Unit 48: Structural Behaviour and Detailing for Construction. Limit State Design

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1 2.1 Introduction Limit State Design Limit state design of an engineering structure must ensure that (1) under the worst loadings the structure is safe, and (2) during normal working conditions the deformation of the members does not detract from the appearance, durability or performance of the structure. Despite the difficulty in assessing the precise loading and variations in the strength of the concrete and steel, these requirements have to be met. Three basic methods using factors of safety to achieve safe, workable structures have been developed over many years; they are 1. The permissible stress method in which ultimate strengths of the materials are divided by a factor of safety to provide design stresses which are usually within the elastic range. 2. The load factor method in which the working loads are multiplied by a factor of safety. 3. The limit state method which multiplies the working loads by partial factors of safety and also divides the materials ultimate strengths by further partial factors of safety. The permissible stress method has proved to be a simple and useful method but it does have some serious inconsistencies and is generally no longer in use. Because it is based on an elastic stress distribution, it is not really applicable to a semi-plastic material such as concrete, nor is it suitable when the deformations are not proportional to the load, as in slender columns. It has also been found to be unsafe when dealing with the stability of structure subject to overturning forces. Jesmond Agius: Chapter 2 Page 1

2 In the load factor method the ultimate strength of the materials should be used in the calculations. As this method does not apply factors of safety to the material stresses, it cannot directly take account of the variability of the materials, and also it cannot be used to calculate the deflections or cracking at working loads. Again, this is a design method that has now been effectively superseded by modern limit state design methods. The limit state method of design, now widely adopted across Europe and many other parts of the world, overcome many of the disadvantages of the previous two methods. It does so by applying partial factors of safety, both to the loads and to the material strengths, and the magnitude of the factors may be varied so that they be used either with the plastic conditions in the ultimate state or with the more elastic range stress range at working loads. This flexibility is particularly important if full benefits are to be obtained from development of improved concrete and steel properties. 2.2 Limit States The purpose of design is to achieve acceptable probabilities that a structure will not become unfit for its intended use that is, that it will not reach a limit state. Thus, any way in which a structure may cease to be fit for use will constitute a limit state and the design aim is to avoid any such condition being reached during the expected life of the structure. The two principle types of limit state are the ultimate limit and the serviceability limit state. (a) Ultimate limit state This requires that the structure must be able to withstand, with an adequate factor of safety against collapse, the loads for which it is designed to ensure the safety of the building occupants and/or the safety of the structure itself. The possibility of buckling or overturning must also be taken into account, as must the possibility of the accidental damage as caused, for example, by an internal explosion. (b) Serviceability limit states Generally the most important serviceability limit states are: 1. Deflection- the appearance or efficiency of any part of the structure must not be adversely affected by deflections nor should the comfort of the building users be adversely affected. 2. Cracking- local damage due to cracking and spalling must not affect the appearance, efficiency or durability of the structure. Jesmond Agius: Chapter 2 Page 2

3 3. Durability this must be considered in terms of the proposed life of the structure and its conditions of exposure. Other limit states that may be reached include: 4. Excessive vibration which may cause discomfort or alarm as well as damage. 5. Fatigue must be considered if cyclic loading is likely. 6. Fire resistance this must be considered in terms of resistance to collapse, flame penetration and heat transfer. 7. Special circumstances any special requirements of the structure which are not covered by any of the more common limit states, such as earthquake resistance, must be taken into account. The relative importance of each limit state will vary according to the nature of the structure. The usual procedure is to decide which is the crucial limit state for a particular structure and base the design on this, although durability and fire resistance requirements may well influence member sizing and concrete class selection. Checks must also be made to ensure that all other relevant limit states are satisfied by the results produced. Except in special cases, such as waterretaining structures, the ultimate limit state is generally critical for reinforced concrete although subsequent serviceability checks may affect some of the details of the design. Prestressed concrete design, however, is generally based on serviceability conditions with checks on the ultimate limit state. In assessing a particular limit state for a structure it is necessary to consider all the possible variable parameters such as the loads, material strengths and all constructional tolerances. 2.3 Characteristic Material Strengths and Characteristic Loads Characteristic Material Strengths The strengths of materials upon which a design is based are, normally, those strengths below which results are unlikely to fall. These are called characteristic strengths. It is assumed that for a given material, the distribution of strength will be approximately normal, so that a frequency distribution curve of a large number of sample results would be of the form shown in the next figure. The characteristic strength is taken as that value below which it is unlikely that more than 5 per cent of the results will fall. This is given by where = characteristic strength, Jesmond Agius: Chapter 2 Page 3

4 = mean strength and S = standard deviation ( ). n = number of tested specimens or samples and x i = individual test results The relationship between characteristic and mean values accounts for variations in results of test specimens and will, therefore, reflect the method and control of manufacture, quality of constituents, and nature of the material Characteristic Actions Figure 1: Normal Frequency Distribution of Strengths In Eurocode terminology the set of applied forces (or loads) for which a structure is to be designed are called actions although the terms actions and loads tend to be used interchangeable in some of the Eurocodes. Actions can also have a wider meaning including the effect of imposed deformations caused by, for example, settlement of foundations. In this text we will standardise on the term actions as much as possible. Ideally it should be possible to assess actions statistically in the same way that material characteristic strengths can be determined statistically, in which case Jesmond Agius: Chapter 2 Page 4

5 Characteristic action = mean action 1.64 standard deviations In most cases it is the maximum value of the actions on a structural member that is critical and the upper, positive value given by this expression is used; but the lower, minimum value may apply when considering stability or the behaviour of continuous members. These characteristic values represent the limits within which at least 90 per cent of values will lie in practice. It is to be expected that not more than 5 per cent of cases will exceed the upper limit and not more than 5 per cent will fall below the lower limit. They are design values that take into account the accuracy with which the structural loading can be predicted. Usually, however, there is insufficient statistical data to allow actions to be treated in this way, and in this case the standard loadings, such as those given in BS EN 1991, Eurocode 1 Actions on Structures, should be used as representing characteristic values. Figure 2: Characteristic Strength f k for two Materials Jesmond Agius: Chapter 2 Page 5

6 Example 1 Calculation of, the characteristic strength of a material. The following table shows the results of 14 compressive strength tests of steel bars made under the same conditions. 1. Determine the characteristic strength of the material tested. 2. Determine the compressive strength above which only 5 percent of the results are likely to fall. 3. Draw the normal distribution curve for the results. Test Values, x i (N/mm 2 ) Answer Sum of Test values = = therefore mean value = x i (N/mm 2 ) ( ) x i (N/mm 2 ) ( ) therefore sum of ( ) S = standard deviation ( ) So the characteristic strength Characteristic Strength above which 5% of results are likely to fall Jesmond Agius: Chapter 2 Page 6

7 Number of Tested Samples Unit 48: Structural Behaviour and Detailing for Construction Continue the following table of the range of compressive strengths and finally draw the normal distribution curve from the results. Range of compressive strength (N/mm 2 ) Number of tested samples Normal Distribution Curve Compressive Strength (N/mm 2 ) Jesmond Agius: Chapter 2 Page 7

8 2.4 Partial Factors of Safety Other possible variations such as constructional tolerances are allowed for by partial factors of safety applied to the strength of the materials and to the actions. It should theoretically be possible to derive values for these from a mathematical assessment of the probability of reaching each limit state. Lack of adequate data, however, makes this unrealistic and, in practise, the values adopted are based on experience and simplified calculations Partial Factors of Safety for Materials ( m ) ( ) ( ) The following factors are considered when selecting a suitable value for m: 1. The strength of the material in an actual member. This strength will differ from that measured in a carefully prepared test specimen and it is particularly true for concrete where placing, compaction and curing area so important to the strength. Steel, on the other hand, is a relatively consistent material requiring a small partial factor of safety. 2. The severity of the limit state being considered. Thus, higher values are taken for the ultimate limit state than for the serviceability limit state. Recommended values for m are given in the next table. The values in the first two columns should be used when the structure is being designed for persistent design situations (anticipated normal usage) or transient design situations (temporary situations such as may occur during construction). The values in the last two columns should be used when the structure is being designed for exceptional accidental design situations such as the effects of fire or explosion. Figure 3: Partial Factors of Safety applied to materials ( m ) Jesmond Agius: Chapter 2 Page 8

9 2.4.2 Partial Factors of Safety for Actions ( f ) Errors and inaccuracies may be due to a number of causes: 1. Design assumptions and inaccuracy of calculation; 2. Possible unusual increases in the magnitude of the actions; 3. Unforeseen stress redistributions; 4. Constructional inaccuracies. These cannot be ignored, and are taken into account by applying a partial factor of safety ( f ) on the characteristic actions, so that Design value of action = characteristics action partial factor of safety ( f ) The value of this factor should also take into account this importance of the limit state under consideration and reflects to some extent the accuracy with which different types of actions can be predicted, and the probability of particular combinations of actions occurring. It should be noted that design errors and constructional inaccuracies have similar effects and are thus sensibly grouped together. These factors will account adequately for normal conditions although gross errors in design or construction obviously cannot be catered for. Recommended values of partial factors of safety are given in tables 2.2 and 2.3 according to the different categorisations of actions shown in the tables. Actions are categorised as either permanent (G k ), such as the self-weight of the structure, or variable (Q k ), such as the temporary imposed loading arising from the traffic of people, wind and snow loading, and the like. Variable actions are also categorised as leading (the predominant variable action on the structure such as an imposed crowd load Q k, 1 ) and accompanying (secondary variable action(s) such as the effect of wind loading, Q k, i, where the subscript i' indicates the i'th action). The terms favourable and unfavourable refer to the effect of the action (s) on the design situation under consideration. For example, if a beam, continuous over several spans, is to be designed for the largest sagging bending moment it will have to sustain any action that has the effect of increasing the bending moment will be considered unfavourable whilst any action that reduces the bending moment will be considered to be favourable. Jesmond Agius: Chapter 2 Page 9

10 Figure 4: Partial Safety Factors at the Ultimate Limit State Figure 5: Partial Safety Factors at the Serviceability Limit State The next example shows how the partial safety factors at the ultimate limit state from tables 2.1 and 2.2 are used to design the cross-sectional area of a steel cable supporting permanent and variable actions. Example 2 SIMPLE DESIGN OF A CABLE AT THE ULTIMATE LIMIT STATE Determine the cross-sectional area of steel required for a cable which supports a total characteristic permanent action of 3.0 kn and a characteristic variable action of 2.0 kn as shown in this figure. Jesmond Agius: Chapter 2 Page 10

11 The characteristic yield stress of the steel is 500 N/mm 2. Carry out the calculation using limit state design with the following factors of safety : Answer G =1.35 for the permanent action, Q =1.5 for the variable action, and m =1.15 for the steel strength. Design value = G permanent action Q variable action = = 7.05 kn = 4.34 N/mm 2 Required cross-sectional area = = 16.2 mm 2 For convenience, the partial factors of safety in the example are the same as those recommended in EC2. Probably, in a practical design, higher factors of safety would be preferred for a single supporting cable, in view of the consequences of a failure. Example 3 shows the design of a foundation to resist uplift at the ultimate limit state using the partial factors of safety from figure 4. It demonstrates the benefits of using the limit state approach instead of the potentially unsafe overall factor of safety design used in part (b). Jesmond Agius: Chapter 2 Page 11

12 Example 3 The next figure shows a beam supported on foundations at A and B. The loads supported by the beam are its own uniformly distributed permanent weight of 20kN/m and a 170kN variable load concentrated at end C. Determine the weight of the foundation required at A in order to resist uplift: (a) By applying a factor of safety of 2.0 to the reaction calculated for the working loads. (b) By using an ultimate limit state approach with partial factors of safety of G = 1.10 or 0.9 for the permanent action and G = 1.5 for the variable action. (c) Investigate the effect on these designs of a 7 per cent increase in the variable action. Figure 6: Uplift Calculation Example Jesmond Agius: Chapter 2 Page 12

13 Answer (a) Factor of safety on uplift = 2.0 Taking moments about B Uplift ( ) Weight of foundation required = R A x safety factor = 3.33 x 2.0 = 6.7 kn With a 7 per cent increase in the variable action Uplift ( ) Thus with a slight increase in the variable action there is a significant increase in the uplift and the structure becomes unsafe. (b) Limit state method ultimate load pattern As the example includes a cantilever and also involves the requirement for static equilibrium at A, partial factors of safety of 1.10 and 0.9 were chosen for the permanent actions as given in the first row of values in figure 4. The arrangement of the loads for the maximum uplift at A is shown in figure 6b. Design permanent action over BC = G 20 2 = = 44 kn Design permanent action over AB = G 20 6 = = 108 kn Design variable action = Q 170 = = 255 kn Taking moments about B for the ultimate actions Uplift ( ) Therefore weight of foundation required = 38 kn. A 7 per cent increase in the variable action will not endanger the structure, since the actual uplift will only be 7.3 kn as calculated previously. In fact in this case it would require an increase of 61 per cent in the variable load before the uplift would exceed the weight of a 38 kn foundation. Jesmond Agius: Chapter 2 Page 13

14 Exercise Explain what is meant by the following: a. Permissible stress, load factor and limit state design methods. b. Characteristic strength c. Characteristic loads 2. Explain why the partial factors of safety for materials are incorporated when the limit state design method is used. 3. Determine the cross-sectional area and the diameter of a circular mild steel bar that is required to safely support a dead load of 10kN and an imposed load of 6kN (see next figure). The yield stress (ultimate strength) of mild steel is 250N/mm 2. Use a factor of safety of 1.8 for the material strength. Use all the three design methods and critically comment on your results. a a Section a-a 10kN 6kN 4. The results shown in the following table are the compressive strengths of 20 steel bars made under the same conditions. a. Determine the characteristic strength of the materials. b. Determine the compressive strength above which only 5 percent of the tested results are likely to fall. c. Draw the normal frequency distribution curve of compressive strength, mark f k and f m on the distribution curve. d. Write a computer program to solve this question. Compressive Strength of Steel Samples (N/mm 2 ) Jesmond Agius: Chapter 2 Page 14

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