11.05 Design strength

Categories: Stone Cladding

Introduction
Test results on natural stones are particularly prone to scatter because of intrinsic variations in the physical and chemical bonding within the material; it is therefore essential to carry out a sufficient number of tests to be confident of the lower limit strength of the stone. The two methods described below can be used to estimate the stone strengths and factors of safety from a programme of stone testing:

• Method 1 is based on North American practice and gives guidance on factors of safety to use, taking into account the variability of the test results and stone type.  For certain applications where durability of the stone is questionable or where a large volume of stone is to be quarried, then aged-strength testing of the stone should be considered within the testing programme.  All the factors of safety specified are based on experience and have been taken at face value.
• Method 2 is based on a statistical approach where the test strength results vary depending on the sample size used for the analysis.  Hence, if a higher safe load is required for a design, a greater number of tests may need to be specified so that one can be certain of the test strength of the stone.  A 95% confidence limit has been taken in common with other testing methods within the construction industry.  No factors of safety are recommended, however; this method of analysis is more transparent than the first method and it is left to the construction professionals to select appropriate factors of safety for the stone design.  No account has been taken of ageing with this method and if strength reduction through ageing is a concern then either aged-strength tests have to be performed or an appropriate factor of safety selected on the basis of experience.

Assessment method 1
For projects with aged-strength testing

This method of analysis is suitable where greater confidence is needed of the test results or for large projects where a greater number of samples are required for the purpose of testing.

The minimum factor of safety to be used for the flexural design of stone and the design of the stone at the fixings should be based on preliminary stone test data, which gives an initial indication of the test strength.  The safety factors incorporated within this design method comprise of two separate entities: the variance factor (VF) and the durability factor (DF).
 

Variance factor
The variance factor (VF) is defined as a measure of the natural range of flexural strength test results.  To evaluate the preliminary tests it is necessary to calculate the coefficient of variance (v) and use the relevant safety factor according to the table below.

The mean (x_m) of the sample tests is calculated based on the number of test samples (n) actually tested and is given by the following expression:

x_m =  ( x₁ + x₂ + x₃  + …. + x_n ) / n

The standard deviation is the positive square root of the variance (v) which is a measure of the dispersion of the test results about the mean.  For a series of (n) tests (x₁,x₂, x₃,  ......  x_n)  with an arithmetic mean (x_m), the square of the standard deviation or variance of the test population is given by the following expression:

s²  = v = [ ( x₁ – x_m )² + … + ( x_n – x_m )² ]  / ( n – 1 )

The variance may be expressed as a percentage of the mean to give the coefficient of variation (v) which is used to obtain a safety factor depending on the type of stone being quarried.  The table below shows the relationship between the coefficient of variation of the test data and the required factors of safety required for design purposes, where:

v = 100 s/ x_m
 
 

Durability factor
The durability factor (DF) is defined as a measure of the effect of accelerated-ageing test results.  To obtain a durability factor for a stone type it is necessary to calculate the flexural strength from the thermal cycling test regime as a proportion of the original strength in accordance with ASTM C 880:

(average after 300 thermal cycles/average before 300 thermal cycles) x 100%.

From this expression a durability factor can be chosen according to the loss in flexural strength as shown in the table below.
 
 

Stone flexural safety factor (F.S.F)
The minimum safety factor to be used in the flexural design of stone panels should be calculated as follows:

F.S.F = (V.F) x (D.F)

Where the safety factors are not required to be high or when specifying exterior stone for small projects the procedure specified below would be more appropriate. In place of aged-strength testing the construction professional may review the production test data and may further modify the safety factors as deemed appropriate.

In the absence of a durability factor based on aged-strength testing the minimum safety factor to be used in the flexural design of stone panels should be specified as shown in the following table
 

These are minimum values to be used in the absence of age-strength testing
 

Stone anchorage safety factor (A.S.F.)
The minimum safety factor to be used in checking stone stresses at anchorage zones should be calculated as follows

A.S.F = (F.S.F) x 1.4

To establish stone allowable stresses, the safety factors calculated, or greater safety factors if deemed appropriate by the engineer responsible for the design of the stone panels shall be applied to the average flexural strength.

To establish allowable stone stresses the safety factors should be applied to the average stone flexural strength.
 

Assessment method 2
The statistical method involves carrying out a specified number of tests as described in the preliminary and production test methodology.  It is important that each batch of test samples are prepared and tested under similar conditions and within a specified time of each other to eliminate the comparison of two sample means and variances within a single batch of stone samples.

For the purpose of the stone tests the construction professional will only use the lower tolerance limits (test strength) for the purpose of design.  For this application it is assumed that the true standard deviation (s) and true mean strength (m) of the stone are unknown and required to be estimated from the sample size for each specific test.

One can assume for statistical purposes that:

x_m =  ( x₁ + x₂ + x₃  + …. + x_n ) / n

s = { [ ( x₁ – x_m ) + … + ( x_n – x_m ) ]  / ( n – 1 ) } ^0.5

Where:
x_m = mean of the sample
s = standard deviation of the sample
n = sample size

To calculate the lower tolerance limit of the sample it is necessary to introduce a K-factor to account for the uncertainty arising from the sample size and the tolerance interval in relation to the true population.  The other uncertainty to account for is the chosen confidence level, which is the probability that the statistical tolerance interval will contain at least a proportion of the population.  For the purpose of the stone test data it may be appropriate to take the values of K that imply a confidence level of 95%.
Knowing the standard deviation (s) and the mean(c_m) of the sample, and assuming the tolerance level and confidence level, one can obtain a representative K-factor depending on the number of tests conducted, using the following table:
 

* values of K are adopted from BS 2846:Part 3 and are specific to a confidence level of 95%
 

Limits of test results
The tolerance limits of the strength tests are defined as the measure of the natural range of flexural strength of the test results and can be described as:

L₉₅ = x_m + K_s

or

L₅ = x_m - K_s

Where:
L₉₅  = Upper limit
L₅ = Lower limit (design strength)
 

Factors of safety
Having calculated the lower tolerance level of the natural stone an appropriate factor of safety should be applied to give a design strength.  The construction professional should use discretion when applying a safety factor, dependent upon the type of stone and level of confidence of the test results.

Design strength = ( x_m – K_s ) / FoS

Where

FoS = Factor of Safety

The construction professional should review the test data and select appropriate safety factors to account for the choice of stone, intended use, durability and effect of ageing on the strength of the stone. The value given to the FoS must also take account of the loads and whether any safety factors have been used in calculating the design loads from the nominal loads (see Section 04.01)