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Actual extreme pressure distributions and LRC formula
Journal of Wind Engineering and Industrial Aerodynamics 90 (2002) 1959–1971
Actual extreme pressure distributions and LRC formula Y. Tamuraa,*, H. Kikuchib, K. Hibib a
Department of Architectural Engineering, Tokyo Institute of Polytechnics, Iiyama 1583, Atsugi, Kanagawa 243-0297, Japan b Research Institute of Technology, Shimizu Corporation, Etchujima 3-4-17, Kohtohku, Tokyo 135-8530, Japan
Abstract This paper discusses the conditionally sampled actual wind pressure distributions causing peak quasi-static internal forces in the structural frames of a low-rise building model with a square plan. The actual extreme pressure distributions are compared with Kasperski’s load– response-correlation (LRC) formula and the quasi-steady pressure distributions. The validity of Kasperski’s LRC formula is proved to estimate the actual extreme wind load for the general location of the frame including the separation bubble region. The only exception was where the frame was immersed in the vicinity of the leading edge, where the non-Gaussian characteristics are predominant. The reason for the discrepancy in the LRC pressure distribution and the actual extreme pressure distribution is attributed to the intermittent characteristics of the fluctuating pressure field in the vicinity of the leading edge, which cannot be represented by the temporal average characteristics. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Low-rise building; Quasi-steady assumption; Quasi-static load effect; LRC formula; Extreme pressure distribution; Internal force
1. Introduction Holmes [1] studied the actual distributions of instantaneous wind pressures along a gabled roof frame producing peak loads and load effects on the frame of a low-rise building model. The load–response-correlation (LRC) formula was proposed by Kasperski [2] to realistically model the spatial distribution of wind loads that *Corresponding author. Fax: +81-46-242-9547. E-mail address: [email protected] (Y. Tamura). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 0 1 - X
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produce accurate peak structural responses. Holmes [3] extended the LRC approach to incorporate proper orthogonal decomposition (POD). Tamura et al. [4] discussed the actual extreme wind load distributions C# p causing the maximum and minimum stresses in members of the structural frames on the basis of 154 samples of 10-min length in full-scale conversion, and compared the results with those (CLRC ) computed from Kasperski’s LRC formula. Although the LRC formula seemed to be valid for general cases, the results suggested that the validity depended on the frame location. The LRC pressure pðjÞY at a point j causing the peak value of a load effect Y is based on the covariance of wind pressures pi pj and mean pressures pðjÞ % as follows: pðjÞY ¼ pðjÞ % þ gY
n X ðpi pj aYi Þ=sY :
ð1Þ
i¼1
Here, sY are the standard deviation of the load effect Y : gY is the peak factor of the load effect Y and aYi is the influence coefficient. Thus, the temporal average characteristics of the wind pressures are strongly reflected in the LRC pressure distribution CLRC : Therefore, if the peak load effects are generated by intermittent or non-stationary conditions that differ from the temporal average characteristics of the pressure field, the LRC pressure distribution can differ from the actual extreme pressure distribution. The mismatch seems to come from the difference of the temporal average characteristics and the extreme condition. This paper compares the three wind load distributions, C# p ; CLRC and GCp for frames set at different locations in the low-rise building models. The wind pressure data are the same as those used in Tamura et al. [4]. One hundred and fifty-four samples of 10-min length of full-scale conversion were analyzed for the urban flow with a power-law index a ¼ 1=4: Each sample is a data set of fluctuating wind pressures at 512 points uniformly distributed on the surface of the low-rise building model with a square plan.
2. Frame locations and pressure distributions The frame locations are indicated in Fig. 1. Frames La, Lb and Lc are set in the along-wind direction, and Frames Ca, Cb, Cc and Cd are set in the across-wind direction. The mean, the fluctuating and the peak pressure coefficients on each frame are shown in Figs. 2–4. The mean pressure for the along-wind Frames La, Lb and Lc has the highest suction at the windward end, showing a slightly different pattern for Frame La located at the periphery of the roof. The mean pressure pattern for acrosswind Frame Ca has peaks at both the ends due to the 3D complex flow generated around the roof corners, and this is more significant than the fluctuating and the minimum pressure distribution shown in Figs. 3 and 4. Across-wind Frames Ca and Cb are probably in the separation bubble region near the leading edge, but the pressure coefficient of Frame Ca is significantly higher (negative) than that of the
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Fig. 1. Frame locations: (a) along-wind frames; (b) across-wind frames.
Fig. 2. Mean pressure distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Fig. 3. Fluctuating pressure (standard deviation) distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Frame Cb and their pressure patterns are quite different, thus suggesting different characteristics of the pressure fluctuations on these two frames. Similar pressure patterns are seen in the mean and the minimum pressure distributions for acrosswind Frame Cb. This is another feature different from Frame Ca. Those for Frames
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Fig. 4. Peak pressure distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Fig. 5. Peak factors of fluctuating pressures acting on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Cc and Cd are uniform and almost the same, showing that they are located in the reattached region. The peak factors of the fluctuating pressures are shown in Fig. 5. The peak factors are higher for the negative side except for the windward wall. The negative side peak factors of the roof pressures on the periphery areas except near the trailing edge of the roof show a high value of more than ()6 as shown in Fig. 5. Especially, those for Frames La and Ca are high and almost constant over the span.
3. Peak factors of internal forces The LRC formula requires the peak factor gY of the load effects, and Kasperski [2] tentatively adopted 3.5. In this study, the peak factor for each load effect could be obtained in the process of calculating the peak load effects. As Kasperski [5] pointed out, positive side roof pressures should also be taken into account in the structural design considering combinations with dead load or snow load. Here the peak internal forces due to the positive and the negative side fluctuations were calculated and the results are indicated by Qþ ; M þ ; Q and M ; where the sign means
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Table 1 Peak factors of internal forces in frames (pin-roller base condition) Frames
Shear force
Q
Bending moment þ
Q
M
Mþ
2.6 2.4 2.5
+4.5 +4.5 +4.7
4.7 5.2 4.4 4.2
+2.2 +2.8 +3.9 +3.7
Along-wind frames (Q and M at windward end) Frame La Frame Lb Frame Lc
5.4 4.4 4.3
+2.4 +2.5 +2.4
Across-wind frames (Q at left end; M at center) Frame Frame Frame Frame
Ca Cb Cc Cd
4.6 5.2 4.3 4.2
+1.9 +2.4 +3.4 +3.4
maximum or minimum load effect. It should be noted that M and M þ do not necessarily mean a negative and a positive bending moment, respectively. Ensemble averaged peak factors of internal forces for 154 samples are shown in Table 1. For the minimum load effects Q and M of Frames Ca, Cb, Cc and Cd, and for the minimum shear force Q of Frames La, Lb and Lc, all of which are caused by the high suction roof pressures, the peak factor changes in the range of 4.2–5.4, while much smaller peak factors in the range 1.9–3.9 are obtained for the maximum load effects Qþ and M þ of Frames Ca, Cb, Cc and Cd, and for the maximum shear force Qþ of Frames La, Lb and Lc, all of which are caused by low suction or positive side roof pressures. The peak factors were less affected by the flow and column base conditions.
4. Extreme pressure distributions The instantaneous pressure distributions causing the peak quasi-static internal forces, i.e. shear force Q and bending moment M in the end and the center of the beam, were extracted from each 10-min sample in full-scale conversion. Then the ensemble averaged instantaneous pressure distribution over 154 samples was defined as ‘‘actual extreme wind load C# p ’’ causing the peak load effect. The extreme wind pressure distribution due to Kasperski’s LRC formula, which is defined as ‘‘LRC extreme wind load CLRC ’’, was also calculated from Eq. (1) using the peak factor gY obtained in the previous section for each load effect. The ‘‘quasisteady wind load GCp ’’, which is the product of the mean pressure Cp and the ensemble averaged gust effect factor for each load effect G (¼ Mmax =Mmean or Qmax =Qmean ) was also evaluated. The gust effect factors of load effects were in the
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range 2.4 3.1 and much less than the square of the gust factor of the approaching flow GV2 ¼ 4:3: The above three extreme wind loads for estimating peak load effect are compared in Figs. 6 and 7 for the along-wind frames (Frames La, Lb and Lc) and across-wind frames (Frames Ca, Cb and Cc), respectively. As expected from Fig. 2(b), the result for Frame Cd was almost the same as that for Frame Cc. As is clearly seen from Fig. 6 for the windward end bending moment M and shear force Q of along-wind frames, the LRC wind load CLRC (K) agrees well with the actual C# p ; especially for the negative side roof pressure (solid lines), as Holmes [3] reported for a fixed base frame model. For shear force Q ; the quasi-steady GCp (J) on the roof is similar to the actual C# p and LRC CLRC ; although CLRC is always closer to the actual C# p : However, the quasi-steady GCp differs from the actual wind load C# p for the bending moment M þ : The ratio of the suction near the leading edge to that in the reattached region is approximately 6 for the actual C# p ; while it is 3 for the quasisteady GCp : Please note that the bending moment M at the windward end of the along-wind frame is caused only by the windward wall pressures in this pin-roller column support condition. For this pin-roller frame, the negative quasi-steady GCp on the roof and leeward wall is much larger than the actual C# p ; although the quasisteady GCp on the windward wall is almost half of the actual C# p : For the pressure
Fig. 6. Extreme wind loads (along-wind frames, Q and M at windward end): (a) Frame La; (b) Frame Lb; (c) Frame Lc.
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Fig. 7. Extreme wind loads (across-wind frames, Q at left end, M at center): (a) Frame Ca; (b) Frame Cb; (c) Frame Cc.
distribution for M ; the LRC wind load (m) becomes smaller than the actual C# p (broken lines) near the leading edge of the roof. These tendencies are commonly observed regardless of the frame location. As seen from Fig. 7 for the across-wind frames, the actual extreme pressure C# p varies significantly with the frame location. For the left end shear force Q ; the actual C# p tends to cluster to the left side, and the LRC CLRC follows this tendency. However, the quasi-steady GCp is symmetric and different from the actual C# p : For Frame Ca, located very close to the leading edge, even the LRC CLRC does not agree with the actual C# p for Q and M : The discrepancy becomes more significant for the beam center bending moment M : As seen from Fig. 7(a), the actual C# p has a gentle peak near the center of the roof, while the quasi-steady GCp has peaks at both ends of the roof. The quasi-steady GCp is much larger than the actual C# p except at the roof center, and the LRC CLRC has a pattern similar to the quasi-steady GCp : For both Frame Cb in the separation bubble region and Frame Cc in the reattached region, LRC CLRC agree very well with the actual C# p :
5. Relative frequency distributions of fluctuating roof pressures and internal forces In order to discuss the reasons of the discrepancy between the LRC CLRC and the actual C# p ; the temporal variations of the roof pressures are first examined. Fig. 8
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Fig. 8. Temporal variations of fluctuating pressures at four points, Nos. 5, 8, 197 and 200 in the separation bubble region.
shows the pressure fluctuation at four points near the leading edge, Nos. 5, 8, 197 and 200. Very high negative peaks are observed in the pressure fluctuations at points Nos. 5 and 197 on Frame Ca. At time ta when the peak pressure was recorded at point No. 5 near the end of Frame Ca, the pressure at point No. 197 near the center of Frame Ca also shows a gentle peak. The time when the peak pressure was recorded at the center point No. 197 is indicated tb in the figure. Fig. 9 shows examples of the relative frequency distributions of the fluctuating pressures. The distributions of Nos. 5 and 197 significantly skew from the Gaussian distribution indicated by dotted line in the figures. The entire span of Frame Ca is immersed in the non-Gaussian pressure field. Thus, non-Gaussian characteristics of the pressure field can be one of the reasons for the discrepancy. Fig. 10 shows examples of the temporal variations of the bending moment M at the beam center of Frames Ca, Cb and Cc. Here only Frame Cc is located in the re-attached region. Compared with Fig. 8, it is obvious that the bending moment of Frame Ca fluctuates following the fluctuation of the pressure at No. 197 near the beam center. The minimum bending moment M of Frame Ca occurred at time tb when the central pressure reached its negative peak. The minimum bending moment of Frame Cb also occurred at time tb reducing its magnitude. The bending moment of Frame Cc had reduced magnitude and showed quite different characteristics.
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Fig. 9. Relative frequency distributions of standardized fluctuating pressures (zero mean, s ¼ 1) at several points.
Fig. 11 shows the relative frequency distributions of the bending moments M of the frames. The bending moments of across-wind Frames Ca and Cb near the leading edge skew from the Gaussian distribution, but that of Frame Cc lying in the re-attached region has a distribution close to Gaussian. An interesting thing is that those of along-wind Frames La, Lb and Lc all skew from the Gaussian distribution. The skewness is most significant for Frame Ca in this case, but it is not necessarily true in other conditions. The load effects have tended to be considered Gaussian because of the summation effects of the randomly fluctuating pressures at many spatial points due to the central limit theorem, even if the pressures show nonGaussian characteristics. However, although the strong non-Gaussian characteristics of pressures are observed only in the region near the leading edge of the roof, the load effects in the frames lying in or including this region, i.e. Frames Ca, Cb, La, Lb and Lc, show strong non-Gaussian characteristics. Here, regarding the load effects in along-wind Frames La, Lb and Lc, only the internal forces at the windward end of the beam were examined. Thus, the pressures in the leading edge region influenced most. The non-Gaussian pressure fluctuation does not seem to be a sufficient condition for the discrepancy between the LRC CLRC and the actual C# p :
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Fig. 10. Temporal variations of bending moments at the beam center of Frames Ca, Cb and Cc.
6. Special features of fluctuating pressure acting on Frame Ca As firstly discussed, the temporal average characteristics of the wind pressures are strongly reflected in the LRC pressure distribution CLRC ; such as the mean value pðjÞ % and covariance pi pj of fluctuating pressures. Regarding the standard deviation spj of the pressures, only Frame Ca shows a special feature having the sharp peaks near both the ends as shown in Fig. 3. Only the mean pressure distribution on Frame Ca also has similar peaks at the ends. These effects are significantly remaining in the LRC pressure distribution CLRC for M as shown in Fig. 7(a), even if the influence coefficient aYi magnifies the contribution of the pressures near the center and reduces those effects near both the ends. However, the actual extreme pressure distribution C# p is extracted at the instance when the peak load effect is observed. For the case of central bending moment of Frame Ca, the extreme instance corresponds to the time when the fluctuating pressure near the beam center reached its minimum, i.e. time tb as shown in Figs. 8 and 10. Fig. 12 may be noteworthy to examine. This figure shows the phase-plane expression of the fluctuating pressures at two points, Nos. 197 and 29. No. 197 is the point near the center of Frame Ca and No. 29 is the point where the peak fluctuating pressure (standard deviation) is observed near the left end of Frame Ca. The
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Fig. 11. Relative frequency distributions of bending moment M of frames.
Fig. 12. Phase-plane expression of fluctuating pressures at two points Nos. 197 and 29.
trajectory shows a quite interesting shape, i.e. an inclined triangular with two horns at the ends of its base. This trajectory implicitly emphasizes the two phenomena: the wind pressure at the center, No. 197, reaches its negative peak and that of the end, No. 29, reaches its negative peak. The temporal average characteristics show some
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Fig. 13. Example of instantaneous pressure pattern causing minimum bending moment M at the beam center of Frame Ca.
correlation by the inclination of the triangular. However, the two special conditions around the tips of the two horns are apparently quite different from the temporal average characteristics. Thus, the trajectory shown in Fig. 12 suggests that the fluctuating pressure field on Frame Ca is very complicated and results from more than two different phenomena. The minimum bending moment M is caused by the extreme condition of one of these phenomena. The temporal average characteristics cannot represent the individual extreme condition. It is also remarkable that the wind pressure No. 29 near the end tends to be very small at the moment when the wind pressure No. 197 near the center reached its negative peak. This moment corresponds to the time when the bending moment of the beam center of Frame Ca becomes its minimum, M : The typical pressure pattern actually causing the minimum bending moment M at the beam center is shown in Fig. 13. The actual pressure pattern C# p was obtained by ensemble averaging of the pressure patterns like Fig. 13 as shown in Fig. 7(a). Here, the main reason of the discrepancy between the LRC CLRC and the actual C# p for the bending moment of Frame Ca can be attributed to the difference of the extreme condition causing the minimum bending moment with the temporal average characteristics of the roof pressures. This extreme condition is limited at near the tip of the right horn of the trajectory in Fig. 12.
7. Concluding remarks The validity of Kasperski’s LRC formula was proved to estimate the actual extreme wind load for the general location of the frame including the separation bubble region. The only exception was where the across-wind frame was immersed in the vicinity of the leading edge, where the non-Gaussian characteristics are
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predominant in pressure fluctuations and very high negative peaks are observed near the windward corners. However, the non-Gaussian characteristics are not a sufficient condition for the discrepancy between the LRC and the actual extreme pressure distributions. The pressure field on Frame Ca shows complex characteristics resulting from more than two different phenomena. The minimum bending moment M at the beam center of the frame is caused by an extreme condition of one of these phenomena, which cannot be represented by the temporal average characteristics. The reason for the discrepancy in the LRC pressure distribution and the actual extreme pressure distribution is attributed to this intermittent characteristics of the fluctuating pressure field on Frame Ca in the vicinity of the leading edge.
References [1] J.D. Holmes, Distribution of peak wind loads on a low-rise building, J. Wind Eng. Ind. Aerodyn. 29 (1988) 59–67. [2] M. Kasperski, Extreme wind load distributions for linear and nonlinear design, Eng. Struct. 14 (1) (1992) 27–34. [3] J.D. Holmes, Optimised peak load distributions, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 267–276. [4] Y. Tamura, H. Kikuchi, K. Hibi, Extreme wind pressure distributions on low-rise building models, Abstract Volume, Fourth International Colloquium on Bluff Body Aerodynamics and Applications, Ruhr University, Bochum, Germany. [5] M. Kasperski, Design wind loads for low-rise buildings: a critical review of wind load specifications for industrial buildings, J. Wind Eng. Ind. Aerodyn. 61 (1996) 169–179.
Actual extreme pressure distributions and LRC formula Y. Tamuraa,*, H. Kikuchib, K. Hibib a
Department of Architectural Engineering, Tokyo Institute of Polytechnics, Iiyama 1583, Atsugi, Kanagawa 243-0297, Japan b Research Institute of Technology, Shimizu Corporation, Etchujima 3-4-17, Kohtohku, Tokyo 135-8530, Japan
Abstract This paper discusses the conditionally sampled actual wind pressure distributions causing peak quasi-static internal forces in the structural frames of a low-rise building model with a square plan. The actual extreme pressure distributions are compared with Kasperski’s load– response-correlation (LRC) formula and the quasi-steady pressure distributions. The validity of Kasperski’s LRC formula is proved to estimate the actual extreme wind load for the general location of the frame including the separation bubble region. The only exception was where the frame was immersed in the vicinity of the leading edge, where the non-Gaussian characteristics are predominant. The reason for the discrepancy in the LRC pressure distribution and the actual extreme pressure distribution is attributed to the intermittent characteristics of the fluctuating pressure field in the vicinity of the leading edge, which cannot be represented by the temporal average characteristics. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Low-rise building; Quasi-steady assumption; Quasi-static load effect; LRC formula; Extreme pressure distribution; Internal force
1. Introduction Holmes [1] studied the actual distributions of instantaneous wind pressures along a gabled roof frame producing peak loads and load effects on the frame of a low-rise building model. The load–response-correlation (LRC) formula was proposed by Kasperski [2] to realistically model the spatial distribution of wind loads that *Corresponding author. Fax: +81-46-242-9547. E-mail address: [email protected] (Y. Tamura). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 3 0 1 - X
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produce accurate peak structural responses. Holmes [3] extended the LRC approach to incorporate proper orthogonal decomposition (POD). Tamura et al. [4] discussed the actual extreme wind load distributions C# p causing the maximum and minimum stresses in members of the structural frames on the basis of 154 samples of 10-min length in full-scale conversion, and compared the results with those (CLRC ) computed from Kasperski’s LRC formula. Although the LRC formula seemed to be valid for general cases, the results suggested that the validity depended on the frame location. The LRC pressure pðjÞY at a point j causing the peak value of a load effect Y is based on the covariance of wind pressures pi pj and mean pressures pðjÞ % as follows: pðjÞY ¼ pðjÞ % þ gY
n X ðpi pj aYi Þ=sY :
ð1Þ
i¼1
Here, sY are the standard deviation of the load effect Y : gY is the peak factor of the load effect Y and aYi is the influence coefficient. Thus, the temporal average characteristics of the wind pressures are strongly reflected in the LRC pressure distribution CLRC : Therefore, if the peak load effects are generated by intermittent or non-stationary conditions that differ from the temporal average characteristics of the pressure field, the LRC pressure distribution can differ from the actual extreme pressure distribution. The mismatch seems to come from the difference of the temporal average characteristics and the extreme condition. This paper compares the three wind load distributions, C# p ; CLRC and GCp for frames set at different locations in the low-rise building models. The wind pressure data are the same as those used in Tamura et al. [4]. One hundred and fifty-four samples of 10-min length of full-scale conversion were analyzed for the urban flow with a power-law index a ¼ 1=4: Each sample is a data set of fluctuating wind pressures at 512 points uniformly distributed on the surface of the low-rise building model with a square plan.
2. Frame locations and pressure distributions The frame locations are indicated in Fig. 1. Frames La, Lb and Lc are set in the along-wind direction, and Frames Ca, Cb, Cc and Cd are set in the across-wind direction. The mean, the fluctuating and the peak pressure coefficients on each frame are shown in Figs. 2–4. The mean pressure for the along-wind Frames La, Lb and Lc has the highest suction at the windward end, showing a slightly different pattern for Frame La located at the periphery of the roof. The mean pressure pattern for acrosswind Frame Ca has peaks at both the ends due to the 3D complex flow generated around the roof corners, and this is more significant than the fluctuating and the minimum pressure distribution shown in Figs. 3 and 4. Across-wind Frames Ca and Cb are probably in the separation bubble region near the leading edge, but the pressure coefficient of Frame Ca is significantly higher (negative) than that of the
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Fig. 1. Frame locations: (a) along-wind frames; (b) across-wind frames.
Fig. 2. Mean pressure distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Fig. 3. Fluctuating pressure (standard deviation) distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Frame Cb and their pressure patterns are quite different, thus suggesting different characteristics of the pressure fluctuations on these two frames. Similar pressure patterns are seen in the mean and the minimum pressure distributions for acrosswind Frame Cb. This is another feature different from Frame Ca. Those for Frames
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Fig. 4. Peak pressure distributions on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Fig. 5. Peak factors of fluctuating pressures acting on frames: (a) along-wind Frames La, Lb and Lc; (b) across-wind Frames Ca, Cb, Cc and Cd.
Cc and Cd are uniform and almost the same, showing that they are located in the reattached region. The peak factors of the fluctuating pressures are shown in Fig. 5. The peak factors are higher for the negative side except for the windward wall. The negative side peak factors of the roof pressures on the periphery areas except near the trailing edge of the roof show a high value of more than ()6 as shown in Fig. 5. Especially, those for Frames La and Ca are high and almost constant over the span.
3. Peak factors of internal forces The LRC formula requires the peak factor gY of the load effects, and Kasperski [2] tentatively adopted 3.5. In this study, the peak factor for each load effect could be obtained in the process of calculating the peak load effects. As Kasperski [5] pointed out, positive side roof pressures should also be taken into account in the structural design considering combinations with dead load or snow load. Here the peak internal forces due to the positive and the negative side fluctuations were calculated and the results are indicated by Qþ ; M þ ; Q and M ; where the sign means
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Table 1 Peak factors of internal forces in frames (pin-roller base condition) Frames
Shear force
Q
Bending moment þ
Q
M
Mþ
2.6 2.4 2.5
+4.5 +4.5 +4.7
4.7 5.2 4.4 4.2
+2.2 +2.8 +3.9 +3.7
Along-wind frames (Q and M at windward end) Frame La Frame Lb Frame Lc
5.4 4.4 4.3
+2.4 +2.5 +2.4
Across-wind frames (Q at left end; M at center) Frame Frame Frame Frame
Ca Cb Cc Cd
4.6 5.2 4.3 4.2
+1.9 +2.4 +3.4 +3.4
maximum or minimum load effect. It should be noted that M and M þ do not necessarily mean a negative and a positive bending moment, respectively. Ensemble averaged peak factors of internal forces for 154 samples are shown in Table 1. For the minimum load effects Q and M of Frames Ca, Cb, Cc and Cd, and for the minimum shear force Q of Frames La, Lb and Lc, all of which are caused by the high suction roof pressures, the peak factor changes in the range of 4.2–5.4, while much smaller peak factors in the range 1.9–3.9 are obtained for the maximum load effects Qþ and M þ of Frames Ca, Cb, Cc and Cd, and for the maximum shear force Qþ of Frames La, Lb and Lc, all of which are caused by low suction or positive side roof pressures. The peak factors were less affected by the flow and column base conditions.
4. Extreme pressure distributions The instantaneous pressure distributions causing the peak quasi-static internal forces, i.e. shear force Q and bending moment M in the end and the center of the beam, were extracted from each 10-min sample in full-scale conversion. Then the ensemble averaged instantaneous pressure distribution over 154 samples was defined as ‘‘actual extreme wind load C# p ’’ causing the peak load effect. The extreme wind pressure distribution due to Kasperski’s LRC formula, which is defined as ‘‘LRC extreme wind load CLRC ’’, was also calculated from Eq. (1) using the peak factor gY obtained in the previous section for each load effect. The ‘‘quasisteady wind load GCp ’’, which is the product of the mean pressure Cp and the ensemble averaged gust effect factor for each load effect G (¼ Mmax =Mmean or Qmax =Qmean ) was also evaluated. The gust effect factors of load effects were in the
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range 2.4 3.1 and much less than the square of the gust factor of the approaching flow GV2 ¼ 4:3: The above three extreme wind loads for estimating peak load effect are compared in Figs. 6 and 7 for the along-wind frames (Frames La, Lb and Lc) and across-wind frames (Frames Ca, Cb and Cc), respectively. As expected from Fig. 2(b), the result for Frame Cd was almost the same as that for Frame Cc. As is clearly seen from Fig. 6 for the windward end bending moment M and shear force Q of along-wind frames, the LRC wind load CLRC (K) agrees well with the actual C# p ; especially for the negative side roof pressure (solid lines), as Holmes [3] reported for a fixed base frame model. For shear force Q ; the quasi-steady GCp (J) on the roof is similar to the actual C# p and LRC CLRC ; although CLRC is always closer to the actual C# p : However, the quasi-steady GCp differs from the actual wind load C# p for the bending moment M þ : The ratio of the suction near the leading edge to that in the reattached region is approximately 6 for the actual C# p ; while it is 3 for the quasisteady GCp : Please note that the bending moment M at the windward end of the along-wind frame is caused only by the windward wall pressures in this pin-roller column support condition. For this pin-roller frame, the negative quasi-steady GCp on the roof and leeward wall is much larger than the actual C# p ; although the quasisteady GCp on the windward wall is almost half of the actual C# p : For the pressure
Fig. 6. Extreme wind loads (along-wind frames, Q and M at windward end): (a) Frame La; (b) Frame Lb; (c) Frame Lc.
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Fig. 7. Extreme wind loads (across-wind frames, Q at left end, M at center): (a) Frame Ca; (b) Frame Cb; (c) Frame Cc.
distribution for M ; the LRC wind load (m) becomes smaller than the actual C# p (broken lines) near the leading edge of the roof. These tendencies are commonly observed regardless of the frame location. As seen from Fig. 7 for the across-wind frames, the actual extreme pressure C# p varies significantly with the frame location. For the left end shear force Q ; the actual C# p tends to cluster to the left side, and the LRC CLRC follows this tendency. However, the quasi-steady GCp is symmetric and different from the actual C# p : For Frame Ca, located very close to the leading edge, even the LRC CLRC does not agree with the actual C# p for Q and M : The discrepancy becomes more significant for the beam center bending moment M : As seen from Fig. 7(a), the actual C# p has a gentle peak near the center of the roof, while the quasi-steady GCp has peaks at both ends of the roof. The quasi-steady GCp is much larger than the actual C# p except at the roof center, and the LRC CLRC has a pattern similar to the quasi-steady GCp : For both Frame Cb in the separation bubble region and Frame Cc in the reattached region, LRC CLRC agree very well with the actual C# p :
5. Relative frequency distributions of fluctuating roof pressures and internal forces In order to discuss the reasons of the discrepancy between the LRC CLRC and the actual C# p ; the temporal variations of the roof pressures are first examined. Fig. 8
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Fig. 8. Temporal variations of fluctuating pressures at four points, Nos. 5, 8, 197 and 200 in the separation bubble region.
shows the pressure fluctuation at four points near the leading edge, Nos. 5, 8, 197 and 200. Very high negative peaks are observed in the pressure fluctuations at points Nos. 5 and 197 on Frame Ca. At time ta when the peak pressure was recorded at point No. 5 near the end of Frame Ca, the pressure at point No. 197 near the center of Frame Ca also shows a gentle peak. The time when the peak pressure was recorded at the center point No. 197 is indicated tb in the figure. Fig. 9 shows examples of the relative frequency distributions of the fluctuating pressures. The distributions of Nos. 5 and 197 significantly skew from the Gaussian distribution indicated by dotted line in the figures. The entire span of Frame Ca is immersed in the non-Gaussian pressure field. Thus, non-Gaussian characteristics of the pressure field can be one of the reasons for the discrepancy. Fig. 10 shows examples of the temporal variations of the bending moment M at the beam center of Frames Ca, Cb and Cc. Here only Frame Cc is located in the re-attached region. Compared with Fig. 8, it is obvious that the bending moment of Frame Ca fluctuates following the fluctuation of the pressure at No. 197 near the beam center. The minimum bending moment M of Frame Ca occurred at time tb when the central pressure reached its negative peak. The minimum bending moment of Frame Cb also occurred at time tb reducing its magnitude. The bending moment of Frame Cc had reduced magnitude and showed quite different characteristics.
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Fig. 9. Relative frequency distributions of standardized fluctuating pressures (zero mean, s ¼ 1) at several points.
Fig. 11 shows the relative frequency distributions of the bending moments M of the frames. The bending moments of across-wind Frames Ca and Cb near the leading edge skew from the Gaussian distribution, but that of Frame Cc lying in the re-attached region has a distribution close to Gaussian. An interesting thing is that those of along-wind Frames La, Lb and Lc all skew from the Gaussian distribution. The skewness is most significant for Frame Ca in this case, but it is not necessarily true in other conditions. The load effects have tended to be considered Gaussian because of the summation effects of the randomly fluctuating pressures at many spatial points due to the central limit theorem, even if the pressures show nonGaussian characteristics. However, although the strong non-Gaussian characteristics of pressures are observed only in the region near the leading edge of the roof, the load effects in the frames lying in or including this region, i.e. Frames Ca, Cb, La, Lb and Lc, show strong non-Gaussian characteristics. Here, regarding the load effects in along-wind Frames La, Lb and Lc, only the internal forces at the windward end of the beam were examined. Thus, the pressures in the leading edge region influenced most. The non-Gaussian pressure fluctuation does not seem to be a sufficient condition for the discrepancy between the LRC CLRC and the actual C# p :
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Fig. 10. Temporal variations of bending moments at the beam center of Frames Ca, Cb and Cc.
6. Special features of fluctuating pressure acting on Frame Ca As firstly discussed, the temporal average characteristics of the wind pressures are strongly reflected in the LRC pressure distribution CLRC ; such as the mean value pðjÞ % and covariance pi pj of fluctuating pressures. Regarding the standard deviation spj of the pressures, only Frame Ca shows a special feature having the sharp peaks near both the ends as shown in Fig. 3. Only the mean pressure distribution on Frame Ca also has similar peaks at the ends. These effects are significantly remaining in the LRC pressure distribution CLRC for M as shown in Fig. 7(a), even if the influence coefficient aYi magnifies the contribution of the pressures near the center and reduces those effects near both the ends. However, the actual extreme pressure distribution C# p is extracted at the instance when the peak load effect is observed. For the case of central bending moment of Frame Ca, the extreme instance corresponds to the time when the fluctuating pressure near the beam center reached its minimum, i.e. time tb as shown in Figs. 8 and 10. Fig. 12 may be noteworthy to examine. This figure shows the phase-plane expression of the fluctuating pressures at two points, Nos. 197 and 29. No. 197 is the point near the center of Frame Ca and No. 29 is the point where the peak fluctuating pressure (standard deviation) is observed near the left end of Frame Ca. The
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Fig. 11. Relative frequency distributions of bending moment M of frames.
Fig. 12. Phase-plane expression of fluctuating pressures at two points Nos. 197 and 29.
trajectory shows a quite interesting shape, i.e. an inclined triangular with two horns at the ends of its base. This trajectory implicitly emphasizes the two phenomena: the wind pressure at the center, No. 197, reaches its negative peak and that of the end, No. 29, reaches its negative peak. The temporal average characteristics show some
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Fig. 13. Example of instantaneous pressure pattern causing minimum bending moment M at the beam center of Frame Ca.
correlation by the inclination of the triangular. However, the two special conditions around the tips of the two horns are apparently quite different from the temporal average characteristics. Thus, the trajectory shown in Fig. 12 suggests that the fluctuating pressure field on Frame Ca is very complicated and results from more than two different phenomena. The minimum bending moment M is caused by the extreme condition of one of these phenomena. The temporal average characteristics cannot represent the individual extreme condition. It is also remarkable that the wind pressure No. 29 near the end tends to be very small at the moment when the wind pressure No. 197 near the center reached its negative peak. This moment corresponds to the time when the bending moment of the beam center of Frame Ca becomes its minimum, M : The typical pressure pattern actually causing the minimum bending moment M at the beam center is shown in Fig. 13. The actual pressure pattern C# p was obtained by ensemble averaging of the pressure patterns like Fig. 13 as shown in Fig. 7(a). Here, the main reason of the discrepancy between the LRC CLRC and the actual C# p for the bending moment of Frame Ca can be attributed to the difference of the extreme condition causing the minimum bending moment with the temporal average characteristics of the roof pressures. This extreme condition is limited at near the tip of the right horn of the trajectory in Fig. 12.
7. Concluding remarks The validity of Kasperski’s LRC formula was proved to estimate the actual extreme wind load for the general location of the frame including the separation bubble region. The only exception was where the across-wind frame was immersed in the vicinity of the leading edge, where the non-Gaussian characteristics are
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predominant in pressure fluctuations and very high negative peaks are observed near the windward corners. However, the non-Gaussian characteristics are not a sufficient condition for the discrepancy between the LRC and the actual extreme pressure distributions. The pressure field on Frame Ca shows complex characteristics resulting from more than two different phenomena. The minimum bending moment M at the beam center of the frame is caused by an extreme condition of one of these phenomena, which cannot be represented by the temporal average characteristics. The reason for the discrepancy in the LRC pressure distribution and the actual extreme pressure distribution is attributed to this intermittent characteristics of the fluctuating pressure field on Frame Ca in the vicinity of the leading edge.
References [1] J.D. Holmes, Distribution of peak wind loads on a low-rise building, J. Wind Eng. Ind. Aerodyn. 29 (1988) 59–67. [2] M. Kasperski, Extreme wind load distributions for linear and nonlinear design, Eng. Struct. 14 (1) (1992) 27–34. [3] J.D. Holmes, Optimised peak load distributions, J. Wind Eng. Ind. Aerodyn. 41–44 (1992) 267–276. [4] Y. Tamura, H. Kikuchi, K. Hibi, Extreme wind pressure distributions on low-rise building models, Abstract Volume, Fourth International Colloquium on Bluff Body Aerodynamics and Applications, Ruhr University, Bochum, Germany. [5] M. Kasperski, Design wind loads for low-rise buildings: a critical review of wind load specifications for industrial buildings, J. Wind Eng. Ind. Aerodyn. 61 (1996) 169–179.