effect of random material variability on seismic design parameters of steel frames

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 20, 101-1 14 (1991)

EFFECT OF RANDOM MATERIAL VARIABILITY ON SEISMIC DESIGN PARAMETERS OF STEEL FRAMES

A. S. ELNASHAI* AND M. CHRYSSANTHOPOULOS' Department of Civil Engineering, Imperial College, Imperial College Road, London, SW7 2BU. U.K.

SUMMARY Whereas an increase in material yield stress beyond the code specified characteristic value enhances plastic capacity, it may cause a reduction in overall ductility and energy absorption capability of steel frames. Since quality control of various shapes of sections used on site is difficult to impose, the effect of this random variability on design response parameters should be accounted for in earthquake-resistant regulations. Moreover, the required weak-beam/strong- column design principle in particular, and failure mode control in general, could be undermined if the yield stresses in beam and column assume two opposite extremes in a random sample. This paper addresses the problem of defining the expected range of response parameters in a steel frame with randomly varying yield stress. A simple portal frame is designed using code specified characteristic values and verified by non-linear transient dynamic analysis. The influence of yield stress variability, including the degree of correlation between beam and column material properties, on several response parameters is assessed through a Monte Carlo simulation study. Results are presented from both univariate and bivariate statistical analyses that quantify the relationship between input (material) and output (response) parameters. Assessment of the interdependence of output parameters given a particular model for yield stress Variability is also undertaken. It is shown that certain response parameters exhibit more favourable statistical properties than others. Thus, the implications for seismic code design are discussed in the light of these results.

INTRODUCTlON

Seismic design of structures makes use of the reserve strength in the inelastic range to ensure that the earthquake energy is absorbed and dissipated with a controllable degree of damage. Therefore, the evaluation of ductility and energy dissipation capacity of structural members and systems is central to the success of the design process. Consequently, the mode of failure of a structure and the sequence in which this mode is realized are of.significance, since this affects the shape of the load-displacement curves and, hence, the two quantities mentioned above. This is manifested in the increasing use of the 'capacity design' concept to define a favourable mode of failure that exhibits a ductile behavi0ur.l It follows from the above that the definition of suitable yield and post-ultimate points in the structural response of a component and/or of a system, and their sensitivity to material yield characteristics, is of importance in earthquake-resistant design.

The use of probabilistic concepts in seismic design and assessment of structures has increased in recent years, following the development of reliability and risk analysis methods in the late 1970's. Both conceptual studies that demonstrate the influence of various uncertainty sources in design asses~ment~.~ and compara- tive studies, in which lifetime reliability levels are computed for similar structures under static and dynamic loading: have been undertaken. It has been suggested that the magnitude in the variability of ground motion parameters determines to a large extent the overall performance variability and hence structural reliability. As a result, a number of investigators have focussed on a statistical description of seismic parameters5 and their use in analytical techniques.6 This conclusion should not, however, hamper the need to improve statistical modelling of other uncertainty sources and, by examining their influence on structural per-

* Lecturer in Earthquake Engineering. f Lecturer in Structural Engineering.

OO98-8847/9 1/020101-14%07.OO 0 1991 by John Wiley & Sons, Ltd.

Received 30 January 1990 Revked 8 August 1990

102 A. S. ELNASHAI AND M. CHRYSSANTHOPOULOS

formance, to assess the adequacy of currently used specifications and quality control procedures. This is emphasized by the observations from a recent study,’ where it was concluded that a simple cumulative damage parameter showed little sensitivity to the characteristics of ground motion used in the analysis. Furthermore, if probability-based criteria are to be adopted in seismic regulations, appropriate statistical models that deal with various uncertainty components must be developed for a range of random variables.

In the treatment of response variability for earthquake-resistant structures designed for a limited amount of plastic deformation, it is essential to consider a number of characteristics, e.g. ductility ratios, maximum and permanent displacements, energy absorption and stiffness measures, that are all-important in assessing structural performance.* A key parameter that affects all of the above is the yield stress of the material. It is widely accepted that nominally identical structural steel products exhibit variations in yield stress not only from one production batch to another but also within the safne batch as well as within the cross-section and along the length of a single member. To account for this inherent variability, structural codes contain material specifications that give ‘lower-bound’ yield stress values for different steel grades usually as a function of the maximum thickness of the section, e.g. Reference 9. These are based on suitably selected characteristic values of probability distributions for yield stress that have been obtained from statistical samples.

This approach is normally adequate for serviceability and ultimate limit states for static design but may adversely affect structural performance under dynamic loading. Moreover, the concept of ‘capacity design’, as mentioned above, requires a well-controlled failure mode in pre-selected members. The implications of this design approach should be investigated, given the degree of variability in yield stress in steel members. In a recent study” it was pointed out that, in certain cases, a high degree of material variability may cause local failure modes which impose severe limitations on the assumed overall ductility of a structure. Results presented in that study are, however, restricted to a single limit state, which may not in some cases be appropriate to seismic design.

The objective of this paper is to examine the effect of random material variability on a wide range of response parameters used in the evaluation of structural performance under earthquake loading. Com- parison of the statistics of these parameters to the underlying material uncertainty enables guidelines to be formulated on their suitability for design assessment for cases with varying degree of member yield stress correlation. In addition, values obtained from deterministic analyses are compared with those obtained from simulation in order to estimate exceedance probabilities, which are of interest in code calibration.

MODEL FRAME DESIGN

Whereas analysis of a realistic multi-storey structure is of interest, the cause of isolating the effect of specific parameters is better served by studying as simple a system as possible, provided it exhibits acceptable sensitivity to the parameters under consideration. It was therefore decided to analyse a single-storey single- bay steel portal frame designed in accordance with seismic code regulations. Preliminary studies confirmed that the chosen system is acceptably sensitive to variations in steel yield strength. In accordance with established practice, the frame was designed for static loading (to BS 5950’) and checked for earthquake ground motion (to draft ECSl). The frame has a span of 8.0 m and a height of 4.0 m, carrying a roof slab of 5-0 m on either side. The dead and live loads used are 45 and 15 kN, respectively. The resulting design comprised a girder with an I Section UB 457 x 152 x 52 and a column I section UC 305 x 305 x 118. Using the characteristic beam and column yield strengths of 275 and 265 N/mm2, consistent with section thcknesses, the column over-design factor is 1.7. The earthquake load is calculated from EC8 for a peak ground acceleration of 0.5g at a site with a deep layer of stiff clay. The design spectrum is hence defined, in terms of corner periods (TI and T2), spectral amplification (Po), site response parameter (Sj and descending branch long period exponent (k). The structural period, calculated by hand and checked by computer analysis, is used to quantify the base shear for damping of 2 per cent critical, as presented in Reference 11.

The behaviour factor ‘q’ defined in EC8 is five times the ratio between the load corresponding to first yield and that corresponding to the development of a mechanism. In the absence of testing or analysis results, the default value is taken as 1.1 and 1.2 for single- and multi-storey buildings, respectively, provided the axial

RANDOM MATERIAL VARIABILITY & DESIGN PARAMETERS 103

load is in the range of 10 per cent of the axial capacity of the column section. For smaller axial loads, the default value is increased to 1-6. This leads to a 'q' factor of 8-0 for the frame under consideration. In Reference 11 details of the local and overall buckling checks for the frame are given alongside member properties and forces.

To verify the design, a series of linear and non-linear analyses were performed using the powerful adaptive dynamic analysis program ADAPTIC'2. The elements used in the latter analysis are quartic beam-columns accommodating very large displacements, and capable of automatic subdivision upon yield detection. Thereafter, plastic hinges are inserted and the analysis proceeds with the modified finite element mesh. To focus attention on the effect of yield strength variability, use is made of a non-hardening plasticity formulation. This is further justified by the dearth of information regarding strain-hardening characteristics, and by the fact that the onset of hardening starts at 10-12 times the yield strain; this is unlikely to affect the failure mode of the analysed frame.

The record chosen for design verification is the Montenegro, Jugoslavia, earthquake of 15 Api-il 1979, selected from the extensive Imperial College strong-motion databank.I3 The selection is based on two considerations; peak ground acceleration and frequency content. The former is 0.4Sg, comparable to the 0-5g used in design. With a fundamental period of the structure of 0.39 sec, the Montenegro earthquake is suitable for the analysis, since it exhibits high energy content in the same period range, as demonstrated by the elastic response spectrum shown in Figure 1. The displacement-time and load-displacement response of the frame are given in Figure 2(a) and Figure 2(b), respectively. It indicates that the design is satisfactory, with an apparent overall ductility ratio (demand) of approximately 2-0, defined as maximum observed to yield

MONTENEGRO EARTHQUAKE 1 5 APRIL 1 9 7 9 PETROVAC NS

RESPONSE SPECTRUM.

ACCELERATION = -0.453 C

PERIOD (SECONDS)

Figure 1. Elastic response spectrum for the 1979 Montenegro earthquake


-4a.

(b)

Figure 2. Frame response from dynamic non-linear analysis: (a) Displacement time-history of node 5; (b) Load-displacement response of node 5


L a t e r a l Displacement ( m )

Figure 3. Load-displacement response of node 5 from static non-linear analysis; insert shows the sequence of plastic hinge formation

displacement. The response displacement time-history, given in Figure 2(a), shows the point at which permanent plastic deformations occurred, and the maximum attained displacement of 0.076 m. It is concluded that the design is realistic and representative of the state-of-practice in steel seismic design.

A pilot static analysis of the frame was performed, using displacement-controlled transverse loading at the top left corner of the structure. The first plastic hinge formed was in the beam, as marked on Figure 3, consistent with the weak-beam/strong-column criterion used in the design. This corresponded to a transverse displacement of 0-0125 m and an applied load of 124 kN. This was followed shortly by yielding at the base of the column, as indicated in the figure. The load-displacement curve shown in Figure 3 implies that softening of the response follows the formation of the fourth plastic hinge in the middle part of the beam, corresponding to a lateral displacement of 0.05 m and a load of 326 kN. If the ratio of loads corresponding to first yield and the development of a mechanism, as defined in EC8, is used to calculate the behaviour factor ‘q’, a value of 13-0 is obtained. This highlights the difficulties in quantifying behaviour factors from simple expressions that are necessary for code provisions.

SELECTION OF SEISMIC DESIGN PARAMETERS

The term ‘ductility’ used in seismic design refers to the ability of a member or a system to sustain deformations beyond its yield point without significant loss of strength. Moreover, the ‘ductility ratio’ is one way of quantifying the ‘deformability’ of a member or a system. This is defined as the ratio between a deformation quantity at ultimate and yield states, and may be used in one of two ways. The elastic forces are scaled by the ductility ratio, or some proportion of it, to arrive at a so-called elasto-plastic force for design. Alternatively, design forces are derived from statistical analysis of the response of inelastic systems with varying characteristics to seismic inputs. Whereas the former is used extensively in existing codes, the latter is under consideration for use in modern design codes.

To study the effect of material variability on the structural response of the steel frame under consideration, twelve parameters were chosen. These represent quantities used extensively in current practice, as well as ‘forward looking’ parameters. Figure 4 illustrates the definition of the selected parameters on a typical load-displacement curve for a single-storey frame. In the following, the parameters are explained and justification of the choice made is given.

There is no universally acceptable definition of ductility ratio;” hence two options are retained herein. For the first definition, the yield displacement is calculated from the intersection of the initial tangent and the horizontal line passing through the peak of the load4isplacement curve. For the second ductility ratio definition, the yield displacement is evaluated from the intersection of the horizontal line as above and the line passing through 75 per cent of the ultimate capacity. In both cases, the ultimate displacement is that

A. S. ELNASHAI A N D M. CHRYSSANTHOPOULOS

a 1 = 6 u / b y 1

P2= 6” / 4 2

K,,==

Figure 4. Definition of response parameters on load-displacement curve

corresponding to a 10 per cent reduction in load-carrying capacity. Clearly, other definitions exist, but the above is adequate for the purposes of the present study.

The ultimate load and the corresponding displacement are chosen as response parameters, for ultimate limit state considerations. Additionally, two resistance values are investigated, corrresponding to a fixed drift of 5 per cent of the height, and to the maximum displacement of 0-5 m used in the analysis.

Two energy parameters are considered, the total energy absorbed by the system up to ultimate displacement and that corresponding to the aforementioned fixed drift value. Finally, the secant stiffness at ultimate displacement is recorded, as a measure of the non-linear vibration period of the structure, and the degree of ‘internal damage’ inflicted by the earthquake. In summary, the following parameters are considered:

d,, (1): Yield displacement corresponding to initial tangent (cm) d,, (2): Yield displacement corresponding to 75 per cent of ultimate load (cm) 6, (3): Ultimate displacement at 90 per cent of maximum capacity (cm) Pl (4): Ductility given by (3)/(1) P2 (5): Ductility given by (3)/(2) d,,, (6): Displacement corresponding to maximum capacity (cm) PmaX (7): Maximum capacity (kN)

(8): Load corresponding to a drift of 5 per cent of frame height (kN) Pa= 5 o (9): Load corresponding to maximum applied displacement (kN) U Ud = 20 (1 1): Strain energy corresponding to a drift of 5 per cent of frame height (kN m) K,,

(10): Total strain energy corresponding to (3) (kNm)

(12): Secant stiffness corresponding to (3) (kN/cm)

MODELLING OF MATERIAL UNCERTAINTY

Yield stress variability has been studied extensively in the U.K. for the purpose of determining the various sources of uncertainty and their impact on structural design.” The data for that study have been obtained from flexural and tensile tests on universal beam sections, as well as on a large number of mill test certificates for various rolled products. By analysing the data in rational subgroups to remove systematic components of


variability, it was concluded that a log-normal distribution can be used for describing random yield stress variability of nominally identical hot-rolled plates obtained from a single manufacturer. Standard deviations of 20 N/mm2 were found representative of Grade 43 steel, for which the nominal values used in codes are a function of the maximum section thickness, as discussed above. The mean was found to be typically two standard deviations above the nominal value, although a somewhat lower value should be used in strength analysis to account for high strain rates used in commercial testing. In a study dealing with the derivation of partial factors used in a recently drafted U.K. bridge code,16 sample mean yield stress values were reduced by 15 N/mm2 to accommodate this effect.

In the current study, a log-normal distribution with a mean of 290 N/mm2 and a standard deviation of 20 N/mm2 is used for describing yield stress in the beam, uyb. The yield stress in the column, u,,,,, is assumed to follow a log-normal distribution with the same coefficient of variation but with a mean value reduced by 10 N/mm2 to reflect the statistically significant thickness trend in steel sections.” In order to assess the influence of interdependence in member material properities, it was decided to examine different degrees of correlation between beam and column yield stress values, namely uncorrelated (p(uy,, uyc) = 00), fully- correlated (p(ayb, uyc) = 1-0) and partially-correlated (p(uyb, uy,) = 0.7).

In terms of the variability in other material properties, it has been established’s~’7 that Young’s modulus has a coefficient of variation less than 0.03 and is not, therefore, expected to have any significant influence on response characteristics. The only other parameter in material modelling that may have an effect is the strain- hardening rate, for which a probabilistic description is needed if strains in excess of about 10-12 times the yield strain are attained. As discussed previously, assuming nominal yield stress values, strains of that magnitude occur when the maximum nodal displacement is equal to about 26,,, and, thus, omission of this parameter will result in the under-estimation of response characteristics that are wholly or partially based on post-ultimate behaviour. In this respect, collection of data on the random variability of strain-hardening parameter@) can enhance the scope of probabilistic modelling in future studies.

GENERATION AND VERIFICATION OF RANDOM SAMPLES

Random samples from the specified log-normal distributions have been created using suitable routines from the NAG library.I8 Following the FE analysis using ADAPTIC, statistical analysis of the results is undertaken by the BMDP series of programs.” Because of the large amount of computed data, a limited selection is presented herein, mainly in tabular form. Histograms and selected bivariate plots are given in Reference 1 1.

In a simulation study, the accuracy of the estimated parameters, in this case the statistics of various response characteristics of the steel frame, depends on the number of trials in the sample. A preliminary decision to use a sample with 200 trials was made on the basis of the computed standard error for the first two moments of the input (yield stress) distributions, i.e. before undertaking any FE analyses.

However, in order to ensure that sample size effects do not have an appreciable influence on the results, the standard error on univariate and correlation statistics of the various output (response) parameters was also examined. Table I shows the estimates of the standard error based on four independently generated sets (200 trials in each set). For the purposes of this sample size study, it has been assumed that beam and column yield stresses are uncorrelated. As can be seen in Table I, the selected sample size gives reliable estimates for all response statistics considered in this study, although it is worth mentioning that higher distribution moments will exhibit larger variabilities. This should be taken into account if fitting of distributions were to be undertaken using the results of the simulation study. Confidence intervals for the estimates at a 95 per cent level can be determined (assuming that sampling distributions are normal) from:

E(E) - 0*98~(E) < 9 < E(E) + 0.98~(=)

where 9 is any of the calculated statistics, E ( 9 ) is the mean value based on four independent sets and s(E) is the standard error. It should be noted that the assumption of normality is not strictly valid for the sampling distributions of the standard deviation and the correlation coefficient but the magnitude of the calculated standard error renders a more detailed analysis unnecessary.


Table I. Sample size study

Response Mean value Std. deviation P (. ,Qyb 1 P ( . P Y A parameter Mean Std. error Mean Std. error Mean Std. error Mean Std. error

6Yl 3.64 002 025 002 065 004 075 0.01 6Y2 4.22 002 0-27 0.01 006 0.05 0.99 001 6, 30.73 0.12 2.27 0,24 0.9 1 0.0 1 0.35 0.02 Pl 8.45 0.03 0.3 1 0.02 0.56 0.03 - 0.71 0-08 P2 7-30 0.04 0.55 0.03 0.84 0.01 - 0.50 0-07 ~m,, 614 0.06 1.05 005 0.9 1 0.01 - 0.10 006 p m a x 352.01 1.34 24.43 1.45 0.65 0.04 0.75 0.01 U 96.88 0.7 1 13.70 1.27 0.83 0.02 0 5 3 0.01

Table 11. Univariate statistical analysis-p(u,,, uycJ = 0.0

Response Code Corresponding parameter Mean Std. deviation 5% fractile value* fractile (%)

S Y l 3.64 672 4-2 1

PI 8.45 PZ 7.30 L a , 6.14 p m a x 352.0 pa=20 332.0 Pa=m 292.0 U 96.9 Ua=m 61.6 K,, 10.3

8, (cm) 307

0.25 0.27 2.27 03 1 0.55 1.05

24.4 24.9 24.2 13.7 4.10 0.38

3.20 3.37 3.80 4-01

8.00 8-33 6.60 7.0 1 4.75 5-00

27.0 28- 1

3 11.0 326-0 29 1 .O 305.0 250.0 265.0 74.5 81.5 54.8 57.2 9.6 10.4

17 26 12 42 40 15 16 16 I5 16 15 57

* Using nominal yield stress values (uyb = 275 N/mmz, uyc = 265 N/mmz).

ANALYSIS O F RESPONSE UNCERTAINTY

The results from all simulation studies are summarized in Tables11 to VI. In the following, the most important features are highlighted and discussed in the context of seismic design. In general, information on the first two statistical moments of the probability distributions of response parameters is presented in order to minimize sample size effects. For the case where the yield stresses in beam and column are assumed to be uncorrelated, results on distribution fractiles are also given, since for this condition four independent sets have been created, as discussed above.

Table I1 presents estimates for the response statistics assuming no correlation between beam and column material properties. This is the situation which will arise in practice if no spatial control is exercised in the use of steel sections in a building. This does not, however, imply that other procedures that ensure quality control are in any way affected. The corresponding values obtained from non-linear static analysis using the nominal values for yield stress given in BS5950’ are also shown for comparison. Response estimates related to the nominal yield stress values are generally below the mean values obtained from simulation, except for Kpc, although in terms of fractiles they correspond to levels higher than the 5 per cent value, which is usually adopted in design. However, the response values obtained for the 5 per cent fractile are at most 9 per cent lower than those obtained assuming nominal yield stress values. Even parameters such as pl, p2 and Kpc, for which the fractile corresponding to the nominal yield stress values is relatively high, are not significantly over-predicted due to the characterisitcs of their distributions (positive skewness and/or low coefficient of


Table 111. Coefficients of variation for response parameters

p(ayb, uyc) = Oa0 P(ayb, uyc) = 0'7 p('yb? = Response parameter Coefficient of variation

p d = 2 0

p d = 5 0 U u d = 2 0

KPC

0.07 0.06 0.07 0.04 0.08 0.17 0.07 008 008 0.14 0.07 0.04

0.10 0.07 0.10 0.02 005 0.19 0 10 010 0.12 0.20 0.09 002

0.1 1 007 0.1 1 0.02 0.04 0.18 011 0.12 0.13 0.22 0.10 001

Table IV. Correlation between yield stress and response parameters

ud = 20

KPC

0.66 0-75 006 099 0 9 1 034 056 - 0.65 0.84 - 0.50 0-92 - 0.09 0 6 5 0.75 0.69 0.7 1 0-70 070 0 8 3 0.53 0.7 1 0.70

- 0 5 6 065

- 0.07 0.93 094 0.07

039 098 084 027 090 027 -0.40 089 0.94 0.83 0.25 084 0.71 0.97 0.66 0.49

- 007 0.93 0.94 006 - 0.02 0.94 0.93 0.09 - 001 094 093 001

0.20 0.96 0.88 018 0.00 0.94 0.92 0.10

- 0.90 - 0.27 040 - 0.89

- 0.66 076 099 -025 1.0 1.0 098

- 0.05 09 1 0.98 1.0 1.0 1.0 0.98 1 .o 005

Table V. Correlation between response parameters-p(oyb, uyc) = 0.0

u*=20

KPC

b y 1 4 2 6, PI P2 amax P m a x Pa=20 Pa=,, u u d = 2 0 Kp 1 .o 0.80 1.0 0.86 0.41 1.0

0.17 - 0.44 064 092 1.0 0.53 - 0.03 0.89 0.73 089 1.0 1.0 080 086 - 017 0.17 0.53 1.0 1 .o 076 0.89 - 0.11 0.23 0.58 1-0 1.0 1 -0 0.75 0.89 - 0.10 0.24 0.59 1.0 1.0 1.0 0.95 0.59 0.98 0.15 046 0.76 0.95 0.96 0.97 1.0 1 .o 0.75 0.89 - 0.11 024 0.58 1.0 1.0 1.0 0.96 1.0 0.19 069 - 0.34 - 1.0 - 091 - 0.72 0-19 0.13 0.12 - 0.13 0.13 1.0

- 0.17 067 0.36 1.0


Table VI. Correlation between response parameters-p(a,,, uyc) = 0.7

4 1

1 .o 0.95 0.97

0.57 0.87 1 .o 1 .o 1 .o 0.99 1.0 0.09

- 0.08

4 2

1 .o 0.86

- 0.37 0.29 068 0-95 0.94 0.94 0.90 0.94 037

6" PI

1 -0 015 1.0 0.14 0.75 096 0.42 0.97 - 0.09 0.98 - 006 0.98 - 005 0.99 0.05 098 - 0.06

- 014 - 1.0

1.0 0.88 1.0 0.57 087 1.0 059 0.88 1.0 1.0 0.59 088 1.0 1.0 1.0 0.67 0.93 0.99 1.0 1.0 0.60 0.88 1.0 1.0 1.0

- 0.74 - 0.42 0.09 0.06 0.06

1 .o 0-99 1-0 0.05 0.06 1.0

Frequency T

L '.2

Figure 5. Histogram of ductility ratio p 2 obtained by simulation-~(cT,,, tryc) = 0

variation), e.g. as shown in Figure 5 for the ductility ratio p2. Within the scope of the present study, this indicates that current specifications for yield stress are adequate for use in the prediction of seismic response parameters. More extensive studies covering a wide range of structures and steel grades would be needed before this conclusion can be generalized.

The above discussion relates the statistical results to a deterministic approach using the same analysis tools. The main objective of the simulation studies, however, is to quantify the effect of yield stress correlation on structural response and to identify the most suitable parameter(s) for design. In this respect, it is worth noting that the mean values for the response parameters are not significantly affected by the degree of yield stress correlation, which implies that the functional relationships between input and output quantities are approximately linear within the range considered, i.e. for random variations within a single steel grade.


The effect on the coefficients of variation, presented in Table 111, is more significant. Thus, the COV and, consequently, the standard deviation (since there are only minor differences in the mean) can be almost doubled (e.g. U) or halved (e.g. p 2 ) as the degree of input correlation increases. On average, the highest COV is associated with the total energy absorption up to the ultimate displacement, since this is based on the entire response characteristics of the frame. By comparing the COV values for Ud= 20 and U, it may be concluded that the energy COV would be further increased had 6, been defined further into the post-ultimate range. This observed lack of COV invariance in energy-related parameters should be taken into account in design assessment, as discussed in the subsequent section of this paper. The lowest coefficient of variation, for fully correlated beam/column yield stress, is that associated with K,,. It is also noteworthy that, between displacement-related parameters, the displacement corresponding to maximum capacity, 6,,,, exhibits the highest COV and, moreover, its value is independent of the degree of material correlation. The COV for load- related parameters is more stable throughout the response regime considered.

Whether the COV value of any parameter increases or decreases depends on the value of the first derivative of the function relating input and output quantities. In general, these relationships cannot be determined analytically owing to geometric and material non-linearities. However, the correlation coefficients between input (yield stress) and output (response) parameters provide an indication of linear dependence. As shown in Table IVY the COV of response parameters decreases if there is negative correlation between one of the two input parameters and a response parameter, i.e. for p i , p2 and Kpc.

From a design viewpoint, it is desirable to reduce the response uncertainty as the degree of yield stress correlation is increased, especially since this has no beneficial effect on the estimated mean values of the parameters. Thus, the conclusion reached from the results shown in Table I11 is that of all the response parameters considered, only the two ductility ratios and the secant stiffness parameter exhibit the desired characteristic. By contrast, the variance associated with both load- and energy-related parameters is increased as yield stress correlation becomes stronger.

Table IV presents the correlation coefficients between the yield stresses and response parameters. An additional variable, the yield stress ratio (r = oyb/oYyc), has been included, since it is directly related to the column over-design factor referred to previously. Considering first the uncorrelated case, p(oyb, oyc) = 0.0, it is observed that load parameters are equally correlated with beam and column yield stresses throughout the response history of the frame. On the other hand, displacement parameters defined prior to the development of a hinge mechanism are more strongly correlated with column rather than beam properties, the opposite being valid for parameters defined within the post-ultimate response regime. Stronger correlation with beam properties is also observed for energy parameters, especially when a larger part of the unloading regime is included, such as in the energy absorbed up to ultimate displacement (U). Finally, it is interesting that the strongest correlation for the two ductility ratios and the secant stiffness parameter is obtained when the independent variable is the yield stress ratio. Figure 6 illustrates schematically the observed correlation structure between input and output variables on a typical load-displacement curve for the frame under consideration, assuming uncorrelated yield stress values.

When the degree of yield stress correlation is increased, the values in Table IV show that, on average, correlations become stronger, except for paramters p2, pl and K p c . In fact, for the limiting case of p(oyyb, oyc) = 1.0 the last two parameters (which are functionally related, i.e. K,, = constant/p,) become practically uncorrelated to material yield stress. In combination with the low variance shown in the last column of Table 111, this implies that the value of these quantities becomes insensitive to random variations in yield stress within a single steel grade.

The ductility ratio p1 has been shown to give poor results for structures and components with no distinct yield point. In the current study, however, there is a clear peak beyond which abrupt stiffness changes occur, thus rendering pl a good measure of ductility, as shown by the results presented in Tables I11 and IV. Nevertheless, this observation is unlikely to hold for other more complicated structural configurations and materials. The same discussion holds for Kpc, leading to the conclusion that p, is likely to be a more suitable design parameter for general use.

Tables V and VI present the correlation matrices between response parameters for the two cases of uncorrelated and partially correlated yield stresses. The purpose of these tables is to clarify the inter-


tB,C1 t0,Cl

, /A"

tB/CI

IB/CI

tB/Cl

Figure 6. Schematic representation of correlation structure on load-displacement curve

dependence of response characteristics and to identify trends that may be used in predicting one parameter from the other. In assessing the significance of the correlation coefficients, it should be stressed that for certain parameters functional dependencies can be obtained directly from their definitions, such as that discussed above between p1 and Kpc, and, thus, the following comments are directed towards inferred, rather than established, relationships.

In this respect, it is of interest to note (Table V) the high correlation of PmaX with U, which indicates that an energy estimate may be obtained without resorting to integration of the load-displacement curve, and of P,,, with a,, which may be used for predicting the slope of the softening part of the response. In this case, P,,, could be estimated by a simplified method, e.g. mechanism analysis, or by performing an inexpensive computer run. It is also of interest to point out the relatively low correlation between p2 and U, which is shown graphically in Figure 7. This implies that, although the two criteria are not strictly independent, attempts to link them functionally may lead to poor results. A probabilistic study on a range of frame designs will enhance the relevance of the above observations in the context of code design.

Finally, the effect of yield stress correlation on the value of one of the response parameters most commonly used for design assessment, i.e. the ductility ratio p2, is discussed. The random variation in this parameter due to the uncertainty in material yield stress can be quantified by the regression lines obtained from the simulation results. Thus, with reference to the correlation coefficients given in Table IV, the mean ductility can be estimated from the following two expressions, depending on the degree of yield stress correlation:

p2 = 2-1 + 4-9r, for p(oyb, uy,) = 0.0

or

p2 = 1.4 + 5 7 r , for p(oyb, oyc) = 0.7

These relationships were obtained for a mean r value of 1.04 and, based on the observed range for r, are valid for 0.9 < r < 1-2. Using also the results of Table I11 that indicate that the ductility COV is reduced by almost 40 per cent when p(oyb, oyc) is increased from 0.0 to 0.7, it is estimated that the value of the characteristic ductility for this frame is about 10 per cent higher when spatial control of material properties is introduced. Although this increase may seem insignificant within the total uncertainty associated with seismic design, it

RANDOM MATERIAL VARIABILITY & DESIGN PARAMETERS

60t I. .

113

6 7 8 9 10 D u c t i I i t y Ratio p2

Figure 7. Correlation between ductility and energy absorption capacity--p(u,, uye) = 0

gives an indication of the benefit arising from the improvement of a single quality control procedure in a structure composed of few members. The advantages would be more substantial in a multi-bay and/or multi- storey structure, susceptible to a number of possible failure modes.

CONCLUSIONS

The main conclusion from this study is that random variability in yield stress and its degree of spatial correlation within a particular structure have a significant effect on a number of strength and deformation parameters used in seismic design. Specific conclusions on individual results presented in preceding sections are given below.

1. It is observed that usinq the nominal yield stress defined by codes for assessment of structural ductility and energy absorption capacity may lead to an erosion of the implied safety margins. However, for the structure studied herein, this reduction does not exceed 9 per cent of the parameter considered.

2. In general, resistance parameters (load) are equally correlated with beam and column yield stress. On the other hand, deformation (displacement) parameters exhibit stronger correlation with either column or beam yield stress, depending on whether they are pre- or post-ultimate quantities, respectively.

3. The scatter of strain energy parameters increases when a larger portion of the unloading regime is considered. This implies that for class H structures in EC8 (highly dissipative) higher quality control on material yield is needed than for the case of class M (moderately dissipative) structures.

4. The maximum displacement shows higher COV than other load-related parameters. Moreover, this is not affected by the degree of correlation between beam and column yield stress. Consequently, drift control in steel structures requires more stringent material control than, say, maximum capacity.

5. The three parameters giving the lowest COV are the two ductility ratios pl and p2, and the secant stiffness corresponding to ultimate displacement KPc. Moreover, the COV of these parameters drops sharply as the degree of beam/column yield stress correlation increases, rendering them suitable parameters for the assessment of non-linear response of structures.

6. The energy absorbed by the system is positively correlated to the ductility ratio, although the observed scatter should be taken into account in expressions that attempt to link the two parameters.

7. The ductility ratio increases by about 10 per cent for an increase in the degree of beam/column yield stress correlation. This modest percentage is likely to increase for more complicated structural configurations where an interaction between several failure modes may occur.


The above concluding remarks serve to highlight, in general, the advantages of more strict quality control procedures leading to tighter limits on the random variability in material characteristics. These conclusions are pertinent to the structural form investigated herein only. Further studies aimed at substantiating conclusions drawn above for multi-storey multi-bay frames using advanced material characterization are currently under consideration at Imperial College.

ACKNOWLEDGEMENTS

The authors wish to thank Pia Alexopulu, past Earthquake Engineering M.Sc. student at Imperial College, for running some of the structural analyses used in this study as part of her M.Sc. dissertation. Thanks are also due to Nick James, Imperial College Computer Centre System Manager, for his help with post- processing programs, and Panos Madas and Ahmed Elghazouli, Ph.D. students, for their help with plotting and frame design.

REFERENCES

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effect of random material variability on seismic design parameters of steel frames

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