Professor Terje Haukaas University of British Columbia, Vancouver Load Combination

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1 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve Load Combination This document descibes the load coincidence method (Wen 1990) and seveal load combination ules that ae utilized in pactice. The load coincidence method employs Poisson pulse pocesses, which ae descibed in anothe document. Load Coincidence Method To undestand the load coincidence method, thee pocesses ae consideed. The two pocesses s 1 and s 2 ae odinay pulse pocesses that ae eithe intemittent o always on. The thid pocess is the coincidence pocess s 12 =s 1 +s 2, which occu only when s 1 and s 2 coincide. It is stessed that s 12 is only on when s 1 and s 2 coincide; othewise it is zeo, even though its intensity is defined by s 12 =s 1 +s 2. Of pimay inteest in engineeing applications is the pobability that the intensity exceeds a theshold,. Exceedance can happen in thee ways: eithe s 1 o s 2 o s 12 exceeds. To addess these thee situations, thee andom vaiables ae defined: =(max{s 1 } fo t (0,T) =(max{s 2 } fo t (0,T) 2 =(max{s 12 } fo t (0,T) The sought esult is essentially the pobability that the ealization of any of these andom vaiables exceed : p f = P( > > 2 > ) (1) Howeve, in keeping with the document on pulse pocesses, it is pefeed to wok with the complementay event, i.e., the one that is expessed in tems of the CDF fo the lifetime maximum intensity, which by de Mogan s ules is witten: F() = 1 p f = 1 P > > 2 > = P 2 Although the thee events ae statistically dependent because R12 is positively coelated with both R1 and R2, independence is assumed to simplify the subsequent deivations. Clealy, this intoduces an eo, but the eo is consevative (it leads to an oveestimation of the pobability of theshold exceedance) and studies have shown that the eo is typically not lage (Wen 1990). Independence yields: F() = P( ) P( ) P( 2 ) (3) = F R1 () F R1 () F R12 () The fist two factos in the ight-hand side ae the CDF fo the lifetime maximum value of the two pocesses s 1 and s 2. Expessions fo these ae povided in the document on pulse pocesses. The CDF fo the lifetime maximum value of s 12 is the subject of the following deivations. As a stating point it is ecognized that, duing any shot time (2) Load Combination Updated Febuay 22, 2014 Page 1

2 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve inteval, Δt, coincidence between s 1 and s 2 can occu in two mutually exclusive ways. The fist is that s 1 is on duing Δt and s 2 comes on duing the occuence of s 1. The duation of the occuence of s 1 is hee denoted by d 1. Accoding to the Poisson distibution, the pobability of occuence of s 1 in the inteval Δt is P(x > 0) = 1 p(0) = 1 exp λ 1 Δt (4) whee λ 1 is the ate of occuences with non-zeo occuences. Similaly, the pobability that s 2 comes on duing d 1 is P(x > 0) = 1 exp λ 2 d 1 (5) Theefoe, the pobability of this fist souce of coincidence events is: P(Coincidence, Case 1) = ( 1 exp( λ 1 Δt) ) 1 exp λ 2 d 1 λ 1 Δt 1 exp λ 2 d 1 ( ) whee the second equality is made unde the assumption that Δt is indeed small. The othe possibility fo coincidence is that s 2 is on duing Δt and that s 1 comes on duing the occuence of s 2. By denoting the duation of the occuence of s 2 by d 2, the following esult analogous to Eq. (6) is obtained: P(Coincidence, Case 2) λ 2 Δt 1 exp( λ 1 d 2 ) (6) (7) Because the two possibilities ae mutually exclusive, the final pobability of coincidence is obtained by simple summation: P(Coincidence) λ 1 Δt 1 exp( λ 2 d 1 ) + λ 2 Δt ( 1 exp( λ 1 d 2 )) (8) Futhemoe, because the pobability of moe than one coincidence in Δt is negligible, the aveage numbe of coincidences in Δt equals P(Coincidence). Dividing this numbe by the time inteval yields the ate of coincidence: = P(Coincidence) Δt + λ 2 ( 1 exp( λ 1 d 2 )) (9) λ 1 1 exp( λ 2 d 1 ) This impotant esult is futhe simplified by noting that: λ 1 λ 2 d 1 + d 2 (10) Howeve, d 1 and d 2 ae andom vaiables, and this is addessed by utilizing the expectation: = λ 1 λ 2 d 1 + d 2 f (d 1 ) f (d 2 )dd 1 dd 1 = λ 1 λ 2 µ d1 + µ d 2 (11) This expession has been veified by Monte Calo sampling analysis and has poven accuate even when λµ d is not vey small fo eithe of the pocesses (Wen 1990). Having the occuence ate fo the coincidence pocess, its mean duation is obtained by the following easoning: The pobability that the coincidence pocess is on is µ d21 and this is equal to the pobability that both of the individual pocesses ae on, namely (λ 1 µ d1 )(λ 2 µ d2 ). As a esult: Load Combination Updated Febuay 22, 2014 Page 2

3 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve Combination of Eqs. (11) and (12) yields: µ d12 = λ 1 λ 2 µ d1 µ d 2 (12) µ d12 = µ d1 µ d 2 µ d1 + µ d 2 (13) The thid and final quantity equied to descibe the coincidence pocess, s 12, is its abitay point in time (APIT) distibution. Essentially, the techniques fom the topic functions of andom vaiables ae employed to obtain this distibution, i.e., the distibution of s 12 =s 1 +s 2. In doing so, F Y1 () and F Y2 () ae given and F Y12 () is sought. Having detemined all thee chaacteistics of the coincidence pocess, the CDF fo its maximum lifetime intensity is, fom the document on pulse pocesses: F R12 ( ) (14) = exp T 1 F Y12 () Consequently, the esult sought in this document, expessed ealie in Eq. (3), is F() F R1 () F R1 () F R12 () ( λ 2 T ( 1 F Y2 ()) T ( 1 F Y12 ())) exp λ 1 T 1 F Y1 () In situations whee one load is always on this expession simplifies. Fo example, if s 1 is always on, then s 2 cannot occu alone, and Eq. (15) simplifies to: F() exp λ 1 T 1 F Y1 () (15) ( T ( 1 F Y12 ())) (16) Load Reduction Facto Rule The objective in the pevious deivations fo the load coincidence method was to compute the pobability that the oveall pocess max{s 1 +s 2 } does o does not exceed. Impotantly, max{s 1 +s 2 } should not be confused with the moe limited pocess s 12, which occu only when s 1 and s 2 coincide. Simplified load combination ules, such as the load eduction facto ule, addesses the same poblem but without employing the coincidence pocess s 12 o its lifetime maximum value 2. Instead, the load combination ules only assume that the lifetime maximum value of the individual pocesses, i.e.,, R 3, etc. ae available. Accodingly, design citeia ae fomulated in tems of, R 3, etc. athe than in tems of max{s 1 +s 2 }. If thee wee no chance of coincidence the ule would be simple: max{s 1 +s 2 }= max{ + }. To account fo the possibility of coincidence, the design citeia usually involve some auxiliay load factos. Fo example, the load eduction facto ule povides the following appoximation fo two load pocesses { } max, γ 1 ( + ) { } (17) Load Combination Updated Febuay 22, 2014 Page 3

4 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve whee typical values fo γ 1 ae in the ange 0.7 to 0.8. The bodeline between what is consideed exceedance of max{s 1 +s 2 } and what is consideed non-exceedance is shown by dashed lines in Figue 1. Fo thee load pocesses the ule genealizes to whee typically γ 2 =0.66. { } max + s 3, R 3, γ 1 ( + ), γ 1 ( ), γ 1 ( ), γ 2 ( + ) (18) 2 = = γ 1 max { s s } = Figue 1: Visualization of the failue domain fo the LRF ule. Squae Root of Sum of Squaes Rule With this load combination ule, exceedance of max above is assumed to occu when the following quantity exceeds : max{ s 1 } The bodeline between exceedance and non-exceedance is shown as a dashed line in Figue 2. Fo thee loads, this ule genealizes to: (19) max{ s 1 + s 3 } (20) Load Combination Updated Febuay 22, 2014 Page 4

5 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve max { s s } Figue 2: Visualization of the failue domain fo the SRSS ule. Companion Action Facto Rule This ule is adopted in ecent vesion of the National Building Code of Canada, whee seveal load cases must be consideed. Each load case has a pimay load and one o moe companion loads. In the case of two loads, the appoximation is: { } max{ + γ 2 + γ 1 } (21) The bodeline between exceedance and non-exceedance is shown as a dashed line in Figue 3. Fo thee loads, this ule genealizes to: { } max + s 3 + γ 2 + γ 3 R 3, + γ 1 + γ 3 R 3, R 3 + γ 1 + γ 2 R 2 (22) = 2 + γ 1 1 = 1 + γ 2 2 max { s s } Figue 3: Visualization of the failue domain fo the CAF ule. Load Combination Updated Febuay 22, 2014 Page 5

6 Pofesso Teje Haukaas Univesity of Bitish Columbia, Vancouve Tuksta s Rule Tuksta s load combination ule is unique in the sense that it intoduces APIT values in the design citeia. As a esult, this ule cannot be visualized in the - plane as the othe ules ealie. Fo two loads, Tuksta s ule eads: { } max{ + s 2 + s 1 } (23) The bodeline between exceedance and non-exceedance is shown as a dashed line in Figue 3. Fo thee loads, this ule genealizes to: { } max{ + s 2 + s 3 + s 1 + s 3, R 3 + s 2 + s 3 } (24) + s 3 (Moe details to be witten.) Refeences Wen, Y. K. (1990). Stuctual load modeling and combination fo pefomance and safety evaluation. Elsevie. Load Combination Updated Febuay 22, 2014 Page 6