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T.Chhay



                                       PaBdabrbs;Fñwm edayviFIfamBl
                                       Deflection of beam: Energy method

1> kmμnþxageRkA W          E   External work
         eRkamGMeBIénbnÞúk muxkat;mYyénr)ar)anpøas;TI. dUcKñaenHEdr m:Um:g;Rtg;muxkat;mYyénr)ar)aneFVI
eGaymuxkat;enaHmanmMurgVil.
         cMeBaHr)arQr EdlmancugesrIenAxageRkam nigmanKl;bgáb;enAxagelI ehIyrgbnÞúk P Rtg;cugesrI
enaHvaeFVIeGaymansac;lUt Δ . kmμnþEdlekIteLIgedaybnÞúk P KWkmμnþxageRkA W = 1 P.Δ .
                                                                               E
                                                                                       2
         cMeBaHm:Um:g; M EdlGnuvtþmkelIFñwmmYyeFVIeGaymanmMurgVil θ . kmμnþxageRkAEdlbegáIteLIgeday
m:Um:g; M KW W = 1 M .θ .
                 E
                      2
2> famBlb:Utg;Esülxagkñúg W               I   Potential energy
        kmμnþxageRkA)anrkSaTukenAkñúgr)ar ehIy)anERbkøayeTACafamBlxagkñúg.
    k> krNIr)arrgbnÞúktamGkS½
                           1
          WI = WE =          P.Δ
                           2
          tamc,ab;h‘Uk eyIg)an
               PL
          Δ=
               AE
                      P2L
          ⇒ WI =
                      2 EI
    x> krNIFñwmrgm:Um:g;Bt; M
        famBlxagkñúgEdlrkSaTukenAelIFatuGnnþtUc dx ehIyeFVIeGaymanmMurgVil dθ KW
                     1
          dWI =        M .dθ
                     2
          b:uEnþeyIgdwgfa dθ = EI dx
                               M

                 1
          dWI =      M 2 dx
               2 EI
                       2
                   L M
          ⇒ WI = ∫       dx
                  0 2 EI


    • cMNaM³
    - smIkar W = ∫ 2EI dx ¬sMrab;Fñwmrgm:Um:g;Bt; M ¦/ M KWCam:Um:g;Bt;enARtg;muxkat;samBaØmYy
                                   2
                      M        L
                       I
                               0


      ehIym:Um:g;enaHbNþalmkBIplbUksrubénbnÞúkxageRkA.
PaBdabrbs;Fñwm edayviFIfamBl                                                                      118
T.Chhay



    - krNI truss Edlr)arrbs;vargEtkMlaMgtamGkS½ famBlb:Utg;EsülTak;TgEtnwgkMlaMgtambeNþay
      GkS½r)arenaHEtb:ueNÑaH.
    - kñúgFñwmeRkamGMeBIénbnÞúk nigRbtikmμTMr eFVIeGayeKmankMlaMgkat;TTwg nigm:Um:g;Bt; Rtg;muxkat;
      samBaØNamYyénmuxkat;. dUcenHtameKalkarN_rkSafamBl famBlb:Utg;EsülEdlrkSaTukkñúg
      r)arenaH KWCaGnuKmn_énplrbs;kMlaMgkat;TTwg V nigm:Um:g;Bt; M . b:uEnþeKsnμt;fa b:Utg;Esülxag
      kñúgEdlekIteLIgedaykMlaMgkat;TTwg V mantMéltUcEdleKGacpat;ecal)an ebIeFobeTAnwgfam-
      Blb:Utg;EsülEdlekIteLIgedaysarm:Um:g;Bt; M .
3> karKNnabMErbMrYlragedayviFIfamBl
          tameKalkarN_rkSafamBl eK)an³
          WE = W I
          WE - CakmμnþxageRkAEdlbegáIteLIgedaykMlaMgxageRkAGnuvtþbnþicmþg²eTAelIrcnasm<n§½.
         W - CafamBlb:Utg;EsülxagkñúgEdlrkSaTukkñúgrcnasm<n§½.
                   I


         smIkarxagelIGnuBaØatieGaymanEtGBaØtimYyb:ueNÑaH dUcenHr)arRtUvrgbnÞúkEtmYyEtb:ueNÑaH.
]TahrN_³ rkbMErbMrYlrag ¬PaBdab¦ Rtg;cMnuc C                               P
sMrab;r)arTMrsamBaØEdlman EI efr ehIyrgbnÞúk
Rtg;cMnuc P Rtg;kNþalElVg.                      A                          C                  B
                                                                l/2                   l/2
dMeNaHRsay³
kmμnþxageRkAEdlbegáIteLIgedaybnÞúk P KW                                                        M
          PΔ
WE =                                                         Px
           2                                                 2
                                                                         Pl
M  Cam:Um:g;EdlekIteLIgedaysarbnÞúk P                                    4

famBlb:Utg;Esül xagkñúgEdlrkSaTukKW
        M2
WI = ∫
          L
             dx
      0 2 EI


Et  M =
        Px
         2
             ⇒M2 =
                   P2 x2
                    4
                            ¬KitcMeBaH 0 ≤ x ≤ 2 ¦l


tamlkçN³qøúHeK)anm:Um:g;srubesμInwgBIrdg m:Um:g;EpñkxageqVg b¤xagsþaM dUcenHeK)an³
                                       l
               l
                   P2 x2      P 2 x3   2       P 2l 3
WI = 2 ∫       2         dx =              =
              0    8 EI       12 EI    0
                                               96 EI
          PΔ P 2l 3       Pl 3
WE = WI ⇒    =       ⇒Δ=
           2   96 EI     48 EI
]TahrN_³ rkmMurgVilRtg;cMnuc B sMrab;r)arTMrsamBaØEdlman
PaBdabrbs;Fñwm edayviFIfamBl                                                                       119
T.Chhay



EI  efr ehIyrgm:Um:g; M Rtg;cMnuc B .
                               B                 A                         B
                                                                                  M
                                                                l
dMeNaHRsay³
kmμnþxageRkAEdlbegáIteLIgedaym:Um:g; M KW
                                       B
                                                                                       MB
    1                                                      M=MlB.x
WE = M B .θ B
    2
famBlb:Utg;Esül xagkñúgEdlrkSaTukKW                                                          M

       M2
WI = ∫
          L
            dx
     0 2 EI


eday M= B
           M .x
             l
         l M .x    1
⇒ WI = ∫ ( B ) 2      dx
        0    l   2 EI
         M2 l
     = 2 B ∫ x 2 dx
       2l EI 0
        M 2l
⇒ WI = B
        6 EI
tameKalkarN_rkSafamBl eK)an
           M B .θ B M B l
                        2
WE = WI ⇒          =
             2       6 EI
       M Bl
⇒ θB =
       3EI
4> kmμnþvaTuyEGl b¤kmμnþminBit (Virtual work)
         kmμnþvaTuyEGl (V W) CakmμnþEdlekIteLIgedaykMlaMgminBit ehIyeFVIeGaymanbMlas;TIBit.
Bakü virtual EdleKeRbIkñúgkmμnþ b¤bMlas;TI sMedAeTAelItMélminBit b¤TMhMRsemIRsém. ebIeKGnuvtþmkelI
RbBn§½manlMnwgmYy Q eK)anbegáIteGaymankMlaMg virtual xagkñúg nigbMlas;TI virtual tUc². eKsnμt;faRb
                           i


Bn§½kMlaMg virtual Q smmUlesμInwg kMlaMg virtual Ékta (1kN ) edIm,IKNnabMlas;TIRtg;cMnucmYyénrcna
                       i


sm<n§½ tameKalkarN_rkSafamBl kmμnþ virtual xageRkAesμInwgkmμnþ virtual xagkñúg.
          We = Wi

5> karKNnabMErbMrYlragedayviFIkmμnþvaTuyEGl (Virtual work method)
         edIm,IKNnabMlas;TIRtg;muxkat;samBaØmYyénrcnasm<n§½eRkamGMeBIénRbBn§½kMlaMgxageRkA. eKGnuvtþ
nUvCMba‘anxageRkam³
    - snμt;lubbM)at;kMlaMgxageRkATaMgGs;ecj
    - dak;Rtg;cMnucEdlcg;rkbMlas;TInUvkMlaMg virtual Ékta (1kN ) tamTisedAénbMlas;TIBit

PaBdabrbs;Fñwm edayviFIfamBl                                                                     120
T.Chhay



    - rkRbtikmμkMlaMg virtual xagkñúg ¬rab;TaMg kMlaMgkat;TTwg kMlaMgbeNþayGkS½ nigm:Um:g;pg¦ Edl
        ekIteLIgedaysarbnÞúkÉktaenaH.
    - rkRbtikmμTMr/ M , V , N EdlekIteLIgedaysarbnÞúkBit
    - tameKalkarN_rkSafamBleK)an W = W   e       i


    k> krNI truss
    kmμnþxageRkAEdlekIteLIgedaysarbnÞúkÉkta (1kN )
    We = 1× δVC
    Edl δ - bMlas;TIBittamTisQrRtg;cMnuc C
            VC


    famBlb:Utg;EsülEdlrkSaTukenAkñúgr)ar truss
    Wi = ∑ p.Δ
    Edl p - kMlaMgkñúgrbs;r)ar truss Edl)anBIbnÞúkÉkta (1kN )
        Δ - bMErbMrYlragBiténr)ar truss

    tameKalkarN_rkSafamBleyIg)an
    We = Wi ⇒ δVC = ∑ p.Δ
    tamc,ab;h‘Uk eyIg)an
          PL
     Δ=
          EA
                 ∑ p.P.L
     ⇒ δVC =
                   EA
    Edl P - kMlaMgkñúgr)arekItBIbnÞúkBit
          L - RbEvgr)ar

          A - muxkat;r)ar

          E - m:UDuleGLasÞic

]TahrN_³ kMNt;PaBdabrbs; truss Rtg;cMnuc C                      P 1 =30KN           P 2 =50KN



ebIeKdwgfamuxkat;rbs;r)arnImYy² A = 1500mm
                                                                  B                   D
                                             2

                                                                                                         2m
nigmanm:UDuleGLasÞic E = 200GPa .                     A
                                                                            C
                                                                                                    E


dMeNaHRsay³                                                      4m                  4m



edaylubecalkMlaMgxageRkATaMgGs;EdlmanGMeBI                         B                 D



mkelI truss edaydak;bnÞúkÉkta (1kN ) Rtg;cMnuc                                                          2m


C manTisedAcuHeRkam CacMnucEdleyIgRtukarrkPaBdab
                                                          A                                     E
                                                                            C
                                                                  4m                 4m

                                                                            p=1KN



PaBdabrbs;Fñwm edayviFIfamBl                                                                                  121
T.Chhay



kMNt;kMlaMgkñúgrbs;r)arEdlekItBIbnÞúkÉkta nigbnÞúkBit
      r)ar RbEvg L(m) kMlaMgkñúgr)ar p(kN ) kMlaMgkñúgr)ar P(kN )              p.P.L

          AB           2.83         − 0.707                 − 49.5            + 99.04

          AC            4            + 0 .5                  + 35               + 70

          BC           2.83         + 0.707                 + 7.07            + 14.15

          BD            4             −1                     − 40              + 160

          CD           2.83         + 0.707                 − 7.07            − 14.15

          CE            4            + 0 .5                  + 45               + 90

          DE           2.83         − 0.707                 − 63.65           + 127.35
                                    ∑ p.P.L                                   + 546.39
               ∑ p.P.L
          δVC =
                  EA
                  546.39 × 1000
          ⇒ δVC =               = 1.8mm ↓
                   1500 × 200
    x> krNIFñwm ¬PaBdab¦
        cg;rkbMlas;TI δ Rtg;cMnuc C eKeFVIdUc truss Edr edayeKlubkMlaMgBitTaMgLayecal rYcehIyenA
                               C


enARtg;cMnuc C EdleKcg;rkbMlas;TIenaH eKdak;kMlaMg virtual Ékta (1kN ) . eKrkm:Um:g;Bt; virtual m .
bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g;BitmkelI dx . enAeBlmanm:Um:g; M naM
eGaymanmMurgVil dθ enAmuxkat;sgxag dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarkMlaMg
                                                  e


virtual Ékta (1kN ) ehIyeFVIeGaymanbMlas;TI δ Rtg;cMnuc C eK)an³
                                              C

          We = 1.δ C
        Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc
                                                      i                                  dx   RtUv)an
begáIteLIgedaym:Um:g; virtual m EdleFVIeGaymuxkat;vil)anmMu dθ .
          dWi = m.dθ

          EteyIgman dθ = EI dx
                         M

                     m.M
          ⇒ dWi =        dx
                      EI
          b¤Wi = ∫
                   L m.M

                  0 EI
                         dx

          tameKalkarN_rkSafamBl
          We = Wi


PaBdabrbs;Fñwm edayviFIfamBl                                                                     122
T.Chhay


                       L       m.M
          ⇒ δC = ∫                 dx
                       0        EI
   K> krNIFñwm ¬mMurgVil¦
   eFVIdUcKñakrNIxagelIEdr EteKRtUvdak;m:Um:g; virtual Ékta (1kN.m) CMnYseGaykMlaMg virtual Ékta
(1kN ) vij. eKrkm:Um:g;Bt; virtual m' . bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g;

BitmkelI dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarm:Um:g; virtual Ékta (1kN.m) ehIyeFVI
                                   e


eGaymanmMurgVil θ Rtg;cMnuc C eK)an³
                           C

          We = 1.θ C
        Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc
                                                  i                                             dx   RtUv)an
begáIteLIgedaym:Um:g; virtual m' EdleFVIeGaymuxkat;vil)anmMu dθ .
          dWi = m'.dθ

          EteyIgman dθ = EI dx
                         M

                     m'.M
          ⇒ dWi =         dx
                      EI
          b¤Wi = ∫
                  0
                   L m'.M

                      EI
                          dx

          tameKalkarN_rkSafamBl
          We = Wi
                       L   m'.M
          ⇒ θC = ∫              dx
                    0       EI
]TahrN_³ eKeGayFñwmsamBaØrgbnÞúkBRgayesμI w = 20          kN
                                                           m
                                                                                  C    w=20kN
                                                                                            m


nig EI = 40000kN.m . rkPaBdabenAkNþalElVg Δ A
                                       2
                                                      C                                              B
                                                                      4m                   4m
nigmMurgVilenATMr A θ .        A


dMeNaHRsay³
    - rkPaBdabenAkNþalElVg Δ               C
                                                                            1kN

    CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C           A
                                                                   4m
                                                                             C
                                                                                      4m
                                                                                           B


    rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA C
              x
     ⇒m=
              2
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA C
           w.L.x w.x 2
     ⇒M =        −
             2       2
                   2
             w.L.x     w.x 3
     ⇒ m.M =         −
                4       4

PaBdabrbs;Fñwm edayviFIfamBl                                                                             123
T.Chhay



    rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI B eTA C
              x
     ⇒m=
              2
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI B eTA C
           w.L.x w.x 2
     ⇒M =        −
             2       2
                   2
             w.L.x     w.x 3
     ⇒ m.M =         −
                4       4
    plKuNm:Um:g; sMrab; 0 ≤ x ≤ 8m BI A eTA B
                w.L.x 2 w.x 3
     ⇒ m.M =            −
                   2        2
                            2
                                w.x 3 1
    dUcenH         0
                    4 w.L.x
             ΔC = ∫ (
                        2
                              −
                                 2 EI
                                     )  dx
                          4        4
              w.L.x 3   wx 4   1
     ⇒ ΔC = (         −      )
                6 0      8 0 EI
                  20 × 8 × 4 3 20 × 4 4     1
     ⇒ ΔC = (                 −         )       = 26.7 mm ↓
                       6          8       40000
    - rkmMurgVilenATMr A θ     A
                                                                     A          B
    CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A                          8m


    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI A eTA B
              x
     ⇒ m' =
              L
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI A eTA B
           w.L.x w.x 2
     ⇒M =         −
              2        2
                   2
               w.x     w.x 3
     ⇒ m'.M =        −
                2        16
                        2
                            w.x 3 1
    dUcenH      8 w.x
          θA = ∫ (
                0    2
                          −      )
                             16 EI
                                    dx
                      8        8
             w.x 3   wx 4   1
     ⇒θA = (       −      )
              6 0 64 0 EI

     ⇒ θ A = 0.01067 rad
]TahrN_³ eKeGayFñwmTMrbgáb;rgkMlaMgcMcMnucRtg;cug                 30kN


r)ar edayman EI = 30000kN .m . kMNt;PaBdab nigmMu
                                       2                      A
                                                                           4m
                                                                                    B


rgVilRtg;cugr)ar.
dMeNaHRsay³

PaBdabrbs;Fñwm edayviFIfamBl                                                            124
T.Chhay



    - rkPaBdabenAcugr)ar δ                  A
                                                                                     1kN



    CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc A
                                                                                 A                                B
                                                                                                4m




    rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B
     ⇒ m = −x
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B
     ⇒ M = −30.x

     ⇒ m.M = 30 x 2
    dUcenH δ = ∫ (30 x ) EI dx
               A
                    0
                     4   1          2


                         4
              30 x 3      1
     ⇒δA = (           )
                3 0 EI
                            1
     ⇒ δ A = (10 × 4 3 )       = 21.3mm ↓
                         30000
    - rkmMurgVilenAcugr)ar θ            A
                                                                1kN.m
                                                                        A                            B

       CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A
                                                                                 4m




    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B
     ⇒ m' = −1
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B
     ⇒ M = −30 x

     ⇒ m'.M = 30 x
    dUcenH θ = ∫ (30 x) EI dx
               A
                    0
                     4  1

                         4
             30 x 2   1
     ⇒θA = (        )
               2 0 EI

     ⇒ θ A = 0.008rad
]TahrN_³ eKeGayFñwmTMrsamBaØrgkMlaMgBRgayesμI w = 20 .            kN
                                                                   m
                                                                                                     w=20kN
FñwmenHman E = 200 nig I = 200 ×10 mm .
                              kN
                             mm 2
                                                    6   4                                                 m


                                                            C
kMNt;PaBdabRtg;cMnuc C / mMurgVilRtg;cMnuc C nig A .
                                                                             A                                B
                                                                        3m                 8m


dMeNaHRsay³
     - rkPaBdabRtg;cMnuc C δ                    C
                                                                1kN


     CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C
                                                            C                A                                B
                                                                        3m                 8m


     rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
PaBdabrbs;Fñwm edayviFIfamBl                                                                                      125
T.Chhay



     ⇒ m = −x
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
                 20 2
     ⇒M =−         .x = −10 x 2
                 2

     ⇒ m.M = 10 x 3
    rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
     ⇒ m = −0.375 x
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
                          20 2
     ⇒ M = 68.75 x −        .x = −10 x 2 + 68.75 x
                          2

     ⇒ m.M = 3.75 x 3 − 25.78 x 2
    dUcenHδC = ∫
                   0
                    L m.M

                        EI
                             dx
                      3
              3 10 x           8 3.75            8 25.78
     ⇒ δC = ∫           dx + ∫        x 3 dx − ∫         x 2 dx
             0 EI             0 EI              0   EI
             1 ⎛ 10 x 4                       25.78 x 3 ⎞
                           4              8              8
                 ⎜              3.75 x 4                   ⎟
     ⇒ δC =                  +              −
            EI ⎜ 4 0               4 0             3       ⎟
                 ⎝                                       0⎠


     ⇒ δ C = 8.94mm ↑
    - rkmMurgVilRtg;cMnuc C θ       C
                                                                      1kN.m


       CMnYsm:Um:g; virtual Ékta (1kN .m) Rtg;cMnuc C
                                                                  C                A        B
                                                                              3m       8m


    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
     ⇒ m' = −1
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
     ⇒ M = −10 x 2

     ⇒ m'.M = 10 x 2
    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
     ⇒ m' = −0.125 x
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
     ⇒ M = −10 x 2 + 68.75 x

     ⇒ m'.M = 1.25 x 3 − 8.59 x 2
    dUcenHθC = ∫
                0
                 L m'.M

                    EI
                          dx



PaBdabrbs;Fñwm edayviFIfamBl                                                                    126
T.Chhay


               1 3                8               8
     ⇒ θC =      ( ∫ 10 x 2 dx + ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx)
               EI 0               0               0
                           3           8            8
            1 10 x 3   1.25 4   8.59 3
     ⇒ θC =   (      +     x −      x
            EI 3 0       4    0   3                 0

     ⇒ θ C = −0.0024 rad



    - rkmMurgVilRtg;cMnuc C θ     A
                                                                           1kN.m


       CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A              C
                                                                      3m
                                                                             A
                                                                                   8m
                                                                                        B


    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A
     ⇒ m'= 0
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A
     ⇒ M = −10 x 2

     ⇒ m'.M = 0
    rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A
     ⇒ m' = −0.125 x
    smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A
     ⇒ M = −10 x 2 + 68.75 x

     ⇒ m'.M = 1.25 x 3 − 8.59 x 2
    dUcenHθA = ∫
                 0
                  L m'.M

                     EI
                          dx
            1 8                   8
     ⇒θA =     ( ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx)
            EI    0               0
                          8           8
            1 1.25 4          8.59 3
     ⇒θA =     (       x −          x )
            EI 4          0     3     0


     ⇒ θ A = −0.00465rad




PaBdabrbs;Fñwm edayviFIfamBl                                                                127

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11e.deflection of beam the energy methode10

  • 1. T.Chhay PaBdabrbs;Fñwm edayviFIfamBl Deflection of beam: Energy method 1> kmμnþxageRkA W E External work eRkamGMeBIénbnÞúk muxkat;mYyénr)ar)anpøas;TI. dUcKñaenHEdr m:Um:g;Rtg;muxkat;mYyénr)ar)aneFVI eGaymuxkat;enaHmanmMurgVil. cMeBaHr)arQr EdlmancugesrIenAxageRkam nigmanKl;bgáb;enAxagelI ehIyrgbnÞúk P Rtg;cugesrI enaHvaeFVIeGaymansac;lUt Δ . kmμnþEdlekIteLIgedaybnÞúk P KWkmμnþxageRkA W = 1 P.Δ . E 2 cMeBaHm:Um:g; M EdlGnuvtþmkelIFñwmmYyeFVIeGaymanmMurgVil θ . kmμnþxageRkAEdlbegáIteLIgeday m:Um:g; M KW W = 1 M .θ . E 2 2> famBlb:Utg;Esülxagkñúg W I Potential energy kmμnþxageRkA)anrkSaTukenAkñúgr)ar ehIy)anERbkøayeTACafamBlxagkñúg. k> krNIr)arrgbnÞúktamGkS½ 1 WI = WE = P.Δ 2 tamc,ab;h‘Uk eyIg)an PL Δ= AE P2L ⇒ WI = 2 EI x> krNIFñwmrgm:Um:g;Bt; M famBlxagkñúgEdlrkSaTukenAelIFatuGnnþtUc dx ehIyeFVIeGaymanmMurgVil dθ KW 1 dWI = M .dθ 2 b:uEnþeyIgdwgfa dθ = EI dx M 1 dWI = M 2 dx 2 EI 2 L M ⇒ WI = ∫ dx 0 2 EI • cMNaM³ - smIkar W = ∫ 2EI dx ¬sMrab;Fñwmrgm:Um:g;Bt; M ¦/ M KWCam:Um:g;Bt;enARtg;muxkat;samBaØmYy 2 M L I 0 ehIym:Um:g;enaHbNþalmkBIplbUksrubénbnÞúkxageRkA. PaBdabrbs;Fñwm edayviFIfamBl 118
  • 2. T.Chhay - krNI truss Edlr)arrbs;vargEtkMlaMgtamGkS½ famBlb:Utg;EsülTak;TgEtnwgkMlaMgtambeNþay GkS½r)arenaHEtb:ueNÑaH. - kñúgFñwmeRkamGMeBIénbnÞúk nigRbtikmμTMr eFVIeGayeKmankMlaMgkat;TTwg nigm:Um:g;Bt; Rtg;muxkat; samBaØNamYyénmuxkat;. dUcenHtameKalkarN_rkSafamBl famBlb:Utg;EsülEdlrkSaTukkñúg r)arenaH KWCaGnuKmn_énplrbs;kMlaMgkat;TTwg V nigm:Um:g;Bt; M . b:uEnþeKsnμt;fa b:Utg;Esülxag kñúgEdlekIteLIgedaykMlaMgkat;TTwg V mantMéltUcEdleKGacpat;ecal)an ebIeFobeTAnwgfam- Blb:Utg;EsülEdlekIteLIgedaysarm:Um:g;Bt; M . 3> karKNnabMErbMrYlragedayviFIfamBl tameKalkarN_rkSafamBl eK)an³ WE = W I WE - CakmμnþxageRkAEdlbegáIteLIgedaykMlaMgxageRkAGnuvtþbnþicmþg²eTAelIrcnasm<n§½. W - CafamBlb:Utg;EsülxagkñúgEdlrkSaTukkñúgrcnasm<n§½. I smIkarxagelIGnuBaØatieGaymanEtGBaØtimYyb:ueNÑaH dUcenHr)arRtUvrgbnÞúkEtmYyEtb:ueNÑaH. ]TahrN_³ rkbMErbMrYlrag ¬PaBdab¦ Rtg;cMnuc C P sMrab;r)arTMrsamBaØEdlman EI efr ehIyrgbnÞúk Rtg;cMnuc P Rtg;kNþalElVg. A C B l/2 l/2 dMeNaHRsay³ kmμnþxageRkAEdlbegáIteLIgedaybnÞúk P KW M PΔ WE = Px 2 2 Pl M Cam:Um:g;EdlekIteLIgedaysarbnÞúk P 4 famBlb:Utg;Esül xagkñúgEdlrkSaTukKW M2 WI = ∫ L dx 0 2 EI Et M = Px 2 ⇒M2 = P2 x2 4 ¬KitcMeBaH 0 ≤ x ≤ 2 ¦l tamlkçN³qøúHeK)anm:Um:g;srubesμInwgBIrdg m:Um:g;EpñkxageqVg b¤xagsþaM dUcenHeK)an³ l l P2 x2 P 2 x3 2 P 2l 3 WI = 2 ∫ 2 dx = = 0 8 EI 12 EI 0 96 EI PΔ P 2l 3 Pl 3 WE = WI ⇒ = ⇒Δ= 2 96 EI 48 EI ]TahrN_³ rkmMurgVilRtg;cMnuc B sMrab;r)arTMrsamBaØEdlman PaBdabrbs;Fñwm edayviFIfamBl 119
  • 3. T.Chhay EI efr ehIyrgm:Um:g; M Rtg;cMnuc B . B A B M l dMeNaHRsay³ kmμnþxageRkAEdlbegáIteLIgedaym:Um:g; M KW B MB 1 M=MlB.x WE = M B .θ B 2 famBlb:Utg;Esül xagkñúgEdlrkSaTukKW M M2 WI = ∫ L dx 0 2 EI eday M= B M .x l l M .x 1 ⇒ WI = ∫ ( B ) 2 dx 0 l 2 EI M2 l = 2 B ∫ x 2 dx 2l EI 0 M 2l ⇒ WI = B 6 EI tameKalkarN_rkSafamBl eK)an M B .θ B M B l 2 WE = WI ⇒ = 2 6 EI M Bl ⇒ θB = 3EI 4> kmμnþvaTuyEGl b¤kmμnþminBit (Virtual work) kmμnþvaTuyEGl (V W) CakmμnþEdlekIteLIgedaykMlaMgminBit ehIyeFVIeGaymanbMlas;TIBit. Bakü virtual EdleKeRbIkñúgkmμnþ b¤bMlas;TI sMedAeTAelItMélminBit b¤TMhMRsemIRsém. ebIeKGnuvtþmkelI RbBn§½manlMnwgmYy Q eK)anbegáIteGaymankMlaMg virtual xagkñúg nigbMlas;TI virtual tUc². eKsnμt;faRb i Bn§½kMlaMg virtual Q smmUlesμInwg kMlaMg virtual Ékta (1kN ) edIm,IKNnabMlas;TIRtg;cMnucmYyénrcna i sm<n§½ tameKalkarN_rkSafamBl kmμnþ virtual xageRkAesμInwgkmμnþ virtual xagkñúg. We = Wi 5> karKNnabMErbMrYlragedayviFIkmμnþvaTuyEGl (Virtual work method) edIm,IKNnabMlas;TIRtg;muxkat;samBaØmYyénrcnasm<n§½eRkamGMeBIénRbBn§½kMlaMgxageRkA. eKGnuvtþ nUvCMba‘anxageRkam³ - snμt;lubbM)at;kMlaMgxageRkATaMgGs;ecj - dak;Rtg;cMnucEdlcg;rkbMlas;TInUvkMlaMg virtual Ékta (1kN ) tamTisedAénbMlas;TIBit PaBdabrbs;Fñwm edayviFIfamBl 120
  • 4. T.Chhay - rkRbtikmμkMlaMg virtual xagkñúg ¬rab;TaMg kMlaMgkat;TTwg kMlaMgbeNþayGkS½ nigm:Um:g;pg¦ Edl ekIteLIgedaysarbnÞúkÉktaenaH. - rkRbtikmμTMr/ M , V , N EdlekIteLIgedaysarbnÞúkBit - tameKalkarN_rkSafamBleK)an W = W e i k> krNI truss kmμnþxageRkAEdlekIteLIgedaysarbnÞúkÉkta (1kN ) We = 1× δVC Edl δ - bMlas;TIBittamTisQrRtg;cMnuc C VC famBlb:Utg;EsülEdlrkSaTukenAkñúgr)ar truss Wi = ∑ p.Δ Edl p - kMlaMgkñúgrbs;r)ar truss Edl)anBIbnÞúkÉkta (1kN ) Δ - bMErbMrYlragBiténr)ar truss tameKalkarN_rkSafamBleyIg)an We = Wi ⇒ δVC = ∑ p.Δ tamc,ab;h‘Uk eyIg)an PL Δ= EA ∑ p.P.L ⇒ δVC = EA Edl P - kMlaMgkñúgr)arekItBIbnÞúkBit L - RbEvgr)ar A - muxkat;r)ar E - m:UDuleGLasÞic ]TahrN_³ kMNt;PaBdabrbs; truss Rtg;cMnuc C P 1 =30KN P 2 =50KN ebIeKdwgfamuxkat;rbs;r)arnImYy² A = 1500mm B D 2 2m nigmanm:UDuleGLasÞic E = 200GPa . A C E dMeNaHRsay³ 4m 4m edaylubecalkMlaMgxageRkATaMgGs;EdlmanGMeBI B D mkelI truss edaydak;bnÞúkÉkta (1kN ) Rtg;cMnuc 2m C manTisedAcuHeRkam CacMnucEdleyIgRtukarrkPaBdab A E C 4m 4m p=1KN PaBdabrbs;Fñwm edayviFIfamBl 121
  • 5. T.Chhay kMNt;kMlaMgkñúgrbs;r)arEdlekItBIbnÞúkÉkta nigbnÞúkBit r)ar RbEvg L(m) kMlaMgkñúgr)ar p(kN ) kMlaMgkñúgr)ar P(kN ) p.P.L AB 2.83 − 0.707 − 49.5 + 99.04 AC 4 + 0 .5 + 35 + 70 BC 2.83 + 0.707 + 7.07 + 14.15 BD 4 −1 − 40 + 160 CD 2.83 + 0.707 − 7.07 − 14.15 CE 4 + 0 .5 + 45 + 90 DE 2.83 − 0.707 − 63.65 + 127.35 ∑ p.P.L + 546.39 ∑ p.P.L δVC = EA 546.39 × 1000 ⇒ δVC = = 1.8mm ↓ 1500 × 200 x> krNIFñwm ¬PaBdab¦ cg;rkbMlas;TI δ Rtg;cMnuc C eKeFVIdUc truss Edr edayeKlubkMlaMgBitTaMgLayecal rYcehIyenA C enARtg;cMnuc C EdleKcg;rkbMlas;TIenaH eKdak;kMlaMg virtual Ékta (1kN ) . eKrkm:Um:g;Bt; virtual m . bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g;BitmkelI dx . enAeBlmanm:Um:g; M naM eGaymanmMurgVil dθ enAmuxkat;sgxag dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarkMlaMg e virtual Ékta (1kN ) ehIyeFVIeGaymanbMlas;TI δ Rtg;cMnuc C eK)an³ C We = 1.δ C Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc i dx RtUv)an begáIteLIgedaym:Um:g; virtual m EdleFVIeGaymuxkat;vil)anmMu dθ . dWi = m.dθ EteyIgman dθ = EI dx M m.M ⇒ dWi = dx EI b¤Wi = ∫ L m.M 0 EI dx tameKalkarN_rkSafamBl We = Wi PaBdabrbs;Fñwm edayviFIfamBl 122
  • 6. T.Chhay L m.M ⇒ δC = ∫ dx 0 EI K> krNIFñwm ¬mMurgVil¦ eFVIdUcKñakrNIxagelIEdr EteKRtUvdak;m:Um:g; virtual Ékta (1kN.m) CMnYseGaykMlaMg virtual Ékta (1kN ) vij. eKrkm:Um:g;Bt; virtual m' . bnÞab;mkeKdak;kMlaMgxageRkATaMgGs;mkelIFñwm rYcehIyeKrkm:Um:g; BitmkelI dx . ebI W Cakmμnþ virtual xageRkAEdlekIteLIgedaysarm:Um:g; virtual Ékta (1kN.m) ehIyeFVI e eGaymanmMurgVil θ Rtg;cMnuc C eK)an³ C We = 1.θ C Fatu differential énb:Utg;Esülxagkñúg dW EdlrkSaTukenAxagkñúgPaKkMNat;GnnþtUc i dx RtUv)an begáIteLIgedaym:Um:g; virtual m' EdleFVIeGaymuxkat;vil)anmMu dθ . dWi = m'.dθ EteyIgman dθ = EI dx M m'.M ⇒ dWi = dx EI b¤Wi = ∫ 0 L m'.M EI dx tameKalkarN_rkSafamBl We = Wi L m'.M ⇒ θC = ∫ dx 0 EI ]TahrN_³ eKeGayFñwmsamBaØrgbnÞúkBRgayesμI w = 20 kN m C w=20kN m nig EI = 40000kN.m . rkPaBdabenAkNþalElVg Δ A 2 C B 4m 4m nigmMurgVilenATMr A θ . A dMeNaHRsay³ - rkPaBdabenAkNþalElVg Δ C 1kN CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C A 4m C 4m B rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA C x ⇒m= 2 smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA C w.L.x w.x 2 ⇒M = − 2 2 2 w.L.x w.x 3 ⇒ m.M = − 4 4 PaBdabrbs;Fñwm edayviFIfamBl 123
  • 7. T.Chhay rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI B eTA C x ⇒m= 2 smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI B eTA C w.L.x w.x 2 ⇒M = − 2 2 2 w.L.x w.x 3 ⇒ m.M = − 4 4 plKuNm:Um:g; sMrab; 0 ≤ x ≤ 8m BI A eTA B w.L.x 2 w.x 3 ⇒ m.M = − 2 2 2 w.x 3 1 dUcenH 0 4 w.L.x ΔC = ∫ ( 2 − 2 EI ) dx 4 4 w.L.x 3 wx 4 1 ⇒ ΔC = ( − ) 6 0 8 0 EI 20 × 8 × 4 3 20 × 4 4 1 ⇒ ΔC = ( − ) = 26.7 mm ↓ 6 8 40000 - rkmMurgVilenATMr A θ A A B CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A 8m rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI A eTA B x ⇒ m' = L smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI A eTA B w.L.x w.x 2 ⇒M = − 2 2 2 w.x w.x 3 ⇒ m'.M = − 2 16 2 w.x 3 1 dUcenH 8 w.x θA = ∫ ( 0 2 − ) 16 EI dx 8 8 w.x 3 wx 4 1 ⇒θA = ( − ) 6 0 64 0 EI ⇒ θ A = 0.01067 rad ]TahrN_³ eKeGayFñwmTMrbgáb;rgkMlaMgcMcMnucRtg;cug 30kN r)ar edayman EI = 30000kN .m . kMNt;PaBdab nigmMu 2 A 4m B rgVilRtg;cugr)ar. dMeNaHRsay³ PaBdabrbs;Fñwm edayviFIfamBl 124
  • 8. T.Chhay - rkPaBdabenAcugr)ar δ A 1kN CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc A A B 4m rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B ⇒ m = −x smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B ⇒ M = −30.x ⇒ m.M = 30 x 2 dUcenH δ = ∫ (30 x ) EI dx A 0 4 1 2 4 30 x 3 1 ⇒δA = ( ) 3 0 EI 1 ⇒ δ A = (10 × 4 3 ) = 21.3mm ↓ 30000 - rkmMurgVilenAcugr)ar θ A 1kN.m A B CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A 4m rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 4m BI A eTA B ⇒ m' = −1 smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 4m BI A eTA B ⇒ M = −30 x ⇒ m'.M = 30 x dUcenH θ = ∫ (30 x) EI dx A 0 4 1 4 30 x 2 1 ⇒θA = ( ) 2 0 EI ⇒ θ A = 0.008rad ]TahrN_³ eKeGayFñwmTMrsamBaØrgkMlaMgBRgayesμI w = 20 . kN m w=20kN FñwmenHman E = 200 nig I = 200 ×10 mm . kN mm 2 6 4 m C kMNt;PaBdabRtg;cMnuc C / mMurgVilRtg;cMnuc C nig A . A B 3m 8m dMeNaHRsay³ - rkPaBdabRtg;cMnuc C δ C 1kN CMnYskMlaMg virtual Ékta (1kN ) Rtg;cMnuc C C A B 3m 8m rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A PaBdabrbs;Fñwm edayviFIfamBl 125
  • 9. T.Chhay ⇒ m = −x smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A 20 2 ⇒M =− .x = −10 x 2 2 ⇒ m.M = 10 x 3 rksmIkarm:Um:g;EdlekItBIkMlaMg virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A ⇒ m = −0.375 x smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A 20 2 ⇒ M = 68.75 x − .x = −10 x 2 + 68.75 x 2 ⇒ m.M = 3.75 x 3 − 25.78 x 2 dUcenHδC = ∫ 0 L m.M EI dx 3 3 10 x 8 3.75 8 25.78 ⇒ δC = ∫ dx + ∫ x 3 dx − ∫ x 2 dx 0 EI 0 EI 0 EI 1 ⎛ 10 x 4 25.78 x 3 ⎞ 4 8 8 ⎜ 3.75 x 4 ⎟ ⇒ δC = + − EI ⎜ 4 0 4 0 3 ⎟ ⎝ 0⎠ ⇒ δ C = 8.94mm ↑ - rkmMurgVilRtg;cMnuc C θ C 1kN.m CMnYsm:Um:g; virtual Ékta (1kN .m) Rtg;cMnuc C C A B 3m 8m rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A ⇒ m' = −1 smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A ⇒ M = −10 x 2 ⇒ m'.M = 10 x 2 rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A ⇒ m' = −0.125 x smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A ⇒ M = −10 x 2 + 68.75 x ⇒ m'.M = 1.25 x 3 − 8.59 x 2 dUcenHθC = ∫ 0 L m'.M EI dx PaBdabrbs;Fñwm edayviFIfamBl 126
  • 10. T.Chhay 1 3 8 8 ⇒ θC = ( ∫ 10 x 2 dx + ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx) EI 0 0 0 3 8 8 1 10 x 3 1.25 4 8.59 3 ⇒ θC = ( + x − x EI 3 0 4 0 3 0 ⇒ θ C = −0.0024 rad - rkmMurgVilRtg;cMnuc C θ A 1kN.m CMnYsm:Um:g; virtual Ékta (1kN.m) Rtg;cMnuc A C 3m A 8m B rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 3m BI C eTA A ⇒ m'= 0 smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 3m BI C eTA A ⇒ M = −10 x 2 ⇒ m'.M = 0 rksmIkarm:Um:g;EdlekItBIm:Um:g; virtual sMrab; 0 ≤ x ≤ 8m BI B eTA A ⇒ m' = −0.125 x smIkarm:Um:g;EdlekItBIkMlaMgBit sMrab; 0 ≤ x ≤ 8m BI B eTA A ⇒ M = −10 x 2 + 68.75 x ⇒ m'.M = 1.25 x 3 − 8.59 x 2 dUcenHθA = ∫ 0 L m'.M EI dx 1 8 8 ⇒θA = ( ∫ 1.25 x 3 dx − ∫ 8.59 x 2 dx) EI 0 0 8 8 1 1.25 4 8.59 3 ⇒θA = ( x − x ) EI 4 0 3 0 ⇒ θ A = −0.00465rad PaBdabrbs;Fñwm edayviFIfamBl 127