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T.Chhay


                                                                              Trusses

             KWCaeRKOgbgÁúMmanragCaRbBn§½RtIekaN EdlekItBIkarpÁúMKñarvagr)arCaeRcInkMNat;. ragRtIekaN
              Truss

CaeKalkarN_lMnwgrbs; truss. xageRkamCaRbePTénrUbragrbs; truss nigPaBmanlMnwgCaragRtIekaN nig
PaBKμanlMnwgCaragctuekaNeRkamGMeBIénkMlaMgxageRkA.


                          Warren truss                                                   Pratt roof truss




                          Howe roof truss                                                 Fink roof truss




                                                         Pratt bridge truss


                                                      rUbTI1³ RbePT Truss
               P                                                   P

                                                                       Original shape                                             Collapsed shape




                 (a) Rigid, stable framework                                             (b) Nonrigid, unstable framework

       kMNat;rbs; truss maneQμaHepSgKñaeTAtamTItaMg xageRkamCarUbbgðajBIkarehAeQμaHrbs;kMNat;
nImYy².
                                             P1   U p p e r c h o rd                P2                                     P3
                                                                                                 P a n e l p o in ts

                                         B                  C                   D                     E                     F

                                                                                                                                W eb m em ber
        W eb m em ber                                                                                                               (v e rtic al)
           (d ia g o n a l)
                                                                                                                                          E n d p o st

                      A                                                                                                               G
                                         H                  I                   J                     K                L

                                                       L o w e r c h o rd
                                                                                                                                        R o lle r su p p o rt
 P in n e d su p p o rt                                                         P4
                                                                            6 bays @ 5m
                                                                                30m




Trusses                                                                                                                                                         28
T.Chhay


        enAeBl truss rgkMlaMgxageRkA EdlmanGMeBIcMRtg;tMNEdlkat;Kñarvagr)arBIrb¤eRcIn kMNat;
nImYy²rbs; truss mantYrnaTIbBa¢ÚnkMlaMgxageRkAenaHmkTMr ehIykMNat;enaHrgnUvkMlaMgBIrepSgKña. kMlaMg
TaMgBIrenaHrt;tamGkS½beNþayrbs;r)ar EdleFVIeGayr)arlUt b¤rYj EdleKehAfa kMlaMgcMGkS½ (axial forces
or direct forces). r)arEdllUt Car)arEdlrgkMlaMgTaj ehIyr)arEdlrYjCar)arEdlrg kMlaMgsgát;.

karviPaK truss rYmmankarkMNt;tMélénkMlaMgEdlr)arrg nigsPaBénGMeBI (Taj b¤sgát;). xageRkamCarUb
bgðajBIsßanPaBénr)arrgkMlaMg.
                                     Longitudinal axis of member




                                            P          P                               P


                  (a) Axial compression                            (b) Axial tension




             edIm,Idwgfa truss mYyCa truss manlMnwgeKRtUvrkGBaØtielIstamrUbmnþxageRkam³
              n = r +b−2f
Edl           n   CacMnYnGBaØtielIs
               r cMnYnRbtikmμ ¬CaTUeTAesμI 3 GaRs½yelITMr¦

                f cMnYntMN

               ebI n < 0 mann½yfa truss minmanlMnwg
               ebI n = 0 mann½yfa truss manlMnwg kMNt;kMlaMgkñúgrbs; truss tamsmIkarsþaTic
               ebI n > 0 mann½yfa truss manlMnwg EtminGackMNt;kMlaMgkñúg rbs; truss tamsmIkarsþaTiceT.
               edIm,IsMrYlkñúgkarviPaK truss eKRtUveFVIkarsnμt;nUvcMnucmYycMnYnxageRkam³
     -    RKb;kMNat;rbs; truss TaMgGs;RtUvsßitenAkñúgbøg;EtmYy.
     -    bnÞúk nigRbtikmμRtUvGnuvtþn¾ EtenAelItMNrrbs; truss.
     -    kMNat;rbs; truss RtUvP¢ab;Kña edaytMNrKμankkit.
     -    RKb;Ggát;TaMgGs;RtUvEtRtg; ehIyCaGgát;rgkMlaMgBIr EdlkMlaMgenAcugsgxag énGgát;RtUvEtmantM
          élesμIKμa pÞúyKña nigrt;Rtg;Kña.
     -    ExSskmμénkMlaMgkñúgénr)arRtUvEtcMGkS½.
     -    bMErbMrYlRbEvgrbs;Ggát;EdlekItBIkMlaMgTaj b¤sgát;minmantMélRKb;RKan;EdleFIVeGaylkçN³
          FrNImaRtrbs; truss ERbRbYleT.
Trusses                                                                                            29
T.Chhay


    - bnÞúkpÞal;rbs;Ggát;RtUv)anecal eRBaHvamantMéltUcNas;ebIeFobeTAnwg bnÞúkxageRkAEdlvaRT.
         EtRbsinebIbnÞúkxageRkArbs;vaRtUv)anrab;bBa©Úl vaRtUv)ansnμt;CabnÞúkcMcMnucEdlmanGMeBIrenAcug
         sgçagkMNat;.
         QrelIeKalkarN¾snμt;xagelI nigeRbInUvsmIkarlMnwgsþaTic eKGackMNt;kMlaMgkñúgrbs;kMNat;
truss )antambIviFIsaRsþ KW viFItMNr method of joint viFImuxkat; method of section vIFIRkahVik Maxwell-

Cremona.

1> viFItMNr method of joint
         sMrab;viFIenH eKcat;Tukfa truss mYymanlMnwgeTA)anluHRtaNaEtRKb;tMNrrbs;vamanlMnwg. enAkñúg
karviPaK truss tamviFItMNr eKRtUvsikSaBItMNrNaEdlmanGBaØtiy:ageRcInBIr. GBaØtiKWCakMlaMgkñúgrbs;
kMNat;r)ar truss.
         vIFIsaRsþkMNt;kMlaMgrbs;r)artamvIFItMNr³
    - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl.
         CMnYskMlaMgeRTt edaybgÁúMkMlaMgbBaÄr b¤kMlaMgedk.
    - pþac;tMNrnImYy² Ca Free body diagram edaybgðajRKb;kMlaMgTaMgGs;EdlmanGMeBIrelItMNrenaH.
         tMNrnImYy²sMEdgedayRbBn§½kMlaMgRbsBVkñúgbøg; nigcab;epþImecjBItMNrNaEdlmanGBaØtimineRcIn
         CagBIr.
    - edayGnuvtþn¾smIkarlMnwgsþaTicTaMgBIr ( ∑ F = 0 nig ∑ F = 0 ) enaHGBaØtiTaMgBIrsMrab;tMNrenaH
                                                  y                 x


         RtUv)ankMNt;. ehIykMlaMgénr)arTaMgenH RtUveRbIsMrab;tMNrbnÞab;EdlP¢ab;edayr)arTaMgenH.
         sMrab;krNIxøH viFanRtIekaNdUcmansar³RbeyaCn¾CaxøaMgkñúgkaredaHRsaysmIkar.
    - eTaHCaminsMxan;k¾eday eKRtUvEtRtYtBinitücMelIyedayepÞógpÞat;tMNrNamYy edayeGay ∑ F = 0                       y


         nig ∑ F = 0 .
               x


    - eRkayeBlEdlkMlaMgRKb;kMNat;r)arRtUv)anrkeXIj KUrnUvdüaRkamkMlaMgsegçbén truss enaH.
]TahrN¾³                                                                P1 =10KN
                                                                                      2m
                                                                                                 P2 =10KN


         Warren truss EdlRTedayTMrsamBaØ rgnUvbnÞúkdUc                    B                        D



 bgðajkñúgrUb. kMNt;RbtikmμTMr nigkMlaMgénr)ar nImYy²
 edayeRbIvIFItMNr. truss RTedayTMr pin Rtg;cMnuc A
                                                                                  °




                                                                                                           °
                                                              60




                                                                                           60°




 nigRTedayTMr roller Rtg;cMnuc E.
                                                                               60




                                                                                                        60
                                                                °




                                                          A                                                    E
                                                                                      C



                                                                                   P3 =10KN
                                                                          2m                       2m




Trusses                                                                                                                30
T.Chhay


          dMeNaHRsay³                                                                     P1 =10KN
                                                                                                           2m
                                                                                                                      P2 =10KN


          truss RtUvmanlMnwg tamrUbmnþ n = r + b − 2 f                                         B                        D


          r = 3 ( H A , V A , VB )
          b = 7 ( AB, AC , BC , BD, CD, CE , DE )
          f = 5 ( A, B, C , D, E )




                                                                                                    60°




                                                                                                                             60°
          ⇒ n = 3 + 7 − (2 × 5) = 0




                                                                                    60°




                                                                                                                60°
                                                                       HA      A                                                      E
                                                                                                          C

          dUcenH truss manlMnwgkMNt;tamsmIkarlMnwgsþaTic                      VA
                                                                                               2m

                                                                                                         P3 =10KN
                                                                                                                        2m
                                                                                                                                      VE


          1> KNnaRbtikmμTMr                                                                              Free body diagram

          ∑ FX = 0 ⇒ H A = 0

          ∑ FY = 0 ⇒ V A − P1 − P2 − P3 + VE = 0
                        ⇒ V A + VE = P1 + P2 + P3 = 10 + 10 + 10 = 30 KN (1)

          ∑ M A = 0 ⇒ −1× P1 − 3 × P2 − 2 × P3 + 4VE = 0
                                    P1 + 3P2 + 2 P3 10 + 30 + 20
                        ⇒ VE =                     =             = 15KN
                                           4              4
          CMnYs V   E   = 15 KN    kñúg (1) eyIg)an
          V A = 15 KN
          2> kMNt;kMlaMgrbs;r)artamviFItMNr
                                      P1 =10KN                                      P2 =10KN

                                       B               FBD        FBD                D



                              FBA                FBC                        FDC                          FDE
                             FAB                       FCB       FCD                                      FED


                         FAC FCA
                                                       60°




                                                                                                          60°




           HA =0
                             60°




                                                                 60°




                                                                                   FCE FEC                        E
                A                                            C
                 VA=15KN                                   P3 =10KN                                                   VE =15KN

tMNr A nigtMNr E CatMNrEdlmanGBaØtiBIr dUcenHtMNrTaMgBIrCatMNrEdleyIgGaccab;epþImedaHRsay)an
       - sikSalMnwgRtg;tMNr A
       edaykMlaMg F CakMlaMgeRTt dUcenHeyIgGacbMEbk)ankMlaMgpÁúMBIrKW V nig H . ehIykMlaMgTaMgbI
                        AB                                                                          AB            AB


pÁúMKña)anCaragRtIekaNEkg EdlmankMlaMg F CaGIub:UetnIs kMlaMg V CaRCugQm nig kMlaMgV CaRCugCab;.
                                                      AB                                  AB                                     AB




Trusses                                                                                                                                    31
T.Chhay


          eyIg)anplbUkkMlaMgRtg;tMNr A                                                                      VAB        FAB




                                                                                                                        60
                                                                                                                             °
          ∑ Fy = 0 ⇒ V A + V AB = 0
                                                                                                                        HAB
          ⇒ V AB = −V A = −15 KN
          ∑ Fx = 0 ⇒ H A + FAC + H AB = 0




                                                                                                                60°
                                                                                                                                        FAC
          ⇒ FAC = − H AB   eday H   A    =0
                                                                                 HA =0          A


                                                      − 15
          eday tan 60 = H
                        V  AB
                                 ⇒ H AB =
                                               V AB
                                                    =
                                              tan 60 tan 60
                                                            = −8.66 KN
                                                                                           VA=15KN
                            AB

          ⇒ FAC = 8.66 KN       rgkMlaMgTaj
                                                     − 15
       eday sin 60 = V ⇒ F
                      F
                           AB
                                    AB   =
                                              V AB
                                                   =
                                             sin 60 sin 60
                                                           = −17.32 KN          rgkMlaMgsgát;
                           AB

     - sikSalMnwgRtg;tMNr B
                                                                                                    P1 =10KN
     ∑ Fy = 0 ⇒ − P1 − V AB − VBC = 0

          ⇒ VBC = − P1 − V AB = −10 − (− 15) = 5 KN
                                                                                                            B                                   FBD


                     VBC      5
          ⇒ H BC =        =       = 2.89 KN
                    tan 60 tan 60
                                                                             HBA=-8.66KN
          ⇒ FBC
                    V
                  = BC =
                             5
                   sin 60 sin 60
                                 = 5.77 KN         rgkMlaMgTaj                                                                   HBC

                                                                                                                                 FBC
                                                                                                VBA=-15KN
                                                                              FBA=-17.32KN                             VBC
      ∑ Fx = 0 ⇒ FBD − H BA + H BC = 0
          ⇒ FBD = H BA − H BC = −8.66 − 2.89 = −11.55 KN          rgkMlaMgsgát;
     - sikSalMnwgRtg;tMNr C                                                     FCB=5.77KN
                                                                                         VCB=5KN
                                                                                                                 VCD
                                                                                                                       FCD

      ∑ Fy = 0 ⇒ VCB + VCD − P3 = 0                                                                                      HCD
                                                                             HCB=2.89KN

          ⇒ VCD = P3 − VCB = 10 − 5 = 5 KN
                                                                                         60°




                                                                                                                60°



                    V          5                                  FCA=8.66KN                                                           FCE
          ⇒ H CD = CD =            = 2.89 KN                                                        C
                   tan 60 tan 60
                   V
          ⇒ FCD = CD =
                              5
                  sin 60 sin 60
                                  = 5.77 KN        rgkMlaMgTaj                             P3 =10KN
      ∑ Fx = 0 ⇒ − FCA − H CB + H CD + FCE = 0
          ⇒ FCE = FCA + H CB − H CD = 8.66 + 2.89 − 2.89 = 8.66 KN rgkMlaMgsgát;
eday truss rgnUvbnÞúksIuemRTI dUcenHkMlaMgkMNat;r)arRtg;tMNr E dUckMlaMgkMNat;r)arRtg;tMNr A ehIy
kMlaMgkMNat;r)arRtg;tMNr D dUckMlaMgkMNat;r)arRtg;tMNr B.                       B               11.55C                  D
                                                                                                                                       17
                                                                       32C




                                                                                          5.7




                                                                                                               7T




                                                                                                                                         .32
                                                                                                            5.7
                                                                   17.




                                                                                               7T




                                                                                                                                            C




                                                              A                8.66T                                    8.66T                     E
                                                                                                        C




Trusses                                                                                                                                          32
T.Chhay


2> viFImuxkat; method of section
              viFImuxkat;CaviFImYysMrab;kMNt;kMlaMgrbs;r)arNamYy EdleyIgcg;dwgeTaHCavaenAq¶ayBITMrk¾
eday edaymincaM)ac;eFVIkaredaHRsaytam lMdab;lMedaydUcviFItMNreT. sMrab;viFIenH truss RtUv)ankat;CaBIr
kMNat;Rtg;r)arNaEdleKcg;dwgkMlaMg.
         vIFIsaRsþkMNt;kMlaMgrbs;r)artamviFImuxkat;³
    - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl.
         CMnYskMlaMgeRTt edaybgÁúMkMlaMgbBaÄr b¤kMlaMgedk.
    - kat;pþac; truss CaBIrkMNat;Rtg;r)arNaEdleKcg;dwg. bøg;kat;minRtUvmanr)arelIsBIbI EdlCaGBaØti
         elIkElgEt ExSskmμénkMNat;r)arkat;enaHRbsBVKñaRtg;cMnucNamYy.
    - edayGnuvtþn¾smIkarlMnwgsþaTicTaMgbI ( ∑ F = 0 , ∑ F = 0 nig ∑ M = 0 ) enaHkMlaMgrbs;kMNat;
                                                         y            x


         r)arEdlcg;dwgRtUv)ankMNt;.
    - eTaHCaminsMxan;k¾eday eKRtUvEtRtYtBinitücMelIyedayepÞógpÞat; ∑ F = 0 , ∑ F = 0 nig ∑ M = 0 .
                                                                                       y                x


         edayeRbIcMnuccab;fμIminEmncas;eT.
                                                                P =5KN              P =10KN
]TahrN¾³
                                                                          1                                 2



                                                                                                D
         kMNt;kMlaMgmanGMeBIenAkñúgkMNat; BD,CD nig CE
                                                                          B                                     F
                                                                                                                      4
                                                                                                                      5 3


 én Howe truss dUcbgðajkñúgrUb edayeRbIviFImux kat;.




                                                                                                                                        3
                                                             A                                                                      H


truss RTedayTMr pin Rtg;tMNr A nig TMr roller
                                                                              C                     E           G



Rtg;tMNr H.                                                             P =15KN             3

                                                                                      4 bays @4m=16m


         dMeNaHRsay³
         truss RtUvmanlMnwg tamrUbmnþ n = r + b − 2 f
          r = 3 ( H A , V A , VH )
          b = 13 ( AB, AC , BC , BD, CD, CE , DE , DF , DG, EG, FG, FH , GH )
           f = 8 ( A, B, C , D, E , F , G, H )               P1 =5KN                                    P2 =10KN
          ⇒ n = 3 + 13 − (2 × 8) = 0
                                                                           Cutting plane
                                                                         a

          dUcenH truss manlMnwgkMNt;tamsmIkar                     B                     D                   F
                                                                                                                    4

          lMnwgsþaTic                                                                                               5 3
                                                                                                                                        3m




          1> KNnaRbtikmμTMr                  HA     A                                                                          H
                                                                      C           a         E               G

          ∑ FX = 0 ⇒ H A = 0                        VA                                                                    VH
                                                                                  P3 =15KN
          ∑ FY = 0 ⇒ VA − P − P2 − P3 + VH = 0
                                                                                  4 bays @ 4m=16m
                            1

                   ⇒ V A + VH = P + P2 + P3 = 5 + 10 + 15 = 30 KN (1)
                                 1




Trusses                                                                                                                        33
T.Chhay



          ∑ M A = 0 ⇒ 4 × P + 8 × P2 + 12 × P3 − 16VH = 0
                           1

                           4 P + 8P2 + 12 P3 20 + 80 + 180
                   ⇒ VH = 1                  =             = 17.5KN
                                  16                 16
          CMnYs V   H   = 17.5 KN   kñúg (1) eyIg)an
          V A = 12.5 KN
          2> kMNt;kMlaMgrbs;r)ar BD,CD nig CE tamviFImuxkat;
          enAeBlEdleKkat; truss Rtg;muxkat; a-a eKnwg)ankMlaMg                                    P1 =5KN
                                                                                                                Cutting plane


          rbs;r)arbI (BD, CD nig CE) KW F , F nig F .
                                                                                                            a

                                                 BD     CD           CE                            B                FBD

          edayr)ar CDCar)areRTtdUcenH kMlaMg F bMEbk)ankM  CD
                                                                                                        VCD         FCD




                                                                                  3m
                                                                                                                            5
                                                                                                                                    3

          laMgpÁúMBIrKW V nig H .
                                                                                                                    HCD
                                                                                                                                4
                          CD         CD
                                                                    A
                                                           H =0           A                                           FCE
                                                                                                    C       a
          Edl F5 = V3 = H4 ¬tamRtIekaNdUc¦.
                   CD       CD       CD
                                                             V =12.5KN        A
                                                                                            4m

          ∑ M C = 0 ⇒ VA × 4 + FBD × 3 = 0
                             4 × 12.5
                    4V
          ⇒ FBD = − A = −
                     3           3
                                      = −16.67 KN               ¬rgkMlaMgsgát;¦
          ∑ Fy = 0 ⇒ VA + VCD − P = 0
                                 1


          ⇒ VCD = P − VA = 5 − 12.5 = −7.5 KN
                   1

                     V        7.5
          ⇒ H CD = 4 CD = −4      = −10 KN
                      3        3
                    V
          ⇒ FCD = 5 CD = −5
                     3
                             7.5
                              3
                                 = −12.5 KN              ¬rgkMlaMgsgát;¦
          ∑ Fx = 0 ⇒ FBD + H CD + FCE = 0
          ⇒ FCE = − FBD − H CD = −(− 16.67 ) − (− 10) = 26.67 KN                  ¬rgkMlaMgTaj¦
          epÞógpÞat;m:Um:g;Rtg;cMnuc A
          ∑ M A = P × 4 + FBD × 3 − VCD × 4 = 0
                   1

          ⇒ 5 × 4 − 16.67 × 3 − (− 7.5)× 4 = 0         epÞógpÞat;!
          dUcenH    FBD = 16.67 KN (C)
                    FCD = 12.5 KN (C)
                    FCE = 26.67 KN (T)
3> viFIRkahVik Maxwell-Cremona
        sMrab;viFIRkahVik k¾dUcviFItMNrEdr KWeKRtUvsikSalMnwgtMNrnImYy². tMNrmYymanlMnwgluHRtaEt
BhuekaNkMlaMg enARtg;tMNrenaHCaBhuekaNbiT.
        vIFIsaRsþkMNt;kMlaMgrbs;r)artamvIFIRkahVik Maxwell-Cremona³
Trusses                                                                                                                         34
T.Chhay


   - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl.
   - EbgEcktMbn;. tMbn;CaEpñkmYyénbøg;EdlxNнedaykMNat;r)ar truss nigExSskmμénkMlaMgRbtikmμ b¤
       ExSskmμénbnÞúkxageRkA.
   - kMNt;maRtdæankMlaMg nigmaRtdæanbøg; truss
   - KUrBhuekaNkMlaMg biTRtg;tMNrNamanGBaØtiy:ageRcInBIrmuneK. rebobehAeQμaHkMlaMg edayeRbI
       eQμaHtMbn;EdlenAsgxagkMlaMgenaH. eKniymehAtamRTnicnaLika.
]TahrN¾³
       Howe truss EdlRTedayTMrsamBaØ rgnUvbnÞúkdUc              P =15KN                2



bgðajkñúgrUb. kMNt;RbtikmμTMr nigkMlaMgénr)ar nImYy² P =15KN C       1       P =15KN                              3


edayeRbIvIFIRkahVik. truss RTedayTMr pin Rtg;cMnuc A




                                                                                                                                                          1.5
nigRTedayTMr roller Rtg;cMnuc E.                       B                         D




                                                                                                                                                          1.5
       dMeNaHRsay³                                 A
                                                                   F
                                                                                         E


       truss RtUvmanlMnwg tamrUbmnþ                                       5                                           5



          n = r +b−2f                                                                                   P2 =15KN

          r = 3 ( H A , V A , VE )                                              P1 =15KN                 C            P3 =15KN
          b = 9 ( AB, AF , BC , BF , CD, CF , DE , DF , EF )




                                                                                                                                                        1.5
          f = 6 ( A, B, C , D, E , F )
                                                                                       B                                           D


          ⇒ n = 3 + 9 − (2 × 6) = 0




                                                                                                                                                        1.5
                                                                HA        A                                                                         E


          dUcenH truss manlMnwgkMNt;tamsmIkar
                                                                                                         F
                                                                                           5                                   5
                                                                         VA                                                                    VE

          lMnwgsþaTic
          1> KNnaRbtikmμTMr
          ∑ FX = 0 ⇒ H A = 0

          ∑ FY = 0 ⇒ V A − P1 − P2 − P3 + VE = 0
                   ⇒ V A + VH = P1 + P2 + P3 = 15 + 15 + 15 = 45KN (1)

          ∑ M A = 0 ⇒ −2.5 × P1 − 5 × P2 − 7.5 × P3 + 10VE = 0
                                 2.5 P1 + 5 P2 + 7.5P3 37.5 + 75 + 112.5
                        ⇒ VE =                        =                  = 22.5 KN
                                          10                  10
          CMnYs V   E   = 22.5 KN   kñúg (1) eyIg)an                                           P1 =15KN
                                                                                                              P2 =15KN

                                                                                                                              P3 =15KN
                                                                                                                  C
                                                                                                          3               4
          V A = 22.5 KN                                                                         B                                      D
                                                                                                              7       8                    5

          2> kMNt;kMlaMgrbs;r)artamviFIRkahVik Maxwell-Cremona
                                                                                   2
                                                                                                    6                              9
                                                                               A                                                                    E
                                                                                                                  F

          - EbgEcktMbn; tamRTnicnaLika                                        VA=22.5KN
                                                                                                                  1
                                                                                                                                               VE=22.5KN




Trusses                                                                                                                                             35
T.Chhay


          - karehAeQμaHkMlaMg V = 1− 2 / F = 2 − 6 / F = 6 − 1
                                        A                AB             AF


          - kMNt;maRtdæanbøg; 50mm=1m                                                       2

          - kMNt;maRtdæankMlaMg 2mm=1KN                                                         VA=22.5KN

          - KUrBhuekaNkMlaMgRtg;tMNr A eRBaHtMNrenHmanGBaØtiBIr                             1


          - KUrkMlaMgtamTisedA nigkarehAeQμaHtamRTnicnaLika                                               FAB
                                                                                                                       6




          - KUrkMlaMg F eGayRsbnwgr)ar AB. edaykMlaMgenHsßit
                           AB
                                                                                                     2

                                                                                                         VA=22.5KN


          enAcenøaHtMbn; 2-6 dUcenHkMlaMg F RtUvKUredayP¢ab;BIcMnuc 2.
                                                    AB
                                                                                                     1



          - KUrkMlaMg F eGayRsbnwgr)ar AF. edaykMlaMgenHsßit
                           AF                                                                                    FAB
                                                                                                                                2

                                                                                                                                    VA=22.5KN

          enAcenøaHtMbn; 6-1 dUcenHkMlaMg F RtUvKUredayP¢ab;BIcMnuc 6
                                                    AB
                                                                                                 6
                                                                                                                   FAF
                                                                                                                                1


           eTAcMnuc 1.
          - bnÞab;mkvas;RbEvgGgát;rYcKuNb¤EckCamYynwgmaRtdæan
          eyIgTTYl)ankMlaMgrbs;kMNat;r)ar.                                                                        2
                                                                                                                           6
                                                                                                                                    B



          - karkMNt;sPaBrbs;kMlaMg KWeKRtUvkMNt;TisedArbs;kMlaMg                                         A
                                                                                                                           1
                                                                                                                                        F
                                                                                                                                             2
                                                                                                                                 KN
          TisedArbs;kMlaMgGaRs½ynwgkarehAeQμaHrbs;kMlaMgtamry³tMbn;.                                                   FA
                                                                                                                          B=4
                                                                                                                              3.7
                                                                                                                                                 VA=22.5KN

          tMbn;EdlehAmunCaKl; ÉtMbn;EdlehAeRkayCacug. kMlaMg F = 2 − 6            AB                         6
                                                                                                                       FAF=37.5KN
                                                                                                                                             1


          manTisedArt;cUltMNr A dUcenH kMlaMg F = 43.7 KN rgkMlaMgsgát; ÉkMlaMg F = 6 − 1
                                                                   AB                                                                   AF


          manTisedArt;ecjBItMNr A dUcenH kMlaMg F = 37.5KN rgkMlaMgTaj. cMNaMBhuekaNkMlaMgRtg;
                                                              AF


          tMNr A CaBhuekaNbiT.
          - KUrBhuekaNkMlaMgRtg;tMNr B edayKñaBIBhuekaNkMlaMgRtg;tMNr A eK)anr)ar AB rgkMlaMg
           F = 43.7 KN rgkMlaMgsgát; r)ar BC rgkMlaMg F = 29.2 KN rgkMlaMgsgát; nigr)ar BF
           BA                                                                BC


          rgkMlaMg F = 14.6KN rgkMlaMgsgát;.
                      BF

                                                     C                                            2
                    P1 =15KN                3                                .7K
                                                                                 N
                                                                           43                            P1 =15KN
                                                                          =
                                B                                      FBA            N              3
                      2                         7                 6                .2K
                                                                               =29                   1
                                                               FBF          FBC
                                                                  =1
                                    6                               4.6
                                                                       KN 7
                A                                        F


          - KUrBhuekaNkMlaMgRtg;tMNr F eK)anr)ar CB rgkMlaMg F = 43.7 KN rgkMlaMgsgát;r)ar CD
                                                                                  CB


          rgkMlaMg F = 29.2KN rgkMlaMgsgát; nigr)ar CF rgkMlaMg F = 15KN rgkMlaMgTaj.
                      CD                                                               CF




Trusses                                                                                                                                           36
T.Chhay




                                                                                              2

                             C                                              8
                                      P2 =15KN                                                   3
                    3                 4                            6
                                                                                                 1
          B              7        8           D           FCF =15KN                              4   P2 =15KN
                                                                            7
                                 F
                                                                       FCD=29.2KN


              - eFIVdUcKñasMrab;tMNr D,E nigtMNr F. eKTTYl)antaragdUc
              xageRkam³
                     l>r> eQμaHr)ar          RbEvg         kMlaMg                   sPaB
                    1.       AB                   2.92m        43.7KN               rgkMlaMgsgát; (C)
                    2.       AF                   5m           37.5KN               rgkMlaMgTaj(T)
                    3.       BC                   2.92m        29.2KN               rgkMlaMgsgát; (C)
                    4.       BF                   2.92m        14.6KN               rgkMlaMgsgát; (C)
                    5.       CD                   2.92m        29.2KN               rgkMlaMgsgát; (C)
                    6.       CF                   3m           15KN                 rgkMlaMgTaj(T)
                    7.       DE                   2.92m        43.7KN               rgkMlaMgsgát; (C)
                    8.       DF                   2.92m        14.6KN               rgkMlaMgsgát; (C)
                    9.       EF                   5m           37.5KN               rgkMlaMgTaj(T)
4> lkçxNÐBiess Special condition                                                            F1
   1. Rtg;tMNrmYypÁúMeLIgedayr)arbI Edlr)arBIrrt;Rtg;Kña

    ehIyRtg;tMNrenaHKμankMlaMgxageRkA dUcenHeK)anr)arBIr
   rt;Rtg;KñamankMlaMgesμIKña F = F Ér)ar ÉekaesμIsUnü
                                          2   3
                                                                                             1            F3
     F = 0 ¬r)arenHeKdak;edIm,IkarBarPøat;¦.
          1                                                                     1
                                                                                                      3

                                                                                             A
                                                                                        2

                                                                       F2

Trusses                                                                                                         37
T.Chhay




     2.  Rtg;tMNrmYypÁúMeLIgedayr)arBIr rt;minRtg;Kña        F1
      ehIymankMlaMgxageRkA P RtYtsIuKña nwgGkS½én                                      P
                                                                          1
      r)arNamYy kMlaMgr)arenaHmantMélesμIkMlaMgxag
     eRkAenaH F = P ÉkMlaMgénr)armYyeTotesμIsUnü
                  2


      F = 0.
          1
                                                                          2

                                                         F2
     3.  Rtg;tMNrmYypÁúMeLIgedayr)arBIr rt;minRtg;Kña            F1
     ehIyKμankMlaMgxageRkAman GMeBIelItMNrenaH                                1
     enaHkMlaMgénr)arnImYy²mantMélesμIsUnü
      F = F = 0.
          1   2                                                               2

                                                              F2


     4.  Rtg;tMNrmYypÁúMeLIgedayr)arbYn                                           F1
                                                    F4
     EdlBIr²rt;Rtg;Kña ehIyKμankMlaMgxag                              1
     eRkAman GMeBIelItMNrenaH enaHkMlaMg                 4
     énr)arBIr² rt;Rtg;KñamantMélesμIKña                                  3
                                                             2
      F = F nig F = F .
          1   2       3   4                                                       F3
                                                  F2




Trusses                                                                                    38
T.Chhay


lMhat;³
   1> edayeRbInUvviFItMNrkMNt;kMlaMgrbs;kMNat;r)arén truss xageRkam³
   1>1>                      100KN                100KN

                             A                    B
             10KN




                                                       8m
                             C                    D
             20KN



                                                       8m
                             E                    F

                                         6m



                                                                   20KN
     1>2>
                                         20KN                      D                 20KN




                                                                                                                  2m
             10KN                        B                                               F               10KN




                                                                                                                  2m
             A                                                                                                H
                                         C                         E                 G

                             4m                           4m                    4m                4m

                                                                       20KN

     1>3>                                             20KN             D
                                                                                     20KN
                 1.5m 1.5m




                              10KN                    B                                  F
                                                                                                       10KN


                                 A                          C                        E                    G


                                                  4m                       4m                4m




     1>4>                            B                         D                 F
                    4m




                                     A                         C                 E                  G


                                                      3m                   3m                3m          120KN
                                                  A

      1>5>                                                              20KN
                             3m




                                                                        C


                                                                                             50KN
                             3m




                                              B                                          E
                                                                       D

                                                            4m                  4m




Trusses                                                                                                                39
T.Chhay


     2> edayeRbInUvviFImuxkat;kMNt;kMlaMgénr)armYycMnYndUcxageRkam³
     2>1>
                             15KN                       20KN                               20KN                  10KN

                             A             B           C                 D             E                F        G


               1.5m
                             N             M           L                 K                 J            I        H
                                 2m            2m           2m               2m                2m           2m

     FCD , FCK , FEI ?
     2>2>                                                                          20KN

                                                                                   C
                                                            15KN
                                 2m




                                                            B
                                 2m




                                       A                    F                      E                        D


                                               3m                3m                        3m


     FBC , FBE , FEF ?
     2>3>                                          20KN              8KN                       12KN

                         B                     D                     E                         G
                                                                                                   2m




                                               C                 F                         H
              4m




                         A


                                  3m                   3m                     3m


     FBD , FDE , FCF ?
     3> edayeRbInUvviFIRkahVikkMNt;kMlaMgénr)arrbs; truss xageRkam³
     3>1>                                    20KN

                             A                     C             E                     G
                                                                                                    40KN
                      2m




                             B                 D                 F                     H

                                      3m               3m                    3m

                                           15KN

Trusses                                                                                                                 40
T.Chhay


     3>2>                                   12KN


                                                C


                                  18KN                    18KN




            3.46m




                                                                       5m
                                   B                        E

                                                D
            5.54m




                                                                       4m
                                            24KN

                         A                                         F

                                  6m                         6m



     3>3>                         24KN

                         G             H              I
                                                            10KN
                    3m




                         E             F              J
                    3m




                                                    16KN
                         C             D
                    3m




                         A             B


                             4m            4m




Trusses                                                                     41

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3.trusses14

  • 1. T.Chhay Trusses KWCaeRKOgbgÁúMmanragCaRbBn§½RtIekaN EdlekItBIkarpÁúMKñarvagr)arCaeRcInkMNat;. ragRtIekaN Truss CaeKalkarN_lMnwgrbs; truss. xageRkamCaRbePTénrUbragrbs; truss nigPaBmanlMnwgCaragRtIekaN nig PaBKμanlMnwgCaragctuekaNeRkamGMeBIénkMlaMgxageRkA. Warren truss Pratt roof truss Howe roof truss Fink roof truss Pratt bridge truss rUbTI1³ RbePT Truss P P Original shape Collapsed shape (a) Rigid, stable framework (b) Nonrigid, unstable framework kMNat;rbs; truss maneQμaHepSgKñaeTAtamTItaMg xageRkamCarUbbgðajBIkarehAeQμaHrbs;kMNat; nImYy². P1 U p p e r c h o rd P2 P3 P a n e l p o in ts B C D E F W eb m em ber W eb m em ber (v e rtic al) (d ia g o n a l) E n d p o st A G H I J K L L o w e r c h o rd R o lle r su p p o rt P in n e d su p p o rt P4 6 bays @ 5m 30m Trusses 28
  • 2. T.Chhay enAeBl truss rgkMlaMgxageRkA EdlmanGMeBIcMRtg;tMNEdlkat;Kñarvagr)arBIrb¤eRcIn kMNat; nImYy²rbs; truss mantYrnaTIbBa¢ÚnkMlaMgxageRkAenaHmkTMr ehIykMNat;enaHrgnUvkMlaMgBIrepSgKña. kMlaMg TaMgBIrenaHrt;tamGkS½beNþayrbs;r)ar EdleFVIeGayr)arlUt b¤rYj EdleKehAfa kMlaMgcMGkS½ (axial forces or direct forces). r)arEdllUt Car)arEdlrgkMlaMgTaj ehIyr)arEdlrYjCar)arEdlrg kMlaMgsgát;. karviPaK truss rYmmankarkMNt;tMélénkMlaMgEdlr)arrg nigsPaBénGMeBI (Taj b¤sgát;). xageRkamCarUb bgðajBIsßanPaBénr)arrgkMlaMg. Longitudinal axis of member P P P (a) Axial compression (b) Axial tension edIm,Idwgfa truss mYyCa truss manlMnwgeKRtUvrkGBaØtielIstamrUbmnþxageRkam³ n = r +b−2f Edl n CacMnYnGBaØtielIs r cMnYnRbtikmμ ¬CaTUeTAesμI 3 GaRs½yelITMr¦ f cMnYntMN ebI n < 0 mann½yfa truss minmanlMnwg ebI n = 0 mann½yfa truss manlMnwg kMNt;kMlaMgkñúgrbs; truss tamsmIkarsþaTic ebI n > 0 mann½yfa truss manlMnwg EtminGackMNt;kMlaMgkñúg rbs; truss tamsmIkarsþaTiceT. edIm,IsMrYlkñúgkarviPaK truss eKRtUveFVIkarsnμt;nUvcMnucmYycMnYnxageRkam³ - RKb;kMNat;rbs; truss TaMgGs;RtUvsßitenAkñúgbøg;EtmYy. - bnÞúk nigRbtikmμRtUvGnuvtþn¾ EtenAelItMNrrbs; truss. - kMNat;rbs; truss RtUvP¢ab;Kña edaytMNrKμankkit. - RKb;Ggát;TaMgGs;RtUvEtRtg; ehIyCaGgát;rgkMlaMgBIr EdlkMlaMgenAcugsgxag énGgát;RtUvEtmantM élesμIKμa pÞúyKña nigrt;Rtg;Kña. - ExSskmμénkMlaMgkñúgénr)arRtUvEtcMGkS½. - bMErbMrYlRbEvgrbs;Ggát;EdlekItBIkMlaMgTaj b¤sgát;minmantMélRKb;RKan;EdleFIVeGaylkçN³ FrNImaRtrbs; truss ERbRbYleT. Trusses 29
  • 3. T.Chhay - bnÞúkpÞal;rbs;Ggát;RtUv)anecal eRBaHvamantMéltUcNas;ebIeFobeTAnwg bnÞúkxageRkAEdlvaRT. EtRbsinebIbnÞúkxageRkArbs;vaRtUv)anrab;bBa©Úl vaRtUv)ansnμt;CabnÞúkcMcMnucEdlmanGMeBIrenAcug sgçagkMNat;. QrelIeKalkarN¾snμt;xagelI nigeRbInUvsmIkarlMnwgsþaTic eKGackMNt;kMlaMgkñúgrbs;kMNat; truss )antambIviFIsaRsþ KW viFItMNr method of joint viFImuxkat; method of section vIFIRkahVik Maxwell- Cremona. 1> viFItMNr method of joint sMrab;viFIenH eKcat;Tukfa truss mYymanlMnwgeTA)anluHRtaNaEtRKb;tMNrrbs;vamanlMnwg. enAkñúg karviPaK truss tamviFItMNr eKRtUvsikSaBItMNrNaEdlmanGBaØtiy:ageRcInBIr. GBaØtiKWCakMlaMgkñúgrbs; kMNat;r)ar truss. vIFIsaRsþkMNt;kMlaMgrbs;r)artamvIFItMNr³ - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl. CMnYskMlaMgeRTt edaybgÁúMkMlaMgbBaÄr b¤kMlaMgedk. - pþac;tMNrnImYy² Ca Free body diagram edaybgðajRKb;kMlaMgTaMgGs;EdlmanGMeBIrelItMNrenaH. tMNrnImYy²sMEdgedayRbBn§½kMlaMgRbsBVkñúgbøg; nigcab;epþImecjBItMNrNaEdlmanGBaØtimineRcIn CagBIr. - edayGnuvtþn¾smIkarlMnwgsþaTicTaMgBIr ( ∑ F = 0 nig ∑ F = 0 ) enaHGBaØtiTaMgBIrsMrab;tMNrenaH y x RtUv)ankMNt;. ehIykMlaMgénr)arTaMgenH RtUveRbIsMrab;tMNrbnÞab;EdlP¢ab;edayr)arTaMgenH. sMrab;krNIxøH viFanRtIekaNdUcmansar³RbeyaCn¾CaxøaMgkñúgkaredaHRsaysmIkar. - eTaHCaminsMxan;k¾eday eKRtUvEtRtYtBinitücMelIyedayepÞógpÞat;tMNrNamYy edayeGay ∑ F = 0 y nig ∑ F = 0 . x - eRkayeBlEdlkMlaMgRKb;kMNat;r)arRtUv)anrkeXIj KUrnUvdüaRkamkMlaMgsegçbén truss enaH. ]TahrN¾³ P1 =10KN 2m P2 =10KN Warren truss EdlRTedayTMrsamBaØ rgnUvbnÞúkdUc B D bgðajkñúgrUb. kMNt;RbtikmμTMr nigkMlaMgénr)ar nImYy² edayeRbIvIFItMNr. truss RTedayTMr pin Rtg;cMnuc A ° ° 60 60° nigRTedayTMr roller Rtg;cMnuc E. 60 60 ° A E C P3 =10KN 2m 2m Trusses 30
  • 4. T.Chhay dMeNaHRsay³ P1 =10KN 2m P2 =10KN truss RtUvmanlMnwg tamrUbmnþ n = r + b − 2 f B D r = 3 ( H A , V A , VB ) b = 7 ( AB, AC , BC , BD, CD, CE , DE ) f = 5 ( A, B, C , D, E ) 60° 60° ⇒ n = 3 + 7 − (2 × 5) = 0 60° 60° HA A E C dUcenH truss manlMnwgkMNt;tamsmIkarlMnwgsþaTic VA 2m P3 =10KN 2m VE 1> KNnaRbtikmμTMr Free body diagram ∑ FX = 0 ⇒ H A = 0 ∑ FY = 0 ⇒ V A − P1 − P2 − P3 + VE = 0 ⇒ V A + VE = P1 + P2 + P3 = 10 + 10 + 10 = 30 KN (1) ∑ M A = 0 ⇒ −1× P1 − 3 × P2 − 2 × P3 + 4VE = 0 P1 + 3P2 + 2 P3 10 + 30 + 20 ⇒ VE = = = 15KN 4 4 CMnYs V E = 15 KN kñúg (1) eyIg)an V A = 15 KN 2> kMNt;kMlaMgrbs;r)artamviFItMNr P1 =10KN P2 =10KN B FBD FBD D FBA FBC FDC FDE FAB FCB FCD FED FAC FCA 60° 60° HA =0 60° 60° FCE FEC E A C VA=15KN P3 =10KN VE =15KN tMNr A nigtMNr E CatMNrEdlmanGBaØtiBIr dUcenHtMNrTaMgBIrCatMNrEdleyIgGaccab;epþImedaHRsay)an - sikSalMnwgRtg;tMNr A edaykMlaMg F CakMlaMgeRTt dUcenHeyIgGacbMEbk)ankMlaMgpÁúMBIrKW V nig H . ehIykMlaMgTaMgbI AB AB AB pÁúMKña)anCaragRtIekaNEkg EdlmankMlaMg F CaGIub:UetnIs kMlaMg V CaRCugQm nig kMlaMgV CaRCugCab;. AB AB AB Trusses 31
  • 5. T.Chhay eyIg)anplbUkkMlaMgRtg;tMNr A VAB FAB 60 ° ∑ Fy = 0 ⇒ V A + V AB = 0 HAB ⇒ V AB = −V A = −15 KN ∑ Fx = 0 ⇒ H A + FAC + H AB = 0 60° FAC ⇒ FAC = − H AB eday H A =0 HA =0 A − 15 eday tan 60 = H V AB ⇒ H AB = V AB = tan 60 tan 60 = −8.66 KN VA=15KN AB ⇒ FAC = 8.66 KN rgkMlaMgTaj − 15 eday sin 60 = V ⇒ F F AB AB = V AB = sin 60 sin 60 = −17.32 KN rgkMlaMgsgát; AB - sikSalMnwgRtg;tMNr B P1 =10KN ∑ Fy = 0 ⇒ − P1 − V AB − VBC = 0 ⇒ VBC = − P1 − V AB = −10 − (− 15) = 5 KN B FBD VBC 5 ⇒ H BC = = = 2.89 KN tan 60 tan 60 HBA=-8.66KN ⇒ FBC V = BC = 5 sin 60 sin 60 = 5.77 KN rgkMlaMgTaj HBC FBC VBA=-15KN FBA=-17.32KN VBC ∑ Fx = 0 ⇒ FBD − H BA + H BC = 0 ⇒ FBD = H BA − H BC = −8.66 − 2.89 = −11.55 KN rgkMlaMgsgát; - sikSalMnwgRtg;tMNr C FCB=5.77KN VCB=5KN VCD FCD ∑ Fy = 0 ⇒ VCB + VCD − P3 = 0 HCD HCB=2.89KN ⇒ VCD = P3 − VCB = 10 − 5 = 5 KN 60° 60° V 5 FCA=8.66KN FCE ⇒ H CD = CD = = 2.89 KN C tan 60 tan 60 V ⇒ FCD = CD = 5 sin 60 sin 60 = 5.77 KN rgkMlaMgTaj P3 =10KN ∑ Fx = 0 ⇒ − FCA − H CB + H CD + FCE = 0 ⇒ FCE = FCA + H CB − H CD = 8.66 + 2.89 − 2.89 = 8.66 KN rgkMlaMgsgát; eday truss rgnUvbnÞúksIuemRTI dUcenHkMlaMgkMNat;r)arRtg;tMNr E dUckMlaMgkMNat;r)arRtg;tMNr A ehIy kMlaMgkMNat;r)arRtg;tMNr D dUckMlaMgkMNat;r)arRtg;tMNr B. B 11.55C D 17 32C 5.7 7T .32 5.7 17. 7T C A 8.66T 8.66T E C Trusses 32
  • 6. T.Chhay 2> viFImuxkat; method of section viFImuxkat;CaviFImYysMrab;kMNt;kMlaMgrbs;r)arNamYy EdleyIgcg;dwgeTaHCavaenAq¶ayBITMrk¾ eday edaymincaM)ac;eFVIkaredaHRsaytam lMdab;lMedaydUcviFItMNreT. sMrab;viFIenH truss RtUv)ankat;CaBIr kMNat;Rtg;r)arNaEdleKcg;dwgkMlaMg. vIFIsaRsþkMNt;kMlaMgrbs;r)artamviFImuxkat;³ - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl. CMnYskMlaMgeRTt edaybgÁúMkMlaMgbBaÄr b¤kMlaMgedk. - kat;pþac; truss CaBIrkMNat;Rtg;r)arNaEdleKcg;dwg. bøg;kat;minRtUvmanr)arelIsBIbI EdlCaGBaØti elIkElgEt ExSskmμénkMNat;r)arkat;enaHRbsBVKñaRtg;cMnucNamYy. - edayGnuvtþn¾smIkarlMnwgsþaTicTaMgbI ( ∑ F = 0 , ∑ F = 0 nig ∑ M = 0 ) enaHkMlaMgrbs;kMNat; y x r)arEdlcg;dwgRtUv)ankMNt;. - eTaHCaminsMxan;k¾eday eKRtUvEtRtYtBinitücMelIyedayepÞógpÞat; ∑ F = 0 , ∑ F = 0 nig ∑ M = 0 . y x edayeRbIcMnuccab;fμIminEmncas;eT. P =5KN P =10KN ]TahrN¾³ 1 2 D kMNt;kMlaMgmanGMeBIenAkñúgkMNat; BD,CD nig CE B F 4 5 3 én Howe truss dUcbgðajkñúgrUb edayeRbIviFImux kat;. 3 A H truss RTedayTMr pin Rtg;tMNr A nig TMr roller C E G Rtg;tMNr H. P =15KN 3 4 bays @4m=16m dMeNaHRsay³ truss RtUvmanlMnwg tamrUbmnþ n = r + b − 2 f r = 3 ( H A , V A , VH ) b = 13 ( AB, AC , BC , BD, CD, CE , DE , DF , DG, EG, FG, FH , GH ) f = 8 ( A, B, C , D, E , F , G, H ) P1 =5KN P2 =10KN ⇒ n = 3 + 13 − (2 × 8) = 0 Cutting plane a dUcenH truss manlMnwgkMNt;tamsmIkar B D F 4 lMnwgsþaTic 5 3 3m 1> KNnaRbtikmμTMr HA A H C a E G ∑ FX = 0 ⇒ H A = 0 VA VH P3 =15KN ∑ FY = 0 ⇒ VA − P − P2 − P3 + VH = 0 4 bays @ 4m=16m 1 ⇒ V A + VH = P + P2 + P3 = 5 + 10 + 15 = 30 KN (1) 1 Trusses 33
  • 7. T.Chhay ∑ M A = 0 ⇒ 4 × P + 8 × P2 + 12 × P3 − 16VH = 0 1 4 P + 8P2 + 12 P3 20 + 80 + 180 ⇒ VH = 1 = = 17.5KN 16 16 CMnYs V H = 17.5 KN kñúg (1) eyIg)an V A = 12.5 KN 2> kMNt;kMlaMgrbs;r)ar BD,CD nig CE tamviFImuxkat; enAeBlEdleKkat; truss Rtg;muxkat; a-a eKnwg)ankMlaMg P1 =5KN Cutting plane rbs;r)arbI (BD, CD nig CE) KW F , F nig F . a BD CD CE B FBD edayr)ar CDCar)areRTtdUcenH kMlaMg F bMEbk)ankM CD VCD FCD 3m 5 3 laMgpÁúMBIrKW V nig H . HCD 4 CD CD A H =0 A FCE C a Edl F5 = V3 = H4 ¬tamRtIekaNdUc¦. CD CD CD V =12.5KN A 4m ∑ M C = 0 ⇒ VA × 4 + FBD × 3 = 0 4 × 12.5 4V ⇒ FBD = − A = − 3 3 = −16.67 KN ¬rgkMlaMgsgát;¦ ∑ Fy = 0 ⇒ VA + VCD − P = 0 1 ⇒ VCD = P − VA = 5 − 12.5 = −7.5 KN 1 V 7.5 ⇒ H CD = 4 CD = −4 = −10 KN 3 3 V ⇒ FCD = 5 CD = −5 3 7.5 3 = −12.5 KN ¬rgkMlaMgsgát;¦ ∑ Fx = 0 ⇒ FBD + H CD + FCE = 0 ⇒ FCE = − FBD − H CD = −(− 16.67 ) − (− 10) = 26.67 KN ¬rgkMlaMgTaj¦ epÞógpÞat;m:Um:g;Rtg;cMnuc A ∑ M A = P × 4 + FBD × 3 − VCD × 4 = 0 1 ⇒ 5 × 4 − 16.67 × 3 − (− 7.5)× 4 = 0 epÞógpÞat;! dUcenH FBD = 16.67 KN (C) FCD = 12.5 KN (C) FCE = 26.67 KN (T) 3> viFIRkahVik Maxwell-Cremona sMrab;viFIRkahVik k¾dUcviFItMNrEdr KWeKRtUvsikSalMnwgtMNrnImYy². tMNrmYymanlMnwgluHRtaEt BhuekaNkMlaMg enARtg;tMNrenaHCaBhuekaNbiT. vIFIsaRsþkMNt;kMlaMgrbs;r)artamvIFIRkahVik Maxwell-Cremona³ Trusses 34
  • 8. T.Chhay - KNnakMlaMgRbtikmμrbs; truss edayeRbI Free body diagram sMrab; truss TaMgmUl. - EbgEcktMbn;. tMbn;CaEpñkmYyénbøg;EdlxNнedaykMNat;r)ar truss nigExSskmμénkMlaMgRbtikmμ b¤ ExSskmμénbnÞúkxageRkA. - kMNt;maRtdæankMlaMg nigmaRtdæanbøg; truss - KUrBhuekaNkMlaMg biTRtg;tMNrNamanGBaØtiy:ageRcInBIrmuneK. rebobehAeQμaHkMlaMg edayeRbI eQμaHtMbn;EdlenAsgxagkMlaMgenaH. eKniymehAtamRTnicnaLika. ]TahrN¾³ Howe truss EdlRTedayTMrsamBaØ rgnUvbnÞúkdUc P =15KN 2 bgðajkñúgrUb. kMNt;RbtikmμTMr nigkMlaMgénr)ar nImYy² P =15KN C 1 P =15KN 3 edayeRbIvIFIRkahVik. truss RTedayTMr pin Rtg;cMnuc A 1.5 nigRTedayTMr roller Rtg;cMnuc E. B D 1.5 dMeNaHRsay³ A F E truss RtUvmanlMnwg tamrUbmnþ 5 5 n = r +b−2f P2 =15KN r = 3 ( H A , V A , VE ) P1 =15KN C P3 =15KN b = 9 ( AB, AF , BC , BF , CD, CF , DE , DF , EF ) 1.5 f = 6 ( A, B, C , D, E , F ) B D ⇒ n = 3 + 9 − (2 × 6) = 0 1.5 HA A E dUcenH truss manlMnwgkMNt;tamsmIkar F 5 5 VA VE lMnwgsþaTic 1> KNnaRbtikmμTMr ∑ FX = 0 ⇒ H A = 0 ∑ FY = 0 ⇒ V A − P1 − P2 − P3 + VE = 0 ⇒ V A + VH = P1 + P2 + P3 = 15 + 15 + 15 = 45KN (1) ∑ M A = 0 ⇒ −2.5 × P1 − 5 × P2 − 7.5 × P3 + 10VE = 0 2.5 P1 + 5 P2 + 7.5P3 37.5 + 75 + 112.5 ⇒ VE = = = 22.5 KN 10 10 CMnYs V E = 22.5 KN kñúg (1) eyIg)an P1 =15KN P2 =15KN P3 =15KN C 3 4 V A = 22.5 KN B D 7 8 5 2> kMNt;kMlaMgrbs;r)artamviFIRkahVik Maxwell-Cremona 2 6 9 A E F - EbgEcktMbn; tamRTnicnaLika VA=22.5KN 1 VE=22.5KN Trusses 35
  • 9. T.Chhay - karehAeQμaHkMlaMg V = 1− 2 / F = 2 − 6 / F = 6 − 1 A AB AF - kMNt;maRtdæanbøg; 50mm=1m 2 - kMNt;maRtdæankMlaMg 2mm=1KN VA=22.5KN - KUrBhuekaNkMlaMgRtg;tMNr A eRBaHtMNrenHmanGBaØtiBIr 1 - KUrkMlaMgtamTisedA nigkarehAeQμaHtamRTnicnaLika FAB 6 - KUrkMlaMg F eGayRsbnwgr)ar AB. edaykMlaMgenHsßit AB 2 VA=22.5KN enAcenøaHtMbn; 2-6 dUcenHkMlaMg F RtUvKUredayP¢ab;BIcMnuc 2. AB 1 - KUrkMlaMg F eGayRsbnwgr)ar AF. edaykMlaMgenHsßit AF FAB 2 VA=22.5KN enAcenøaHtMbn; 6-1 dUcenHkMlaMg F RtUvKUredayP¢ab;BIcMnuc 6 AB 6 FAF 1 eTAcMnuc 1. - bnÞab;mkvas;RbEvgGgát;rYcKuNb¤EckCamYynwgmaRtdæan eyIgTTYl)ankMlaMgrbs;kMNat;r)ar. 2 6 B - karkMNt;sPaBrbs;kMlaMg KWeKRtUvkMNt;TisedArbs;kMlaMg A 1 F 2 KN TisedArbs;kMlaMgGaRs½ynwgkarehAeQμaHrbs;kMlaMgtamry³tMbn;. FA B=4 3.7 VA=22.5KN tMbn;EdlehAmunCaKl; ÉtMbn;EdlehAeRkayCacug. kMlaMg F = 2 − 6 AB 6 FAF=37.5KN 1 manTisedArt;cUltMNr A dUcenH kMlaMg F = 43.7 KN rgkMlaMgsgát; ÉkMlaMg F = 6 − 1 AB AF manTisedArt;ecjBItMNr A dUcenH kMlaMg F = 37.5KN rgkMlaMgTaj. cMNaMBhuekaNkMlaMgRtg; AF tMNr A CaBhuekaNbiT. - KUrBhuekaNkMlaMgRtg;tMNr B edayKñaBIBhuekaNkMlaMgRtg;tMNr A eK)anr)ar AB rgkMlaMg F = 43.7 KN rgkMlaMgsgát; r)ar BC rgkMlaMg F = 29.2 KN rgkMlaMgsgát; nigr)ar BF BA BC rgkMlaMg F = 14.6KN rgkMlaMgsgát;. BF C 2 P1 =15KN 3 .7K N 43 P1 =15KN = B FBA N 3 2 7 6 .2K =29 1 FBF FBC =1 6 4.6 KN 7 A F - KUrBhuekaNkMlaMgRtg;tMNr F eK)anr)ar CB rgkMlaMg F = 43.7 KN rgkMlaMgsgát;r)ar CD CB rgkMlaMg F = 29.2KN rgkMlaMgsgát; nigr)ar CF rgkMlaMg F = 15KN rgkMlaMgTaj. CD CF Trusses 36
  • 10. T.Chhay 2 C 8 P2 =15KN 3 3 4 6 1 B 7 8 D FCF =15KN 4 P2 =15KN 7 F FCD=29.2KN - eFIVdUcKñasMrab;tMNr D,E nigtMNr F. eKTTYl)antaragdUc xageRkam³ l>r> eQμaHr)ar RbEvg kMlaMg sPaB 1. AB 2.92m 43.7KN rgkMlaMgsgát; (C) 2. AF 5m 37.5KN rgkMlaMgTaj(T) 3. BC 2.92m 29.2KN rgkMlaMgsgát; (C) 4. BF 2.92m 14.6KN rgkMlaMgsgát; (C) 5. CD 2.92m 29.2KN rgkMlaMgsgát; (C) 6. CF 3m 15KN rgkMlaMgTaj(T) 7. DE 2.92m 43.7KN rgkMlaMgsgát; (C) 8. DF 2.92m 14.6KN rgkMlaMgsgát; (C) 9. EF 5m 37.5KN rgkMlaMgTaj(T) 4> lkçxNÐBiess Special condition F1 1. Rtg;tMNrmYypÁúMeLIgedayr)arbI Edlr)arBIrrt;Rtg;Kña ehIyRtg;tMNrenaHKμankMlaMgxageRkA dUcenHeK)anr)arBIr rt;Rtg;KñamankMlaMgesμIKña F = F Ér)ar ÉekaesμIsUnü 2 3 1 F3 F = 0 ¬r)arenHeKdak;edIm,IkarBarPøat;¦. 1 1 3 A 2 F2 Trusses 37
  • 11. T.Chhay 2. Rtg;tMNrmYypÁúMeLIgedayr)arBIr rt;minRtg;Kña F1 ehIymankMlaMgxageRkA P RtYtsIuKña nwgGkS½én P 1 r)arNamYy kMlaMgr)arenaHmantMélesμIkMlaMgxag eRkAenaH F = P ÉkMlaMgénr)armYyeTotesμIsUnü 2 F = 0. 1 2 F2 3. Rtg;tMNrmYypÁúMeLIgedayr)arBIr rt;minRtg;Kña F1 ehIyKμankMlaMgxageRkAman GMeBIelItMNrenaH 1 enaHkMlaMgénr)arnImYy²mantMélesμIsUnü F = F = 0. 1 2 2 F2 4. Rtg;tMNrmYypÁúMeLIgedayr)arbYn F1 F4 EdlBIr²rt;Rtg;Kña ehIyKμankMlaMgxag 1 eRkAman GMeBIelItMNrenaH enaHkMlaMg 4 énr)arBIr² rt;Rtg;KñamantMélesμIKña 3 2 F = F nig F = F . 1 2 3 4 F3 F2 Trusses 38
  • 12. T.Chhay lMhat;³ 1> edayeRbInUvviFItMNrkMNt;kMlaMgrbs;kMNat;r)arén truss xageRkam³ 1>1> 100KN 100KN A B 10KN 8m C D 20KN 8m E F 6m 20KN 1>2> 20KN D 20KN 2m 10KN B F 10KN 2m A H C E G 4m 4m 4m 4m 20KN 1>3> 20KN D 20KN 1.5m 1.5m 10KN B F 10KN A C E G 4m 4m 4m 1>4> B D F 4m A C E G 3m 3m 3m 120KN A 1>5> 20KN 3m C 50KN 3m B E D 4m 4m Trusses 39
  • 13. T.Chhay 2> edayeRbInUvviFImuxkat;kMNt;kMlaMgénr)armYycMnYndUcxageRkam³ 2>1> 15KN 20KN 20KN 10KN A B C D E F G 1.5m N M L K J I H 2m 2m 2m 2m 2m 2m FCD , FCK , FEI ? 2>2> 20KN C 15KN 2m B 2m A F E D 3m 3m 3m FBC , FBE , FEF ? 2>3> 20KN 8KN 12KN B D E G 2m C F H 4m A 3m 3m 3m FBD , FDE , FCF ? 3> edayeRbInUvviFIRkahVikkMNt;kMlaMgénr)arrbs; truss xageRkam³ 3>1> 20KN A C E G 40KN 2m B D F H 3m 3m 3m 15KN Trusses 40
  • 14. T.Chhay 3>2> 12KN C 18KN 18KN 3.46m 5m B E D 5.54m 4m 24KN A F 6m 6m 3>3> 24KN G H I 10KN 3m E F J 3m 16KN C D 3m A B 4m 4m Trusses 41