imperfections of steel structural frames
5.2 Statistical data for initial geometric imperfection
The methods for modelling geometric imperfections can be classified as deterministic or random. For deterministic modelling, the maximum amplitude of an initial geometric imperfection is typically determined from a steel structural specification (Table 2-2). The pattern for the initial out-of-straightness is often assumed to be a half-sine wave and the frame out-of-plumb follows a linear pattern with all columns leaning in the same direction. In probabilistic modelling, the initial geometric imperfections (both shape and the magnitude) are treated as random variables. The probabilistic modelling requires statistical information for the geometric imperfection, such as distribution type, mean and standard deviation. Ideally, the probabilistic models should be established on the basis of sufficient experimental data.
5.2.1 Initial out-of-straightness
Although a great number of experimental results on column strength can be found in the literature, very few studies report the detailed measurements of initial imperfections along the length of the member. Some of these measurements are back to 1960s and 1970s and published in form of research reports with no electronic recourses to easily access. In most studies, out-of-straightness is assumed to follow a half-sine shape and only the value for mid-span is reported which does not provide information about the contribution of higher order buckling modes with multiple half-waves. In this case only the magnitude of imperfection at mid-height can be modelled randomly while the shape of initial out-of- straightness is treated as deterministic. The statistical data for the non-dimensional out-of- straightness at mid-height of steel I-section members are summarized in Table 2-4. The presented results in this table show that a significant difference exists between measured
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imperfections from different regions. It appears that on average, Japanese sections have smaller initial out-of-straightness compared to those from Europe or North America.
In this study, both the shape and magnitude are treated as random variables. Thus, detailed measurements of initial imperfections along the length of the member are required to obtain the statistics of initial out-of-straightness. The approach is based on the superposition of elastic buckling modes. For a member in compression the buckling modes are assumed to take the form of sin where i=1, 2, 3, … and 0,1 is the non-dimensional coordinate measured along the length of the member (L) (see Figure 5-1 for three buckling modes).
Figure 5-1: First three buckling modes of simply supported, axially loaded column
In general, the initial out-of-straightness of the member can be expressed in terms of a linear superposition of a given number of these eigen buckling modes:
∑ sin 0,1 5-1 in which is the initial out-of-straightness at location x, is the scale factor for the ith mode and is the number of buckling modes included. In the following, it is assumed that a sample of N members is available and that for each member, the out-of straightness at locations along the length of member are measured.
2 3 Mode 1 Mode 2 Mode 3 0 0 0 1 1 1 L
This study is based on the initial out-of-straightness measurements of nine (N=9) IPE 160 columns carried out at the University of Politecnico di Milano (C.E.A.C.M. 1966) and published by ECCS Committee 8.1 (Sfintesco 1970). The reported data comprises geometric imperfection measurements at mid-length and quarter points. First, the actual measurements are non-dimensionalised by dividing the measured imperfection by the length of the member (see Table 5-1 .
Table 5-1: Measured initial out-of-straightness (d) by Sfintesco (1970)
Sample No . . .
1 0.000431 0.000507 0.000278
2 -0.00018 -0.00032 -0.0002
3 -7.27E-07 8.87E-05 0.000193
4 5.01E-05 -4.80E-05 -7.34E-05
5 -9.74E-05 -0.00031 -0.00025
6 -0.00011 -7.99E-05 -4.00E-05
7 -3.92E-05 1.45E-05 3.92E-05
8 -0.00016 -0.00022 -0.00022
9 -0.00026 -0.00023 -0.00036
μ(| |) 0.000148 0.000204 0.000183
As three readings of out-of-straightness are available for each sample, the out-of- straightness can be expressed as a linear combination of the first three buckling modes as shown in Figure 5-1. The scale factors, or contributions of each mode ( , i=1, 2, 3), can be determined by solving a set of three equations for each member (Equation 5-2) which is the expanded form of Equation 5-1.
. sin sin sin . sin sin sin . sin sin sin
5-2
Subsequently the statistical information of the scale factors , and (e.g. mean and coefficient of variation) can be obtained.
As shown in Table 5-1, the mean of the absolute values of measured out-of-straightness at mid-height, as reported by Sfintesco (1970), is equal to 0.000204 (1/4910), which appears
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to be small compared to the mean provided by Fukumoto and Itoh (1983) based on 437 measurements and mean values obtained from other measurements around the world (Table 2-4). Thus, while the COVs of obtained scale factors remain unchanged (Table 5-2), the mean values are scaled up by a factor of 2.62 to match with the mean (1/1996) provided by Fukumoto and Itoh (1983) which appears to be a reliable representative value of initial out-of-straightness at member mid-might. The scale factor of 2.62 was obtained based on the fact that if the first three buckling modes are used to model the initial geometric imperfection of a single member, at mid-span only the first and the third modes contribute (see Figure 5-1). Therefore, the mean of the non-dimensional initial out-of- straightness at mid-span of a single member may be calculated as the difference between the mean of scale factors corresponding to mode one and three ( ), which is 0.00048 (1/2000) after scaling by factor of 2.62. The final statistical characteristics, i.e. mean (μ), standard deviation (σ) and COV of the scale factors ( , i=1, 2, 3), are summarized in Table 5-3. The distribution of the scale factors, while based on a small number of data, was found to be approximately normal.
`
Table 5-2: Scale factors of the first three buckling modes Sample No
1 0.000504 7.63E-05 2.86E-06
2 0.000294 1.02E-05 2.88E-05
3 0.000112 9.67E-05 2.35E-05
4 3.22E-05 6.18E-05 1.58E-05
5 0.000279 7.56E-05 3.47E-05
6 9.44E-05 3.71E-05 1.45E-05
7 7.27E-06 3.92E-05 7.27E-06
8 0.000246 2.91E-05 2.17E-05
9 0.000337 5.16E-05 0.000103
μ 0.000212 5.31E-05 2.8E-05
COV 0.76771 0.511367 1.065306
Table 5-3: Statistic characteristics of scale factors
Statistics
Mean (μ) 0.000556 0.000139 0.000073
COV 0.76771 0.511367 1.065306
Using these statistics and combining the first three modes, random initial imperfection can be generated for different frame members. Since the absolute values of scale factors are considered to find the statistical characteristics, a random sign is generated and assigned to each scale factor. An example of randomly generated member out-of- straightness is presented in Figure 5-2.
5.2.2 Initial out-of-plumb
The out-of-plumb can also be treated as a random variable and modelled as all columns leaning in same direction (Buonopane 2008) (Figure 5-3 (a)) or as each column leaning in its own direction (Lindner 1984 ; Lindner and Gietzelt 1984 ) (Figure 5-3 (b)).
Figure 5-2: Example of random shape of initial out-of-straightness for a simply-supported column Δ Δ (b) (a)
Figure 5-3: Initial out-of-plumb along the height (a) same direction (b) different direction
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Horizontal displacements in both in-plane and out-of-plane directions of a multistorey steel frame has been reported by ECCS (1976) for the first, sixed and eleventh floor. Beaulieu and Adams (1978) measured the out-of-plumb, ∆ ⁄ , of 916 columns in both directions and reported the mean as almost zero and the standard deviation as 0.00162. Extensive research in this area was undertaken by Linder (Lindner 1984 ; Lindner and Gietzelt 1984) who reported two groups of measurements from different buildings with different column heights. Thefirst group of out-of-plumb measurements (approximately 725) were a second Canadian study by Beaulieu and Adams (1978) on two high rise buildings with a storey height of 3.6m. In the second group, the out-of-plumb of more than 900 German buildings with different heights between 3m and 125m were measured by Lindner and Gietzelt (1984 ). The mean and standard deviation of the total number of 1760 measurements recorded in Canada and Germany was ∆ ⁄ =0.0002 and
∆ ⁄ =0.000173 respectively. The histogram of these data is presented in Figure 5-4 and appears to be normally distributed. Since these data are based on the measured out- of-plumb of excising structures in various parts of the world, they represent realistic values and are used in this study. It is assumed that all columns lean in a same direction and a single value of random out-of-plumb is applied to the whole frame.
Figure 5-4: Out-of-plumb statistics reported by Lindner and Gietzelt (1984 )(a) with sign (b) absolute . 10 (rad) 500 400 300 200 100 n -40 -20 0 20 40 0.00002 0.000173 1760 200 400 600 800 1000 n 10 20 30 40 50 . 10 (rad) 0.0013 0.00114 1760 COV 0.88 (b)