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					                                                                             International Journal of Computer Information Systems,
                                                                                                                 Vol.4 , No. 4, 2012

   Proposal for a biphasic model of Young modulus of
         the high performance concretes (hpc)
                    Mbessa Michel1                                                  Tamo Tatietse Thomas4
          University of Yaoundé I - Cameroon                        University of Yaoundé I - Cameroon – Laboratory of civil
   Laboratory of civil engineering and design sciences,               engineering and design sciences, National Advanced
       National Advanced School of Engineering:                                      School of Engineering
                   Yaoundé, Cameroon                                                  Yaoundé, Cameroon

                  Bwemba Charles2                                                        Agu Henry Tata5
University of Yaoundé I - Cameroon – Laboratory of civil                             Ministry of Public Works,
  engineering and design sciences, National Advanced                               Technical Standardization Unit,
                 School of Engineering                                                  Yaoundé, Cameroon
                  Yaoundé, Cameroon                                          

                        Pera Jean3
          Laboratory of materials and Structures,
          National Institute of Apllied Sciences,
                      Lyon - France

                                                                    as dispersed phase. From these considerations, mathematical
         Abstract                                                   models had been proposed taking into account the elasticity
          In this article, the Young modulus is established by      modulus and the voluminal proportion of each phase.
considering the high performance concrete (hpc) as a biphasic                The diversity of these formulations shows that non of
material (mortar and coarse aggregate) and, on the basis of a       them is perfectly satisfactory.
representative cell of hpc with defined sizes, on the basis of
                                                                             In this study, while considering the hpc as a biphasic
Voigt’s model [9]. The coefficient of form is integrated (to take
into account moreover the original shape of the coarse              material (mortar and coarse aggregate), and leaving from a
aggregate).                                                         representative cell of hpc with fixed dimensions, on the basis
         The representative cell is a cube with a spherical         of the Voigt’s model [9], it is integrated the coefficient of the
aggregate of diameter D0 at its center, surrounded by mortar of     shape (to take into account the original shape of the coarse
thickness eM.                                                       aggregate).
         Each phase is regarded as homogeneous. No sliding is                The representative cell is a cube with a spherical
possible between the phases so that the deformations inside the     aggregate of diameter D0 at its center, surrounded by mortar of
solid do not cause any discontinuity (compatibility of              thickness eM.
deformation) [8].

   I.    Key words : high performance concrete, Young modulus,
          biphasic, thickness of mortar, coarse aggregate.              1.     Justification of the biphasic model

                      II.   INTRODUCTION                                     Figure 1 presents the resistance of a hpc made with
         It is necessary to know the Young modulus of a             the slag in variable proportions of coarse aggregates [7]
material when studying its mechanical behavior. While               according to the dosage of coarse aggregate, translating a
studying the characteristics of hpc made with different types       variation of the thickness of the mortar. As already shown by
of aggregates, some authors [9] had shown that the coarse           De Larrard and Belloc (1997) [3], the resistance decreases
aggregate governs the shape of the constraint/deformation           with an increase in the proportion of the mortar.
curves of the hpc. In other words, the Young modulus of the
hpc depends largely on that of the coarse aggregate.
         On the other hand, other authors [1, 2, 6] had already
considered the concrete like a two-phase material (dough and
aggregate) to study its mechanical behavior. The concrete had
also been presented like an intimate association of two phases
[9]: the mortar as a continuous phase and the coarse aggregate

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                                                                               International Journal of Computer Information Systems,
                                                                                                                     Vol. 4, No. 4, 2012
                                                                                 H2 : Each phase is regarded as homogeneous. No
                                                                                 sliding is possible between the phases so that the
                                                                                 deformations inside the solid do not cause any
                                                                                 discontinuity (compatibility of deformation);


                    80                                                           H3 : The aggregates are all spherical and one-
                    40                                                  dimensional (D0) ;
                                                                                 H4 : the volume of the stacking (dry stacking V0, and
                         0      0,2    0,4    0,6    0,8    1
                                                                        dilated stacking Vb) is proportional to the volume of the
                             % voluminal proportion of mortar
                                    % volumique du mortier
                                                                        correspondent coarse aggregate (respectively D0 and Dd).
                                                                                 H5 : The representative cell is a cube with a spherical
  Figure 1. Influence of the thickness of the mortar on the             aggregate of D0 diameter at its center, surrounded of mortar
             mechanical resistances of the BHP,                         with a thickness eM (Figure 2).
The tests have been made on cylindrical test-tubes 7x14 at 28
                          days age

         Besides, 0,35 being roughly the mass proportion of
water necessary for the total hydration of the cement [5], in the
hpc, some grains of cement remain anhydrous even at long-
term, and can be accounted for in the granular phase. The
cement in addition to its role of binder plays then the role of
inert and small dimensional aggregate.
         It then becomes very difficult or impossible to isolate
the actual granular phase (sand and gravel) from the cement
paste (binder). Considering the dimensional difference
between the sand and gravel, it becomes justifiable to consider
the hpc as a biphasic material: mortar and coarse aggregate.
                                                                                Figure 2. a) representative cell; b) cut in the representative
    2. Position of the problem

         Let Vg be the volume of aggregate having been used                 4. Parameters
for the composition of a concrete of volume Vb. Vg is placed in
a container where it is well packed (Figure 3.a). The whole
                                                                                            D0, the efficient middle diameter of the
content forms a non handy stacking which will allow the
                                                                        coarse aggregate;
composition of a handy concrete. This is achieved by injecting
                                                                                            Dd, the diameter of the dilated coarse
into the aggregate stacking an excess of mortar more than the
Vm necessary to fill the inter granular voids. The grains can
                                                                                          V0, the volume of the stacking;
then move against each other in the course of putting in place
                                                                                          Vg, the volume of the dry granular skeleton;
                                                                                          g, the voluminal proportion of the coarse
                                                                        aggregate in the concrete.
                                                                                          The H4 hypothesis allows to write:
                                                                                                              Dd 3
                                                                                                     Vd = (      ) V0
           a) stacking               b) suspension
the material (Figure 3.b). The skeleton is dilated and one is in                          Let ρa et ρr be the apparent and real
             Figure 3. Dilation of the granular skeleton
the presence of a concentrated suspension.                              voluminal masses of the coarse aggregate :
                                                                                          V0 = m/ρa (where m is the masse of the dry
                                                                        coarse aggregate.

    3. Hypotheses                                                                           According to equation (1), one can write :
                                                                                                     Dd 3 m       D       Vg ρ r
         H1 : The hpc is considered as a biphasic material                                  Vd = (      ) *    = ( d )3 *
(mortar and coarse aggregate)                                                                        D0     ρa    D0       ρa

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                                                                                        International Journal of Computer Information Systems,
                                                                                                                            Vol. 4, No. 4, 2012
                      Vd    D      ρ
                         = ( d )3 * r                                                              eM = D0[(g*/g)1/3-1] = D0[(ρa/gρr)1/3-1)]
                      Vg    D0     ρa                                          (6)

             1   D + eM 3 ρ r       e       ρ                                                              with       g*           =        ρa/ρr
               =( 0    ) *    = (1 + M ) 3 * r                                 (7).
             g      D0     ρa       D0      ρa                                            In the formulas proposed by F. De Larrard et al. [3,
                                                                               4] :
From where the relation below:                                                                     g* = 1-0,39(d/D)0,22          for the rolled
                                                                                                   et g* = 1-0,45(d/D)0,19 for the ground
                                        1 ρ
                             e M = D0 [( * a ) 3 − 1]
                                        g ρr                                           Formula proposed by Caquot [3, 4] :
                                                                                                g* = 1-0,47(d/D)1/5
                                   (2)                                         Where d and D are respectively the smallest and biggest
                                                                               dimension of the granular extent of the coarse aggregate (6
                                                                               mm and 10 mm in this study).
      5.    Young Modulus expression                                                   The values of the apparent and real densities of the
                                                                               coarse aggregate (respectively 1.53 et 2.64) have been
      5.1. Preamble                                                            determined according to norm NF P 18-554 (December 1979).
                                                                               Table 2 presents the values of g* and eM issued from the
                                                                               formulas above :
          Let g and m be respectively the coarse aggregate and
the mortar indications, the Young Modulus Eb of the hpc can
                                                                                            Table 2. Estimated Values of g* and eM
be written from the Voigt' model [9] as below :
                                                                                         Expression        g*         eM (mm ; 10-3ft)
          Eb = Cg Eg + CmEm = Cg(Eg - Em) + Em (loi des
mélanges)                                                                            F. DE LARRARD           0,59          (1,52 ; 4,98)
where Cg and Cm are the voluminal proportions of the phases
                                                                                     CAQUOT                  0,58          (1,48 ; 4,85)
(in the cell) : aggregate and mortar.
                                                                                     ρa/ρr       (modèle     0,58          (1,48 ; 4 ;85)
                       πD0 3
           Cg =                                               (3)                    actuelle)
                  6( D0 + eM ) 3
         While taking into account the sphericity index "i" (or
coefficient of shape) of the coarse aggregate, the Young                              5.3. Consequences
Modulus of the hpc is given by the formula below :
                                                                                       The results in table 2 show that the three expressions
                         igρ r π ( E g − E m )                                 are quasi-equivalent, what entails a simplification of the
       Eb = E m +                                                   (4)        determination of some parameters of the model.
                                6ρ a
                                                                                                a) The apparent and real coarse aggregate
                                                                               voluminal mass (ρa et ρr)
      5.2. Determination of the parameters                                             Knowing one, the other can be determined either by
                                                                               the formula proposed by F. De Larrard, or by the one proposed
      5.2.1.      The sphericity index "i"                                     by Caquot. One can then write:
                                                                                                         ρa/ρr = g*
         It is the report of the efficient volume of the grain and                                          (8)
the volume of the circumscribed sphere. in general, one has :
0.30 < i < 0.40 [5].                                                                               b) Expression of young modulus of the
      5.2.2.      The compactness of the mixture (g*)
                                                                                       Finally, the formula below is proposed to determine
                                             1/3                               the young modulus of the hpc :
            eM           =        D0[(g*/g) -1]         [3,               4]
(5)                                                                                                    igπ ( E g − E m )
                                                                                           Eb = Em +
                                                                                                              6g *
            (4) and (5) allow to write :

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                                                                                        International Journal of Computer Information Systems,
                                                                                                                            Vol. 4, No. 4, 2012
                                  igπ         igπ                                  7.1. Test
                     E b = (1 −        ) Em +      Eg
                                  6g *        6g *
                                                                                       The secant modulus has been determined on
                                      (9)                                     cylindrical test-tubes 11x22 cm after simple sawing on a
                                                                              diamond-tipped milstone machine, the average maximal load
                                                                              being 50 MPa (1044271 psf), the test was conducted at ten
                                                                              months age of the concretes on a press of 2500 KN.
                                                                                       Table 3 presents the results obtained from the tests.
                                                                              They are compared to those from the literature [9] (table 4 and
    6. Experimentation

    6.1. Materials

    6.1.1.  cements
          Two types of cement have been used : the artificial
Portland cement (pac) CEM I 52.5 R from Rochefort (France)
   and the blast-furnace cement (bfc) CEM A 52.5 from
                      Ebange (France).
                                                                                                Figure 3. Diamond-tipped milstone machine
    6.1.2.          The slag
                      The slag was a ferro-alloy from France.                      8.       Results and validation of the model

    6.1.3.          The superplasticizer
                                                                               Tableau 3. Values of young modulus of hpc made with slag
                                                                                            (10 months after the removing)
             The superplasticizer was the Glénium 51 made with
                                                                             g                          0,386       0,290      0,193      0,00
                          polycarboxylate                                    Eg (Gpa) (*)                 75          75         75        75
                                                                             psf                      15664073    15664073   15664073   15664073
    6.1.4.          Aggregates                                               Em (GPa)                     41          41         41        41
                                                                                                      8563026     8563026    8563026    8563026
                                                                             i                           0.35        0.35       0.35      0.35
              The gravel was a 6/10 ground Comblanchien, with                g*                          0,6         0,6        0,6        0,6
                            density 2,64                                     measured E (GPa)             48         45.5        44        41
               The sand was a 0/4 limestone with density 2,54                psf                      10025006    9502871    9189589    8563026
                                                                             our model(GPa)               45          44         43        41
                                                                             psf                      9398444     9189589    8980735    8563026
    6.2. Preparation of the samples                                          Voigt (GPa)                  54          50         47        41
       7.     Table 1. Dosage of one cube meter of materials [7]             psf                     119278132    10442715   9816152    8563026
                                                                             Reuss(GPa)                   57          60         64        41
                                                                             psf                     125904695    13253125   14136667   8563026
             apc        bfc    Slag     sand 0/4   gravel 6/10 (kg)   w/b
MATERIA                                                                (*)                                             8         5
             (kg)      (kg)    (kg)       (kg)
  LS                                                                         Hashin(GPa)                 51           47         46       41
                                                                             psf                      10651569    9816152    9607298    8563026
                                                                             Hirsh(GPa)                  52           46         45       41
 Mass        387       129     52.5         662         1021          0,26   psf                      10860424    9607298    9398444    8563026
                                                                             Counto(GPa)                 55           54         54       41
                                                                             psf                     121486987    11927813   11927813   8563026
                                                                                                                       2         2
                  : b (binder) = cement + ultrafine particules               Bache(GPa)                  52           49         46       41
                                                                             psf                      10860424    9398443    9607298    8563026
                                                                             Popovics(GPa)               56           55         56       41
        The mixing was done following a very precise                         psf                     123695841    12148698   12369584   8563026
method [7]                                                                                                             7         1
        The concrete was then sunk in 11x22 cylindrical                                (*) Eg is the modulus pulled from the literature [10], it
mould and vibrated during 10 seconds on a vibrating table.                    is the Young modulus of the chalky rocks..
The removal was done 24 hours later.
                                                                                1) Tests done by B. Tighiouart et al [9]

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                                                                          International Journal of Computer Information Systems,
                                                                                                               Vol. 4, No. 4, 2012
         Table 4. Some results of elasticity modulus of hpc,                i                0,35      0,35     0,35        0,35
                                                                           g                0,393     0,404      (*)       0,435
                              E/L=0,27                                     g*                0,6        0,6      0,6         0,6
                                Types of aggregates                   measured E              48        45       42          31
                                                                         (MPa)            1002501    112781 87718         647448
                     Limest    Granite   Quartz   Sandstone               psf                                     9
                       one                                         our model (MPa)          45,12       45       (*)       36,33
     Eg (MPa)           68       66        44           37                psf             1130319    112781               758768
        psf          150202    142784    918959       772761          Voigt(MPa)              52        51       43          40
                        09       38                                       psf             1086042    106515 89807         835417
    Em (MPa)            39       39        39            39                                              7        3
       psf           814532    814532    814532       814532          Reuss(MPa)              49        49       43          39
         i            0,35      0,35      0,35          0,35              psf             1023386    102338 89807         814532
        g             0,393     0,404      (*)         0,435                                             6        3
       g*              0,6       0,6       0,6          0,6          Hashin(MPa)              50        50       43          39
   measured E           42       42        45            30               psf             1044272    104427 89807         814532
     (MPa)           877188    877188    939844       626563                                             2        3
       psf                                                            Hirsh(MPa)              50        50       43          39
   Our model          42,5   42,3          (*)         36,7               psf             1044272    104427 89807         814532
     (MPa)           887631 883454                    766495                                             2        3
       psf                                                           Counto(MPa)              50        50       43          39
   Voigt (MPa)       50       49        40          38                    psf             1044272    104427 89807         814532
       psf         104427 102338 835417          793646                                                  2        3
                      2        6                                      Bache(MPa)              50        50       43          39
   Reuss (MPa)       47       46        40          38                    psf             1044272    104427 89807         814532
        psf        981615 960730 835417          793646                                                  2        3
       Hashin        48       47        40          38              Popovics(MPa)             50        50       43          39
       (MPa)       100250 981615 835417          793646                   psf             1044272    104427 89807         814532
        psf           1                                                                                  2        3
   Hirsh (MPa)       48       48        40          38                        (*) : The voluminal mass of the quartz used has not
        psf        100250 100250 835417          793646             been given.
                      1        1
      Counto         48       48        40          38
       (MPa)       100250 100250 835417          793646                 9. Conclusion/analysis
        psf           1        1
   Bache (MPa)       48       48        40          38                  9.1. Contribution of the model
        psf        100250 100250 100250          793646
                      1        1         1                                  The model allows to the evaluation of Young
     Popovics        48       48        40          38                      modulus with very limited errors. It takes into
       (MPa)       100250 100250 835417          793646                     account the sphericity index which is not taken into
        psf           1        1                                            account in the other models;
         (*) : The voluminal mass of quartz used has not been               The contribution of the mortar phase is very great.
given.                                                                      The Young modulus increases with the thickness of
                                                                            the mortar. Hence, a great Young modulus is
                                                                            obtained when the coarse aggregate is well chosen
     Table 5. some results of elasticity modulus of hpc,                    and the tests well led;
                       E/L=0,22 [9]                                          The model shows that, the Young modulus of hpc
                               Types of aggregates                          depends more on the voluminal proportion of the
                  Limestone Granite Quartz Sandstone                        coarse aggregate than the maximal dimension, limited
   Eg (MPa)           68             66       44         37                 by the requirements of putting in place. However, the
      psf         15020210        145784 91895        772761                thickness of the mortar having an influence on the
                                     38       9                             characteristics of the material, for a same volume, the
   Em (MPa)           39             39       39         39                 characteristics will be better in the case of small
      psf           814532        814532 81453        814532                dimensions (the thickness of the mortar being
                                              2                             reduced);

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                                                                          International Journal of Computer Information Systems,
                                                                                                                Vol. 4, No. 4, 2012
        The present model is more explicit because, it leaves      [6] Maso , J.C.. La nature minéralogique des aggrégats,
        from the physical hypotheses that allow to better          facteur essentiel de la résistance des béons à la rupture et à
        understand the nature of the parameter (g*). Indeed,       l’action du gel, Thèse de doctorat ès sciences , Toulouse 1967
        the value of g* can be calculated from the measurable      [7] Mbessa, M.. Rôle des ultrafines dans les bétons industriels
        physical sizes ( the apparent and real voluminal           à hautes performances, Editions Universitaires Européennes,
        masses).                                                   139 p., 2011
                                                                    [8] Recho, N.. Rupture par fissuration des structures, Paris,
                                                                   Hermes,. 363 p, 1995
    9.2. The limits /weaknesses of the model                       [9] Tighiouart, B., Benmokrane, B., Baalbaki, W..
                                                                   Caractéristiques mécaniques et élastiques des bétons à hautes
                                                                   performances confectionnés avec différents types de gros
        The model has been tested only for small dimensions
                                                                   granulats, Materials and Structures, vol. 27, n°168, p. 211-
        of the aggregates and only for one granular class.
                                                                   227, May 1994
        Tests on larger granular extents and for several
                                                                   [10] Tourenq C., Primel, L.. Propriétés des roches et des
        granular classes are certainly necessary.;
                                                                   granulats, Par G. Arquie et C. Tourenq. Paris : Presse de
        The model doesn' take in account nor the effect of
                                                                   l’Ecole Nationale des Ponts et Chaussées (ENPC), 1990, p 71
        stamping, nor the porosity led by the putting in place,
                                                                   - 130
        nor the other physico-mechanical interactions that
        would exist at the interface coarse aggregate/mortar.
        Besides, the model considers homogeneous the
        distribution of the coarse aggregate, which is not
        absolutely true in practice. These parameters would
        be certainly at the origin of the difference between
                                                                                          Michel Mbessa is a Permanent teacher at the
        the measured value and the estimated value;
                                                                                          University of Yaoundé I in Cameroon; Civil
        The hypothesis of the homogeneity of the granular
        distribution requires a particular attention at the time                          Engineer from the National Advanced
        of the putting in place: to avoid the segregation being                           School of Engineering of Yaoundé, he
        the main rule to respect when applying the model;                                 holds a Diploma (DEA) and a PhD from the
        Finally, the model is only valid for the hpc of which                             INSA Lyon in France. He was, while
        the Young modulus of the coarse aggregate is                                      preparing his thesis, Winner of the first class
        superior to the one of the mortar. Indeed the Young                               of the Eiffel scholarship in France. He also
        modulus of the high performance mortar being                                      holds a degree of Expert in the Procurement
        generally not too different from 39 MPa, the coarse                               of World Bank contracts. Mbessa Michel is
        aggregate with Young modulus lower than this value                                the author of 04 scientific publications in
        would not fulfill the conditions required for the                                 international journals of references, 02
        confection of the hpc.                                                            papers in international conferences, one with
                                                                                          acts and the other refereed.
                                                                                          Member of the committee (Reviewer), for
                        References                                                        ACI Journal. Consultant Expert to
[1] Calvet, H.. Etude sur modèle plan des déformations sous                               international      organizations       (African
charge autour d’un granulat, Colloque géotechnique,                                       Development Bank) and national (ministries,
Toulouse, 1969                                                                            Consultancy Firms, Enterprises, Public
[2] Dantu, P.. Etude des contraintes dans les milieux                                     contracts Regulatory board). Michel Mbessa
hétérogènes, Anales de l’ITBTP. , N° 121, p 53-98, janvier                                is the focal point of the scientific committee
1958                                                                                      of the University of Yaounde I in the Civil
[3] De Larrard, F. and Belloc, A.. The influence of aggregate                             Engineering Department of the Polytechnic
on the compressive strength of normal and high strength                                   National High School of Yaoundé.
concrete, ACI Materials journal, vol. 94, n° 1, p. 417-426,
Sept.-Nov. 1997
 [4] De Larrard, F. et Tondat, P.. Sur la contribution de la
topologie du squelette granulaire à la résistance en
compression du béto, Materials and structures, vol. 26, p.
505-516, 1993
 [5] De Larrard, F.. Formulation et propriétés des bétons à
très hautes performances, Rapport de recherche Paris, LCPC,.
335 p, 1988

       April Issue                                        Page 57 of 58                                  ISSN 2229 5208
                                                              International Journal of Computer Information Systems,
                                                                                                  Vol. 4, No. 4, 2012

                                                                                     Thomas TAMO TATIETSE is
                   Charles BWEMBA is Civil                                           Professor of the Universities, Civil
                   Engineer from the National                                        Engineer from the National
                   Advanced         School        of                                 Advanced School of Engineering,
                   Engineering,      University    of                                University of Yaoundé I in
                   Yaoundé I in Cameroon. He holds                                   Cameroon, he holds a Master of
                   a Diploma of Advanced Studies in                                  Science and a PhD from the INSA of
                   Engineering Sciences from the                                     Lyon in France. He completed his
                   same school; he is a Researcher at                                Post Doctoral at Laval University
                   the     Laboratory     of    Civil                                (Canada) holds a HDR of the
                   Engineering and Science of                                        Polytechnic Institute of Grenoble.
                   Design, and author of a scientific                                Professor and Visiting Scholar at the
                   publication. Consultant Expert to                                 Universities of Grenoble, Lyon,
                   international and national bodies                                 Liege and Quebec. Winner of the
                   (ministries    and    Consultancy                                 Excellence Award of the AUF and
                   Firms), Charles is BWEMBA is                                      Project Scientist for Scientific
                   Head of Technical Standardization                                 Cooperation           Inter-University
                   Unit at the Ministry of Public                                    (HPIC), Prof. T. Tamo Tatietse is the
                   Works since 2002 in Cameroon,                                     author of 35 scientific publications in
                   after having been Director of the                                 international journals of references
                   Centre of Trades of Public Works                                  and 40 papers in international
                   of Garoua in the same country.                                    conferences      with     proceedings.
                                                                                     Member       of     the     committee
                                                                                     (Reviewer) magazines i) "The
              AGU Henry TATA is a                                                    International Journal ENERGY", ed.:
              Civil Engineer from the                                                Elsevier ii) "Journal of Decision
              National       Advanced                                                Systems", ed. Hermes & Lavoisier.
              School of Public Works                                                 Chairman       of    the     Scientific
              Yaoundé. Assistant in                                                  Committee of several international
              Charge of Studies at the                                               conferences. Consultant Expert to
              Technical                                                              international agencies (AUF, IRD
              Standardization Unit at                                                (France), NSF (United States), EU
              the Ministry of Public                                                 (European Union ),..) and national
              Works in Cameroun,                                                     (ministries and Consultancy Firms).
              person in charge of                                                    T. Tamo Tatietse is Deputy Director
              technical standardization                                              of the Polytechnic National High
              in building and road                                                   School of Yaoundé in 2004 after
              works.                                                                 leading the Civil Engineering
                                                                                     Department for 05 years.

April Issue                                   Page 58 of 58                                  ISSN 2229 5208

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