Symbols in table:
e02 the greater of the two first order eccentricities e01 the lesser eccentricity fo/h slenderness corresponding to the limit for 10 % moment increase at y = 0
n relative normal force N/Actcd nu0 load capacity for the current slenderness and y = 0
nL long-term load yet effective creep ratio = y- nL / a; here y = 3 has been assumed nu1 load capacity including the effect of creep according to 1- step method nu2 load capacity including the effect of creep according to 2- step method
The agreement between the 1-step and 2-steps methods is in most cases good. For e01/e02 = -0,9 (double curvature bending) the 1-step method is generally slightly conservative compared to the 2-steps method. There are also a few cases where the opposite is true, but in these cases the long-term load is close to the limit where instability would occur with 9ef = 9, and then the 2-steps method becomes uncertain. In these cases, the result would have been more representative with a somewhat lower long-term load.
The use of an extended stress-strain diagram in the 2-steps method can be discussed. In principle it means that creep deformations will correspond to the stresses in the final stage. In a more accurate calculation they should be integrated from 0 to 9, with increasing second order moment. However, the error will be small, since the stresses are normally not very high under long-term load, and since second order moments are small at these low slenderness ratios.
In most cases a first order eccentricity e02/h = 0,32 has been used, with the aim of having a moderate normal force. For the sake of completeness, one case with a high normal force is also included (nu0 = 1,022, e02/h = 0,08) and one with a low normal force (nu0 = 0,122, e02/h = 1,28). Even with the low normal force, there is a significant effect of creep. (The 2-steps method in this case seems to give more effect of creep for low than for high normal force. This is misleading, however; it is a consequence of the long-term load being close to the instability load for 9ef = 9, see discussion in the first paragraph above.)
Conclusion concerning the effect of creep on the slenderness limit
In these examples, creep reduces the load capacity by 5 to 30% (average 15%). If second order moments are neglected, which is allowable at these slenderness ratios, the result is in principle already 10% on the unsafe side. If creep would also be neglected, the results would be another 5 to 30% on the unsafe side.
The conclusion is that creep can not be neglected in the slenderness limit. 5.8.4.5 Safety under long-term load only
The effective creep ratio is based on moments under quasi-permanent load which, according to its definition in EN 1990, is an SLS load with no load factors (except ^2 < 1 for variable loads). Thus, in the extreme case of permanent load only, assuming first order moments proportional to the load as in the above examples, the highest possible effective creep ratio is
9ef = 9- Mol/Mod = 9- 1,0/1,35 = 0,749 The following question now arises:
Can load Nl together with 9ef = 9 be more severe than Nd = 1,35- Nl with 9^ = 0,749 ?
The example in 4.3.2 is used again. The load capacity for ^ef = 0,749 = 2,22 is found to be nRd = 0,159. The corresponding long-term load is n. = 0,159 / 1,35 = 0,118. The load capacity for full creep, 9ef = 9 = 3, is found to be nRdL = 0,141. This is higher than the current long term load, and the "safety factor" is
This safety factor may be considered somewhat low, although it should be observed that it is not the whole safety factor as the normal material safety factors are already included in the calculated capacities. A reasonable lower limit for the load safety factor could be 1,35.
As shown above, it is conservative to use an effective creep ratio based on first order moments.
The "extra" safety can be estimated by comparison with more accurate calculations, e.g. a two-step calculation or a one-step calculation with 9ef based on total moments.
A two-step calculation according to the above scheme is done with different values of a, until a value is found for which nRd = 0,159. This happens for n. = 0,134. Thus, one could say that the additional "built-in" safety is 0,134/0,118 = 1,14, and the total safety against creep failure would be
This is considered to be sufficient, and it can be shown that this factor will be higher for lower values of slenderness, higher first order moments and higher amounts of reinforcement. In this respect, the current example is rather extreme, in the unfavourable direction.
Furthermore, long-term load = 74 % of design load is the worst possible case for consideration of the effect of creep. In normal cases there is always some variable load. The percentage of longterm load then decreases, since variable loads are included in Ql with ^2- Qk, where ^2 < 18, whereas in Qd they are included with yq- Qk, where yo > 1. Therefore, the more variable the load, the higher the safety against "creep failure" will be.
The conclusion is that a one-step calculation, using an effective creep ratio based on first order moments, will give sufficient safety against failure under quasi-permanent load with full creep. Therefore, this case need not be checked separately, and it is not necessary to include any safety factor on M0L in the definition of 9ef.
In 5.8.4 (3) the alternative of using total moments in the definition of effective creep ratio is given. This is less conservative, however, and most of the extra safety against "creep failure" is then lost. Therefore, 5.8.4 (3) states that a separate check should then be made for 1,35 Ql and with yef = 9. This may become the governing factor in cases where the percentage of long-term moment is moderate or high, more precisely when first order moment ratio Mol / Mod > 0,5.
5.8.5 Methods of analysis C5.8.5. Methods of analysis
Three basic methods are described in 5.8.5. Of the simplified methods, (b) is basically the "model column method" in the ENV, with some modifications. The old name is not used here, since it tells nothing about the method (all methods are based on models). A more suitable name is "curvature method", since the method is based on the estimation of a curvature.
This name will be used here, together with "stiffness method" for method (a), which is based on the estimation of stiffness.
There are simplified methods other than those mentioned in EC2. One such method, combining analysis and cross section design in one step, will be shortly described here as an example (it is currently used in the Swedish code).
It can be used for isolated columns with formally centric load, i.e. no other first order effect than the prescribed imperfection. The load bearing capacity is given as
where kc and ks are coefficients depending on slenderness ratio, imperfection, concrete grade, effective creep ratio etc, calibrated against calculations with the general method.
A method of this type works ideally if the imperfection, an eccentricity or an initial deflection, is proportional to the buckling length of the column. This is the case in some codes, but not in the Eurocodes. If the imperfection is proportional to the effective length, the coefficients can be given in one simple table or diagram with slenderness as the basic parameter.
If the imperfection is not proportional to the effective length, then the absolute value of this length must be added as a separate parameter, which complicates the presentation (for example, one diagram or table would only be valid for one length). However, with some simplifications this type of method could be useful also under EN 1992, particularly for storey high pin-ended columns, which are common as interior columns in buildings.
If there are first order moments other than that due to the imperfection, a separate design for normal force and (magnified) moment must be made. A special moment magnification factor is included in the method for such cases, but the simplicity is lost and the method no longer has any particular advantages over the "stiffness" or "curvature" methods in EC2.
In the following chapters, the general method and the simplified methods (a) and (b) are described.
5.8.6 General method C58 6. General method
The most accurate of the methods described in 5.8.5 is the "general method". It is based on nonlinear analysis, including both material and geometric non-linearity (second order effects).
"General" here refers to the fact that the method can be used for any type of cross section, any variation of cross section, axial load and first order moment, any boundary conditions, any stressstrain relations, uniaxial or biaxial bending etc. The limiting factor is the capability of the available computer program. The method rests on a few simple assumptions:
• linear strain distribution
• equal strains in reinforcement and concrete at the same level
• stress-strain relationships for concrete and steel
8 EN 1990 gives values for ^2. For some loads, e.g. wind, = 0. A common value is 0,3 (office and residential areas). The highest value given is 0,8.can be based on axial loads as well as moments. In the general case only moments should be used.
Conditions of equilibrium and deformation compatibility are satisfied in a number of cross sections, and the deflection is calculated by double integration of the curvature, having an assumed variation between the selected sections. This may be self-evident, but it is mentioned in 5.8.6 (6) as a reference for a simplified version, in which only one cross section (or certain critical sections) is studied, and the curvature is pre-assumed to have a certain variation in other parts of the member. This gives simpler computer programs and faster calculation, but less accuracy. See figure 5.24.
Any stress-strain relations can be used. A continuous curve with a descending branch is considered to be the most realistic alternative for the concrete; it is also convenient for computational reasons. Creep can be considered in different ways; the simplest way is to multiply all concrete strains by (1+9ef), see clause 6.4.
Tension stiffening (i.e. the contribution from concrete in tension between cracks) can easily be taken into account in the general method, e.g. by using a descending branch of the concrete stressstrain curve in tension, by modifying the stress-strain curve of the reinforcement or by any other suitable model. In the calculations presented in this report, however, all contributions from concrete in tension have been ignored; this is always more or less conservative.
The safety format in non-linear analysis has been much debated, and different models have been proposed. The safety format is particularly important in second order analysis, where the absolute magnitude of deformations has a direct influence on the ultimate load.9
The safety format should satisfy two basic criteria.
1. It should be possible to use the same set of material parameters in all parts of the member, in order to avoid discontinuities and computational problems.
The model in ENV 1992-1-1 (Appendix 2) does not comply with this, since it assumes mean values of material parameters for the calculation of deformations and design values for the check of resistance in critical sections. This also means that there will be no "material safety" at all in the calculated resistance, in cases where failure occurs before reaching the design cross section resistance (stability failure) - unless "critical section" is substituted by some "critical length" (which then remains to be defined, however).
2. The safety format should be compatible with the general design format based on partial safety factors.
The model in ENV 1992-2 (Appendix B) does not comply with this, since it uses mean values for the analysis and a global safety factor yr = 1,3 to reduce the ultimate load resulting from the analysis. This gives the same results as using design values fcm/1,3, fyk/1,3, Ecm/1,3 and Esm/1,3. Thus, it makes no difference whether the ultimate load is governed by concrete or steel, resistance or stiffness. The reduction of the reinforcement strength is too severe, as is also the reduction of the material stiffness parameters, particularly for reinforcement (Esm/1,3). A non-linear analysis using this safety format will be conservative, and the potential benefits of using a refined method are lost.
The safety format defined in 5.8.6, based on using design values in the analysis, satisfies both criteria. A design value of the ultimate load will be obtained as a direct result of the analysis, and the problems associated with the above-mentioned safety formats are avoided. Since the E-moduli vary less than the corresponding strengths, the partial safety factors given for E should be lower than for f:
9 The absolute magnitude can be of importance also in e.g. continuous beams, but only in the check of rotation capacity, and it would normally not have the same direct influence on the ultimate load as in 2nd order analysis.
10 This diagram is taken from [1], which primarily deals with high strength concrete according to Swedish rules, but this makes no difference for what the diagram is intended to show.
For concrete, yc = 1,5 for strength takes into account not only strength variation, but also geometrical deviations in the cross section. Assuming a factor 1,1 for these deviations, and considering the relationship between strength and E-modulus, a reasonable value of the factor for Ec is ycE = 1,1- (1,5/1,1)1/3 « 1,2.
For steel, ys = 1,15 includes a factor of about 1,05 for geometrical deviations. Thus, a design value Esd=Esm/1,05 would be logical, considering the fact that variations in the E-modulus are negligible. However, a factor 1,0 has been chosen as a simplification, and in order not to deviate from 3.2.3; differences in terms of calculated result are negligible.
The resistance of slender columns resulting from a general analysis can be shown in a practical form with interaction curves, figure 5.25. One such curve shows the maximum first order moment M0 (or eccentricity eo = Mo/N) for a certain axial load N.
The thin curves in figure 5.25 show the total moment M as a function of N for a given e0. The higher the slenderness, the more the total moment M increases over the first order moment M0. (Note that the diagram gives axial load and moment in relative terms n and m.) One point on the interaction curve for a given slenderness is obtained by plotting the maximum value of n on the line representing m0 or e0. This is demonstrated in figure 5.25 for one relative eccentricity e0/h = 0,1 and different slenderness values X = 35, 70, 105 and 140.
The difference Mu - M0 between the cross section resistance (curve X = 0) and the first order moment at maximum load represents the second order moment. However, in some cases there is a stability failure before any cross section reaches its ultimate moment, and then the "true" second order moment is less than Mu - M0. This occurs for X = 105 and 140 in figure 5.25.
This nominal second order moment Mu - M0 is useful as a basis for simplified methods; see clause
This nominal second order moment Mu - M0 is useful as a basis for simplified methods; see clause
Figure 5.25. Interaction curves for columns of different slenderness, calculated with the general method. Rectangular cross section. n and m0 are relative axial force and first order moment respectively, i.e. n = N/bhfcd, m0 = M0/bh2fcd. All curves are based on ro = 0,2 and yef = 0. Concrete grade is C80.10 First order moment is constant, e.g. caused by equal end eccentricities
Figure 5.25. Interaction curves for columns of different slenderness, calculated with the general method. Rectangular cross section. n and m0 are relative axial force and first order moment respectively, i.e. n = N/bhfcd, m0 = M0/bh2fcd. All curves are based on ro = 0,2 and yef = 0. Concrete grade is C80.10 First order moment is constant, e.g. caused by equal end eccentricities
Creep can be taken into account in different ways. The most accurate model would be to increase load and time in steps, for each step taking the stresses, strains (and corresponding deflections) from the previous step as starting values for the next increment. For each step, strains would be calculated taking into account their time-dependence.
A simplified model is to multiply all strain values in the concrete stress-strain function with the factor (1+9ef), see figure 5.26, where yef is an effective creep ratio relevant for the load considered. With this model, the analysis can be made either in steps for loads of different duration, or directly for the design load combination in one step, see chapter 4. For creep in slender members in particular, see clause 4.3.
Figure 5.26. Simple way of taking into account creep in general method
Figure 5.26. Simple way of taking into account creep in general method
Figure 5.27 is calculated in this way, using qef = 2,0 and other parameters the same as in figure 5.25. Curves according to figure 5.25 are also included (dashed), showing the reduction of the load capacity resulting from creep. The relative reduction increases with slenderness.
Figure 5.27 is calculated in this way, using qef = 2,0 and other parameters the same as in figure 5.25. Curves according to figure 5.25 are also included (dashed), showing the reduction of the load capacity resulting from creep. The relative reduction increases with slenderness.
Figure 5.27. Interaction curves for qef = 2. Other parameters are the same as in figure 5.25 Dashed curves are the corresponding curves from figure 6-2, i.e. for q>ef = 0. The difference represents the effect of creep
Another question is whether one and the same effective creep ratio should be used along a compression member (or in different parts of a structure), or if it should vary as the ratio Meqp/MEd may vary. The latter would be the most correct alternative, but normally it is reasonable to use one representative value of yef for a member or even a whole structure.
Figure 5.27. Interaction curves for qef = 2. Other parameters are the same as in figure 5.25 Dashed curves are the corresponding curves from figure 6-2, i.e. for q>ef = 0. The difference represents the effect of creep
Another question is whether one and the same effective creep ratio should be used along a compression member (or in different parts of a structure), or if it should vary as the ratio Meqp/MEd may vary. The latter would be the most correct alternative, but normally it is reasonable to use one representative value of yef for a member or even a whole structure.
In a simplified calculation method one can use the difference between cross section resistance and first order moment, Mu - Mo in figure 5.25, as a nominal second order moment. When this moment is added to the first order moment, a design moment is obtained for which the cross section can be designed with regard to its ultimate resistance. As pointed out above, this nominal second order moment is sometimes greater than the "true" second order moment.
However, it can give correct end results, even in cases where the load capacity is governed by a stability failure before reaching the cross section resistance, if given appropriate values.
For practical design, there are two principal methods to calculate this nominal second order moment:
1. estimation of the flexural stiffness EI to be used in a linear second order analysis (i.e. considering geometrical non-linearity but assuming linear material behaviour); this method is here called stiffness method, see chapter 5.8.7
2. estimation of the curvature 1/r corresponding to a second order deflection for which the second order moment is calculated; this method is here called curvature method, see chapter 5.8.8.
Before entering into details of the two methods in chapters 5.8.7 and 5.8.8, their common basis will be shortly described.
The total moment including second order moment for a simple isolated member is:
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