Poisson’s Ratio of Soilcrete Blocks

Poisson’s Ratio of Soilcrete Blocks

D. O. Onwukaa, C. E. Okere b, N. N. Osadebec,

aCivil Engineering, Federal University of Technology, Owerri bCivil Engineering, Federal University of Technology, Owerri cCivil Engineering, University of Nigeria, Nsukka

Abstract

Tropical countries are subject to extreme weather conditions and as such require special building materials to accommodate this. Soilcrete blocks (made with laterite, cement and water) can effectively be used in these regions because of their thermal insulating properties and greater resistance to extreme weather conditions. Poissons ratio, which is required for structural computations, has been ignored to an extent with regards to block moulding technology. In this work, Poissons ratio was determined for the soilcrete blocks produced. Modified regression theory was used to generate a model for prediction of the Poissons ratio of soilcrete blocks. The model was subjected to statistical tests which proved its adequacy.

1. Introduction

   Block can be generally described as a solid mass used in construction. It can be made from a wide variety of materials ranging from binder, water, sand, laterite, coarse aggregates, and clay to admixtures. The constituent materials determine the type of block which includes soilcrete, sandcrete, mud blocks, clay bricks, etc. Soilcrete blocks are made of cement, laterite and water. Sandcrete blocks (made of river sand, cement and water) are used in most places but they are not considered the best for building in tropical countries because of their poor environmental and thermal insulating properties as a result of high degree of porosity. Soilcrete blocks can effectively be used in tropical areas because of their good thermal insulating properties. They are advantageous in hot dry climates where extreme temperature can be moderated inside buildings of compressed stabilized earth blocks [1].

   Several researchers have reported that laterite can be used in good quality block production, road and building construction. Boeck et. al., produced cement stabilized laterite blocks using 4-6% cement [2]. Good laterite blocks were produced from different sites in Kano when laterite was stabilized with 3 to 7% cement [3]. Laterite stabilized with cement was used successfully to produce bricks in Sudan [1]. Aguwa produced laterite cement blocks using 0-10% cement content by weight of the soil [4-5]. Alutu and Oghenejobo used 3% to 15% of cement to produce cement-stabilised laterite hollow blocks [6]. It is worthy of note here that most of these researchers work revolved around compressive strength of blocks and the cost effectiveness of using laterite in block production. Other properties/characteristics like Poissons ratio have not been handled adequately. Model for prediction of Poissons ratio using mix ratio and vice versa has not been formulated.

   Knowledge of Poissons ratio (an elastic constant) is necessary for structural design computations. In all engineering materials, the elongation produced by an axial tensile load in the direction of the force is accompanied by a contraction in any transverse (lateral) direction. The ratio of the lateral contractive strain to axial strain in a material is referred to as Poissons ratio

   [7] and it is given as:

   µ = lateral strain/axial strain = l / (1) where l is the lateral strain, is the axial strain and µ, the Poissons ratio.

   Neville proposed another way of estimating Poissons ratio. It is the ratio of tensile stress at cracking in flexure to compressive stress at cracking in compression specimen [8]. Hence,

   µ = t / c (2)

   Y(n) = izi (n) +

   z (n) z (n) (5)

   ij

   i

   j

   where µ = Poissons ratio

   t = tensile stress at cracking in flexure

   c = compressive stress at cracking in compression specimen

   Ability to predict this vital structural characteristic/ property of blocks is of utmost importance. Modified regression method proposed by Osadebe [9] was used in this work to formulate a model for prediction of

   Poissons ratio of soilcrete blocks using specified mix

   where 1 i j 4 and n = 1,2,3, 10

   Eqn (5) can be put in matrix from as

   [Y(n)] = [Z(n) ] {} (6)

   Rearranging Eqn (6) gives:

   {} = [Z (n) ]-1 [Y (n)] (7)

   i

   The actual mix proportions, si(n) and the corresponding fractional portions, z (n) are presented on Tables 1 and 2.

   These values of the fractional portions Z(n) were used to

   (n)

   develop Z(n) matrix and the inverse of Z matrix. The

   ratio. The model can also yield all the possible mix ratios for a desired Poissons ratio. The formulation of the regression equation was done from first principles using the so-called absolute volume (mass) as a necessary condition. This principle assumes that the volume (mass) of a mixture is equal to the sum of the absolute volume (mass) of all the constituent components. Osadebe assumed that the response function is continuous and differentiable with respect to its predictors. By making use of Taylors series, the response function could be expanded in the neighborhood of a chosen point. The modified regression theory has been applied successfully with good results by various scholars [9-14].

2. Methodology

   The materials used for this work are Eagle cement brand of Ordinary Portland Cement, laterite, sourced from Ikeduru LGA, river sand from Otamiri river in Imo State and potable water. All the materials conform to British Standard/ specifications [15-17].

   Here, analytical and experimental procedures were used in formulating a mathematical model for predicting the Poissons ratio of soilcrete blocks. The model is based on the modified regression theory.

   1. Formulation of model based on modified regression theory

      The polynomial equation as given by Osadebe [9] is

      Y = 1z1 + 2z2 + 3z3 + 12z1z2 + 13z1z3 + 23z2z3 (3) In general, Eqn (3) is given as:

      Y = izi+ij zizj (4)

      where 1 i j 3

      Eqns (3) and (4) are the optimization model equations. Y is the response function at any point of observation, zi, the predictors are the ratios of the actual portions to the quantity of soilcrete (fractional portions) and i are the coefficients of the optimization model equations.

      i

      Different points of observation will have different responses with different predictors at constant coefficients. At the nth observation point, Y(n) will correspond with Z (n). That is to say that:

      solution of Eqn (7) with known Z(n) matrix and Y(n) matrix from laboratory tests gives the unknown constant coefficients i.

      Table 1. Values of actual mix proportions and their corresponding fractional portions for a 3- component mixture

      N	S1	S2	S3	RESPONSE	Z1	Z2	Z3
      1	0.8	1	8	Y1	0.081633	0.102041	0.816327
      2	1	1	12.5	Y2	0.068966	0.068966	0.862069
      3	1.28	1	16.67	Y3	0.067546	0.05277	0.879683
      4	0.9	1	10.25	Y12	0.074074	0.08305	0.843621
      5	1.04	1	12.335	Y13	0.072348	0.069565	0.858087
      6	1.14	1	14.585	Y23	0.068161	0.059791	0.872048

      S1 = Actual water cement ratio Z1 = Fractional water/cement ratio S2 =Actual cement quantity Z2 = Fractional portion of cement S3 = Actual laterite quantity Z3 = Fractional portion of laterite

      Table 2. Z(n) matrix for a 3-component mixture

      Z1	Z2	Z3	Z1Z2	Z1Z3	Z2Z3
      0.081633	0.102041	0.816327	0.00833	0.066639	0.083299
      0.068966	0.068966	0.862069	0.004756	0.059453	0.059453
      0.067546	0.05277	0.879683	0.003564	0.059419	0.046421
      0.074074	0.082305	0.843621	0.006097	0.06249	0.069434
      0.072348	0.069565	0.858087	0.005033	0.062081	0.059693
      0.068161	0.059791	0.872048	0.004075	0.05944	0.05214

   2. Experimental Investigation

      The mix proportions from Table 1 were used to measure out the quantities of water (S1), cement (S2), laterite (S3), for production of soilcrete blocks. A total of twelve mix ratios were used to produce thirty six solid blocks that were cured and tested on the 28th day. Six out of the twelve mix ratios were used as control mix ratios to produce eighteen blocks for the confirmation of the adequacy of the mixture design

      model. The initial cracking load in flexure was recorded and used to calculate tensile stress at cracking in flexure.

      The initial cracking load in compression specimen was recorded and used to calculate compressive stress at cracking in compression specimen. With these two parameters known, Poissons ratio was calculated using

      Eqn (2). Three blocks were tested for each point and the average taken as the Poissons ratio of the point.

3. Results and Discussions

   The experimental values of Poissons ratios of the soilcrete blocks are presented on Table 3 while the replication variances of the test result are presented on Table 4.

   Table 3. Experimental values of Poissons ratio of soilcrete blocks

   Exp. No	Mix ratios (w/c: cement: laterite)	Repli- Cates	Initial Cracking Load in Flexure (KN)	Tensile Stress at Cracking in Flexure t (N/mm2)	Initial Cracking Load in Compression (KN)	Compressive Stress at Cracking in Flexure c (N/mm2)	Poissons Ratio µ = t/c	Average Poissons Ratio µ
   1	0.8:1:8	A	15.5	0.230	80	1.185	0.194	0.174
   		B	17.5	0.259	90	1.333	0.194
   		C	16.0	0.237	120	1.778	0.133
   2	1:1:12.5	A	2.5	0.037	20	0.296	0.125	0.135
   		B	3.5	0.052	25	0.370	0.141
   		C	2.8	0.041	20	0.296	0.139
   3	1.28:1:16.67	A	2.5	0.037	20	0.296	0.125	0.110
   		B	2.3	0.034	40	0.593	0.057
   		C	3.0	0.044	20	0.296	0.149
   4	0.9:1:10.25	A	3.5	0.052	30	0.444	0.117	0.095
   		B	3.0	0.044	35	0.518	0.085
   		C	2.9	0.043	35	0.518	0.083
   5	1.04:1:12.335	A	2.1	0.031	30	0.444	0.070	0.074
   		B	4.0	0.059	50	0.741	0.080
   		C	2.9	0.043	40	0.593	0.073
   6	1.14:1:14.585	A	2.2	0.033	20	0.296	0.111	0.091
   		B	2.5	0.037	40	0.593	0.062
   		C	3.0	0.044	30	0.444	0.099
   7	1.09:1:13.46	A	2.5	0.037	20	0.296	0.125	0.099
   		B	2.0	0.030	20	0.296	0.101
   		C	2.1	0.031	30	0.444	0.070
   8	1.02:1:12.417	A	2.1	0.031	30	0.444	0.070	0.105
   		B	2.1	0.031	20	0.296	0.105
   		C	2.8	0.041	20	0.296	0.139
   9	0.866:1:9.485	A	2.5	0.037	50	0.741	0.050	0.054
   		B	2.5	0.037	50	0.741	0.050
   		C	2.8	0.041	45	0.667	0.061
   10	1.0924:1:13.8761	A	4.0	0.059	40	0.593	0.099	0.079
   		B	2.5	0.037 /td>	45	0.667	0.055
   		C	3.8	0.056	45	0.667	0.084
   11	1.052:1:12.818	A	4.0	0.059	30	0.444	0.133	0.092
   		B	2.5	0.037	30	0.444	0.083
   		C	3.8	0.031	35	0.518	0.060
   12	1.1:1:13.685	A	2.0	0.030	20	0.296	0.101	0.098
   		B	2.0	0.030	20	0.296	0.101
   		C	2.3	0.034	25	0.370	0.092

   Table 4. Poissons ratio test results and replication variance

   Expt. No.	Replicates	Response Yi	Response Symbol	Yi	Y	Y 2 i	S 2 i
   1	1A 1B 1C	0.194 0.194 0.133	Y1	0.521	0.174	0.093	0.0012
   2	2A 2B 2C	0.125 0.141 0.139	Y2	0.405	0.135	0.055	0.000
   3	3A 3B 3C	0.125 0.057 0.149	Y3	0.331	0.110	0.041	0.002
   4	4A 4B 4C	0.117 0.085 0.083	Y12	0.285	0.095	0.028	0.000
   5	5A 5B 5C	0.070 0.080 0.073	Y13	0.223	0.074	0.017	0.000
   6	6A 6B 6C	0.111 0.062 0.099	Y23	0.272	0.091	0.026	0.001
   Control
   7	7A 7B 7C	0.125 0.101 0.070	C1	0.296	0.099	0.031	0.001
   8	8A 8B 8C	0.070 0.105 0.139	C2	0.314	0.105	0.035	0.0012
   9	9A 9B 9C	0.050 0.050 0.061	C3	0.161	0.054	0.0087	0.000
   10	10A 10B 10C	0.099 0.055 0.084	C4	0.238	0.079	0.020	0.001
   11	11A 11B 11C	0.133 0.083 0.060	C5	0.276	0.092	0.0282	0.001
   12	12A 12B 12C	0.101 0.101 0.092	C6	0.294	0.098	0.0289	0.000
   							0.0084
   Legend: y = y/n S 2= [1/(n-1)]{y 2 [1/n(y )2]} where 1in y i i yi = the responses y = the mean of responses for each control point n = control points, n-1 = degree of freedom

   Considering all the design points, number of degrees of freedom,

   Ve = (Ni-1) (8)

   where 1i 12

   Ve = 12 1=11

   Replication variance,

   S 2 = 1/ V S 2 (9)

   where is the estimated standard deviation or error,

   y e i

   y

   S 2 = 0.0084/11 = 0.0007636

   i

   where S 2 is the variance at each point

   y

   Replication error, Sy = S 2 (10)

   = 0.0007636 = 0.028

   This replication error value was used below to determine the t-statistics values for the model.

   1. Determination of Osadebes mathematical model for Poissons ratio of soilcrete blocks

      Substituting the values of Y(n) from test results (given in Tables 3 and 4) into Eqn (7) gives the values of the coefficients, as:

      1 = 8790.55199, 2 = 1164.1268, 3 = 27.3908, 4 = – 16111.1579, 5 = -9787.0468, 6 = -866.3843

      Substituting the values of these coefficients, into Eqn

      (3) yields:

      Y = 8790.55199Z1 + 1164.1268Z2 + 27.3908Z3

      16111.1579Z4 9787.0468Z5

      866.3843Z6 (11)

      Eqn (11) is the Osadebes mathematical model for optimisation of Poissons ratio of soilcrete block based on 28-day strength.

      3.1.1 Test of adequacy of Osadebes model for Poissons ratio of soilcrete blocks

      t is the t-statistics,

      n is the number of parallel observations at every point Sy is the replication error

      ai and aij are coefficients while i and j are pure components

      ai = Xi(2Xi-1) aij = 4XiXj

      Yobs = Y(observed) = Experimental results Ypre = Y(predicted) = Predicted results

      Using Eqns (12), (13), (14), the students t-test computations are presented on Table 5.

      T-value from table

      For a significant level, = 0.05, t/l(ve) = t 0.05/6(5) = t 0.01(5) = 3.365. The t-value is obtained from standard t- statistics table.

      This value is greater than any of the t-values obtained by calculation (as shown in Table 5). Therefore, we accept the Null hypothesis. Hence the model equation is adequate.

      (ii) Fisher Test

      For this test, the parameter y, is evaluated using the following equation:

      y = Y/n (15)

      where Y is the response and n the number of responses. The Fisher test computations are presented on Table 6.

      Using variance, S2 = [1/(n1)][ (Y-y)2] and y = Y/n for 1in, S2 and S2 are calculated as follows:

      The model equation was tested for adequacy against the S2

      (obs)

      (pre)

      2

      controlled experimental results. The statistical hypothesis for this mathematical model is as follows: Null Hypothesis (H0): There is no significant difference

      (obs) = 0.001763/5 = 0.0003526 and S (pre) =

      0.000695/5 = 0.000139

      The fisher test statistics is given by:

      F = S 2/ S 2 (16)

      between the experimental and the theoretically 1 2

      expected results at an -level of 0.5.

      where S 2 is the larger of the two variances.

      Alternative Hypothesis (H1): There is a significant

      difference between the experimental and theoretically

      1

      1

      Hence, S 2

      = 0.0003526 and S 2

      = 0.000139

      2

      expected results at an -level of 0.05.

      The students t-test and fisher test statistics were used for this test. The expected values (Ypredicted) for the test control points were obtained by substituting the values of Z1 from (Table 2) into the model equation i.e. Eqn (11). These values were compared with the experimental result (Yobserved) given in (Table 3).

      (i) Students t-test

      For this test, the parameters y, and t are evaluated using the following equations respectively

      Y = Y(observd) – Y(predicted) (12)

      = ( 2 + a 2) (13)

      Therefore, F = 0.0003526/ 0.000139 = 2.54

      From standard Fisher table, F0.95(5,5) = 5.1 which is higher than the calculated F-value. Hence the regression equation is adequate.

      i ij

      t = yn / (Sy1+ ) (14)

      Table 5. T-statistics test computations for Osadebes Poissons ratio model

      N	CN	i	j	i	ij	2 i	ij2		Y(observed)	y(predicted)	Y	t
      		1	2	-0.125	0.25	0.01562	0.0625
      1	C1	1 2	3 3	-0.125 -0.125	0.5 0.5	0.01562 0.01562	0.25 0.25
      		3	–	0	–	0	–
      						0.04686	0.5625	0.6094	0.099	0.075	0.024	1.170
      Similarly
      2		–	–	–	–	–	–	0.6094	0.105	0.079	0.026	1.268
      3		–	–	–	–	–	–	0.899	0.054	0.101	0.047	2.110
      4		–	–	–	–	–	–	0.8476	0.079	0.096	0.017	0.774
      5		–	–	–	–	–	–	0.640	0.092	0.073	0.019	0.918
      6		–	–	–	–	–	–	0.6208	0.098	0.078	0.020	0.972

      Table 6. F-statistics test computations for Osadebes Poissons ratio model

      Response Symbol	Y(observed)	Y(predicted)	Y(obs) y(obs)	Y(pre)-y(pre)	(Y(obs) -y(obs))2	(Y(pre) y(pre))2
      C1	0.099	0.075	0.011167	-0.00867	0.000125	7.51E-05
      C2	0.105	0.079	0.017167	-0.00467	0.000295	2.18E-05
      C3	0.054	0.101	-0.03383	0.017333	0.001145	0.0003
      C4	0.079	0.096	-0.00883	0.012333	7.8E-05	0.000152
      C5	0.092	0.073	0.004167	-0.01067	1.74E-05	0.000114
      C6	0.098	0.078	0.010167	-0.00567	0.000103	3.21E-05
      	0.527	0.502			0.001763	0.000695
      	y(obs)=0.087833	y(pre)=0.083667

      Legend: y =Y/n

      where Y is the response and n the number of responses.

4. Conclusion

1. Poissons ratio of soilcrete blocks was determined

2. Modified regression theory proposed by Osadebe was used to generate a model for prediction of Poissons ratio of soilcrete blocks

3. The efficacy of the model was proved using students t test and fisher test

4. The model can predict the Poissons ratio of soilcrete blocks if the mix ratio is specified and vice versa

5. References

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16. British Standard Institution, BS 882, Specification for aggregates from natural sources for concrete, 1992

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