Strength Parameters and Their Correlation

1.1 Mechanical Characteristics of Concrete

1.1.2 Strength Parameters and Their Correlation

Concrete strength is deducted from codiﬁed tests. The representativeness of the values obtained is strictly related to the correct testing procedures. First of all the size of the specimen has to be correlated to one of the aggregates used: l 5d_a, where l is the minimum dimension of the specimen and d_a is the maximum aggregate size.

Compression tests are carried out loading specimens placed between the plates of a press up to failure. The quantity measured on cubic specimens is calledcubic strength (in compression) and it is indicated with R_c. Failure usually occurs as indicated with dashed lines in Fig.1.7, with lateral spalling of the material and the formation of a residual double-cone shape.

The stress state of a cubic specimen compressed between the plates of a press is influenced by the friction on the faces of the specimen itself. In addition to the longitudinal component of stresses, a transversal component is induced, in compression too, that opposes the transversal expansion and increases the strength.

To overcome the effect of friction, prismatic (or cylindrical) specimens have to be used that are slender enough. In this way, between the end portions roughly as long as the transverse dimension, where the effects caused by the friction are signiﬁcant, an intermediate portion remains subject to a pure longitudinal stress flow. The strength measured on prismatic or cylindrical specimens whose length is at least 2.5 times the transverse dimension is called prismatic or cylinder strength (or more simplycompressive strength) and indicated withf_c(see Fig. 1.7b).

The correlation between the two strength values deﬁned above is given by the formula

fc0:83Rc

largely veriﬁed experimentally. This allows to adopt, in the practice of reinforced concrete constructions, the test on the more manageable cubic specimens and to derive then from the results the prismatic strength required for structural design calculations.

Fig. 1.7 Compression failure modes

Strength Classes

As better speciﬁed further on there are correlations between strength parameters that permit to identify the concrete class associating it to a unique quantity, the one corresponding to the lead parameter. The lead parameter is chosen as the compressive strength, the one that derives from the most elementary and direct test on the material.

The extent of the possible codiﬁed classes depends on the production technological capabilities: one starts from the lower bound with the lowest strength class compatible with the structural use of concrete; the upper limit is imposed by the level attained by the industrial production of the concrete itself.

The discretization introduced in identifying aﬁnite number of classes within an upper and lower bound is based on the minimum step that would have a practical meaning on site in relation to the precision allowed by the calibration capabilities of the production itself.

The minimum strength for structural use is set around 8 MPa. The maximum one, achievable with modern industrial technologies, can be higher than 70 MPa.

This limit does not take into account concretes aged in autoclaves, whose strength can be largely higher than 100 MPa. These concretes represent a different material not treated in this textbook. The minimum signiﬁcant step is around 5 MPa.

Concrete normalized classes are indicated with the symbol Cfollowed by the nominal values of cylinder and cubic strength. With these premises, the following strength groups can be codiﬁed, where the ones indicated as superior are currently admitted by national regulations only under some additional conditions for quality control.

Strength Classes

•very low C8/10–C12/15

•low C16/20–C20/25–C25/30

•medium C30/37–C35/43–C40/50–C45/55

•high C50/60–C55/67–C60/75–C70/85

•superior C80/95–C90/105

In the following section it is to be noted that a signiﬁcant random variability of strength values is associated to every single production event. The values mentioned above have to be considered as the characteristic ones mentioned hereafter.

With this clariﬁcation, the introduced classiﬁcation shows

• very low strength classes, minimum for plain and lightly reinforced concrete structures;

• low strength classes, minimum for reinforced concrete structures;

• mediumstrength classes, minimum for pre-stressed concrete structures;

• high-strength classes, for which a special prior experimentation is required;

• superiorstrength classes, presently done only for experimental purposes.

The so-deﬁned classes univocally identify the product according to its principal mechanical characteristics: compressive strength, tensile strength and modulus of elasticity. They do not identify other technological characteristics, such as worka- bility of fresh concrete that, for the same strength, can be improved for example with the use of plasticizers, and the maximum aggregate size which relates to the elements’ thicknesses and to the spacing between reinforcement bars. Those additional characteristics will have to be explicitly speciﬁed in the design docu- mentation together with the strength class.

In Table1.2 data relative to the three main mechanical parameters mentioned above are reported for all concrete classes.

Tensile Strength

Tensile tests are mainly carried with the following two methods. Theﬁrst one leads to direct strength (in tension) f_ct measured inducing a ﬁeld of pure longitudinal stresses in a specimen subject to tension between the clamps of a testing machine.

Conventional prismatic or cylindrical specimens are used for this test, having glued with epoxy resin the metal articulatedﬁxtures required for clamping device of the testing machine (see Fig.1.8a). Glueing can be avoided using friction grips, directly applied at the ends of the specimens.

The relationship between tensile and compressive strength can be given by the formula

fct¼0:27 ffiffiffiffi f_c² p3

forfc 58 MPa f_ct¼2:12 ln 1þ fc

forf_c [58 MPa:

The indirect strength in tension f′ct (splitting strength) is measured with the Brazilian test, which consists of inducing a linearly concentrated compression in the

Fig. 1.8 Tests for tensile strength

specimen (v. Fig.1.8b, c). The diffusion of stresses in the specimen leads, in addition to aflux of vertical compressive stresses, to a distribution of transversal tension stresses more or less constant throughout the intermediate part of the specimen.

Cylindrical specimen can be used, placed horizontally between the plane plates of a press, or more simply cubic specimens, same as the ones for the compressive test, having inserted loading strips to concentrate the load. Solving the problem of plane elasticity, the value of the transversal tensile component is obtained which, for the fracture loadP, gives the strength value

f⁰_ct¼ 2P pU1;

where l is the length of the specimen and U is its diameter (U= l for cubic specimens). As it will be mentioned further on, the presence of the vertical compressive components does not influence signiﬁcantly the tensile strength. The crack lines along which rupture occurs are indicated with dashed lines in Fig.1.8.

The tensile strength measured indirectly with the Brazilian test coincides with the direct one; the correlation formula can therefore be

f_ct⁰ f_ct:

The standards give the conservative valuef_ct0.9f′ct.

Theflexural test (see Fig.1.9) gives another method for the indirect evaluation of the tensile strength. It consists of applying a bending load on a concrete beam in order to induce triangular distributions of normal stressr, in tension at one side and in compression at the other side. Given the lower strength in tension of concrete, the part in tension will fail, from which theflexural strengthf_ctfcan be obtained.

The test has to be conducted with appropriate measures to isolate the central part of the beam outside the zones involved by stress concentrations due to loads and

Fig. 1.9 Test forflexural strength

reactions and to avoid parasite stresses (due to torsion for example). Assuming a linear distribution of stresses, the strength value is obtained at the extremeﬁbre in the central part subject to tension under the failure bending momentM= Pl:

fctf ¼6P1 bh²;

wherebis the width and his the depth of the rectangular section of the beam.

Theflexural strength obtained is systematically higher than the tensile strength obtained directly. This is due to the fact that close to failure, the distribution of stressesrin the section is not linear, as assumed the formula that interprets the test.

The part in tension is outside the elastic range, with a distribution similar to the one indicated in Fig.1.9b.

Very uncertain is the correlation with the direct tensile strength:

fctf¼bfct;

where very different values (b= 1.3–1.9) are proposed for b, whilst CEB–FIP Model Code 2010 sets it as a function of the beam depth h, deducing it from fracture theory as

b¼25þ1:5h⁰^:⁷

1:5h⁰^:⁷ ðhin mmÞ;

with values between 1.1 and 1.7 indicatively.

Modulus of Elasticity

The test for the evaluation of concrete modulus of elasticity Ecis carried out on prismatic specimens subject to compression, measuring, for a given load, the contraction of the central part of the specimen itself. The loading is assumed equal to 0.4 times the predicted material strengthf_c, and the measurement of shortening is done with four extensometers placed on the faces to compensate, with the mean value of readings, the possible eccentricity of the load itself (see Fig.1.10a).

The following ratio is therefore evaluated Ec¼rp=ep

that represents the secant modulus of elasticity (see Fig.1.10b) and is a little smaller than the tangentE_oat the origin.

The correlation between modulus of elasticity and compressive strength can be set according to the formula

Ec¼22000½fc=10⁰^:³:

With this value the deformation parameters of the constitutive model as reported in Table1.3can be deducted.

The determination of the Poisson ratio m (of transversal contraction) requires more complex testing procedures. Values between 0.16 and 0.20 are obtained for concrete. Those values are valid if high levels of compression are excluded, higher than 0.5 times the material strength, for which high increments of apparent transverse expansion are measured, because of the formation, when progressively approaching the rupture load, of macroscopic longitudinal cracks in the specimen.

The values of mechanical characteristics presented above are reported, for var- ious concrete strength classes, in Table1.2.

Mean and Characteristic Values

Tests, repeated on several specimens of the same concrete, show a dispersion of results, quite significant if related to the entire production cycle on site of a construction from foundation to the roof. If related to the continuous industrialized production of precast elements in industrial plants, given that the production procedures themselves are subject to an efficient system of quality control, the dispersion of results can be significantly smaller.

Extensive surveys have been conducted on construction sites and industrial plants. Analysing data, for example the ones relative to cubic strength R_c, with statistical procedure,mean valueshave been calculated:

Rcm¼ Pn

i¼1R_ci n and standard deviations

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn

i¼1ðRciRcmÞ² n1 s

; obtaining thecharacteristic value

Fig. 1.10 Tests for the modulus of elasticity

Rck¼Rcmks

to be used in resistance veriﬁcations, which corresponds to the 5% probability of having a more unfavourable value. Using a suitable model of the distribution function, for a sufﬁcient number of measurements (n 30), the deviation was found as

DR¼R_cmR_ck¼ks

quite homogeneous across all controlled sites and plants, for which the following value can be assumed on average

DR9:6 MPa independently from the concrete strength class.

Such ﬁxed deviation penalizes less-resistant concretes more, as indicatively reported in the following table (values expressed in MPa):

R_ck 20 37 55 75

DR/R_ck 0.48 0.26 0.17 0.13

R_cm/R_ck 1.48 1.26 1.17 1.13

If for example a cubic characteristic strength of 30 MPa is prescribed, the mix design of the relative concrete production will have to refer to a mean value Rcm= 30 + 9.640 MPa about 1/3 higher than the characteristic one required. At the same time, such production can guarantee a characteristic strength equal to 0.75 times the mean value itself.

The model that ﬁts in the best way to the random distribution of concrete strength throughout its site production is the lognormal expressed by (see Fig.1.11)

fðxÞ ¼ 1 ðxxdÞ ffiffiffiffiffiffi

p2p re

ð Þnn² 2r2 ;

f(x)

ox x

k x

d x x

Fig. 1.11 Strength distribution curve

wheren= ln(x−x_d) forx x_d, and wherenis the mean value ofn andr is its standard deviation. The lower bound value of the possible interval of random variability of the quantityxis indicated withx_d, calculated on the basis its meanx and its standard deviationswith

xd¼xbs:

For the reliability index, le regulations assume the value b= 3.8. The characteristic valuexk¼xkscorresponding to the 5% fractile is calculated with values ofkthat vary with the coefﬁcient of variation d¼s=x:

k= 1.579 for d¼0:05

k= 1.514 for d¼0:10

k= 1.451 for d¼0:15

k= 1.390 for d¼0:20

If referred to prefabrication plants with an efﬁcient quality control system, the deviation between mean and characteristic values of a given continuous concrete production is limited to

DR6:0 MPa:

The management cost of the control is therefore compensated by a reduction of cement quantity in the mix design, with less penalizing mean values. If for example a cubic characteristic strength equal to 55 MPa is prescribed, the mix can be designed for a mean valueR_cm= 55 + 6.0 61 MPa about 11% higher than the characteristic one, whilst such production will be able to guarantee a characteristic strength equal to about 0.92 times the mean value itself.

If reported in terms of prismatic strength values (Df0.83DR), such differences become

Df ¼8:0 MPa and Df ¼5:0 MPa; respectively, for ordinary and industrial controlled productions.

At the design stages, for calculations done according to the semi-probabilistic ultimate limit state method, previsions have to be based on thecharacteristic value f_ckof strength. Therefore, in order to deduct the other mechanical characteristics necessary for calculations, based on the type of ordinary or industrial production of the relative site or plant, the designer will have to estimate the mean value of compressive strength, respectively, withf_cm= f_ck+ 8 or withf_cm=f_ck+ 5 and on these values he can apply the correlation formulas reported in the previous pages.

The numerical values of related quantities are reported in Tables1.2a and b for the two types of production, deduced from the mentioned correlations. In particular the characteristic (lower) values of tensile strength and modulus of elasticity are

calculated assuming their ratio to the corresponding mean values equal to 0.7 and 0.8, respectively, for ordinary and industrial productions. The mean value of cubic strength is reported as it is the most immediate reference for the tests that will be carried out during production, unless more articulated elaborations of results are required for acceptance veriﬁcations.

For the control of continuous concrete production-speciﬁc charts and diagrams can be used such as the ones reported in Table1.11, whereshifting mean valuesare used based on tests on the last three weeks of production.

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