The resistance of a structural member is defined as the:

capacity of a member or component, or cross-section of a member or component of a structure, to withstand actions without mechanical failure

In structural engineering, resistance is a function of the structure's material strengths and its dimensions, but not of the magnitude of any actions applied to the structure, i.e.:

where the notation R{...} denotes that the design resistance (Rd) depends solely on design material strengths (Xdi) and design dimensions (ad,j). This holds true for non-prestressed beams, but not for pre-stressed beams and columns.

Returning to the example of Figure 2.13 may help to illustrate the ideas behind this equation. Figure 2.14 shows a cross-section through the simply-supported beam. The load on the beam causes it to bend, with

stress blocks Figure 2.14. Material properties and resistance of a concrete beam the top part going into compression and the bottom part into tension (which is carried by reinforcing steel).

Assuming plane sections remain plane allows the beam's bending resistance to be determined as:

Asfyd ( fyA

2 fcbd where As is the area of steel reinforcement; b and d are the breadth and depth of the beam, respectively; fy is the yield strength of steel; and fc is the compressive strength of concrete.

In Eurocode terms: MRd = function {fy, fc, As, b, d} = E{xdi, ad j} 2.10.2 Material properties

Material properties are introduced into design as characteristic values (Xk), with a prescribed probability of not being exceeded in a hypothetically unlimited test series. [en 1990 §1.5.4.1 and 4.2(1)]

The results of tests on man-made materials, such as concrete and steel, often follow a normal (a.k.a. 'Gaussian') probability density function (PDF), as shown in Figure 2.15. The normal distribution arises when a physical property depends on the combination of a large number of individual, random effects.7 It is encountered frequently in nature and is one of the most important PDFs in the field of statistics. The horizontal axis of Figure 2.15 measures deviation of the variable X from its mean value and the vertical axis gives the probability density of X.

kk,inf "m -Vsup Strength

Figure 2.15. Normal strength distribution for man-made materials (e.g. concrete)

The lower (or 'inferior') characteristic value Xk,inf is defined as the value of X below which 5% of all results are expected to occur. In other words, there is a 95% probability that X will be greater than Xkinf. This value is used in situations where overestimating the magnitude of a material property may be unsafe. For example, the lower characteristic value should be used to check that a material is strong enough to carry a particular load. Since strength-checks are a very common design requirement, the qualifier 'lower/inferior' is normally dropped from the description and symbol, leaving Xk as 'the characteristic value'.

Likewise, the upper (or 'superior') characteristic value Xk is defined as the value of X above which 5% of all results are expected to occur. There is a 95% probability that X will be lower than Xk . Although used less frequently than its lower counterpart, the upper characteristic value is important in situations where underestimating the magnitude of a material property may be unsafe. For example, since the force acting on a retaining wall depends on the weight density of the soil behind it, the wall should be designed to withstand an upper estimate of that weight density. Since the upper characteristic value is not used as often as the lower one, it should always be qualified as 'upper/superior' and denoted Xk .

With prior knowledge of the standard deviation: in situations where the standard deviation oX (or variance oX2) of the population is known from prior knowledge (and hence does not need to be determined from the sample), the statistical definitions of Xkinf and Xk are:

where is the mean value of X, the standard deviation of the population, ÔX its coefficient of variation (COV), and kn is a statistical coefficient that depends on the size of the population N.

These terms are defined as follows:

where t495% is Student's t-value8 for infinite degrees of freedom at a confidence level of 95% (see Figure 2.16).

The statistical coefficient kn is given by:

The statistical coefficient kn is given by:

Numerical values of kn are given in Figure 2.17 by the lower line, labelled 'variance known', and vary between «1.645 for a population size of a hundred and «2 for a population size of two.

To illustrate the use of this diagram, imagine that a series of forty concrete strength tests measured a mean compressive strength fc = 38.6 MPa. Based on prior experience, the standard deviation of the concrete's strength is assumed to be ofc = 4.56 MPa. With k40 = 1.665 calculated from the equation for kn (or taken from Figure 2.17), the concrete's lower characteristic strength is then given by:

fck = fc - KNvfc = 38.6 -1.665 X 4.56 = 31.0 MPa and its upper characteristic strength by: 7k = 7c + KN°fc = 38.6 +1.665 X 4.56 = 46.2 MPa

With no prior knowledge of the standard deviation: in situations where the variance of the population is unknown ab initio (and hence must be determined from the sample), the statistical definitions of Xkinf and Xk change to:

where mX is the mean value of X, sX the sample's standard deviation, VX its coefficient of variation, and kn is a statistical coefficient that depends on the sample size 'n'. (The use of Latin symbols distinguishes this equation from its variance-known counterpart, which uses the Greek symbols o, and 5.)

These terms are defined as follows:

YjXi ±(- mx)2 s mx = ——, sx = —-:-, and Vx =-y-

Note: the divisor in the expression for standard deviation is (n - 1) not n. The statistical coefficient kn is given by:

\ n where tn-195% is Student's t-value for (n - 1) degrees of freedom at a confidence level of 95% (see Figure 2.16).

Numerical values of kn are given in Figure 2.17 by the upper line, labelled 'variance unknown', and vary between . 1.645 for a sample size of a hundred and > 3 for a sample size of three. An important feature of this curve is the rapid rise in kn that occurs as the sample size decreases below about ten. This will have serious implications for the use of statistics for geotechnical problems, as discussed fully in Chapter 5.

Returning to the example of concrete strength tests, in this instance we must calculate the standard deviation sfc from the test results. Imagine that this calculation produced a value of sfc = 4.56 MPa identical to the previously assumed value for ofc. With k40 = 1.706 calculated from the equation for kn (or taken from Figure 2.17), the concrete's lower characteristic strength is now given by:

J ck J c n fc and its upper characteristic strength by: f, = f + ksc = 38.6 +1.706 x 4.56 = 46.4 MPa ck c n fc

The greater uncertainty in the standard deviation results in slightly more pessimistic values of fck.

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