A General Equation for the Flexural – Membrane Behaviour of Rigid – Plastic RC Square Slabs Having Variously Restrained Edges

Uniformly loaded rigid – plastic reinforced concrete (RC) square slabs having six different cases of boundary restraints have been recently analyzed (1) under the combined effect of bending and membrane action, and a separate load – deflection relationship for each slab case has b een obtained. In this paper, th e load – deflection behaviour of all these six slab cases is expressed in one single compact equation as a function of the slab material properties and the configuration of the slab boundary restraints. The application of the proposed equation in the analysis of a typical RC Square slab indicates that when two or more edges of the slab are restrained against r otation a nd horizontal tra nslation the live load c arrying capacity of the slab can reach up to five times that suggested by the simple yield line theory.


Simply supported edge
Fixed edge

1-Introduction
Previous studies (2,3) of the phenomenon of membrane action in RC slabs showed that the presence of restraining conditions at the edges of a slab can considerably enhance the load carrying capacity of the slab.Values of ultimate loads which are many times greater than that suggested by the simple Johansen's yield line theory (4) have been recorded both theoretically (2,5,6) and experimentally (2,7,8) .Even in cases of unrestrained slabs, the self balanced inplane membrane forces which develop inside the slab as a result of the applied loading have been found (9,10) to help in producing higher yield loads with continuing deflection.
In a recently submitted M.Sc thesis (1) , the effect of boundary restraints on the ultimate capacity of rigidplastic RC square slabs has theoretically been established.It has been found that the combination of restrained and unrestrained slab edges give six possible cases of such slabs as shown in Fig. ( 1).Accordingly, each slab case has been analyzed separately to include the combined effect of bending and membrane action and different loaddeflection relationships have been obtained for the six different slab cases.
It is aimed from the present research to express the loaddeflection behaviour of all the six cases of RC square slabs in one single compact equation.The equation is general and is a function of the slab material properties and the configuration of the slab boundary restraints.

2-Analysis of RC
are parameters defining the shape of the compressive stress block of concrete which are given by Hognested (11) as functions of the concrete cylinder strength ' c f (in units of MPa ); It can be seen that the yield line pattern of slab cases (1) and ( 6), being the well known cross yield line pattern, has no variable parameter.The yield line pattern of slab case (3) and (4) has one variable parameter (which is o x in slab case 3 and o y in slab case 4) while that for slab case (2) and ( 5) has two variable parameters (which is 1 x , 2 x in slab case 2 and o x , o y in slab case 5).
By applying the principle of virtual work for each slab case the following values of Johansen's uniform load ( J w ) will be obtained;

Slab Case (1)
Differentiating Eq.( 4) with respect to 1 x and 2 x each at a time and setting the derivatives equal to zero; The solution of Eqs. ( 5a) and (5b) gives the true values of 1 x and 2 x that fix the actual yield line pattern A substitute of these values of 1 x and 1 x into Eq.( 4) gives; which when substituted into Eq.( 6) gives; 9

Analysis by the Modified Yield Line Theory (Including Membrane Action):
The effect of including membrane action in the yield line analysis of the deferent slab cases under study has been carried out in ref.
(1).It was found that, due to the assumption of rigidplastic consideration, the maximum value of the yield load in the restrained slab cases (1) to (5) occurs at zero deflection and decreases nonlinearly with increasing slab deflection.A reverse behaviour was obtained for the unrestrained slab case (6) where the yield load at the start of collapse (i.e at zero deflection) is Johansen's load and increases non -linearly with increasing slab deflection.However, the loaddeflection relationships for the different slab cases were derived separately and were given in complex forms that require lengthy computations before they can be used.
In this paper, these loaddeflection equations are reexamined, simplified further and re arranged in form of ordinary quadratic equations.By doing so, all the equations become identical in form but they differ only in the coefficients of the terms.Therefore, a single compact general equation is proposed to express the loaddeflection behaviour of all the six cases of the RC square slabs.The proposed equation is simple to use and illustrates that the loaddeflection behaviour of a particular slab is affected by the slab material properties and the configuration of the slab boundary restraints.This equation, written in nondimensional form, is: 14) where  It can be seen from this figure that the increase in the maximum yield load above the simple yield line theory load is more pronounced in slab case 1 (where all the edges of the slab are fixed).As the number of the fixed edges in the slab decreases, the enhancement in the maximum yield load (due to membrane action) will decrease, and when the slab has all edges simply supported the initial collapse load of the slab will simply be the well known Johansen's load.

3-Determination of the Actual Load Carrying Capacity of RC Square Slabs:
Previous studies (2,3,8) of the effect of membrane action in elasticplastic RC restrained slabs show that the real loaddeflection relationship of the slab consists of an initial ascending part (representing both the elastic deformations and the subsequent elasticplastic stage which is characterized by the development of compressive membrane action in the slab) followed by a descending part (representing the spread of full depth cracking and tensile membrane action in the slab).Therefore, the ultimate load is reached at a certain value of deflection and is many times greater than Johansen's load due to the combined effect of bending and membrane action.
PDF created with pdfFactory Pro trial version www.pdffactory.comThe proposed Equation ( 14), being based on the assumption of considering the slab to behave in a rigidperfectly plastic manner, predicts a continuous descending loaddeflection curve starting from maximum value of load at zero slab deflection.Therefore to make use of equation ( 14) in estimating the actual load carrying capacity of a particular RC square slab, the value of the slab central deflection corresponding to the slab ultimate load has to be specified first and then can be inserted in the equation to get the actual value of the ultimate load.
Existing experimental tests (12,13) on restrained RC slabs indicated an average value of such deflection to be 0.3 times the slab thickness.Therefore inserting in Eq. ( 14) will enable determining the actual load carrying capacity of any RC square slab as illustrated in the following typical examined slab.The slab is assumed to be reinforced with the minimum amount of reinforcement (required for shrinkage and temperature) as specified by ACI-05 Code (14) , giving .Therefore the steel ratio ρ at any slab section is; The constant α and β of the slab section (as defined in Eq. 14) can be calculated as follows;

Properties of the Examined
From Eq. (2a); The yield moment of the slab section (given by Eq. 1) is;

Estimation of the live load Carrying Capacity Using the Simple Yield Line Theory:
According to the simple yield line theory, the values of the collapse uniform load for the six cases of the RC square slab under examination are given by Eqs.(3), ( 6), ( 8), ( 10), ( 12) and ( 13) respectively.The values of J w obtained from these equations are listed in column (3) of Table ( periphery of a RC square slab gives the six possible cases shown in Fig. (1).Under the applied uniform load these slabs will collapse in the form of the yield line patterns shown in the same figure.
the value of the collapse uniform load 's simple yield line theory for the six cases of RC square slabs is shown in Fig. (2).
Figs. (3) and (4) to show respectively the effects of the slab material properties and the configuration of the slab boundary restraints on the loaddeflection behaviour of RC square slabs.Figure(3) shows the effect of varying the slab material properties (

Figure
Figure (4) shows the effect of varying the slab boundary restraints on the load deflection

.
are placed at the bottom face only at positive moment regions (slab central zone) and at the top face only at negative moment regions (zones of fixed edges if present)

o
By assuming the slab to be a panel in an intermediate floor of a RC residential building, the total dead load of the slab ( Figure (1) The Six Possible Cases of R.C Square Slabs With Their Collapse Modes

Estimation of the live load carrying capacity using the proposed modified yield line equation (14) of the present
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-Membrane Behaviour of Rigid - Plastic RC Square Slabs Having Variously Restrained Edges 2056 Table (1) Values of the Coefficients
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-Membrane Behaviour of Rigid - Plastic RC Square Slabs Having Variously Restrained Edges 2057 Table (2) Estimation of the Uniform Live Load Carrying Capacity of the Examined RCSquare Slab by: (1) The Simple Yield Line Theory (2) The Proposed Modified Yield Line Equation (14) of the Present Research
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