# Optimization design of structures subjected to transient loads using first and second derivatives of dynamic displacement and stress

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3 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 447 Fig.. Elmnt cross-sction and lmnt dsign variabls. ( ) = ± d 3 + d 2 2 Th first momnt of th shadd ara with rspct to th nutral axis is Sz = d d d 3 + d d 2 d 2 2 d Th structur is dividd with N lmnts. Th dsign variabl vctor is dfind as. (3) (4) d = d,d 2,d 3,d 4,,d,d 2,d 3,d 4,,d N,d N 2,d N 3,d N ] T 4 (5) Considr th quations of motion for a linar systm subjctd to dynamic forcs Mẍ + Cẋ + Kx = f (t) (6) with th following initial conditions: { x (0) = x0 (7) ẋ (0) = ẋ 0 whr K, M, andc ar stiffnss matrix, mass matrix, and damping matrix, rspctivly. x (t), ẋ (t) and ẍ (t) ar unknown nodal displacmnt, vlocity and acclration vctors, and f(t) is th load vctor. Suppos that th dynamic loads act on th nods. Rayligh damping is usd in this work, th structural damping matrix is C = α M + α 2 K (8) whr ( ) 2 ς ω ς2 ω 2 α = ω 2 ω2 2 (9) α 2 = 2(ς 2ω 2 ς ω ) ω 2 2 ω2 (0) whr ω and ω 2 ar th first and scond natural frquncy of th structur, rspctivly. ς and ς 2 ar th damping ratio. In this work, ς = ς 2 =0.02. α and α 2 ar constants. Equations (6) and (7) must b satisfid for all tim priod t 0, Γ]. Γ is th duration of th dynamic loads. In practic, th solution of this initial-boundary-valu problm (IBVP) rquirs intgration through tim. This is achivd numrically by discrtising in tim th continuous tmporal drivativs that appar in th quation. Any on of th tim intgration procdurs can b usd for this purpos. Th most widly usd family of dirct tim intgration mthods for solving Eq. (6) is th Nwmark family of mthods. Th Nwmark mthod can b formulatd by considring quilibrium at any discrt tim t +Δt, and is givn by th following quation: Mẍ +Cẋ +Kx =f ()

4 448 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Th nodal displacmnts, nodal acclrations and nodal vlocitis can b achivd at any tim t +Δt by solving Eq. (). Prhaps th most widly usd dirct mthod for th quation of motion (6) is th Nwmark mthod, an implicit tchniqu, which consists of th following finit diffrnc assumptions with rgard to th volution of th approximat solution: ( ) ] x =x (t)+δtẋ (t)+δt 2 2 β ẍ (t)+βẍ (2) ẋ =ẋ (t)+δt ( δ) ẍ (t)+δẍ ] (3) whr any particular choic of th paramtrs β and δ dtrmins th stability and accuracy charactristics of th solution. In this work, paramtrs δ 0.5 and β =0.25 (0.5+δ) 2.Wdfin th intgral constants: a 0 = βδt, ( 2 a = δ βδt, a 2 = βδt, a 3 = 2β, a 4 = δ β,a 5 = Δt δ 2 β ), 2 a 6 =Δt ( δ), a 7 = δδt. Th paramtrs β and δ will b rplacd by thos constants in th following formulas. In addition to Eqs (2) and (3) th quilibrium Eq. () at tim station t +Δt is considrd. This way a systm of quations is formd for th dtrmination of th thr unknowns x, ẋ and ẍ, assuming that th displacmnt, vlocity, and acclration vctors at th prvious tim station t hav alrady bn computd. Thus, th solution for th displacmnt vctor is K x =F (4) whr K = K + a 0 M + a C (5) and F =f +M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] (6) Th acclrations, ẍ, which ar rquird for th computations at th nxt tim station, can b computd as ẍ =a 0 x x (t)] a 2 ẋ (t) a 3 ẍ (t) (7) whil th vlocitis, ẋ, can b obtaind dirctly from quation (3), as follows. ẋ =ẋ (t)+a 6 ẍ (t)+a 7 ẍ (8) Whn th dynamic loads act on th nods, w only nd to dtrmin th intrnal forcs and intrnal coupls at th cntr and two nds of th lmnt. Th lmnt nodal forc vctor, F,is F = K T δ (9) whr K is th lmnt stiffnss matrix in a local coordinat systm, T is th lmnt coordinat transformation matrix, δ is th lmnt nodal displacmnt vctor in a local coordinat systm. Th intrnal forc vctor at th cntr of lmnt, F M,is F M = F F 2 l 2 F 2 F 3 ] T (20) whr l is th lngth of lmnt. Th thr trms, i.., F, F 2 and l F 2 2 F 3, ar axial forc, shar forc and bnding momnt, rspctivly. F, F 2 and F 3 ar th first, scond and third trm of th lmnt nodal forc vctor, F, rspctivly. Th imum strss at th cntr and two nds, i.. th i nd and th j nd, of th lmnt ar calculatd as follows. Th imum normal strss at th i nd of th lmnt is σix = F A + F 3 y m (2)

5 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 449 Th imum normal strss at th j nd of th lmnt is σjx = F 4 A + F 6 y m (22) whr F 4 and F 6 ar th fourth and sixth trm of th lmnt nodal forc vctor, F, rspctivly. Th imum normal strss at th cntr of th lmnt is σmx = F M A + F M3 y m (23) whr F M and F M3 ar th first and third trm of th intrnal forc vctor at th cntr of lmnt, F M, rspctivly. Th imum shar strss at th i nd of th lmnt is τixy = F 2 Sz d (24) 4 I z Th imum shar strss at th j nd of th lmnt is τjxy = F 5 S d 4 I z whr F 5 is th fifth trm of th lmnt nodal forc vctor, F. Th imum shar strss at th cntr of th lmnt is τmxy = F M2 S d 4 I z whr F M2 is th scond trm of th intrnal forc vctor at th cntr of lmnt, F M. (25) (26) 2... Formulas for first drivativs of dynamic displacmnt and strss Now w will driv th formulas for th first drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Diffrntiating Eq. (4) with rspct to th dsign variabl d i,whav x K = F K x (27) whr K and = K + a 0 M + a C (28) F a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M = M + C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C a 0 x (t) ẋ (t) + a 2 ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 ] ẍ (t) + a 3 Aftr th first drivativs of displacmnt vctor at tim t +Δt is obtaind from Eq. (27), through diffrntiating Eq. (7) with rspct to dsign variabls d i,whav ] ẍ x x (t) ẋ (t) ẍ (t) = a 0 a 2 a 3 (30) Thn, th first drivativs of acclration vctor at tim t +Δt is obtaind from Eq. (30). Diffrntiating Eq. (8) with rspct to dsign variabl d i,whav ẋ = ẋ (t) + a 6 ẍ (t) (29) + a 7 ẍ (3)

6 450 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Diffrntiating Eq. (9) with rspct to th dsign variabl d i,whav F whr δ (t+δt) = K T δ + K T δ (32) can b calculatd from x(t+δt). Diffrntiating Eq. (20) with rspct to th dsign variabl d i,whav F M = d F (t+δt) i F 2 (t+δt) l 2 F 2 (t+δt) F ] T 3 (t+δt) (33) Diffrntiating Eq. (2) with rspct to th dsign variabl d i,whav σix = F + F 3 A F + F 3 A A 2 F 3 y m EI 2 z Diffrntiating Eq. (22) with rspct to th dsign variabl d i,whav σjx = F 4 + F 6 A F 4 + F 6 A A 2 F 6 y m EI 2 z Diffrntiating Eq. (23) with rspct to th dsign variabl d i,whav σmx = F M A F M A 2 A I z (34) I z (35) + F M3 + F M3 F M3 y m Iz 2 (36) Diffrntiating Eq. (24) with rspct to th dsign variabl d i,whav τixy = F 2 S d 4 I z + S F 2 d 4 I z d 4 F 2 S d 2 I z F 2 4 I z S d 4 I2 z Diffrntiating Eq. (25) with rspct to th dsign variabl d i,whav τjxy = F 5 d 4 F 5 S d 4 I z S d 2 I z F 5 4 I z + S F 5 d 4 I z S d 4 I2 z Diffrntiating Eq. (26) with rspct to th dsign variabl d i,whav τmxy d 4 F M2 = F M2 S d 4 I z S d 2 I z F M2 4 I z + S F M2 d 4 I z S d 4 I2 z (37) (38) (39)

7 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Formulas for scond drivativs of dynamic displacmnt and strss Now w will driv th formulas for th scond drivativs of dynamic rspons, i.., dynamic displacmnt and strss. Furthr diffrntiating Eq. (27) with rspct to th dsign variabl d j,whav K 2 x whr 2 K = and 2 F = 2 F 2 K x K x K x (40) 2 K 2 M 2 C + a 0 + a (4) x (t) ẋ (t) ẍ (t) a 0 + a 2 + a 3 ] = 2 M a 0 x (t)+a 2 ẋ (t)+a 3 ẍ (t)] + M + M ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ẍ (t) a 0 + a 2 + a 3 + M a 0 + a 2 + a C a x (t)+a 4 ẋ (t)+a 5 ẍ (t)] + C ] x (t) ẋ (t) ẍ (t) a + a 4 + a 5 + C ] x (t) ẋ (t) ẍ (t) 2 x (t) 2 ẋ (t) 2 ] ẍ (t) a + a 4 + a 5 + C a + a 4 + a 5 Aftr 2 x(t+δt) is computd from Eq. (40), with furthr diffrntiating Eq. (30) with rspct to th dsign variabls d j,whav 2 ẍ 2 ] x = a 0 2 x (t) 2 ẋ (t) 2 ẍ (t) a 2 a 3 (43) Thn 2 ẍ(t+δt) is obtaind from Eq. (43). Furthr diffrntiating Eq. (3) with rspct to th dsign variabls d j, w obtain 2 ẋ = 2 ẋ (t) + a 6 2 ẍ (t) + a 7 2 ẍ (44) Furthr diffrntiating Eq. (32) with rspct to th dsign variabl d j,whav 2 F (t+δt) 2 = K T δ + K (45) + K T δ (t+δt) + K T 2 δ (t+δt) T δ (t+δt) Furthr diffrntiating Eq. (33) with rspct to th dsign variabl d j, w obtain 2 F M = F 2 (t+δt) 2 F 2 (t+δt) l 2 2 F 2 (t+δt) F ] T 2 3 (t+δt) (46) Furthr diffrntiating Eq. (34) with rspct to th dsign variabl d j,whav 2 σix + 2 F A A A 3 F 3 F 3 = 2 F A F A F A 2 Iz + F 3 Iz y m F 3 A F A 2 2 A + 2 F 3 EI z y m Iz 2 + I z I z F 3 EI 2 z A + F 3 y m + F 3 EI z ] 2 (42) (47) 2 F 3 y m Iz EI 3 F 3 y m 2 Iz EI 2 z z

8 452 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Furthr diffrntiating Eq. (35) with rspct to th dsign variabl d j, w obtain 2 σjx + 2 F 4 F 6 = 2 F 4 A A A A 3 F 4 F I z + F 6 I z y m A 2 F 4 A 2 A 2 A + 2 F 6 I z F 6 F 6 EI 2 z F 4 A 2 + F F 6 Iz + I z 2 F 6 EI 3 Furthr diffrntiating Eq. (36) with rspct to th dsign variabl d j,whav 2 σmx = 2 F M A F M A 2 z A + 2 F M A A A 3 F M 2 A A F M3 F M3 (t+δt) F M3 2 2 Iz + F M3 (t+δt) Iz y m F M3 EI z EI 2 z I z F M3 (t+δt) EI 2 I z z A EI z 2 (48) Iz F 6 2 Iz EI 2 F M A 2 y m z A + F M3 + F M3 (t+δt) 2 + I z 2 F M3 y m Iz 3 F M3 y m 2 Iz 2 (49) Furthr diffrntiating Eq. (37) with rspct to th dsign variabl d j, w obtain 2 τixy F 2 Iz S = 2 F 2 d 4 I2 z S d 4 F 2 d 2 4 I z d 4 F 2 Sz d 2 d 4 4 I z + d 4 Iz F 2 Sz d 2 4 I2 z Sz d + F 2 4 I z + 2 Sz F 2 d 4 I z S Iz S S d 4 I z + S F 2 F 2 d 4 I z F 2 d 2 d 4 F 2 S 4 I2 d 2 z F 2 d 2 + d 4 d 4 2 F 2 4 I z 2 Iz F 2 S d 4 I2 z 4 I z EI z d 4 Sz d 2 S d 3 4 I z I z F 2 S d 4 I2 z I z Sz F 2 d + I z d 4 F 2 S 4 I2 z d 2 + I z Iz 2 F 2 S 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (38) with rspct to th dsign variabl d j,whav 2 τjxy F 5 Iz S = 2 F 5 d 4 I2 z S d 4 I z + F Sz F 5 d 4 I z S d 4 I z + S F 5 4 I z F 5 d 4 Sz d 2 d 4 I z 4 I z (50)

9 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 453 S d 4 d 4 F 5 + d 4 I z F 5 F 5 d 2 4 I z S Iz F 5 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F 5 d 4 I2 z 2 d 4 F 5 F 5 d 2 + d 4 d 4 2 F 5 4 I z S d I z F 5 4 I2 z S d 2 4 I z S d 3 4 I z S d 4 I2 z I z Sz F 5 d + I z d 4 F 5 Sz 4 I2 z d 2 + I z Iz 2 F 5 Sz 4 I2 z d 4 I3 z Furthr diffrntiating Eq. (39) with rspct to th dsign variabl d j, w obtain 2 τ Mxy (t+δt) = 2 F M2 (t+δt) F M2 Iz S S d 4 F M2 d 4 I2 z d 2 4 I z d 4 F M2 + d 4 I z F M2 I z S Sz d 4 I z + F M2 (t+δt) + 2 Sz F M2 d 4 I z S Iz F M2 Sz d 2 d 4 4 I z S d 2 4 I2 z S 2 I z F M2 F M2 d + I z d 4 F M2 4 I2 z d 4 I2 z S d 4 I z + S F M2 (t+δt) d 4 F M2 2 d 4 F M2 F M2 d 2 + d 4 d 4 2 F M2 4 I z S d I z 4 I2 z S d 2 4 I2 z d 4 I z S d 2 4 I z S d 3 4 I z F M2 + I z I z 2 F M2 Sz d 2 S d 4 I2 z S d 4 I3 z Computation procdur of th first and scond drivativs of dynamic displacmnt and strss In this work, w suppos that th initial conditions ar x (0) = 0, ẋ (0) = 0, ẍ (0) = 0, and th dynamic rsponss (nodal displacmnts and strsss), thir first and scond drivativs with rspct to dsign variabls ar qual to zro. Sction 2..2 givs th formulas for calculating th first and scond drivativs of th dynamic rspons. This sction provids th dtaild computation procdur as follows. Procdur of calculating dynamic rspons first and scond drivativs: Stp Initial calculations: Stp. x (0) = 0,ẋ (0) = 0,ẍ (0) = 0,σix (0) = 0,σ jx (0) = 0,σ Mx (0) = 0,τ ixy (0) = 0,τ jxy (0) = 0, τ Mxy (0) = 0. Stp.2 x(0) =0, ẋ(0) τ jxy (0) =0, τ Mxy (0) =0. Stp.3 2 x(0) =0, 2 ẋ(0) =0, ẍ(0) =0, σ ix (0) =0, σ jx (0) =0, σ Mx (0) =0, τ ixy (0) =0, 2 τ ixy (0) =0, 2 τ jxy (0) =0, 2 τ Mxy (0) =0. =0, 2 ẍ(0) =0, 2 σ ix (0) =0, 2 σ jx (0) =0, 2 σ Mx (0) =0, 4 I z (5) (52)

10 454 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Stp.4 St Δt. Stp.5 Comput intgral constants: a 0 = βδt 2,a = δ βδt,a 2 = βδt,a 3 = 2β,a 4 = δ β,a 5 = Δt 2 a 7 = δδt. whr intgral paramtrs δ 0.5 and β =0.25 (0.5+δ) 2. Stp.6 Solv Eq. (5) K. ( ) δ β 2,a 6 =Δt ( δ), Stp 2 Calculations for ach tim stp t +Δt: Stp 2. Solv Eq. (4) x. Stp 2.2 Solv Eq. (7) ẍ, solv Eq. (2) σix, solv Eq. (22) σ jx, solv Eq. (23) σmx, solv Eq. (24) τ ixy, solv Eq. (25) τ jxy, and solv Eq. (26) τmxy. Stp 2.3 Solv Eq. (8) ẋ. Stp 2.4. Solv Eq. (27) x(t+δt). Stp 2.5 Solv Eq. (30) ẍ(t+δt) solv Eq. (35) σ jx (t+δt) solv Eq. (37) τ ixy (t+δt) and solv Eq. (39) τ Mxy (t+δt). Stp 2.6 Solv Eq. (3) ẋ(t+δt). Stp 2.7 Solv Eq. (40) 2 x(t+δt)., solv Eq. (34) σ ix (t+δt), solv Eq. (36) σ Mx (t+δt),,, solv Eq. (38) τ jxy (t+δt), Stp 2.8 Solv Eq. (43) 2 ẍ(t+δt), solv Eq. (47) 2 σix (t+δt), solv Eq. (48) 2 σ jx (t+δt), solv Eq. (49) 2 σ Mx (t+δt), solv Eq. (50) 2 τixy (t+δt), solv Eq. (5) 2 τjxy (t+δt), and solv Eq. (52) 2 τ Mxy (t+δt). Stp 2.9 Solv Eq. (44) 2 ẋ(t+δt). Stp 3 Rptition for th nxt tim stp. Rplac t by t +Δt and implmnt stps 2. to 2.9 for th nxt tim stp First and scond drivativs of structural mass W us structural mass as th objctiv function in optimization. Th objctiv function or th structural mass can b xprssd as w (d) = N ρa l = Th first drivativs of th structural mass ar obtaind by diffrntiating Eq. (53) with rspct to th dsign variabls, w (d) N A = ρl (54) = Th scond drivativs of th structural mass can b obtaind by furthr diffrntiating Eq. (54) with rspct to th dsign variabls, (53)

11 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs w (d) = N 2 A ρl = (55) 3. Optimization mathmatical modl In gnral, w aim to minimiz th total mass w (d) of th structur. At th sam tim, th normal strsss, shar strsss, and nodal displacmnts should satisfy th constraints in th duration of transint loads. W formulat th optimization problm of th structurs subjctd to transint loads as follows. Find d min w (d) s.t. σix + (d,t) ( =, 2,,N,t 0, Γ]) σ ix (d,t) ( =, 2,,N,t 0, Γ]) σ + jx (d,t) ( =, 2,,N,t 0, Γ]) σ jx (d,t) ( =, 2,,N,t 0, Γ]) σ + Mx (d,t) ( =, 2,,N,t 0, Γ]) σ Mx (d,t) ( =, 2,,N,t 0, Γ]) τ] τixy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τjxy (d,t) τ] ( =, 2,,N,t 0, Γ]) τ] τmxy (d,t) τ] ( =, 2,,N,t 0, Γ]) x k ] x k (d,t) x k ] (k =, 2,,N f ) d J d J d J (J =, 2,, 4N) (56) whr + and ar tnsion and comprssion, rspctivly. is allowabl normal strss. τ] is allowabl shar strss. Γ is th duration of transint loads. N f is th numbr of dgr of frdom. x k ] is allowabl displacmnt on th kth dgr of frdom. d J is th lowr limit of th Jth dsign variabl. dj is th uppr limit of th Jth dsign variabl. Normalizing th constraints of th optimal modl Eq. (56), w obtain th nw quivalnt mathmatic modl as follows. Find d min w (d) s.t. g (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ+ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= σ ix (d,t) 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= σ+ jx (d,t) 0( =, 2,,N; t 0, Γ])

12 456 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs g +6N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= σ jx (d,t) 0( =, 2,,N; t 0, Γ]) g +8N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +9N (d,t)= σ+ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +0N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +N (d,t)= σ Mx(d,t) 0( =, 2,,N; t 0, Γ]) g +2N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +3N (d,t)= τ ixy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +4N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +5N (d,t)= τ jxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +6N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g +7N (d,t)= τ Mxy(d,t) τ ] 0( =, 2,,N; t 0, Γ]) g k+8n (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g k+nf +8N (d,t)= x k(d,t) x k ] 0(k =, 2,,N f ) g J (d) = dj d J 0(J =, 2,, 4N) g J+4N (d) = dj d J 0(J =, 2,, 4N) (57) 4. Transformation of mathmatical modl Th ky fatur of th transformation mthod is to transform a constraind problm into an unconstraind problm. Thus, w minimiz only on function in th transformation mthod. This is an attractiv aspct in that many tim-dpndnt constraints and an objctiv function can b mrgd into a singl function. Th rprsntativs of th transformation mthod ar th augmntd Lagrang multiplir mthod 6] and th xtrior pnalty function mthod 20] in structur optimal dsign undr dynamic loads. Howvr, th augmntd Lagrang multiplir function and th xtrior pnalty function ar discontinuous functions, thrfor, th gradint and Hssian matrix calculations of ths functions ar difficult whn th dirct diffrntiation mthod is mployd to obtain th first and scond drivativs. Compard to th augmntd Lagrang multiplir function and th xtrior pnalty function, th intrior pnalty function is a continuous function, so th gradint and Hssian matrix calculations of th intrior pnalty function ar rlativly asy whn th dirct diffrntiation mthod is usd to obtain th first and scond drivativs. Th intrior pnalty function mthod rquirs a fasibl initial dsign point. Typically, it may b difficult to obtain a fasibl initial dsign in a complx problm. Howvr, in structural optimization problms, a fasibl dsign point can b found in th structurs with th larg cross-sctional aras. Thrfor, in this papr th intrior pnalty function is mployd to transform th inquality constraint optimization problm.

13 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Intrior pnalty function mthod In this papr th inquality constraint optimal mathmatic modl Eq. (57) is convrtd to a squnc of appropriatly formd unconstraind intgral mathmatic modl using th intrior pnalty function mthod. Th intrior pnalty function mthod is adoptd as follows. 2N f +8N Γ P (d,r k )=w (d) r k 8N g J (d,t) dt + (58) g I (d) J= 0 In Eq. (58), pnalty paramtr r k is a squnc of numbrs which ar dgrssiv. Whn r k 0, th minimum of pnalty function P (d,r k ) approachs th minimum of th constraint problm. So th solution of th inquality constraint optimization modl, Eq. (57), is transformd into a squnc of unconstraint problms: Find d 2N f +8N Γ min P (d,r k )=w (d) r k 8N g J (d,t) dt + (59) g I (d) J= 0 Initial pnalty paramtr r can b calculatd by th following quation, 2N f +8N Γ r 8N g J (d 0,t) dt + = p 0 g I (d 0 ) 00 w (d 0) (60) J= 0 I= whr d 0 is th initial dsign point and p 0 = 50. In this work, w choos p 0 = 50. Th pnalty paramtr r k dcrass according to th following rul: r k+ = r k (6) c whr c =0 50 and c = 0 in this work. k is th numbr of pnalty paramtr which will b usd in th procss of sarch Calculation of gradint and Hssian matrix of intrior pnalty function Now w calculat th first and scond drivativs of th pnalty function with rspct to th structural dsign variabls. Th tim stp and duration of dynamic loads ar Δt and Γ, rspctivly. Lt a = Γ Δt. Th first drivativs of pnalty function can b obtaind by diffrntiating Eq. (58) with rspct to th dsign variabl d i, P (d,r k ) = w (d) + r k 2N f +8N Γ J= 0 I= I= g J (d,t) 8N gj 2 (d,t) dt + I= g I (d) gi 2 (d) (62) Th scond drivativs of pnalty function is calculatd by furthr diffrntiating Eq. (62) with rspct to th dsign variabl d j, 2 P (d,r k ) 0 = 2 w (d) + r k 2N f +8N J= Γ 0 2 g J (d,t) gj 3 (d,t) g J (d,t) + 2 ] g J (d,t) gj 2 (d,t) dt 8N 2 g I (d) g I (d) + r k g 3 I= I (d) (63) Th intgral trms in Eqs (58), (59), (60), (62), (63) ar computd by using th trapzoidal form intgral formula: Γ a g J (d,t) dt = ] Δt 2 g J (d,zδt) + (64) g J (d, (z +)Δt) z=0

14 458 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs Γ 0 g J (d,t) gj 2 (d,t) a a dt= z=0 Δt 2 Γ 2 g J (d,t) g J (d,t) 0 gj 3 (d,t) + { Δt 2 g J (d,zδt) 2 g 3 z=0 J (d,zδt) 2 g J (d, (z +)Δt) gj 3 (d, (z +)Δt) g 2 J (d,zδt) g J (d,zδt) + 2 ] g J (d,t) gj 2 (d,t) dt = g J (d,zδt) + g 2 c (d,zδt) g J (d, (z +)Δt) + ] g J (d, (z+)δt) gj 2 (d, (z+)δt) (65) 2 ] g J (d,zδt) + (66) ]} 2 g J (d, (z +)Δt) gj 2 (d, (z +)Δt) Th first and scond drivativs of th pnalty function with rspct to th structural variabls ar calculatd. Thn th gradint and Hssian matrix can b achivd. 5. Solving optimization problms Th inquality constraint optimization modl, Eq. (57), is convrtd into a squnc of th appropriatly formd unconstraind intgral modl, Eq. (59). Marquardt s mthod, a gradint-hssian matrix-basd algorithm, is adoptd to solv th unconstraind problm, taking advantags that th gradint and Hssian matrixs ar fully usd in this optimal mthod. Marquardt s mthod combins Cauchy s and Nwton s mthods in a convnint mannr that xploits th strngths of both but dos rquir scond-ordr information. Th major mrit of Marquardt s mthod is its simplicity, dscnt proprty, xcllnt convrgnc rat nar th optimum, and absnc of a lin sarch. Basd on Marquardt s mthod, th computation procdur of solving th mathmatic modl Eq. (56) is as follows. Th computr procdur of solving th mathmatic modl Eq. (56): Stp. Chos th initial fasibl dsign point d 0, calculat r by solving Eq. (60), dfin convrgnc critrion ε,ltk =. Stp 2. Start from dsign point d k, solv th mathmatic modl Eq. (59) with Marquardt s mthod to obtain th optimum dsign d k.th stps of solving th mathmatic modl Eq. (59) with Marquardt s mthod is from Stp 2.. to Stp 2..: Stp 2.. Lt d (0) k = d k. Dfin M I = imum numbr of itrations allowd ε 2 =convrgnc critrion I =idntity matrix Stp 2.2. St i =0. λ (0) ( =0 5. ) Stp 2.3. Calculat P d (i) k,r k. ( Stp 2.4. Is P d (i) k k),r ε2? Ys: Go to stp 2.. No: Continu. Stp 2.5. Is i M I? Ys: Go to stp 2.. No: Continu. Stp 2.6. Calculat S Stp 2.7. St d (i+) k ( ) d (i) k = ( k + S = d(i) ( d (i+) k,r k Stp 2.8. Is P Ys: Go to stp 2.9. ) <P 2 P d (i) k ). ( d (i) k,r k ( ) ] ( ) d (i) k,r k + λ (i) I P d (i) k,r k. )?

15 Q. Liu t al. / Optimization dsign of structurs subjctd to transint loads using first and scond drivativs 459 No:Gotostp2.0. Stp 2.9. St λ (i+) = 2 λ(i) and i = i +.Gotostp2.3. Stp 2.0. St λ (i) =2λ (i).gotostp2.6. Stp 2.. St d k = d (i) k. Go to stp 3. Stp 3. Is P (d k,r k ) P (d k,r k ) ε? Ys: d k is th bst dsign. Printrsultandstop. No: Continu. Stp 4. Calculat r k+ by solving Eq. (6), k = k +.Gotostp2. 6. Numrical xampl In this sction, optimization dsign of th plan fram shown in Fig. 2 is prformd using th mthod proposd in this papr. Th plan fram is dividd with thr plan bam lmnts. Th cross-sction of th lmnt is H structural-stl shap, shown in Fig.. W us typical stl matrials proprtis, i.., lastic modulus E = 20 GPa and matrial dnsity ρ = 7850 kg/m 3, for all lmnts. Th structural damping ratios ar ς = 0.02, ς 2 = Th horizontal dynamic load, F h (t) = 500 sin 3π 4 t (kn), acts on th nod 2. A duration of 3 s and an incrmntal tim stp of 0.0 s ar considrd in optimization procdurs. Th convrgnc critrion: ε = 0 3, ε 2 = 0 3 and th imum numbr of itrations allowd: M I = 5. Numrical tsts show that ths critria ar sufficint for achiving convrgnc in a rasonabl tim. Fig. 2. Plan fram. In th optimal mathmatic modl Eq. (57), th allowabl normal strss =200 MPa and allowabl shar strss τ] =00 MPa; th nodal allowabl displacmnts ar: x ]=0.00 m, x 2 ]=0.00 m, x 3 ]=0. rad, x 4 ] = 0.00 m, x 5 ] = 0.00 m, x 6 ] = 0. rad; th dsign variabl vctor is dfind as: d = d,d 2, d 3,d 4,d2,d2 2,d2 3,d2 4,d3,d3 2,d3 3,d3 4 ]T. Th dsign spac is shown in Tabl. Tabl Dsign spac of fram (unit: mm) Dsign variabls d d 2 d 3 d 4 d 2 d 2 2 d 2 3 d 2 4 d 3 d 3 2 d 3 3 d 3 4 Lowr limit Uppr limit Th optimum dsigns of th plan fram ar sarchd from two diffrnt initial fasibl dsign points. Squncs of optimum dsigns ar shown in Tabls 2 and 3. From th optimum rsults shown in Tabls 2 and 3, th masss of th optimum dsigns approach to th final targts, 26 kg and 6 kg, from th initial fasibl dsign (i.., 53 kg) and 2 (i.., 45 kg), rspctivly. It indicats that th optimization mthod prsntd in this papr is ffctiv. Howvr, th optimum dsigns ar diffrnt if th initial dsigns ar not sam. Thrfor, th optimum dsigns obtaind with th optimization mthod in this papr ar local solutions, but not global solutions. W should find th diffrnt local solutions from as many initial dsigns as possibl. Thn w can choos th bst dsign from th local dsigns as th ffctiv dsign.

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