensure stresses in the farthest reinforcement layer do not exceed permissible limits.
• Welding: Permitted as per IS 456-1978 (Clause 5.2.1).
• Reinforcement Distribution: Bars should be closely spaced for uniform steel distribution. Minimum 8 mm diameter bars at ~200 mm c/c in compression zones (Clause 12.2.5).

Typical Reinforcement Design Formula (from IS 456):

[ A_s = \frac{M}{0.87 f_y z} ]

Where:

• (A_s) = area of steel required
• (M) = bending moment
• (f_y) = yield strength of steel
• (z) = lever arm (approx. 0.95d, d = effective depth)

Minimum Reinforcement (IS 456):

Reinforcement Placement:

• Compression zone: 8 mm bars @ 200 mm c/c minimum.
• Edge beams: Multiple layers as required.
• Uniform distribution to avoid stress concentration.

flowchart TD
    A[Loads on Shell] --> B[Edge Beam Longitudinal Tensile Forces (Nx)]
    B --> C[Design Reinforcement for Nx]
    C --> D[Check Stress in Farthest Layer ≤ Permissible Stress]
    D --> E[Provide Multiple Layers if Needed]
    E --> F[Place Bars Close, Min 8mm dia @ 200mm c/c]
    F --> G[Welding as per IS 456 Allowed]

References:

• IS 2210 Clause 12.3, 11.1.2, 12.2.5
• IS

IS 2210: Detailed Classification of Stressed Skin Surfaces (Appendix A & Clauses)

1. Classification of Stressed Skin Surfaces

2. Key Notations (Clause 3.1)

• d = Thickness of shell
• a, b = Semi-major and semi-minor axes (elliptical shells)
• H = Total depth of shell
• Mx, My, Mxy = Bending and twisting moments
• Nx, Ny, Nxy = Membrane forces
• Ec, Es = Modulus of elasticity of concrete and steel
• fck = Characteristic compressive strength of concrete
• v = Poisson's ratio
• R, Rc, RI = Radii of curvature

3. Reinforcement Spacing (Clause 12.3.2)

• Maximum spacing in any direction ≤ 5 × thickness (d)
• Area of unreinforced panel ≤ 15 × (thickness)²

4. Buckling and Stress

• Permissible compressive stress fac from buckling considerations (see IS 456 for reinforced concrete edge members)
• Use stress function (F) for in-plane stress with bending in doubly-curved shells

5. Tables for Analysis (Clause 8.1.1.3 & Appendix B)

• Simplified tables are provided for circular cylindrical shells to aid in membrane and bending stress calculations.
• These tables use parameters like Pc (semi-central angle), P, and K (Jakobsen’s parameters).

Summary Diagram of

IS 2210: Analysis of Circular Cylindrical Shells & Folded Plates (Appendix B Highlights)

Key Points:

• Simplification via Tables (Clause 8.1.1.3):
  Use precompiled tables (Appendix B) for stress resultants and moments in circular cylindrical shells to simplify calculations.

• Two-Stage Analytical Method (Clause 8.1.3.1):

  1. Assume shell simply supported over one span; calculate stress resultants.
  2. Apply continuity corrections and superimpose for final stresses.

• Folded Plate Approximation:
  Long cylindrical shells can be approximated as folded plates, analyzed by standard folded plate methods (see Appendix B).

• Finite Element & Finite Strip Methods (Clause 8.0):
  Recommended for complex geometries, boundary conditions, openings, variable loads, and large spans (>30 m).
  FSM is efficient for prismatic folded plates and cylindrical shells by discretizing into strip/ring elements.

Typical Formulas (from classical shell theory):

• Membrane Stress Resultants:

[ N_\theta = \frac{pR}{t}, \quad N_x = \frac{pR}{2t} ]

Where:

• (p) = external pressure

• (R) = shell radius

• (t) = shell thickness

• Bending Moments:

[ M_\theta = \frac{pR^2}{t}, \quad M_x = \frac{pR^2}{2t} ]

Tables in Appendix B (summary):

Recommended Analysis Workflow:

flowchart TD
    A[Start: Define shell geometry & loads] --> B{Is shell simple & span < 30m?}
    B -- Yes --> C[Use classical tables & methods (Appendix B)]
    B -- No --> D[Use Finite Strip or Finite Element Method]
    C --> E[Calculate stresses & moments]
    D --> E
    E --> F[Apply continuity corrections]
   

Governing Equations for Doubly-Curved Shells
(IS 2210: Clause 8.2.3.1, Appendix C)

C-1. Membrane Analysis

• Projected vs Real Stress Resultants:

[ \begin{aligned} N_x &= N_{xp} \sqrt{1 + q^2} \sqrt{1 + p^2} \ N_y &= N_{yp} \sqrt{1 + p^2} \sqrt{1 + q^2} \ N_{xy} &= N_{xyp} \sqrt{1 + p^2} \sqrt{1 + q^2} \end{aligned} ]

• Fictitious Forces (X, Y, Z) vs Real Forces (W_x, W_y, W_z):

[ \begin{aligned} X &= \frac{W_x}{\sqrt{1 + p^2 + q^2}} \ Y &= \frac{W_y}{\sqrt{1 + p^2 + q^2}} \ Z &= \frac{W_z}{\sqrt{1 + p^2 + q^2}} \end{aligned} ]

• Equilibrium Equations:

[ \begin{cases} \frac{\partial N_{xyp}}{\partial y} + X = 0 \ \frac{\partial N_{yp}}{\partial y} + 2 \frac{\partial N_{xyp}}{\partial x} + Y = 0 \ r N_{xp} + 2 s N_{xyp} + t N_{yp} = p_x + q_y - Z \end{cases} ]

• Stress Function (\Phi):

[ \begin{aligned} N_{xp} &= \frac{\partial^2 \Phi}{\partial y^2} \ N_{yp} &= \frac{\partial^2 \Phi}{\partial x^2} \ N_{xyp} &= - \frac{\partial^2 \Phi}{\partial x \partial y} \end{aligned} ]

Using (\Phi), the equilibrium reduces to a single PDE involving (\Phi).

C-2. Bending Analysis

• Governed by bending theory equations for shallow doubly-curved shells (Clause