This Technical Note describes how the program calculates the moment capacity of a noncomposite steel beam, including a cover plate, if applicable.

Size: px
Start display at page:

Download "This Technical Note describes how the program calculates the moment capacity of a noncomposite steel beam, including a cover plate, if applicable."

Transcription

1 COPUTERS AND STRUCTURES, INC., BERKEEY, CAIORNIA DECEBER 001 COPOSITE BEA DESIGN AISC-RD93 Techical te This Techical te descibes how the ogam calculates the momet caacit of a ocomosite steel beam, icludig a cove late, if alicable. Oveview The ogam ol calculates the momet caacit,, if the beam is comact o ocomact. It does ot calculate if the sectio is slede. The lastic momet,, fo a ocomosite olled steel beam sectio without a cove late is calculated as Z. The exact methodolog used to comute the lastic momet caacit i the othe cases deeds o whethe the beam, icludig the cove late if it exists, is doubl o sigl smmetic, ad whethe the beam web is classified as comact o ocomact. igue 1 shows a flowchat that diects ou to the aoiate sectio i this chate fo calculatig the momet caacit of the steel sectio aloe. The figue has boxes labeled a though g; stat i the box labeled a. te that the citeia used b the ogam to detemie if a sectio is comact o ocomact fo the AISC-RD93 secificatio is descibed i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. Steel Beam Poeties If oeties fo the steel sectio aloe ae available diectl fom the ogam's sectio database, those oeties ae used to comute the momet caacit. o othe cases such as a use-defied sectio o a sectio with a cove late, the sectio oeties ae calculated i a mae simila to that descibed i Techical te Tasfomed Sectio omet of Ietia Comosite Beam Desig AISC-ASD89, excet that thee is o cocete o eifocig steel to coside. Oveview Page 1 of 13

2 Comosite Beam Desig AISC-RD93 Is sectio doubl smmetic o a chael? Is the beam web comact? Is the beam web ocomact? a b c Beam sectio is classified as slede ad is ot desiged. Go to ext tial sectio. Refe to omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio i this Techical te. Refe to omet Caacit fo a Sigl Smmetic Beam with a Comact Web i this Techical te. Refe to omet Caacit fo a Sigl Smmetic Beam with a comact Web i this Techical te. d igue 1: e f g lowchat o Detemiig Which Sectio of this Chate Alies i Calculatig Plastic omet fo Steel Sectio Aloe Afte the momet of ietia has bee calculated, the sectio moduli ad adius of gatio ae calculated usig stadad fomulas. This ocess is eeated to get oeties about both axes. The tosioal costat is detemied b summig u the tosioal costats fo the vaious comoets of the sectio. o examle it ma be detemied b summig the J's of a olled sectio ad the cove late, if alicable, o i a use-defied sectio, b summig the J's fo the to flage, web, bottom flage ad cove late, if alicable. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio igue shows a flowchat that detemies the equatios the ogam uses to calculate fo a doubl smmetic steel sectio aloe o a chael sectio aloe. The figue has boxes labeled a though k; stat i the box labeled a. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page of 13

3 Comosite Beam Desig AISC-RD93 Ae the web, flage ad cove late comact? a Is the web ocomact? Is b? g Beam sectio ot desiged. Go to ext tial sectio. k Is b? Is b? h b based o ieldig citeia i AISC- RD93 Sectio 1.1. c d Ae the flage ad cove late comact? i Beam sectio ot desiged. Go to ext tial sectio. e of ieldig citeia i AISC-RD93 Sectio 1.1 ad lateal tosioal bucklig citeia i AISC-RD93 Sectio 1.a. f of ieldig citeia i AISC-RD93 Sectio 1.1, lateal tosioal bucklig citeia i AISC-RD93 Sectio 1.a ad flage ad web local bucklig citeia i AISC- RD93 Aedix 1(b) equatio (A- 1-3). j igue : lowchat o Calculatig fo a Doubl Smmetic Steel Sectio Aloe o a Rolled Chael Steel Sectio Aloe Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The followig subsectio discusses the ubaced legth checks i the ogam that ae used to detemie how to calculate fo a doubl smmetic beam o a chael sectio. Subsequet subsectios discuss each of the code sectios metioed i igue that ae used to calculate the momet caacit. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 3 of 13

4 Comosite Beam Desig AISC-RD93 ateal Ubaced egth Checks The ubaced legths listed i igue ae b, ad. Defiitios of each of these items ae listed below. b ateall ubaced legth of beam; legth betwee oits which ae baced agaist lateal dislacemet of the flage, i. imitig lateall ubaced legth of beam fo full lastic bedig caacit, i. imitig lateall ubaced legth of beam fo ielastic lateal-tosioal bucklig, i. The ubaced legth of a beam, o a beam segmet, b is detemied fom the iut data. The limitig ubaced legth fo full lastic caacit,, is detemied fom Equatio 1 which is also Equatio 1-4 i AISC-RD Eq. 1 f I Equatio 1, is take fo the steel beam sectio icludig the cove late, if alicable. The f tem i Equatio 1 is fo the flage. The limitig ubaced legth fo lateal tosioal bucklig,, is detemied fom Equatio which is also Equatios 1-6 though 1-8 i AISC-RD93. X X 1 1 π S x 1+ EGJA 1+ X smalleof ( f ad, whee X ) ad C 4 I w w Sx GJ Eq. I Equatio,, the comessive esidual stess i the flage, is take as 10 ksi fo olled shaes ad 16.5 ksi fo use-defied shaes. The waig costat, C w, is based o the steel beam aloe igoig the cove late if it exists. o olled sectios, icludig chaels, the ogam takes C w fom its built-i database. o use-defied sectios C w is calculated usig Equatio 3. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 4 of 13

5 Comosite Beam Desig AISC-RD93 te that Equatio 3 actuall alies to smmetical sectios but it is also used whe the flages have diffeet dimesios. C w I d t f to 4 t f bot Eq. 3 Yieldig Citeia i AISC-RD93 Sectio 1.1 The ieldig citeia is that. The ocess fo detemiig has bee eviousl descibed i the sectio etitled "Oveview" i this techical ote. ateal Tosioal Bucklig Citeia i AISC-RD93 Sectio 1.a The lateal tosioal bucklig citeia i AISC RD 1.a is based o AISC- RD93 Equatio 1-. I this case is give b Equatio 4. b Cb ( ) Eq. 4 I Equatio 4, C b is calculated usig Equatio 5, which is also AISC-RD93 Equatio 1-3. C b max Eq. 5.5 max A B + 3 C Refe to the otatio i Techical te Geeal ad tatio Comosite Beam Desig AISC-RD93 fo a exlaatio of the tems i Equatio 5. I Equatio 4, is calculated usig Equatio, is calculated fom Equatio 1 ad comes fom Equatio 6. S Eq. 6 x whee is as descibed fo Equatio. AISC-RD Aedix 1(b) Equatio A-1-3 The limit state fo flage ad web local bucklig is based o AISC-RD93 Equatio A-1-3, which is show heei as Equatio 7. omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 5 of 13

6 Comosite Beam Desig AISC-RD93 λ λ Eq. 7 ( ) λ λ Equatio 7 alies to both flage local bucklig ad web local bucklig. lage ocal Bucklig o flage local bucklig usig Equatio 7:! is calculated e Equatio 6.! λis equal to b f /(t f ) fo I-sectios ad b f /t f fo chaels. The b f ad t f tems ae fo the flage.! λ is give b Equatio 8a if the sectio is a olled o use-defied I- sectio, o Equatio 8b if the sectio is a olled chael. The f i these equatios is fo the flage. bf 65 Eq. 8a t f f bf 65 Eq. 8b t f f! λ is give b Equatio 9a if the sectio is a olled beam o chael, o Equatio 9b if it is a use-defied sectio. 141 λ, fo olled shaes Eq. 9a 16 λ, fo use-defied shaes Eq. 9b k c I Equatio 9a ad 9b, is as defied fo Equatio. I Equatio 9b, k 4 h but ot less tha 0.35 k c Equatios 9a ad 9b ae c t w take fom AISC-RD93 Table A-1.1. Web ocal Bucklig o web local bucklig usig Equatio 7: omet Caacit fo a Doubl Smmetic Beam o a Chael Sectio Page 6 of 13

7 Comosite Beam Desig AISC-RD93! is calculated usig Equatios 10 ad 11 fo both the to ad bottom flages seaatel. The smalle value of is used. R e f S x Eq. 10 I Equatio 10, R e is equal to 1.0 fo olled shaes ad is give b Equatio 11 fo use-defied shaes. Equatio 10 is take fom AISC-RD93 Table A R e 3 ( 3m m ) a Eq a Equatio 11 comes fom the defiitio of R e give with Equatio A-G-3 i AISC-RD93 Aedix G. I Equatio 11 the tem a is the atio of the web aea (ht w ) to the flage aea (b f t f ), but ot moe tha 10, ad m is the atio of the web ield stess to the flage ield stess.! λ is equal to h/t w.! λ is give b Equatio 5a, o 5b i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93 deedig o the axial load i the membe, if a. See the descitio accomaig these equatios fo moe ifomatio.! λ is give b oe of Equatios 6 ad 7 i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93 deedig o the te of membe ad the amout of axial, if a. See the descitio accomaig these equatios fo moe ifomatio. omet Caacit fo a Sigl Smmetic Beam with a Comact Web igue 3 shows a flowchat that detemies the equatios the ogam uses to calculate fo a sigl smmetic steel sectio aloe with a comact web. The figue has boxes labeled a though ; stat i the box labeled a. ost of the fomulas associated with this flowchat ae based o AISC- RD93 Secificatio Aedix sectio 1ad Table A-1.1. omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 7 of 13

8 Comosite Beam Desig AISC-RD93 Is web comact? a Ae the flage ad cove late comact? b This is the wog flowchat. See igue 1. e Beam sectio ot desiged. Go to ext tial sectio. te: Ae the flage ad cove late ocomact? f WB Web local bucklig B lage local bucklig TB ateal tosioal bucklig Beam sectio ot desiged. Go to ext tial sectio. Is beam comact fo TB? c g Is beam ocomact fo TB? h Is beam comact fo TB? j l Is beam ocomact fo TB? m AISC-RD93 Aedix A-1-1 fo WB A-1-1 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-1 fo B A-1- fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-3 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-1 fo WB A-1-3 fo B A-1- fo TB. d i k igue 3: lowchat o Calculatig fo a Sigl Smmetic Steel Sectio Aloe with a Comact Web Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The followig subsectio descibes the lateal tosioal bucklig (TB) checks i the ogam that ae used to detemie how to calculate fo a sigl smmetic beam with a comact web. Subsequet subsectios descibe each of the AISC-RD93 Secificatio Aedix equatios metioed i igue 3 that ae used to calculate the momet caacit. AISC-RD93 Equatio A-1-1 fo WB o this case is equal to, the lastic momet caacit of the sectio. omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 8 of 13

9 Comosite Beam Desig AISC-RD93 AISC-RD93 Equatio A-1-1 fo B o this case is equal to, the lastic momet caacit of the sectio. AISC-RD93 Equatio A-1-3 fo B AISC-RD93 Equatio A-1-3 fo flage local bucklig is iteeted b the ogam as show i Equatios 1a though 1f. λ λ λ λ ( ) Eq. 1a whee S λ b f t f x Eq. 1b Eq. 1c λ 65 f Eq. 1d 141 λ, olled beams ad chaels Eq. 1e λ 16, use-defied beams Eq. 1f k c I Equatio 1b, ad S x ae fo the beam flage (ot cove late). I Equatios 1c ad 1d, b f, t f ad f ae fo the beam flage (ot cove late). I Equatio 1e, is fo the beam flage (ot cove late). I Equatio 1f, is fo the beam flage (ot cove late), ad k 4 h but ot less tha 0.35 k c c t w omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 9 of 13

10 Comosite Beam Desig AISC-RD93 AISC-RD93 Equatio A-1-1 fo TB o this case is equal to, the lastic momet caacit of the sectio. AISC-RD93 Equatio A-1- fo TB AISC-RD93 Equatio A-1- fo lateal tosioal bucklig is iteeted b the ogam as show i Equatios 13a though 13d ad Equatios 14a though 14c. C b λ λ λ λ ( ) Eq. 13a whee, S S Eq. 13b xc f xt λ b c Eq. 13c λ 300 f Eq. 13d The tem λ i Equatio 13a is the value of λ fo which c as defied b Equatios 14a though 14c is equal to the smalle of S xc ad f S xt whee is the smalle of ( f - ) ad w. Whe calculatig, the tem f is the ield stess of the flage ad whe calculatig f S xt, the tem f is the ield stess of the tesio flage. whee, B 1 c ( 57000)( 1) b I J B B + B 1 Eq. 14a I c h I.5 1 Eq. 14b I b J I c I c h B 5 1 I J Eq. 14c b omet Caacit fo a Sigl Smmetic Beam with a Comact Web Page 10 of 13

11 Comosite Beam Desig AISC-RD93 To calculate λ fo Equatio 13a, the ogam detemies the value of b fo which c is equal to the smalle of S xc ad f S xt. The it divides that value of b b c to get λ. omet Caacit fo a Sigl Smmetic Beam with a comact Web igue 4 shows a flowchat that detemies the equatios the ogam uses to calculate fo a sigl smmetic steel sectio aloe with a ocomact web. The figue has boxes labeled a though ; stat i the box labeled a. Is web ocomact? a Ae the flage ad cove late comact? b This is the wog flowchat. See igue 1. e Beam sectio ot desiged. Go to ext tial sectio. te: Ae the flage ad cove late ocomact? f WB Web local bucklig B lage local bucklig TB ateal tosioal bucklig Beam sectio ot desiged. Go to ext tial sectio. Is beam comact fo TB? c g Is beam ocomact fo TB? h Is beam comact fo TB? j l Is beam ocomact fo TB? m AISC-RD93 Aedix A-1-3 fo WB A-1-1 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-1 fo B A-1- fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-3 fo B A-1-1 fo TB. AISC-RD93 Aedix A-1-3 fo WB A-1-3 fo B A-1- fo TB. d i k igue 4: lowchat fo Calculatig fo a Sigl Smmetic Steel Sectio Aloe with a comact Web omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 11 of 13

12 Comosite Beam Desig AISC-RD93 ost of the fomulas associated with this flowchat ae based o AISC- RD93 Secificatio Aedix sectio 1ad Table A-1.1. Ifomatio elatig to how the ogam calculates the comact ad ocomact sectio equiemets is i Techical te Comact ad comact Requiemets Comosite Beam Desig AISC-RD93. The lateal tosioal bucklig checks ad all but oe of the Aedix equatios metioed i igue 4 ae descibed i the evious sectio etitled, "omet Caacit fo a Sigl Smmetic Beam with a Comact Web." Refe to that sectio fo moe ifomatio. The oe equatio that has ot bee descibed eviousl is AISC-RD93 Secificatio Aedix Equatio A-1-3. This equatio is descibed i the followig subsectio. AISC-RD93 Equatio A-1-3 fo WB AISC-RD93 Equatio A-1-3 fo web local bucklig is iteeted b the ogam as show i Equatios 15a though 15g. λ λ λ λ ( ) Eq. 15a I Equatio 15a:! is calculated usig Equatios 15b ad 15c fo both the to ad bottom flages seaatel. The smalle value of is used. R e f S x Eq. 15b I Equatio 15b, R e is give b Equatio 15c. Equatio 15b is take fom AISC-RD93 Table A-1.1. R e 3 ( 3m m ) a Eq. 15c 1 + a Equatio 15c comes fom the defiitio of R e give with Equatio A-G-3 i AISC-RD93 Aedix G. I Equatio 15c, the tem a is the atio of the web aea (ht w ) to the flage aea (b f t f ), but ot moe tha 10, ad m is the atio of the web ield stess to the flage ield stess. omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 1 of 13

13 Comosite Beam Desig AISC-RD93! λ is equal to h/t w.! λ is give b Equatio 15d, o 15e deedig o the axial load i the membe, if a. λ P P 1 u u, fo P P φ b φb 0.15 Eq. 15d λ 191 Pu.33 φbp 53, Pu fo φ P b > 0.15 Eq. 15e! λ is give b eithe Equatio 15f o Equatio 15g. Equatio 15f defies λ fo beams with equal sized flages. λ P 1 φ b P u Eq. 15f I Equatio 15f, the value of used is the lagest of the values fo the beam flages ad the web. Equatio 15g defies the ocomact sectio limit fo webs i beams with uequal size flages: λ whee, h h h h c c 0.74P 1 φ b P 3 u, Eq. 15g I Equatio 15g, the value of used is the lagest of the values fo the beam flages ad the web. Equatio 15g is based o Equatio A-B5-1 i the AISC-RD93 secificatio. omet Caacit fo a Sigl Smmetic Beam with a comact Web Page 13 of 13

DESIGN OF BEAMS FOR MOMENTS

DESIGN OF BEAMS FOR MOMENTS CHAPTER Stuctual Steel Design RFD ethod Thid Edition DESIGN OF BEAS FOR OENTS A. J. Clak School of Engineeing Deatment of Civil and Envionmental Engineeing Pat II Stuctual Steel Design and Analysis 9 FA

More information

General. Eqn. 1. where, F b-bbf = Allowable bending stress at the bottom of the beam bottom flange, ksi.

General. Eqn. 1. where, F b-bbf = Allowable bending stress at the bottom of the beam bottom flange, ksi. COMPUERS AND SRUCURES, INC., BERKELEY, CALIORNIA DECEMBER 2001 COMPOSIE BEAM DESIGN AISC-ASD89 echnical Note Allowale Bending Stesses Geneal his echnical Note descies how the pogam detemines the allowale

More information

C. C. Fu, Ph.D., P.E.

C. C. Fu, Ph.D., P.E. ENCE710 C. C. Fu, Ph.D., P.E. Shear Coectors Desig by AASHTO RFD (RFD Art. 6.10.10) I the egative flexure regios, shear coectors shall be rovided where the logitudial reiforcemet is cosidered to be a art

More information

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3

More information

Check the strength of each type of member in the one story steel frame building below.

Check the strength of each type of member in the one story steel frame building below. CE 33, Fall 200 Aalysis of Steel Baced Fame Bldg / 7 Chec the stegth of each type of membe the oe stoy steel fame buildg below. A 4 @ 8 B 20 f 2 3 @ 25 Side Elevatio 3 4 Pla View 32 F y 50 si all membes

More information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.

More information

Overview. Overview Page 1 of 8

Overview. Overview Page 1 of 8 COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIORNIA DECEMBER 2001 COMPOSITE BEAM DESIGN AISC-LRD93 Tecnical Noe Compac and Noncompac Requiemens Tis Tecnical Noe descibes o e pogam cecks e AISC-LRD93 specificaion

More information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

More information

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r. Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p

More information

Supplementary materials. Suzuki reaction: mechanistic multiplicity versus exclusive homogeneous or exclusive heterogeneous catalysis

Supplementary materials. Suzuki reaction: mechanistic multiplicity versus exclusive homogeneous or exclusive heterogeneous catalysis Geeal Pape ARKIVOC 009 (xi 85-03 Supplemetay mateials Suzui eactio: mechaistic multiplicity vesus exclusive homogeeous o exclusive heteogeeous catalysis Aa A. Kuohtia, Alexade F. Schmidt* Depatmet of Chemisty

More information

THE ANALYTIC LARGE SIEVE

THE ANALYTIC LARGE SIEVE THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig

More information

On composite conformal mapping of an annulus to a plane with two holes

On composite conformal mapping of an annulus to a plane with two holes O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy

More information

2012 GCE A Level H2 Maths Solution Paper Let x,

2012 GCE A Level H2 Maths Solution Paper Let x, GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs

More information

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If

More information

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae

More information

At the end of this topic, students should be able to understand the meaning of finite and infinite sequences and series, and use the notation u

At the end of this topic, students should be able to understand the meaning of finite and infinite sequences and series, and use the notation u Natioal Jio College Mathematics Depatmet 00 Natioal Jio College 00 H Mathematics (Seio High ) Seqeces ad Seies (Lecte Notes) Topic : Seqeces ad Seies Objectives: At the ed of this topic, stdets shold be

More information

Using Counting Techniques to Determine Probabilities

Using Counting Techniques to Determine Probabilities Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe

More information

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

More information

seventeen steel construction: columns & tension members ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture

seventeen steel construction: columns & tension members ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture ARCHITECTURAL STRUCTURES: ORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS Co-Ten Steel Sculptue By Richad Sea Museum of Moden At ot Woth, TX (AISC - Steel Stuctues of the Eveyday) SUMMER 2014 lectue seventeen

More information

Lecture 24: Observability and Constructibility

Lecture 24: Observability and Constructibility ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio

More information

A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS

A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com

More information

See the solution to Prob Ans. Since. (2E t + 2E c )ch - a. (s max ) t. (s max ) c = 2E c. 2E c. (s max ) c = 3M bh 2E t + 2E c. 2E t. h c.

See the solution to Prob Ans. Since. (2E t + 2E c )ch - a. (s max ) t. (s max ) c = 2E c. 2E c. (s max ) c = 3M bh 2E t + 2E c. 2E t. h c. *6 108. The beam has a ectangula coss section and is subjected to a bending moment. f the mateial fom which it is made has a diffeent modulus of elasticity fo tension and compession as shown, detemine

More information

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n! 0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet

More information

ME311 Machine Design

ME311 Machine Design ME311 Machine Desin Lectue 7: Columns W Donfeld 6Oct17 Faifield Univesit School of Enineein Column Bucklin We have aad discussed axiall loaded bas. Fo a shot ba, the stess /A, and the defction is L/AE.

More information

Using Difference Equations to Generalize Results for Periodic Nested Radicals

Using Difference Equations to Generalize Results for Periodic Nested Radicals Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =

More information

Cross section dependence on ski pole sti ness

Cross section dependence on ski pole sti ness Coss section deendence on ski ole sti ness Johan Bystöm and Leonid Kuzmin Abstact Ski equiment oduce SWIX has ecently esented a new ai of ski oles, called SWIX Tiac, which di es fom conventional (ound)

More information

APPLIED THERMODYNAMICS D201. SELF ASSESSMENT SOLUTIONS TUTORIAL 2 SELF ASSESSMENT EXERCISE No. 1

APPLIED THERMODYNAMICS D201. SELF ASSESSMENT SOLUTIONS TUTORIAL 2 SELF ASSESSMENT EXERCISE No. 1 APPIED ERMODYNAMICS D0 SEF ASSESSMEN SOUIONS UORIA SEF ASSESSMEN EXERCISE No. Show how the umeti effiiey of a ideal sigle stage eioatig ai omesso may be eeseted by the equatio ( / Whee is the leaae atio,

More information

LESSON 15: COMPOUND INTEREST

LESSON 15: COMPOUND INTEREST High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed

More information

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P. Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a

More information

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you

More information

Chapter 8 Complex Numbers

Chapter 8 Complex Numbers Chapte 8 Complex Numbes Motivatio: The ae used i a umbe of diffeet scietific aeas icludig: sigal aalsis, quatum mechaics, elativit, fluid damics, civil egieeig, ad chaos theo (factals. 8.1 Cocepts - Defiitio

More information

Electron states in a periodic potential. Assume the electrons do not interact with each other. Solve the single electron Schrodinger equation: KJ =

Electron states in a periodic potential. Assume the electrons do not interact with each other. Solve the single electron Schrodinger equation: KJ = Electo states i a peiodic potetial Assume the electos do ot iteact with each othe Solve the sigle electo Schodige equatio: 2 F h 2 + I U ( ) Ψ( ) EΨ( ). 2m HG KJ = whee U(+R)=U(), R is ay Bavais lattice

More information

Design of Eccentrically Braced Frames

Design of Eccentrically Braced Frames Design of Eccentricall Braced Frames Aninda Dutta, Ph.D., S.E. What is an eccentricall braced frame? In an eccentric braced frame the braces are eccentric to the beam-column connection, i.e. the do not

More information

The Discrete Fourier Transform

The Discrete Fourier Transform (7) The Discete Fouie Tasfom The Discete Fouie Tasfom hat is Discete Fouie Tasfom (DFT)? (ote: It s ot DTFT discete-time Fouie tasfom) A liea tasfomatio (mati) Samples of the Fouie tasfom (DTFT) of a apeiodic

More information

The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005

The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005 The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used

More information

GRAVITATIONAL FORCE IN HYDROGEN ATOM

GRAVITATIONAL FORCE IN HYDROGEN ATOM Fudametal Joual of Mode Physics Vol. 8, Issue, 015, Pages 141-145 Published olie at http://www.fdit.com/ GRAVITATIONAL FORCE IN HYDROGEN ATOM Uiesitas Pedidika Idoesia Jl DR Setyabudhi No. 9 Badug Idoesia

More information

eighteen steel construction: column design ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2019 lecture

eighteen steel construction: column design ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2019 lecture ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SPRING 2019 letue eighteen steel onstution: olumn design Co-Ten Steel Sulptue By Rihad Sea Museum of Moden At Fot Woth,

More information

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist

More information

Failure Theories Des Mach Elem Mech. Eng. Department Chulalongkorn University

Failure Theories Des Mach Elem Mech. Eng. Department Chulalongkorn University Failure Theories Review stress trasformatio Failure theories for ductile materials Maimum-Shear-Stress Theor Distortio-Eerg Theor Coulomb-Mohr Theor Failure theories for brittle materials Maimum-Normal-Stress

More information

Multivector Functions

Multivector Functions I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed

More information

Week 03 Discussion. 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 2% die.

Week 03 Discussion. 30% are serious, and 50% are stable. Of the critical ones, 30% die; of the serious, 10% die; and of the stable, 2% die. STAT 400 Wee 03 Discussio Fall 07. ~.5- ~.6- At the begiig of a cetai study of a gou of esos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the five-yea study, it was detemied

More information

Ground Rules. PC1221 Fundamentals of Physics I. Uniform Circular Motion, cont. Uniform Circular Motion (on Horizon Plane) Lectures 11 and 12

Ground Rules. PC1221 Fundamentals of Physics I. Uniform Circular Motion, cont. Uniform Circular Motion (on Horizon Plane) Lectures 11 and 12 PC11 Fudametals of Physics I Lectues 11 ad 1 Cicula Motio ad Othe Applicatios of Newto s Laws D Tay Seg Chua 1 Goud Rules Switch off you hadphoe ad page Switch off you laptop compute ad keep it No talkig

More information

Chapter 2 Sampling distribution

Chapter 2 Sampling distribution [ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize

More information

On ARMA(1,q) models with bounded and periodically correlated solutions

On ARMA(1,q) models with bounded and periodically correlated solutions Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,

More information

Lecture 6: October 16, 2017

Lecture 6: October 16, 2017 Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of

More information

ME 354, MECHANICS OF MATERIALS LABORATORY MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS: TORSION TESTING*

ME 354, MECHANICS OF MATERIALS LABORATORY MECHANICAL PROPERTIES AND PERFORMANCE OF MATERIALS: TORSION TESTING* ME 354, MECHANICS OF MATEIALS LABOATOY MECHANICAL POPETIES AND PEFOMANCE OF MATEIALS: TOSION TESTING* MGJ/08 Feb 1999 PUPOSE The pupose of this execise is to obtai a umbe of expeimetal esults impotat fo

More information

CE 562 Structural Design I Midterm No. 1 Closed Book Portion (25 / 100 pts)

CE 562 Structural Design I Midterm No. 1 Closed Book Portion (25 / 100 pts) CE 56 Structural Desig I Name: Midterm No. 1 Closed Book Portio (5 / 100 pts) 1. [6 pts / 5 pts] Two differet tesio members are show below - oe is a pair of chaels coected back-to-back ad the other is

More information

physicsandmathstutor.com

physicsandmathstutor.com physicsadmathstuto.com physicsadmathstuto.com Jauay 2009 2 a 7. Give that X = 1 1, whee a is a costat, ad a 2, blak (a) fid X 1 i tems of a. Give that X + X 1 = I, whee I is the 2 2 idetity matix, (b)

More information

1. C only. 3. none of them. 4. B only. 5. B and C. 6. all of them. 7. A and C. 8. A and B correct

1. C only. 3. none of them. 4. B only. 5. B and C. 6. all of them. 7. A and C. 8. A and B correct M408D (54690/54695/54700), Midterm # Solutios Note: Solutios to the multile-choice questios for each sectio are listed below. Due to radomizatio betwee sectios, exlaatios to a versio of each of the multile-choice

More information

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito

More information

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio

More information

Bernoulli, poly-bernoulli, and Cauchy polynomials in terms of Stirling and r-stirling numbers

Bernoulli, poly-bernoulli, and Cauchy polynomials in terms of Stirling and r-stirling numbers Novembe 4, 2016 Beoulli, oly-beoulli, ad Cauchy olyomials i tems of Stilig ad -Stilig umbes Khisto N. Boyadzhiev Deatmet of Mathematics ad Statistics, Ohio Nothe Uivesity, Ada, OH 45810, USA -boyadzhiev@ou.edu

More information

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of

I PUC MATHEMATICS CHAPTER - 08 Binomial Theorem. x 1. Expand x + using binomial theorem and hence find the coefficient of Two Maks Questios I PU MATHEMATIS HAPTER - 08 Biomial Theoem. Epad + usig biomial theoem ad hece fid the coefficiet of y y. Epad usig biomial theoem. Hece fid the costat tem of the epasio.. Simplify +

More information

The program calculates the required thickness of doubler plates using the following algorithms. The shear force in the panel zone is given by: V p =

The program calculates the required thickness of doubler plates using the following algorithms. The shear force in the panel zone is given by: V p = COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001 STEEL FRAME DESIGN AISC-ASD89 Tehial Note Oe aspet of the esig of a steel framig system is a evaluatio of the shear fores that exist i

More information

On a Problem of Littlewood

On a Problem of Littlewood Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995

More information

1. Show that the volume of the solid shown can be represented by the polynomial 6x x.

1. Show that the volume of the solid shown can be represented by the polynomial 6x x. 7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends

More information

P1 Chapter 8 :: Binomial Expansion

P1 Chapter 8 :: Binomial Expansion P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework

More information

Technical Report: Bessel Filter Analysis

Technical Report: Bessel Filter Analysis Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we

More information

Zero Level Binomial Theorem 04

Zero Level Binomial Theorem 04 Zeo Level Biomial Theoem 0 Usig biomial theoem, epad the epasios of the Fid the th tem fom the ed i the epasio of followig : (i ( (ii, 0 Fid the th tem fom the ed i the epasio of (iii ( (iv ( a (v ( (vi,

More information

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A

More information

The Pigeonhole Principle 3.4 Binomial Coefficients

The Pigeonhole Principle 3.4 Binomial Coefficients Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple

More information

BENDING OF BEAM. Compressed layer. Elongated. layer. Un-strained. layer. NA= Neutral Axis. Compression. Unchanged. Elongation. Two Dimensional View

BENDING OF BEAM. Compressed layer. Elongated. layer. Un-strained. layer. NA= Neutral Axis. Compression. Unchanged. Elongation. Two Dimensional View BNDING OF BA Compessed laye N Compession longation Un-stained laye Unchanged longated laye NA Neutal Axis Two Dimensional View A When a beam is loaded unde pue moment, it can be shown that the beam will

More information

Case Study in Steel adapted from Structural Design Guide, Hoffman, Gouwens, Gustafson & Rice., 2 nd ed.

Case Study in Steel adapted from Structural Design Guide, Hoffman, Gouwens, Gustafson & Rice., 2 nd ed. Case Std i Steel adapted from Strctral Desig Gide, Hoffma, Gowes, Gstafso & Rice., d ed. Bildig descriptio The bildig is a oe-stor steel strctre, tpical of a office bildig. The figre shows that it has

More information

Minimal order perfect functional observers for singular linear systems

Minimal order perfect functional observers for singular linear systems Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig

More information

4. PERMUTATIONS AND COMBINATIONS

4. PERMUTATIONS AND COMBINATIONS 4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3)

More information

Advanced Physical Geodesy

Advanced Physical Geodesy Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig

More information

Principle Of Superposition

Principle Of Superposition ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give

More information

Lower Bounds for Cover-Free Families

Lower Bounds for Cover-Free Families Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set

More information

This web appendix outlines sketch of proofs in Sections 3 5 of the paper. In this appendix we will use the following notations: c i. j=1.

This web appendix outlines sketch of proofs in Sections 3 5 of the paper. In this appendix we will use the following notations: c i. j=1. Web Appedix: Supplemetay Mateials fo Two-fold Nested Desigs: Thei Aalysis ad oectio with Nopaametic ANOVA by Shu-Mi Liao ad Michael G. Akitas This web appedix outlies sketch of poofs i Sectios 3 5 of the

More information

Appendix: The Laplace Transform

Appendix: The Laplace Transform Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio

More information

EXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI

EXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y

More information

NONLOCAL THEORY OF ERINGEN

NONLOCAL THEORY OF ERINGEN NONLOCAL THEORY OF ERINGEN Accordig to Erige (197, 1983, ), the stress field at a poit x i a elastic cotiuum ot oly depeds o the strai field at the poit (hyperelastic case) but also o strais at all other

More information

RECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.

RECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia. #A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad

More information

18.01 Calculus Jason Starr Fall 2005

18.01 Calculus Jason Starr Fall 2005 Lecture 18. October 5, 005 Homework. Problem Set 5 Part I: (c). Practice Problems. Course Reader: 3G 1, 3G, 3G 4, 3G 5. 1. Approximatig Riema itegrals. Ofte, there is o simpler expressio for the atiderivative

More information

Ma 530 Introduction to Power Series

Ma 530 Introduction to Power Series Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power

More information

7.2.1 Basic relations for Torsion of Circular Members

7.2.1 Basic relations for Torsion of Circular Members Section 7. 7. osion In this section, the geomety to be consideed is that of a long slende cicula ba and the load is one which twists the ba. Such poblems ae impotant in the analysis of twisting components,

More information

Lecture 3 : Concentration and Correlation

Lecture 3 : Concentration and Correlation Lectue 3 : Cocetatio ad Coelatio 1. Talagad s iequality 2. Covegece i distibutio 3. Coelatio iequalities 1. Talagad s iequality Cetifiable fuctios Let g : R N be a fuctio. The a fuctio f : 1 2 Ω Ω L Ω

More information

Kepler s problem gravitational attraction

Kepler s problem gravitational attraction Kele s oblem gavitational attaction Summay of fomulas deived fo two-body motion Let the two masses be m and m. The total mass is M = m + m, the educed mass is µ = m m /(m + m ). The gavitational otential

More information

Discussion 02 Solutions

Discussion 02 Solutions STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied

More information

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a

More information

The Substring Search Problem

The Substring Search Problem The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is

More information

Regenerative Property

Regenerative Property DESIGN OF LOGIC FAMILIES Some desirable characteristics to have: 1. Low ower dissiatio. High oise margi (Equal high ad low margis) 3. High seed 4. Low area 5. Low outut resistace 6. High iut resistace

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +

More information

THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES

THE ANALYSIS OF SOME MODELS FOR CLAIM PROCESSING IN INSURANCE COMPANIES Please cite this atle as: Mhal Matalyck Tacaa Romaiuk The aalysis of some models fo claim pocessig i isuace compaies Scietif Reseach of the Istitute of Mathemats ad Compute Sciece 004 Volume 3 Issue pages

More information

Minimization of the quadratic test function

Minimization of the quadratic test function Miimizatio of the quadatic test fuctio A quadatic fom is a scala quadatic fuctio of a vecto with the fom f ( ) A b c with b R A R whee A is assumed to be SPD ad c is a scala costat Note: A symmetic mati

More information

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function. MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied

More information

CMSC 425: Lecture 5 More on Geometry and Geometric Programming

CMSC 425: Lecture 5 More on Geometry and Geometric Programming CMSC 425: Lectue 5 Moe on Geomety and Geometic Pogamming Moe Geometic Pogamming: In this lectue we continue the discussion of basic geometic ogamming fom the eious lectue. We will discuss coodinate systems

More information

ARCH 631 Note Set 21.1 S2017abn. Steel Design

ARCH 631 Note Set 21.1 S2017abn. Steel Design Steel Desig Notatio: a = ame or width dimesio A = ame or area Ag = gross area, equal to the total area igorig a holes Areq d-adj = area required at allowable stress whe shear is adjusted to iclude Aw sel

More information

Client: Client No 1. Location: Location No 1

Client: Client No 1. Location: Location No 1 1 of 3 JOINT CONIGURATION AND DIMENSIONS: bolted ed plate coectio (ustiffeed) Referece clauses: Resistaces for the exteded ed plate coectio. Assumptio: The desig momets i the to beams are equal ad opposite.

More information

MATH Midterm Solutions

MATH Midterm Solutions MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca

More information

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity

More information

physicsandmathstutor.com

physicsandmathstutor.com physicsadmathstuto.com physicsadmathstuto.com Jue 005 5x 3 3. (a) Expess i patial factios. (x 3)( x ) (3) (b) Hece fid the exact value of logaithm. 6 5x 3 dx, givig you aswe as a sigle (x 3)( x ) (5) blak

More information

The number of r element subsets of a set with n r elements

The number of r element subsets of a set with n r elements Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I Uivesity of Hawaii ICS141: Discete Mathematics fo Compute Sciece I Dept. Ifomatio & Compute Sci., Uivesity of Hawaii Ja Stelovsy based o slides by D. Bae ad D. Still Oigials by D. M. P. Fa ad D. J.L. Goss

More information

cosets Hb cosets Hb cosets Hc

cosets Hb cosets Hb cosets Hc Exam, 04-05 Do a total of 30 oits (moe if you wat, of couse) Show you wok! Q: 0 oits (4 ats, oits each fo the fist two ats, 3 oits each fo the secod two ats) Q: 0 oits ( ats, oit each, max cedit 0 oits)

More information

AP CALCULUS - AB LECTURE NOTES MS. RUSSELL

AP CALCULUS - AB LECTURE NOTES MS. RUSSELL AP CALCULUS - AB LECTURE NOTES MS. RUSSELL Sectio Number: 4. Topics: Area -Sigma Notatio Part: of Sigma Notatio Upper boud Recall ai = a+ a + a3 + L + a idex i= Lower boud Example : Evaluate each summatio.

More information

2.4 - Sequences and Series

2.4 - Sequences and Series 2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,

More information