FEMA 451B Topic 4 Handouts Structural Dynamics of MDOF 1. Symbol Styles Used in this Topic M U. Matrix or vector (column matrix) m u

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1 Structural Dyacs of Lear Elastc Multple-Derees-of-Freedo (MDOF) Systes u u u Structural Dyacs of Elastc MDOF Systes Equatos of oto for MDOF systes Ucoupl of equatos throuh use of atural ode shapes Soluto of ucoupled equatos Recobato of coputed respose Modal respose hstory aalyss Modal respose spectru aalyss Equvalet lateral force procedure Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - M U u Sybol Styles Used ths opc Matrx or vector (colu atrx) Eleet of atrx or vector or set (ofte show wth subscrpts) Relevace to ASCE 7-5 ASCE 7-5 provdes udace for three specfc aalyss procedures: Equvalet lateral force (ELF) aalyss Modal superposto aalyss (MSA) Respose hstory aalyss (RHA) C s ELF usually allowed ELF ot allowed W Scalars See ASCE 7-5 able.6- s.5 s Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Plaar Frae wth 6 Derees of Freedo Majorty of ass s floors 9 uy rz ux ypcal odal DOF Plaar Frae wth 6 Statc Derees of Freedo But wth Oly HREE Dyac DOF u Moto s predoatly lateral u u U u u u Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

2 Developet of Flexblty Matrx f kp d, d, d, d, Developet of Flexblty Matrx (cotued) d, f kp d, d, d, d, d, d, d, d, d, d, Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 f kp Developet of Flexblty Matrx (cotued) d, d, d, d, d, d, d, d, d, d, d, d, Cocept of Lear Cobato of Shapes (Flexblty) d, d, d, f U d, d, d, f d, d, d, f d, d, d, U d f + d f + d f d, d, d,,,, D F U - K D K U F Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Statc Codesato K, K, U F K, K, U {} K U + K U F,, K U K U + {},, DOF wth ass Massless DOF Rearrae Plu to Splfy Statc Codesato (cotued) U K K U,, K U K K K U F,,,, K K K K U F,,,, ˆK K K K K,,,, Codesed stffess atrx Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

3 Idealzed Structural Property Matrces f (t), u (t) k -k K -k k + k -k k -k k +k f (t), u (t) k k f( t) u() t F f U u f() u() t t M f (t), u (t) Note: Dap to be show later Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Coupled Equatos of Moto for Udaped Forced Vbrato MU && () t + KU() t F() t u && k k u f u && + k k+ k k u f + u && k k k u f DOF u && + ku ku f DOF u() && t ku() t + ku() t + ku() t ku() t f() t DOF u() && t ku() t + ku() t + ku() t f() t Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Develop a Way o Solve the Equatos of Moto hs wll be doe by a trasforato of coordates fro oral coordates (dsplaceets at the odes) o odal coordates (apltudes of the atural Mode shapes). Solutos for Syste Udaped Free Vbrato (Natural Mode Shapes ad Frequeces) MU && () t () + KU t { } Assue U φ sωt he Kφ ω Mφ { } U( && t) ω φ sωt has three () solutos: Because of the orthooalty property of the atural ode shapes, the equatos of oto becoe ucoupled, allow the to be solved as SDOF equatos. After solv, we ca trasfor back to the oral coordates. φ, φ φ,, ω φ, Natural ode shape φ, φ φ,, ω φ, Natural frequecy φ, φ φ,, ω φ, Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Solutos for Syste Udaped Free Vbrato (cotued) For a SINGLE Mode φ, Mode Shapes for Idealzed -Story Frae φ, φ, KΦ MΦΩ For ALL Modes Where: Φ [ φ φ ] φ ω Ω ω Kφ ω Mφ ω φ,. Note: Mode shape has arbtrary scale; usually or Φ MΦ I φ, φ, φ, φ, Node φ, φ, Node Node MODE MODE MODE Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

4 Cocept of Lear Cobato of Mode Shapes (rasforato of Coordates) φ, φ, φ, y U φ, φ, φ, y φ φ φ,,, y U ΦY Mode shape φ, φ, φ, U φ y + φ y + φ y φ φ φ,,,,,, Modal coordate apltude of ode shape Orthooalty Codtos Φ [ φ φ ] φ Geeralzed ass Geeralzed stffess k Φ MΦ Φ KΦ k k Geeralzed dap Geeralzed force c f() t Φ CΦ c Φ F f c f() t Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Developet of Ucoupled Equatos of Moto MDOF equato of oto: rasforato of coordates: Substtuto: Preultply by Φ : MU+ && CU+KU & F U ΦY MΦY+C && ΦY+K & Φ Y F Φ Φ && +Φ Φ & M Y C Y +Φ KΦ Y Φ F () t Us orthooalty codtos, ucoupled equatos of oto are: && & y c y k y f ( t ) && & y + c y + k y f ( t ) && & y c y k y f ( t ) Developet of Ucoupled Equatos of Moto (Explct For) Mode Mode Mode y+cy+ky && & f y+cy+ky && & f() t y+cy+ky && & f() t Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Developet of Ucoupled Equatos of Moto (Explct For) c Splfy by dvd throuh by ad def ξ ω Mode Mode Mode y&& + ξωy& + ω y f / y&& + ξωy& + ωy f / y&& + ξωy& + ωy f / Earthquake Load for MDOF Syste u& & U&& r, u&& + && () t u r,() t F( && + && I t) M u u r, && + && u() t u r,() t. u && r, M. u&& () + M u && t r,() t. u && r, Move to RHS as F - M R u&& EFF Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 4

5 Modal Earthquake Load u u u + u u u u&& (t) u&& () t M M () F t Φ MRu&& () t R R Defto of Modal Partcpato Factor For earthquakes: ypcal odal equato: f () t φ MRu() && t Modal partcpato factor p f φ && + ξω & () t MR y + ω && y y u ( ) t Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Cauto Reard Modal Partcpato Factor p φ MR Varato of Frst Mode Partcpato Factor wth Frst Mode Shape p. p.4 p.6... φ Mφ Its value s depedet o the (arbtrary) ethod used to scale the ode shapes. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 Cocept of Effectve Modal Mass p For each Mode I, he su of the effectve odal ass for all odes s equal to the total structural ass. Varato of Frst Mode Effectve Mass wth Frst Mode Shape /M.. /M /M he value of effectve odal ass s depedet of ode shape scal. Use eouh odes the aalyss to provde a total effectve ass ot less tha 9% of the total structural ass. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 5

6 Dervato of Effectve Modal Mass (cotued) For each ode: y&& + ξωy& + ω y pu&& SDOF syste: q&& + ξωq& + ω q u&& Recall Substtute Dervato of Effectve Modal Mass (cotued) Fro prevous slde y() t pq() t u() t φ y u() t pφ q() t Modal respose hstory, q (t) s obtaed by frst solv the SDOF syste. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Dervato of Effectve Modal Mass (cotued) Appled statc forces requred to produce u (t): V() t Ku () t PKφ q () t Recall: Substtute: Kφ ω Mφ V() t MφPω q () t Dervato of Effectve Modal Mass (cotued) otal shear ode: V V R φ ω φ ω V ( M ) RP q () t MRP q () t Defe effectve odal ass: M φ MRP ad V Mω q () t Accelerato ode Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 φ MR M φ MRP φ MφP φ Mφ M Dervato of Effectve Modal Mass (cotued) P Developet of a Modal Dap Matrx I prevous developet, we have assued: c Φ CΦ c c wo ethods descrbed here: Rayleh proportoal dap Wlso dscrete odal dap Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 6

7 MASS PROPORIONAL DAMPER Rayleh Proportoal Dap (cotued) SIFFNESS PROPORIONAL DAMPER C α M + βk Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Rayleh Proportoal Dap (cotued) C α M + βk For odal equatos to be ucoupled: ω ξ φ Cφ Us orthooalty codtos: ω ξ α + βω ω α + ω ξ Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 β Assues Φ MΦ I Rayleh Proportoal Dap (cotued) Select dap value two odes, ξ ad ξ Copute coeffcets α ad β: α ωω ω ω ξ β ω ω ω ω / / ξ For dap atrx C α M + βk Rayleh Proportoal Dap (Exaple) 5% crtcal Modes ad Structural frequeces Mode ω Modal Dap Rato.5..5 YPE MASS SIFFNESS OAL α.4487 β Frequecy, Radas/Sec. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Rayleh Proportoal Dap (Exaple) 5% Dap Modes &, &, & 4, or & 5 Wlso Dap Proportoalty factors (5% each dcated ode) Modes α β & & & & Modal Dap Rato.5..5 MODES,,,4,5 Drectly specfy odal dap values c Φ CΦ c ωξ c ω ξ ξ ωξ 4 6 Fequecy, Radas/sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 7

8 Forato of Explct Dap Matrx Fro Wlso Modal Dap (NO Usually Requred) ξω ξω Φ CΦ c ξ ω ξω ( Φ ) cφ C C M ξ ωφ φ M Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Wlso Dap (Exaple) 5% Dap Modes ad, % Mode 5, Zero Mode 4 Modal Dap Rato Frequecy, Radas per secod Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-44 Wlso Dap (Exaple) 5% Dap all Modes Soluto of MDOF Equatos of Moto Modal Dap Rato Explct (step by step) terato of coupled equatos Explct terato of FULL SE of ucoupled equatos Explct terato of PARIAL SE of ucoupled Equatos (approxate) Modal respose spectru aalyss (approxate) Frequecy, Radas per secod Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-45 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-46 Force, V(t) V t δ V t t Coputed Respose for Pecewse Lear Load t t e t δ Vt V t t t e, t ft ft ft Exaple of MDOF Respose of Structure Respod to 94 El Cetro Earthquake. k-s /.5 k-s /. k-s / u (t) k 6 k/ u (t) k k/ u (t) k 8 k/ Exaple Assue Wlso dap wth 5% crtcal each ode. N-S copoet of 94 El Cetro earthquake Maxu accelerato.5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-47 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-48 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 8

9 Exaple (cotued) Exaple (cotued). k-s / u (t) k 6 k/.5 k-s / u (t) k k/. k-s / u (t) k 8 k/ For property atrces:. M.5 kp s /. 6 K kp/. k-s / u (t) k 6 k/.5 k-s / u (t) k k/. k-s / u (t) k 8 k/ Solve eevalue proble: KΦ MΦΩ. Ω Φ sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-49 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Noralzato of Modes Us Φ MΦ I Exaple (cotued) Mode Shapes ad Perods of Vbrato Φ vs MODE MODE MODE ω 4.58 rad/sec ω 9.8 rad/sec ω 4.57 rad/sec.7 sec.69 sec.4 sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Exaple (cotued) Exaple (cotued). k-s /.5 k-s / u (t) k 6 k/ u (t) k k/. k-s / u (t) M k 8 k/ ω rad/sec Φ M Φ sec.4 Copute Geeralzed Mass: kp sec /.. k-s / u (t) k 6 k/.5 k-s / u (t) k k/. k-s / u (t) k 8 k/ Copute eeralzed load: V V () t Φ MRv&& () t v&&.8 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-54 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 9

10 Exaple (cotued) Wrte ucoupled (odal) equatos of oto: Modal Partcpato Factors. k-s / u (t) k 6 k/.5 k-s / u (t) k k/. k-s / u (t) k 8 k/ y&& + ξωy& + ω y V / y&& + ξωy& + ωy V / y&& + ξωy& + ωy V / y&& +.458y& +.y.45 v&& y&& +.98y& y.5 v&& y&& +.457y& +.4y.9 v&& Mode Mode Mode Modal scal φ,. φ M φ. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-55 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-56 Modal Partcpato Factors (cotued) us φ M us, φ φ M P Effectve Modal Mass Mode Mode Mode M % Accu% % 4.5 % Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-57 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-58 Exaple (cotued) Solv odal equato va NONLIN: For Mode : y && & + ξ ω y + ω y V /.y&& +.458y& +.y.45 v&& M. kp-sec / C.458 kp-sec/ K. kps/ch Scale roud accelerato by factor.45 Exaple (cotued) Modal Dsplaceet Respose Hstores (fro NONLIN) MODE MODE MODE e, Secods.7 sec.64.4 Maxa Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-59 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

11 Modal Dsplaceet, Iches Exaple (cotued) Modal Respose Hstores: - MODE MODE -4 MODE e, Secods Exaple (cotued) Copute story dsplaceet respose hstores: u (t) u (t) u (t) u Φy( t) e, Secods x Mode x Mode +.47 x Mode Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Exaple (cotued) Copute story shear respose hstores: u (t) Exaple (cotued) Dsplaceets ad forces at te of axu dsplaceets (t 6.4 sec) 4.8 k u (t) u (t) k.5 k e, Secods k [u (t) - u (t)] Story Shear (k) Story OM (ft-k) Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs Exaple (cotued) Dsplaceets ad forces at te of axu shear (t.8 sec) 8. k 4.7 k 8.7 k Modal Respose Respose Spectru Method Istead of solv the te hstory proble for each ode, use a respose spectru to copute the axu respose each ode. hese axa are eerally ococurret. Cobe the axu odal resposes us soe statstcal techque, such as square root of the su of the squares (SRSS) or coplete quadratc cobato (CQC). Story Shear (k) 45.6 Story OM (ft-k) 5567 he techque s approxate. It s the bass for the equvalet lateral force (ELF) ethod. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-65 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-66 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

12 Exaple (Respose Spectru Method) Spectral Dsplaceet, Iches Perod, Secods Mode.4 sec Dsplaceet Respose Spectru 94 El Cetro,.5, 5% Dap Modal respose Mode.69 sec Mode.7 sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-67 Spectral Dsplaceet, Iches Modal Equatos of Moto y&& + & + &&.458y.y.45 v y&& + & + &&.98y 96.6y.5 v y&& +.457y& +.4y.9 v&& Exaple (cotued) Perod, Secods Modal Maxa y " y " y.9..8" Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-68 Spectral Dsplaceet, Iches x.45.4 x.5. x Perod, Secods Mode.7 Exaple (cotued) he scaled respose spectru values ve the sae odal axa as the prevous te Hstores Exaple (cotued) Coput Nococurret Story Dsplaceets Mode Mode Mode.64 Mode Mode e, Secods Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-69 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Exaple (cotued) Modal Cobato echques (for Dsplaceet) Su of Absolute Values: At te of axu dsplaceet Exact Square Root of the Su of the Squares: Evelope of story dsplaceet Mode Mode Mode Exaple (cotued) Coput Iterstory Drfts (.9).48.9 (.48) (.78) Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

13 Mode Mode Mode Exaple (cotued) Coput Iterstory Shears (Us Drft).759(6) ().9.48(8) (6) () 4..48(8) (6)..545() (8) 48. Exaple (cotued) Coput Iterstory Shears: SRSS Cobato Exact At te of ax. shear Exact 5 97 At te of ax. dsplaceet Exact 7 46 Evelope axu per story Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-74 Cauto: Do NO copute story shears fro the story drfts derved fro the SRSS of the story dsplaceets. Calculate the story shears each ode (us odal drfts) ad the SRSS the results. Modal Dsplaceet, Iches Us Less tha Full (Possble) Nuber of Natural Modes Modal Respose Hstores: e, Secods MODE MODE MODE Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-75 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-76 Us Less tha Full Nuber of Natural Modes y( t) y( t) y( t) y( t) e-hstory for Mode y( t) y( t) y( t4) y( t5) y( t6) y( t7) y( t8)... y( t) y( t) y( t) y( t4) y( t5) y( t6) y( t7) y( t8)... y ( ) t y( t) y( t) y( t4) y( t5) y( t6) y( t7) y( t8)... y ( ) t Us Less tha Full Nuber of Natural Modes e Hstory for Mode y() t y() t y() t y(4) t y(5) t y(6) t y(7) t y(8) t... y( t) yt () y() t y() t y() t y(4) t y(5) t y(6) t y(7) t y(8) t... y( t) NOE: Mode NO Aalyzed rasforato: [ φ φ φ ] y( ) u t x t x x t rasforato: ut () φ φ yt () x t x x t x t x t e Hstory for DOF e hstory for DOF u( t) u( t) u( t) u( t) u( t) u u u( t) u u u( t4) u ( t4) u ( t4) u( t5) u ( t5) u ( t5) u( t6) u ( t6) u ( t6) u( t7) u ( t7) u ( t7) u( t8) u ( t8) u ( t8)... u( t)... u ( ) t... u ( ) t u( t) u( t) u( t) u( t) u( t) u u u( t) u u u( t4) u ( t4) u ( t4) u( t5) u ( t5) u ( t5) u( t6) u ( t6) u ( t6) u( t7) u ( t7) u ( t7) u( t8) u ( t8) u ( t8)... u( t)... u ( ) t... u ( ) t Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-77 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-78 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

14 Us Less tha Full Nuber of Natural Modes (Modal Respose Spectru echque) Su of absolute values: Square root of the su of the squares: odes odes At te of axu dsplaceet Exact : ft ft ft Exaple of MDOF Respose of Structure Respod to 94 El Cetro Earthquake.5 k-s /.5 k-s /.5 k-s / u (t) k 5 k/ u (t) k 5 k/ u (t) k 5 k/ Exaple Assue Wlso dap wth 5% crtcal each ode. N-S copoet of 94 El Cetro earthquake Maxu accelerato.5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-79 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 Exaple (cotued) Exaple (cotued).5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/ For property atrces:.5 M.5 kp s / K 5 5 kp/ 5.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/ Solve eevalue proble:.9 Ω KΦ MΦΩ 9. sec Φ Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 Noralzato of Modes Us Φ MΦ I Exaple (cotued) Mode Shapes ad Perods of Vbrato Φ vs Mode Mode Mode ω.44 rad/sec ω 9.66 rad/sec ω.96 rad/sec.8 sec.65 sec.45 sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-84 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 4

15 Exaple (cotued) Exaple (cotued).5 k-s /.44.8 ω 9.66 rad/ sec.65 sec u (t) k 5 k/.5 k-s / u (t) Copute eeralzed ass: k 5 k/.5 k-s / u (t) M 4.6 Φ MΦ 7.58 kp sec /.4 k 5 k/.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/ Copute eeralzed load: V () t Φ MRv&& () t 5.67 V &&.5 v.88 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-85 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-86 Exaple (cotued) Wrte ucoupled (odal) equatos of oto: Modal Partcpato Factors.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/.5 k-s / u (t) k 5 k/ y&& + ξωy& + ω y V / y&& + ξωy& + ωy V / y&& + ξωy& + ωy V / y&& +.45y& +.88y. v&& y&& +.966y& + 9.9y.8 v&& y&& +.95y& y.6 v&& Mode Mode Mode Modal scal φ,. φ M φ. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-87 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-88 M P Effectve Modal Mass Mode Mode Mode M Accu% % % Solv odal equato va NONLIN: For Mode : y && & + ξ ω y + ω y V /.y&& +.45y& +.88y. v&& M. kp-sec / C.45 kp-sec/ Exaple (cotued) K.88 kps/ch Scale roud accelerato by factor. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-89 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 5

16 Exaple (cotued) Modal Dsplaceet Respose Hstores (fro NONLIN) Mode Mode C Mode Maxa Modal Dsplaceet, Iches Exaple (cotued) Modal Respose Hstores MODE MODE MODE e, Secods Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Exaple (cotued) Copute story dsplaceet respose hstores: u Φy( t) u (t) u (t) u (t) x Mode.47 x Mode +.8 x Mode Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Exaple (cotued) Copute story shear respose hstores: u (t) u (t) u (t) e, Secods k [u (t)-u (t)] Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs Exaple (cotued) Dsplaceets ad Forces at te of Maxu Dsplaceets (t 8.96 secods). k 75.9 k.9 k Exaple (cotued) Dsplaceets ad Forces at e of Maxu Shear (t 6.6 sec). k 8.45 k 5. k Story Shear (k) Story OM (ft-k) Story Shear (k) Story OM (ft-k) Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-95 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-96 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 6

17 Modal Respose Respose Spectru Method Istead of solv the te hstory proble for each ode, use a respose spectru to copute the axu respose each ode. hese axa are eerally ococurret. Cobe the axu odal resposes us soe statstcal techque, such as square root of the su of the squares (SRSS) or coplete quadratc cobato (CQC). he techque s approxate. Spectral Dsplaceet, Iches Exaple (Respose Spectru Method) Dsplaceet Respose Spectru 94 El Cetro,.5, 5% Dap MODAL RESPONSE Perod, Secods It s the bass for the equvalet lateral force (ELF) ethod. Mode.45 sec Mode.65 sec Mode.8 sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-97 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-98 Spectral Dsplaceet, Iches Modal Equatos of Moto y&& + & + &&.45y.88y. v y&& + & + &&.966y 9.9y.8 v y&& +.95y& y.6 v&& Exaple (cotued) Perod, Secods Modal Maxa y " y " y " Spectral Dsplaceet, Iches x... x x Perod, Secods Mode.8 Mode.65 Mode.45 Exaple (cotued) he scaled respose spectru values ve the sae odal axa as the prevous te hstores C Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-99 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Exaple (cotued) Coput Nococurret Story Dsplaceets Mode Mode Mode Exaple (cotued) Modal Cobato echques (For Dsplaceet) Su of absolute values: Square root of the su of the squares At te of axu dsplaceet Exact Evelope of story dsplaceet Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 7

18 Mode Exaple (cotued) Coput Iterstory Drfts Mode Exaple (cotued) Coput Iterstory Shears (Us Drft).8(5) (5) 7.9.(5) 465. Mode Mode.845 (.469) (.5) (.) Mode Mode.4(5) (5) (5) (5) (5) (5) 5.4 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 Exaple (cotued) Coput Iterstory Shears: SRSS Cobato Exact Exact Exact At te of ax. shear At te of ax. dsplaceet Evelope axu per story ASCE 7 Allows a Approxate Modal Aalyss echque Called the Equvalet Lateral Force Procedure Eprcal perod of vbrato Soothed respose spectru Copute total base shear, V, as f SDOF Dstrbute V alo heht assu reular eoetry Copute dsplaceets ad eber forces us stadard procedures Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Equvalet Lateral Force Procedure Method s based o frst ode respose. Hher odes ca be cluded eprcally. Has bee calbrated to provde a reasoable estate of the evelope of story shear, NO to provde accurate estates of story force. May result overestate of overtur oet. Equvalet Lateral Force Procedure Assue frst ode effectve ass total Mass M W/ Use respose spectru to obta total Accelerato, S a Perod, sec W VB ( Sa) M ( Sa) SaW Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 8

19 Equvalet Lateral Force Procedure d r Assue lear frst ode respose ELF Procedure Exaple h h x W x f x hx W fx() t dr() t ω h stores stores dr () t ω VB() t f() t hw h Porto of base shear appled to story fx() t hxwx stores VB () t hw x h. k-s / k 6 k/.5 k k/. k 8 k/ Recall.7 sec Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - ELF Procedure Exaple otal weht M x ( ) kps Spectral accelerato w S D (p/.7) x /sec ELF Procedure Exaple (Story Forces) 86( h) f.8vb.75(7) 5 kps 86( h) + 579( h) + 7( h) 5 k W86 k Spectral Dsplaceet, Iches sec h W579 k W7 k 5 k 77 k 5 k ( k) 5 k (5 k) Perod, Secods Base shear S a W.88 x 78 7 kps 7 k Story Shear (k) 7 k ( k) Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - ELF Procedure Exaple (Story Dsplaceets) e Hstory (Evelope) Uts ches Modal Respose Spectru ELF ELF Procedure Exaple (Suary) ELF procedure ves ood correlato wth base shear (7 kps ELF vs kps odal respose spectru). ELF story force dstrbuto s ot as ood. ELF uderestates shears upper stores. ELF ves reasoable correlato wth dsplaceets. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF 9

20 Equvalet Lateral Force Procedure Hher Mode Effects + ASCE 7-5 ELF Approach Uses eprcal perod of vbrato Uses soothed respose spectru Has correcto for hher odes Has correcto for overtur oet Has ltatos o use st Mode d Mode Cobed Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-6 Approxate Perods of Vbrato Ch x a t C t.8, x.8 for steel oet fraes C t.6, x.9 for cocrete oet fraes C t, x.75 for eccetrcally braced fraes C t, x.75 for all other systes Note: For buld structures oly! a.n For oet fraes < stores heht, u story heht of feet. N uber of stores. Adjustet Factor o Approxate Perod acu coputed S D C u > <..7 Applcable oly f coputed coes fro a properly substatated aalyss. Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-7 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-8 ASCE 7 Soothed Des Accelerato Spectru (for Use wth ELF Procedure) SDS CS V C S W R I Spectral Respose Accelerato, S asds Short perod accelerato S D Lo perod accelerato SD CS R I Vares R s the respose odfcato factor, a fucto of syste elastc behavor. hs s covered the topc o elastc behavor. For ow, use R, whch ples lear elastc behavor. I s the portace factor whch depeds o the Sesc Use Group. I.5 for essetal facltes,.5 for portat hh occupacy structures, ad. for oral structures. For ow, use I... Perod, Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-9 Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF

21 Dstrbuto of Forces Alo Heht C vx F x C V vx wh x k x wh k k Accouts for Hher Mode Effects k.. k (sloped porto oly).5.5 Perod, sec k k Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - ELF Procedure Exaple (Story Forces) V 7 kps.7 sec k.5(.7) ASCE 7 ELF Procedure Ltatos W 86k 46 k (5 k) Applcable oly to reular structures wth less tha.5 s. Note that s S D /S DS. h W 579k W 7k k (5 k) 6 k (77 k) 46 k 66 k Adjacet story stffess does ot vary ore tha %. Adjacet story streth does ot vary ore tha %. Adjacet story asses does ot vary ore tha 5%. 7 k 7 k If volated, ust use ore advaced aalyss (typcally odal respose spectru aalyss). Story Shear (k) Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4 - Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-4 ASCE 7 ELF Procedure Other Cosderatos Affect Load Orthooal load effects Redudacy Accdetal torso orsoal aplfcato P-delta effects Iportace factor Ductlty ad overstreth Istructoal Materal Copleet FEMA 45, Des Exaples MDOF Dyacs 4-5 FEMA 45B opc 4 Hadouts Structural Dyacs of MDOF