Tensile Stress-Strain Relationship For Ferro cement Structures

Tests on fibrocement discs were carried out to measure tensile stressstrain relationship including testing wide ranges of matrix strength. Equations were proposed for calculating tensile strength based on the present test results and other results in the literature. A trilinear idealized model was proposed for calculating tensile stress – strain response of Ferro cement. The model was found to be accurate especially for higher strength Ferro cement. Later an analysis was carried out for calculating the loaddeflection relationship of Ferro cement beams based on moment – curvature relationship and bending theory of elastic beam. The analysis which based on the proposed tensile model was found to be accurate for Ferro cement beams and slab strips provided that the span / depth ratio less than 22. KEYWARDS: Analysis, Beam, Elastic, Ferro cement, Flexure, Plastic, Slab, Strain, Tensile Stress ذشناداهصا تللاع تيخًُسوريفنا ثآشًُهن لاعفَلاا يصاع ىظاك دازايَد ذًحي ردامناذبع دازا.د تيَاًيهسنا تعياص / تسذُهنا تيهك تيَاًيهسنا تعياص / تسذُهنا تيهك تصلاخنا شايمن ثاصىحف ءارصا ىح يناحنا ذحبنا يف تللاع ذشنا داهصا – لاعفَلاا تيخًُسوريف سراًَ ًهع جًُسنا تَىي عاىَلا . تفهخخي طاغضَا تيوامًب يىصمنا ذشنا تيوامي باسحن ثلاداعي ريىطح ىح تيهًعنا شئاخُنا ٍي ةدافخسلااب جًُسوريفن . يرخا دىحب شئاخَو يناحنا ذحبهن باسحن ًرلازنا يطخنا عىَ ٍي تللاع حارخلا ىَ ذشنا داهصا تللاع لاعفَلاا . جًُسوريفن تميرطنا ٌاب ثذصو ذشنا داهصا تللاع باسحن تميلد لاعفَلاا دادسح ايذُع اصىصخ تيوامي طاغضَلاا . جًُسنا تَىًن مًحنا تللاع باسحن ًطخنا ريغ ميهحخنا تميرط ىيذمح ىح اهذعب – ثاطلابناو ثاضراعهن دولاا تيخًُسوريفنا وسعنا تللاع قامخشا ٍي ةدافخسلااب – نا حئافصهن دولاا شئاخَ و عطمًهن شىمخنا . تَرً ٌأب ثذصو ذلو مًحنا تللاع باسحن تميلد ميهحخنا تميرط وسعنا تللاعو دولاا – شىمخنا تيخًُسوريفنا ثاطلابناو ثاضراعهن ايذُع ٌىكي ٍي ملا عطمًنا كًع / ءاضفنا تبسَ 22 . Received: 9 – 1 2010 Accepted: 15 – 6 2011 Al-Rafidain Engineering Vol.20 No. 2 March 2012


1-Introduction
Ferro cement is considered as one of the nonconvential types of reinforced concrete usually known by its thin section and high tensile strength to weight ratio. Application of Ferro cement in which the section subjects to uniaxial tension can be seen in the case of silos, liquid retaining tanks, pipes, etc. Although the invention of Ferro cement turned back to the period of middle of the nineteenth century , researches for understanding mechanical and structural properties begin at the early 1970s. Nowadays the basic properties of Ferro cement were well understood and equations for design were proposed, that basically depend on the knowledge obtained from tests carried out during the last forty years.
ACI Committee 549 [ 1 ] prepared a state-of -the-art report to be a guidance for practical applications of Ferro cement. According to ACI 549 recommendations the role of Ferro cement mortar on tensile strength is neglected and the tensile strength is calculated from the load carried by the wire meshes divided by the sectional area. It should be noted that this recommendation is based on the results of tests carried out during the period from 1971 to 1977 [3,5,7] for Ferro cement sections made from a cement mortar of medium strength. Such recommendation is acceptable for calculating tensile strength of Ferro cement section made from normal strength mortar and greatly underestimates the tensile strength of high strength mortar Ferro cement. This is because the tensile strength of plain matrix alone is high and may be considerably greater than that calculated from the recommendations of ACI 549. For example, for a Ferro cement section of 25.4 mm thickness reinforced with five layers of wire mesh of 0.6 mm diameter, 12 mm spacing, and 412 MPa yield stress, the tensile stress will be 1.91 MPa. If the compressive strength of matrix be 80 MPa and cracking tensile stress is calculated from the relationship given by Shah and Balaguru [ 8 ] the result which becomes 4.91 MPa. Therefore , one can found that the ultimate tensile stress calculated from ACI 549 recommendation is only 39% of the cracking stress. The later ratio is quite small and can be considered not reviewed. Therefore developing equations for predicting ultimate tensile stress of Ferro cement made of mortar of high strength is necessary. The aim of the present research is to provide an alternative equation for calculating tensile strength of Ferro cement applicable for wide range of matrix strength based on tensile stress tests and regression analysis. In the present study an attempt was made to develop a model for calculating tensile stressstrain relationship depending on the idealized trilinear response. Later, an analysis was introduced for calculating the complete loaddeflection relationship of Ferro cement beams or slab strips based on derivation of moment-curvature relationship and results of elastic beam theory. The predictions of analysis were compared with the flexural test data carried out on ferrocement from literature to test the accuracy of the analysis.

2-Experimental Work
Ordinary Portland cement ( Type I ) was used for casting disc and compression specimens. Medium graded clean river sand passed by 100% on 2.36 mm sieve was used. For high strength mixes a high range water reducer (super plasticizer) of Gellunum type which is commercially available was used. Welded galvanized steel wire mesh fabric of 0.6 mm diameter and 12 mm spacing between wires and of 458 MPa yield stress was used as reinforcement. Details of mixes and arrangement of reinforcement in each tensile test specimen can be found in Table 1. The tensile test specimens were discs of 152 mm and 25 mm thickness. Such type of specimens were cast inside the standard cylinders after putting wooden spacers to cancel a height of about 275 mm and to leave only 25 mm for thickness of the specimen. Figure ( 1 ) shows the mould used and the arrangement of Ferro cement wire mesh at casting period. All specimens were cured in the potable water at about 25 o C for the period of 28 days and then left to dry in the laboratory for another 14 days before testing. For measuring the strain a foil of electrical strain gage was paste to the concrete surface in the direction of wire meshes but perpendicular to the direction of applied load . Figure 2 shows the specimen after testing which illustrates the position of strain gage . Disc specimens were tested in a manner similar to that of Brazilian split cylinder test by applying a load of a rate equal to 5 kN/min. Using the following relationship given in the textbooks of concrete technology, the tensile stress can be calculated as follows In which f tu is the splitting tensile stress, P is the failure load, L is the length of specimen, and D is the diameter of specimen.

3-Test Results and Regression Analysis
Test results of compressive strength and tensile strength are shown in Table ( 1 ). Tensile stressstrain relationship for the disc specimens are shown in Figures ( 3 ) to ( 13 ). It is shown that the slope of the curves ( or stiffness ) is high for the initial portion and the slope is reduced with increasing in stress. The curve in general become horizontal in the later stage which is considered to be a plastic stage. Three different stages are clearly observed in the most of the tensile stressstrain relationships. Such observation is not new and observed by others like Shah and Baluguru [ 8 ] and Baluguru et al [ 2 ] . Figure (   where f tu is the tensile strength of Ferro cement, f tm is the tensile strength of the matrix, and f ts is the tensile stress carried by the wire mesh. The tensile strength of matrix is the cracking stress and can be calculated from the expression given by Oluokun [ 6 ] , as follows: Later the value of f ts is calculated from the result of subtracting f tm from f tu obtained from test results. In order to obtain the dependent variable ( f ts ) a regression analysis was carried out. The independent variable is the reinforcement index ( ). The two variables are combined in the following linear equation Statistical regression analysis based on least square method was carried out using the 14 data points and the values of constants α and β was found to be 0.56 and 0.72, respectively. Hence the final form of the tensile strength equation is as follows The above equation has a correlation coefficient equal to 0.81, mean value ( test / theoretical ) equal to 0.99 and standard deviation equal to 17%. Figure ( 16 ) shows the relationship between the calculated and test tensile strength of Ferro cement. It is shown that there is a fairly good agreement between the two values.

4-Tensile Stress -Strain Model
The proposed model for the tensile stressstrain relationship is based on experimental data obtained from present test results. The relationship is based on three portions : elastic, elasticplastic and plastic portion . Elastic stage terminates at cracking of the matrix and plastic stage begins at a value of stress equal to ultimate tensile stress and the other stage locates between

-------------------------------------------------------------------------------------------(5)
where E c is the elastic modulus of the composite given by [ 8 ] ) and n is the ratio of elastic modulus of reinforcement to that of matrix ( E R /E m ), and V R is the ratio of reinforcement in the section given by where A sf is the area of the reinforcement in the section and bh is the cross sectional area . The last variable needs to be assessed for constructing the tensile stressstrain relationship is the strain corresponding to the ultimate stress ε tu . For this purpose an approximation is made utilizing the trend of the cracking zone of the actual tensile stressstrain relationship. With increasing stress the stiffness is reduced due to cracking and here it is assumed that the ratio of the slope of the curve for the elastic -plastic range, ψ, to that of elastic stage, E c , [ Figure ( 17 ) ] is equal to the ratio of tensile stress of matrix to the ultimate tensile stress, or

5-Flexural Analysis of Ferro cement
Having the Proposed idealized tensile model and an idealized compressive stressstrain given by Walraven and Spierenburg [ 10 ] shown in Figure ( 18 ), Ferro cement structural member can be analyzed for momentcurvature and loaddeflection relationships. Calculation of momentcurvature relationship is based on the equilibrium of forces and compatibility of strains. According to the compressive stress-strain and tensile stress-strain relationships different cracking stages exist and as a result nonlinear response is obtained. Hence, the material nonlinearity is included in the present analysis.

5-1 Moment-Curvature Relationship 5-1-1 Elastic Stage
Stress and strain distributions for this stage are shown in Figure ( 19-a ). Depth of the compression zone c is calculated from equilibrium of compressive and tensile forces acting on the section, or It is assumed that the elastic modulus calculated from Eq. ( 6 ) is valid for compression and tension zone of the section and as a result the depth of compression zone for the elastic stage is equal to that in tension zone and equal to h/2. For this stage the momentcurvature relationship is given by Where b is the width of the section,  is the curvature, h is the depth of the section and c is the depth of compression zone. Substituting c= h/2 the moment-curvature relationship becomes

5-1-2 Elastic-Plastic Stage (1)
Stress and strain distributions for this stage are shown in Figure ( 19-b ). For this stage the depth of compression zone is given by ψ is the slope of elasticplastic portion of the tensilestress strain relationship given by Eq. ( 10 ) . Moment-curvature relationship for this stage is as follows

5-1-3 Elastic-Plastic Stage ( 2 ):
Stress and strain distributions for this stage are shown in Figure ( 19-c ). Equilibrium of forces lead to the following value of c for this stage is given by Test Calculated

5-1-4 Elastic-Plastic Stage (3):
Stress and strain distributions for this stage are shown in Figure ( 19 The relationship between moment and curvature is given by The terminating point for this stage or the collapse occurs when the compressive strain in concrete reaches 0.0038. The corresponding curvature becomes

5-2 Load-Deflection Relationship
Having the calculated moment-curvature relationship given in the previous section, loaddeflection relationship can be calculated for any type of beams loaded by various type of loads. For example, for a simply supported beam loaded by two central point load each P/2, the value of central deflection is given by Where EI is the value of flexural rigidity which varies with the cracking stages.

5-3 Procedure for Analysis
1-Specify an initial value of curvature. 2-Calculate the depth of compression zone. 3-Check the cracking limit. 4-Calculate the moment. 5-Calculate EI from the moment divided by curvature. 6-Calculate value of the load from Eq.( 25 ) 7-Calculate the deflection from Eq. ( 24 ). 8-Repeat the steps 2 to 7 for another curvature increment and other cracking stages until the complete load-deflection response is obtained .

5-4 Results of Analysis and Role of Span / Depth Ratio
Test results from researches carried out on Ferro cement beams and slab strips [ 4,9,10 ] are used here to assess the accuracy of the proposed analytical procedure. From the comparison between test and predicted ultimate load, it is found that the analytical method is accurate for Ferro cement having wide ranges of compressive strength of mortar and wire mesh ratios but significantly sensitive to the span / depth ratio of the beam or slab strip. Figure (20) shows the test and calculated momentcurvature relationship for Ferro cement specimens of different span / depth ratios tested by Mansur [ 4 ] . From a comparison between test and calculated momentcurvature relationship it is observed that there is a good agreement between the two responses. Figure( 21 ) shows the loaddeflection relationship calculated from the analytical procedure presented in this study and test results obtained by Walraven and Spierenburg [ 10 ] . It is observed from the figure that the analysis overestimates the load for all values of deflection. This occurs because the tested specimen has a span / depth ( L/h ) ratio equal to 35.7 and considered to be thin section. It is suggested [ 11 ] that any slab strips having span / depth ratio larger than 22 considered to be very thin slabs and between 18 and 22 defined as thin slabs. In fact the analysis presented in this paper is not applied to very thin slabs and its predictions are accurate ( as shown from the comparison with test results ) only for the case of thick Ferro cement section ( L/h < 18 ) and thin Ferro cement sections ( 18< L/h < 22 ) because very thin slabs suffer from large deflections and such effect should be incorporated in the analysis. Figure ( 22 ) shows the variation of the ratio of test to calculated ultimate load with the span / depth ratio. It is clearly shown that for Ferro cement specimens having L/h ratio smaller than 22 the prediction is safe and accurate, regardless the wire mesh ratio and mortar strength. Walraven and Spierenburg [ 10 ] in their analytical model on Ferro cement section assumed a constant value of the strain at which plastic stage commences, ε ut , equal to 1.2x10 -3 . The authors believe that the value of ε ut given in the present study using Eq.( 11 ) is better and accurate for use in flexural analysis . Because the proposed value of ε ut depends on the vital parameters of Ferro cement, namely tensile strain of matrix, tensile strength of matrix and ultimate tensile stress of Ferro cement which found to depend on both the matrix strength and reinforcement index ( V R f yR ).

6-CONCLUSIONS
From the present study which contains test data and analytical procedures, the following conclusions can be drawn 1-Test results on Ferro cement disc specimens indicate that the tensile strength depends on both matrix strength and wire mesh ratio. An equation was proposed to calculate the tensile strength of Ferro cement depending on the matrix strength and reinforcement index ( V R f yR ) [ Eq. ( 4 ) ].
2-Tensile strain at which plastic stage of stress-strain response commences was found to depend on tensile strain of matrix, tensile strength of matrix and ultimate tensile stress of Ferro cement [ Eq.( 11 ) ]. 3-Flexural analysis was carried out to calculate the loaddeflection relationship of Ferro cement beams and slab strips. The predictions of the analysis were compared with the test data from literature, and was found that the analysis is accurate and safe for calculating the load-deflection relationship, provided that the span/depth ratio not more than 22.