Circular Slabs

Circular Slabs

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Introduction: They are commonly used as:Roof of a circular room.Base slab or cover slab for circular water tanks.Cover slabs for wells.Circular footing for a circular column.Roof of a traffic control post.

When loaded, these slabs bend like a saucer producing tensile stresses at bottom and compressive stresses on top.

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Analysis:Analysis of circular slabs is based on theory of elasticity assuming Poissons ratio concrete as zero.

For convenience, circular slabs are usually analysed in polar coordinates so that the bending moments are expressed as radial moments (Mr) and tangential moments (M ).

Any point in a circular plate is conveniently represented by radial distance r from centre and the angel made by that radius with respect to fixed direction.

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Radial and Circumferential(tangential) moments in circular slab:

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If the boundary conditions and loads are axisymmetrical, the condition of a strip represents the condition of entire slab.

In other words, there is no twisting and variation with . It means Mr= Mr=Q=0.

Thus, the element is subjected to Mr,M and Qr only.

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Support conditions:Simply supported at edgesFixed at edges Partially fixed at edges

Equations for moment and shear for a few standard cases:Slab simply supported at edges and loaded with U.D.L.Fixed at edges and loaded with U.D.L.Partially fixed at edges and loaded uniformly Simply supported at edges with U.D.L. along the circumference of a concentric circleSimply supported at edges with U.D.L. inside a concentric circle.

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Notations: w = uniformly distributed load (ie load per unit area) a = radius of the slab Mr= radial bending moment per unit width at any radius r (Mr)c and (Mr)e= radial bending moment per unit witdth at centre and edge respectively M= circumferential bending moment per unit width at any radius r (M)c and (M)e= circumferential bending moment per unit width at centre and edge respectively

(1) S.S. Slab with U.D.L

Both the moments vary parabolically as can be seen from their expressions.Refer punmia8

(2) Fixed Slab with U.D.L

Refer punmia9

(3) Partially fixed Slab with U.D.LThis case is intermediate between the case of a simply supported slab and a fixed slab. Hence the moment may be taken as the average of the moment of the corresponding moment of the two cases. For radial moment the point of contraflexure occurs at a radius

(4) SS Slab with U.D.L. along the circumference of a concentric circle

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(5) SS Slab with U.D.L. inside a concentric circle

Derivation of expressions for moments and shear force using plate bending theory: Overview:

Bending of plates or plate bending refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections.

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Detailing of reinforcements:Logically, we should provide steel in radial direction for radial moment and steel in circumferential direction for circumferential moment.But such a provision of steel will lead to congestion of steel at the centre.Hence, it is preferrable to adopt rectangular grid arrangement of reinforcementsIn such arrangement,reinforcement is designed for the maximum of Mr and M.For positive circumferential moment at edges, circumferential steel is provided at the bottom of slabs to a distance of anchorage length.To resist negative radial moment at edges,radial steel is provided at top of slab to a sufficient anchorage length.

In grid arrangement, same reinforcement is provided in two mutually perpendicular directions.Such mesh cant take circumferential moment at the edge due to lack of available development length.

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For annular slabs ie circular slabs with a hole at centre, there is no problem of congestion of reinforcements at the centre and hence we can provide radial and circumferential reinforcements.19

Example:

Data:

This is the first example given in the book by Punmia and AK Jain20

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Shear check:v =.5*10.688*2.5/(1000*103)=0.129 N/mm2