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PARAMETRIC ANALYSIS OF SPAN TO RISE RATIO OF SHALLOW FUNICULAR

CONCRETE SHELLS WITH OPENINGS LARGE ENOUGH TO DISTURB GLOBAL

STRESS DISTRIBUTION

A. Leema Rose1, S. Gunaselvi 2, M. Jeganathan3

1Associate Professor, Department of Civil Engineering, Valliammai Engineering College, Chennai, 603203.Tamilnadu, India.

2AssistantProfessor, Department of Civil Engineering, Valliammai Engineering College, Chennai, 603203.Tamilnadu, India.

3Associate Professor, Prime Nest College of Architecture and Planning, Siruganur, Thiruchirappalli.

ABSTRACT

Shells of diverse types are used for structural purposes due to their lightweight, swish shape

and excessive load resisting capacity. Those burdened pores and skin structures regularly

consist openings on their floor to necessitate ventilation and connections for in addition

systems. The objective of this project is to have a look at the behavior of Shallow Funicular

Concrete Shells over square floor Plan of various arbitrary rises RI and RII with 0% — 16%

openings. Analysis of Shallow Funicular Concrete Shells over square floor Plan of diverse

rises- RI and RII with 0% most of 16% starting is carried the use of the Finite element

software ANSYS and offered on this report. Fabric properties are defined and assigned

Restraints with all stages of freedom have been furnished at helps. From the analytical end

result the Membrane stresses, Bending pressure and Deflections are in comparison for the

diverse arbitrary rises. From the evaluation results Conclusions are armed for various rises for

the applied load.

KEY WORDS: concrete shell, shallow funicular shell, span to rise ratio.

INTRODUCTION

Systems can be categorized in many methods in keeping with their form, their

characteristic and the materials from which they're made. A form or structural detail can be a

fully 3 dimensional strong object like a monolithic pyramid or it would have a few

dimensions amazing smaller than a ball bearing. Shells belong to the elegance of forced-pores

and skin structures which, because of their geometry and small stress of the pores and skin,

have a tendency to carry loads basically via direct stresses acting in their plane. Shells impart

electricity thru its shape in desire to energy via mass. Concrete shells are systems which

because of their shape offer an terrific load bearing capability.

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Volume 9, Issue 1, January 2020

Concrete shells are structures which due to their form offer a first-rate load bearing

functionality. Reinforced concrete shell systems are significantly hired because of the fact

they offer each brilliant performance and architectural beauty. Although a primary form in

nature, a whole lot remains to be learnt about its utility in production enterprise. Ideally a

concrete shell in its membrane state consists of the external masses through way of natural

compression, unaccompanied by means of way of shear stresses in order that no tensile

stresses develop. Its miles advantageous to select the form of shell in any such way that,

below the circumstance of loading, the shell is subjected to pure compression without

bending. This will be completed by shaping the shell in the form of catenaries which the

funicular shape is similar to the dead weight. Shell of square and square ground plans are

very frequent in present day construction and architectural exercise.

Classification of Shells

Shells may be widely labeled as ‘singly-curved’ and ‘doubly-curved’. This is based on Gauss

curvature that is the made from the 2 essential curvatures, 1/R1 and l/R2 at any point at the

floor of the shell. Therefore, evolved doubly curved shells are non-evolved and are

categorized as synclastic or anticlastic in accordance as their Gauss curvature is advantageous

or poor.

The governing equations of membrane theory of singly curved shells are parabolic. It is -

elliptic for synclastic shells and hyperbolic for anticlastic shells. If z = f (x, y) is the equation

to the surface of a shell, the surface will be synclastic, developable or anticlastic according as

s2-rt ≠ 0 where

• r= δ2z/ δx2

• T= δ2z/δ y2

• s= δ2z/ δx δy

There are other special types of doubly curved shells, such as funicular shells, which

are synclastic and anticlastic in parts and corrugated shells which are alternately synclastic

and anticlastic. The gauss curvature for such shells is positive where they are synclastic and

negative where they are anticlastic

Shells are used for lots structural purposes due to its light weight, swish shape and

maximum load resisting capability. Shallow funicular shells usually offer a better ultimate

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power than the shells of single curvature including cylindrical shells. The data collection of

the funicular shells with openings of various rises. Funicular concrete shells may be hired for

mass creation of roofs and floors subjected to most hundreds. Sustainability is finished

because of the minimal constituent materials used while in comparison to conventional

methods.

The present study at investigates the effect of several probabilities of openings on

shallow funicular concrete shells over rectangular floor plan with openings large sufficient to

disturb globally strain distribution. The concrete funicular shells with two unique spans to

upward push ratio are taken into consideration for this investigation. The analyses of shallow

funicular concrete shells over square floor plan of span to upward thrust ratio (1) of twelve

with rise RI and span to rise ratio (2) of nine with upward thrust RII having 0%, 2%, 4%,

6%, 8%, 12% & 16% openings are carried using the standard program ANSYS by executing

the subsequent steps.

1. The choice of detail – 4 noded quadrilateral shell elements.

2. Joint co-ordinates are taken from investigational specimen with the aid of ordinary station.

3. Discretization of shell – The shell is discretized into twenty element along x-route and

twenty elements along y-direction.

4. Material properties are described and assigned as proven in table 1.

5. Restraints – Given at the boundary nodes.

6. Loading circumstance- Focused load

LITERATURE REVIEW

Josip kacmarcik (2011) widely as compared three design techniques for openings in

cylindrical shells under internal strains. The calculation strategies used in the 3 strategies are

notably one of a kind, so the assessment has been mostly on the most permissible format

pressure for remarkable geometries. sachithanantham et al (2011) concluded that the

deflections of shallow funicular concrete shells lower with growth in upward thrust within

elastic variety and also concluded that the ultimate load wearing capability will increase with

boom in rise.

zhang et al (2006) representated the first-recognized actual solutions for vibration of

open round cylindrical shells with more than one step wise thickness versions based on the

skinny shell concept. An open cylindrical shell is believed to be in reality supported

alongside the two instantly edges and the closing contrary curved edges can also have any

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aggregate of vicinity help conditions. The shell is subdivided into segments at the locations of

thickness versions. The kingdom-location method is adopted to derive the homogeneous

differential equations for a shell section and the area decomposition method is employed to

impose the equilibrium and compatibility requirements alongside the interfaces of the shell

segments. The correctness of this approach is checked in opposition to present effects and

results generated from finite detail bundle ANSYS and exceptional agreement is finished.

Several open shells with numerous combos of quit boundary conditions are studied.

Ayman et al (2002) investigated the openings in curved-field-girder bridges which are

commonly furnished in the backside flange right now before or after a variety joint. this paper

discussed both of these troubles and proposes procedures that are appropriate for figuring out

appropriate locations for openings. they investigated seven curved-box-girder bridges

positioned in the kingdom of florida. one of the bridges became selected for observe the use

of an in depth finite element version.

Ayman et al (2002) investigated the openings in curved-box-girder bridges which are

usually provided in the bottom flange immediately before or after an expansion joint. This

paper discussed both of these issues and proposes approaches that are suitable for identifying

appropriate locations for openings. They investigated seven curved-box-girder bridges

located in the state of Florida. One of the bridges was chosen for study using a detailed finite

element model. Gus et al (2001) reported the results of stress strain distribution in shells with

curvilinear openings. Numerical results obtained from thin and non-thin shells are analyzed

with regard for the features of deformations in the structures.

Indian Standard Code of Practice for Construction of Reinforced Concrete Shell Roof

(1995) stipulates that openings in shells shall preferably be avoided in zones of critical

stresses. Small openings of size not exceeding five times the thickness in shells may be

treated in the same way as in the case of reinforced concrete structures. For larger openings,

detailed analysis should be carried out to arrive at stresses due to the openings.

Ziya Aktas et al. (1970) outlined a numerical method to determine stress distribution

around arbitrarily shaped openings in shells. In this method, shell governing differential

equations must first be written for an arbitrary orthogonal coordinate system. Later, this

coordinate system is transformed into another coordinate system using a proper

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transformation function to treat any arbitrary opening shape. Then a numerical method is

used to solve the resulting equations. Using the equations of shallow thin shell theory, the

method is applied to a circular cylindrical shell with a circular hole along perimeter of which

a unit moment is applied. The variations of stress resultants are given and they are compared

with an available solution.

THEORETICAL FINITE ELEMENT ANALYSIS

Shell Elements

The theoretical Finite Element Analysis and the computation of membrane stresses

and deflections corresponding to the applied loads on shallow funicular shells are carried

using the standard program ANSYS Version 12.Shell elements are a special class of elements

that are designed to efficiently model thin structures. They take advantage of the fact that the

only shear on the free surfaces is in-plane. The element coordinate system for all the shell

elements has the z-axis normal to the plane. Nodes are normally located on the center plane

of the element.

Element Description

SHELL63 has both bending and membrane capabilities. Both in-plane and normal

loads are permitted. The element has six degrees of freedom at each node. Translations in the

nodal x, y, and z directions and rotations about the nodal x, y, and z-axes shown in figure 1.

Stress stiffening and large deflection capabilities are included. A consistent tangent stiffness

matrix option is available for use in large deflection (finite rotation) analyses.

The thickness is assumed to vary smoothly over the area of the element, with the

thickness input at the four nodes. If the element has a constant thickness, only need be input.

If the thickness is not constant, all four thicknesses must be input. The elastic foundation

stiffness (EFS) is defined as the pressure required producing a unit normal deflection of the

foundation. The elastic foundation capability is bypassed if EFS is less than, or equal to, zero.

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Figure 1. SHELL63

ANALYSIS OF SHALLOW FUNICULAR SHELLS

Modeling of Funicular Concrete Shells

The Shallow Funicular Concrete Shells are analysed to determine the influence of

openings in deflections (w), membrane stresses (SX), (SY) and bending stresses (MX), (MY)

using the Finite Element Software ANSYS. The Shallow Funicular Shells with various rises

are modelled by taking the x,y and z coordinates of the shell specimens prepared

experimentally by the following procedure.Concrete funicular pre-moulds of size 100cm x

100cm in plan are prepared using cement concrete. A form consisting of square steel frame

and polyurethane membrane are used for casting the pre-moulds. The polyurethane

membrane is stretched between the boundaries of the square steel frame of square ground

plan of size 100cm x 100cm and clamped. Fresh concrete with high workability is poured on

the polyurethane membrane which results in sag of the membrane. Funicular shell geometry

is formed by the sag of this membrane. When the concrete is hardened, it is removed from the

clamped membrane. The sag of the concrete becomes rise of the shell geometry when it is

inverted. The maximum sag of the membrane is predetermined and adjusted by clamping the

membrane with the steel frames and therefore the rise of the shell is determined when it is

inverted. The surface of the experimentally prepared shell specimens are painted and the

shells are discritized with twenty elements in X direction and twenty elements in Z direction

as shown in figure 2.

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Figure 2. Discretized specimen with elements and nodes.

The intersections are taken as coordinates for the theoretical shell models. These x, y and z

coordinates of the shell specimens are determined by using the total station. The shallow

funicular shells are modelled by assigning the values of x, y and z coordinates in the software

program. The discretized model of shallow funicular shells with rise RI and RII are shown in

figure 3.

Rise - RI

Rise – RII

Figure 3. Discretized shallow funicular shell without opening

Material Property

Material properties of the shallow funicular shells are defined and the values are

assigned to the elements. The values of the material properties assigned to the elements are

shown in table 1. Constrains are given for the supports and a concentrated load of 15 kN is

applied at the crown of the shell.

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Table 1. Material Property

Property Value

Modulus of Elasticity E, N/mm2 2 .17 x 104

Poisson’s ratio, 0.17

Maximum thickness, mm 25

From the results of the analysis, Stress contour, SX of shallow funicular concrete

shells of rise RI and RII without opening (0% opening) is obtained and shown in figure 4.

Rise - RI

Rise - RII

Figure 4. Stress contour, SX- Shell without opening

From the analysis results, a typical stress contour, SX of

shells of rise RI and RII with 16% opening is obtained as shown in figure 5.

Rise - RI

Rise - RII

Figure 5. Stress contour, SX – Shell with opening

Deflection (w) along X,Z directions

From the analysis results, the deflections of the shallow funicular shell along X, Z and

J axis are calculated and a plot is made between the span of the shell and the shallow

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funicular shell deflections 0%,2%,4%,6%,8%,12%, and 16% as shown in figure 6 and figure

7.

Figure 6. Deflected shape along X,Z directions rise RI

Figure 7. Deflected shape along X,Z directions rise RII

Membrane stress (SX) along X,Z directions

From the analysis results, a plot is made between the distance from the concentrated

load and membrane stress(SX) distribution along the X, Z and J axis for shallow funicular

concrete shell 0%, 2%, 4%, 6%, 8%, 12%, and 16% as shown in figure 8 and figure 9.

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Figure 8. Membrane stress along X,Z directions, SX rise RI

Figure 9. Membrane stress along X,Z directions, SX rise RII

Bending stress (MX) along X,Z directions

From the analysis results, a plot is made between the distance from the concentrated

load and Bending stress(MX) distribution along the X, Z and J axis for shallow funicular

concrete shell 0%, 2%, 4%, 6%, 8%, 12%, and16% as shown in figure 10 and figure 11.

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Figure 10. Bending stress along X,Z directions, MX rise RI

Figure 11. Bending stress along X,Z directions, MX rise RII

CONCLUSIONS

From the analysis of shallow funicular concrete shells of span to rise ratio (1) of twelve

and nine the following conclusions are made

The deflection of shallow funicular concrete decreases with increase in rise.

The membrane stress of shallow funicular concrete decreases with increase in rise.

The bending stress of shallow funicular concrete decreases with increase in rise.

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