Μητρωική Ανάλυση Φορέων Με Επιρροή Στρέβλωσης...

of 52 /52
ΓΕΝΙΚΟ ΕΠΙΤΕΛΕΙΟ ΣΤΡΑΤΟΥ ΣΧΟΛΗ ΤΕΧΝΙΚΗΣ ΕΚΠΑΙΔΕΥΣΗΣ ΑΞΙΩΜΑΤΙΚΩΝ ΜΗΧΑΝΙΚΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΜΗΤΡΩΙΚΗ ΑΝΑΛΥΣΗ ΦΟΡΕΩΝ ΜΕ ΕΠΙΡΡΟΗ ΔΙΑΤΜΗΤΙΚΗΣ ΚΑΙ ΣΤΡΕΠΤΙΚΗΣ ΣΤΡΕΒΛΩΣΗΣ Λγός (ΜΧ) Πάνος Στέργιος Επιβλέπων καθηγητής: Σαπουντζάκης Ευάγγελος, Καθηγητής Στατικής ΣΤΕΑΜΧ, Τακτικός Καθηγητής ΕΜΠ ΠΕΡΙΛΗΠΤΙΚΗ ΈΚΔΟΣΗ Αθήνα, Ιανουάριος 2015

Embed Size (px)

description

Περιληπτική έκδοση (και ελαφρώς πιο κατανοητή) της διπλωματικής εργασίας στη ΣΤΕΑΜΧ.

Transcript of Μητρωική Ανάλυση Φορέων Με Επιρροή Στρέβλωσης...

  • ()

    :

    , ,

    , 2015

  • II

    '

    67

    , .

    1 3,

    , Euler Bernoulli, Timoshenko,

    .

    4

    , 5

    .

    6

    , ,

    .

    ..,

    . ,

    .

    ... .

    ,

    . ,

    ... . .

    ,

    , , ,

    ,

    . ,

    , .

    , 2015

    () [-02]

  • III

    1o - i i ................................................1

    1.1 ..........................................................................................................1

    1.2 ..............................................1

    1.3 .....................................1

    1.3.1 ( ): .....1

    1.3.2 (

    Hooke ): ...............................................................2

    1.3.3 ( ) : .....3

    1.3.4 .....................................................4

    2o - ..............................................5

    2.1 ..........................................................................................................5

    2.2 Euler - Bernoulli .....................................5

    2.2.1 Euler Bernoulli ..............................................5

    2.2.2 Euler Bernoulli .......................................6

    2.2.3 Euler Bernoulli ................................6

    2.2.4 .....................................................6

    2.2.5 .......................................................................................7

    2.3 Timoshenko ...........................................7

    2.3.1 Timoshenko ...................................7

    2.3.2 Timoshenko ..............................................8

    2.3.3 Timoshenko ......................................8

    2.3.4 .....................................................9

    2.3.5 .......................................................................................9

    3o - ............................................10

    3.1 (Saint-Venant) .........................10

    3.1.1 .........................................10

    3.1.2 ............................11

    3.1.3 ....................11

    3.1.4 ...................................................12

    3.1.5 .....................................................................................12

    3.2 ...................................12

  • IV

    3.2.1 ...............................12

    3.2.2 ........................13

    3.2.3 .................13

    3.2.4 ...................................................14

    3.2.5 .....................................................................................14

    4o -

    .............................................................15

    4.1 - .....................15

    4.2 ..........................16

    4.3 ...................18

    4.4 ..........................................................................................19

    5o - ................20

    5.1 .........................................................20

    5.1.1 . ....................20

    5.1.2 . ..................................20

    5.1.3 .....................................21

    5.1.4

    , ......................................................21

    5.2 .....................................................22

    6o ..........................................................23

    6.1 .....................................23

    6.1.1 ..............................................................................................23

    6.1.2 RHS 500X300X20 .....................24

    6.1.3 ..................................................................25

    6.2 ...............................................32

    6.2.1 ..............................................................................................32

    6.2.2 ............................33

    6.2.3 ..................................................................35

    6.3 ........................................................................40

    6.4 ..........................................................................................41

  • V

    1 Oxy ........................ 2

    2 ............................................. 3

    3 Euler - Bernoulli .................................................... 5

    4 Euler - Bernoulli Timoshenko ............ 7

    5 ....................................... 9

    6 .... 10

    7 ()

    () ................................................................... 15

    8 ........................... 16

    9 ,

    Euler-Bernoulli (Saint-Venant)[

    1212] ............................................................................................... 20

    10

    ( 2020) ................................................................................ 21

    11 ............................. 24

    12 RHS500X300X20 ................... 24

    13 w(x) , ........................... 26

    14 v(x) , ..... 26

    15 u(x) ,

    .......................................................................................... 27

    16 v(x) ,

    .......................................................................................... 27

    17 xP ..... 28

    18 yP ( y) .......... 28

    19 zP ( z) .......... 29

    20 xS ... 29

    21

    .......................................................................................................... 30

    22 max=4.90MPa,

    .............................................................................................. 31

    23

    ................................................................................. 31

    24 ........................ 32

  • VI

    25 .......................................................... 33

    26 ... 34

    27 w(x) , .................... 35

    28 x(x), ........ 35

    29 y(x) ......... 36

    30 xP ..... 36

    31 yP ( y) .......... 37

    32 zP ( z) .......... 37

    33 xS ... 37

    34 1,85.

    ............................................................................................ 38

    35 1,85.

    ....................................................................................... 39

    36 CPU RAM ,

    (131.262

    solid elements) .................................................................................. 40

    37 CPU RAM ,

    (30 )

    .......................................................................................................... 40

  • & ()

    - 1 -

    11oo -- ii ii

    1.1

    .

    ,

    . ,

    15 15 ,

    .

    1.2

    u=u (x,y,z)

    v=v (x,y,z)

    w=w (x,y,z)

    xx

    yy

    zz

    xy

    yz

    xz

    xx

    yy

    zz

    xy

    yz

    xz

    , , :

    , :

    1.3

    1.3.1 ( ):

    =

    =

    =

    =

    +

    =

    +

    =

    +

  • 1o - i i

    - 2 -

    1 Oxy

    ,

    . ,

    (u,v,w),

    ,

    .

    1.3.2 (

    Hooke ):

    =

    (1 + ) (1 2)(1 ) + +

    =

    (1 + ) (1 2)(1 ) + ( + )

    =

    (1 + ) (1 2)(1 ) + +

    =

    2 (1 + )

    =

    2 (1 + )

    =

    2 (1 + )

  • & ()

    - 3 -

    ,

    . , :

    ,

    ( [](x,y,z)= [E] ).

    , ,

    ( xx= Eyy= Ezz= E ).

    ( ),

    (

    , )

    ,

    .

    1.3.3 ( ) :

    = 0

    +

    +

    + = 0

    = 0

    +

    +

    + = 0

    = 0

    +

    +

    + = 0

    2

  • 1o - i i

    - 4 -

    1.3.4

    (

    ). ,

    ,

    , ,

    .

    ,

    :

    Neumann,

    .

    Dirichlet,

    .

    , .

    ,

    .

    , ,

    .

  • & ()

    - 5 -

    22oo --

    2.1

    ,

    (dx, dy, dz), (L, b,

    h). Ox ().

    (ij),

    (Pij)

    .

    2.2 Euler - Bernoulli

    2.2.1 Euler Bernoulli

    , :

    3 Euler - Bernoulli

    ()

    ( )( xxyy, zz yyzz 0 ).

    , ( i,i=)

    , , ( yy zz0 ).

  • 2o -

    - 6 -

    ( Ox)

    , ( , = () (, , )), (

    xyxz0 ,

    =

    ,

    =

    ) (

    yz0 ) .

    2.2.2 Euler Bernoulli

    ,

    , :

    =

    = ()

    = ()

    =

    =

    :

    =

    = z(x)

    =

    = y(x)

    2.2.3 Euler Bernoulli

    15 , ,

    , () :

    = () ( yy)

    = () ( zz)

    : = z2 dA

    yy

    = y2 dA

    zz

    2.2.4

    Euler-Bernoulli,

    , .

    , ,

    :

  • & ()

    - 7 -

    =

    =

    =

    =

    2.2.5

    Euler Bernoulli

    . ,

    . ,

    Euler Bernoulli.

    2.3 Timoshenko

    2.3.1 Timoshenko

    Stephen Timoshenko, ,

    Euler Bernoulli,

    . ,

    xy ( Oy) xz ( Oz).

    4 Euler - Bernoulli Timoshenko

  • 2o -

    - 8 -

    2.3.2 Timoshenko

    T , :

    =

    = ()

    = ()

    =

    =

    +

    =

    +

    =

    =

    2 (1 + )

    =

    2 (1 + )

    :

    = z(x)

    = y(x)

    2.3.3 Timoshenko

    15 ,

    , () :

    = ()

    ( yy)

    = ()

    ( zz)

    : A :

    G : [ =

    ()]

    : Poisson

    =

    =

    : (0

  • & ()

    - 9 -

    ,

    .

    2.3.4

    ,

    :

    = = +

    = = +

    = =

    =

    =

    2.3.5

    , , (G),

    xy xz .

    5

  • 3o -

    - 10 -

    33oo --

    3.1 (Saint-Venant)

    3.1.1

    :

    ()

    ( )( xxyy, zz yyzz 0 ).

    , ( i,i=)

    , , ( yy zz0 ).

    ( yz0 ) .

    ( Ox)

    .

    . ,

    (),

    ( xx0 ) ( xx 0 ).

    ( x

    x = ).

    6

  • & ()

    - 11 -

    3.1.2

    s(y,z) S .

    s u(y,z)

    (

    = 1) :

    =

    =

    T , :

    =

    (, )

    =

    =

    =

    +

    =

    +

    =

    2 (1 + )

    =

    2 (1 + )

    :

    :

    x ()

    3.1.3

    s

    ( =0), :

    +

    =

  • 3o -

    - 12 -

    3.1.4

    ,

    :

    = + =

    = + +

    ,

    Saint-Venant .

    3.1.5

    .

    3.2

    3.2.1

    Saint-Venant . ,

    , , .

    ( xx 0 xx 0 )

    (

    = f(x) ),

    .

    , .

    y,z ( ), x,y,z. ,

    ,

    .

  • & ()

    - 13 -

    3.2.2

    T , :

    =

    +

    =

    =

    =

    =

    +

    =

    (

    + ) +

    =

    =

    2 (1 + )

    =

    2 (1 + )

    ,

    x ()

    =

    (, ),

    S .

    u(y,z) , (

    = 1)

    .

    =

    (, , ),

    S .

    u(x,y,z) ,

    (

    = 1) .

    3.2.3

    ,

    ( =0) It, Cs, :

    +

    =

    ,

    ( Saint-Venant)

  • 3o -

    - 14 -

    ,

    ( )

    = + +

    ,

    Saint-Venant .

    = ()

    ,

    = () ,

    3.2.4

    ,

    :

    =

    + =

    =

    + =

    3.2.5

    ,

  • & ()

    - 15 -

    44oo --

    4.1 -

    ,

    ,

    , .

    , .

    , , .

    (),

    (),

    (), ... ( 7).

    ()

    Pxy

    Pxz

    ()

    PS

    Pxx

    E

    0SP Sxyxx xz

    x y z

    Sxy

    Sxz

    ()

    SS

    Sxx

    E

    0TS Txyxx xz

    x y z

    Txy

    Txz

    ()

    Pxx

    E

    0PP Pxyxx xz

    x y z

    Pxy

    Pxz

    ()

    PCY ,

    PC

    Sxx

    E

    0SS Sxyxx xz

    x y z

    Sxy

    Sxz

    7 ()

    ()

    :

    ()

    ( )( xxyy, zz yyzz 0 ).

    , ( i,i=)

    , , ( yy zz0 ).

    ( yz0 ) .

  • 4o -

    - 16 -

    ()

    . ,

    (), .

    8

    4.2

    T ,

    ( ).

    Sxyz, S

    ( )

    . ()

    . CXYZ

    C

    .

    ,

    ( )

    . ,

    , :

  • & ()

    - 17 -

    ( S) (, )

    ( S) (, , )

    ( CZ) (, )

    ( C) (, )

    , ( ,

    ). (, ),

    (, ),

    (, ) . ,

    , :

    (, ) = +

    (, )

    (, ) = +

    (, )

    (, ) =

    (, ) + (, )

    (, ) =

    (, ) + (, )

    (, ) =

    (, ) + (, )

    (, ) =

    (, ) + (, )

    ,

    , :

    = () + () () + (, ) ()

    + (, ) () +

    (, ) () + (, ) ()

    = () ()

    = () + ()

    =

    +

    +

    +

    +

    +

  • 4o -

    - 18 -

    =

    +

    =

    w

    x+

    y

    + v

    x Z

    y

    +x

    x +

    y

    ()

    + w

    x

    y

    +

    x+

    y

    + ( x

    )

    y

    + ( +x

    )

    y

    =

    +

    =

    +

    +

    +

    +

    ()

    +

    +

    +

    + (

    + +

    = E

    =

    2 (1 + )

    =

    2 (1 + )

    4.3

    ,

    .

  • & ()

    - 19 -

    4.4

    Euler - Bernoulli

    Saint-Venant,

    , , ,

  • 5o -

    - 20 -

    55oo --

    5.1

    5.1.1 .

    ,

    ,

    .

    5.1.2 .

    (. ),

    .

    , .

    ,

    , :

    {} = [] {} (5.1.1)

    {}

    , {}

    , []

    .

    9 ,

    Euler-Bernoulli (Saint-Venant)[ 1212]

  • & ()

    - 21 -

    10

    ( 2020)

    5.1.3

    ,

    (5.1.1),

    {} ,

    {} , []

    .

    5.1.4

    ,

    , ,

    .

    , . ,

    Euler Bernoulli,

    :

    =

    (5.1.2)

    i = y, z, j=z, y

  • 5o -

    - 22 -

    5.2

    ,

    ,

    . , .

    =

    + = +

    + = +

    {} = [] {}

    : .

    : .

    : .

    : .

    : .

    : (

    , , .).

    : o i

    .

  • & ()

    - 23 -

    66oo

    6.1

    6.1.1

    (= 210GPa, v= 0.3) ,

    (

    ux, uy, uz) ( 11),

    RHS500X300X20 ( 12), o ,

    10 /m

    1m/m.

    Euler-Bernoulli Timoshenko,

    (solid elements).

    astran v.8.5 FEMAP for Windows v11.0.1.

    Euler-Bernoulli Timoshenko,

    30 (10 ),

    131.262 , , (solid elements).

    , ,

    , ,

    500 .

    .

  • 6o

    - 24 -

    11

    6.1.2 RHS 500X300X20

    RHS 500X300X20, b=300mm, h=500mm 20mm

    ( 12),

    ( 1).

    12 RHS500X300X20

  • & ()

    - 25 -

    1 RHS500X300X20

    3.03340 -2 m2

    Iyy 1.01042 -3 m4

    Izz 4.49194 E-4 m4

    Ay 8.94076 E-3 m2

    Az 1.82493 E-2 m2

    It 9.80216 E-4 m4

    Cs 9.03594E-7 m6

    8.94667 E-7 m6

    2.85448 E-6 m4

    1.48675 E-7 m4

    -7.57998 E-9 m5

    7.80773 E-10 m5

    -4.27323 E-10 m5

    -5.82498 E-10 m5

    3.14321 E-8 m6

    6.1.3

    Euler-Bernoulli

    ,

    ,

    ,

    ( 13 16).

  • 6o

    - 26 -

    13 w(x) ,

    14 v(x) ,

    -0.00016

    -0.00014

    -0.00012

    -0.0001

    -0.00008

    -0.00006

    -0.00004

    -0.00002

    0

    0 1 2 3 4 5

    w

    (x)

    [m]

    [m]

    FEM-131.262 solid elements (hexa, 8-noded)

    Bernoulli BT

    Timoshenko BT

    GWBT

    -0.00045

    -0.0004

    -0.00035

    -0.0003

    -0.00025

    -0.0002

    -0.00015

    -0.0001

    -0.00005

    0

    0 1 2 3 4 5

    v

    (x)

    [m]

    [m]

    FEM-131.262 solid elements (hexa, 8-noded)

    Bernoulli BT

    Timoshenko BT

    GWBT

  • & ()

    - 27 -

    15 u(x) ,

    16 v(x) ,

    ( 17 20),

    .

    ,

    (

    x, y, z, x ) ( ) .

    -0.00007

    -0.00006

    -0.00005

    -0.00004

    -0.00003

    -0.00002

    -0.00001

    0

    0 0.5 1 1.5 2 2.5 3

    u

    (x)

    [m]

    [m]

    FEM-131.262 solid elements (hexa, 8-noded)

    Bernoulli BT

    Timoshenko BT

    GWBT

    -0.0003

    -0.00025

    -0.0002

    -0.00015

    -0.0001

    -0.00005

    0

    0 0.5 1 1.5 2 2.5 3

    v

    (x)

    [m]

    [m]

    FEM-131.262 solid elements (hexa, 8-noded)

    Bernoulli BT

    Timoshenko BT

    GWBT

  • 6o

    - 28 -

    17 x

    P

    18 y

    P ( y)

    xP

  • & ()

    - 29 -

    19 z

    P ( z)

    20 x

    S

  • 6o

    - 30 -

    ()

    Euler Bernoulli Timoshenko,

    , ,

    .

    ,

    ( 21). ,

    ,

    .

    , ,

    , Euler-Bernoulli

    37%,

    11% ( 21).

    21

  • & ()

    - 31 -

    22 max=4.90MPa,

    23

    ux yy zz nx ny nz x

    xx -0.34946 3.46621 0.222543 0.596707 0.829138 -0.00276 0.13961

    -1

    0

    1

    2

    3

    4

    m

    ax[M

    Pa]

  • 6o

    - 32 -

    24

    6.2

    6.2.1

    , (= 32GPa,

    v= 0.2), ( 25), (

    26), , 50 /m

    600

    .

    :

    = 80

    = 20 (4 + 4 ), t[0 , 1]

    Euler-Bernoulli Timoshenko,

    (solid elements).

    astran v.8.5 FEMAP for Windows v11.0.1.

    Euler-Bernoulli Timoshenko,

    60 ,

  • & ()

    - 33 -

    85.317 ,

    , (solid elements).

    .

    25

    6.2.2

    ,

    15,20. 3,45 ( 26),

    ( 2).

    h1=20.0m

    l2=20.0 m

    l1=80.0 m

    Pz= 600.0 KN

    15.0 m

    l2=20.0 m

    15.0 m x

    y

    A

    B

    C

    D

    E

  • 6o

    - 34 -

    26

    2

    1.12800000E+01 m2

    Iyy 1.86582458E+01 m4

    Izz 1.69128155E+02 m4

    Ay 6.77123923E+00 m2

    Az 2.51949388E+00 m2

    It 4.27703700E+01 m4

    Cs 6.28917900+01 m6

    5.93901373E+01 m6

    1.03406453E+00 m4

    3.31586366E+00 m4

    1.91801014E-02 m5

    -2.52900456E+00 m5

    4.30514488E-03 m5

    3.60455513E+00 m5

    1.05797152E+01 m6

  • & ()

    - 35 -

    6.2.3

    Euler-Bernoulli

    ,

    ,

    ,

    ( 27 29).

    27 w(x) ,

    28 x(x),

    -0.03

    -0.025

    -0.02

    -0.015

    -0.01

    -0.005

    0

    0 20 40 60 80

    w

    (x)

    [m]

    [m]

    FEM-85.317 solid elements (tetra, 10-noded)Bernoulli BTTimoshenko BTGWBT

    -0.0009

    -0.0008

    -0.0007

    -0.0006

    -0.0005

    -0.0004

    -0.0003

    -0.0002

    -0.0001

    0

    0 10 20 30 40 50 60 70 80 90

    x(

    x) [

    rad

    ]

    [m]

    FEM-85.317 solid elements (tetra, 10-noded)

    Bernoulli BT

    Timoshenko BT

    GWBT

  • 6o

    - 36 -

    29 y(x)

    ( 30 33),

    .

    (

    x, y, z, x ) .

    30 x

    P

    -0.0008

    -0.0006

    -0.0004

    -0.0002

    0

    0.0002

    0.0004

    0.0006

    0.0008

    0 10 20 30 40 50 60 70 80 90

    y(x)

    [ra

    d]

    [m]

    Bernoulli BT

    Timoshenko BT

    GWBT

    2683.2 2683.2

    -1689.7

  • & ()

    - 37 -

    31 y

    P ( y)

    xP

    32 z

    P ( z)

    33 x

    S

    741.7 741.7

    -142.0

    88.9 88.9

    -69.3

    -35.7 -35.7

    8.47

  • 6o

    - 38 -

    ()

    ,

    Euler-Bernoulli

    18%,

    6,5% ( 34 35).

    34 1,85.

  • & ()

    - 39 -

    35 1,85.

  • 6o

    - 40 -

    6.3

    , , ,

    .

    ,

    , , .

    .

    ,

    , ,

    .

    ,

    . ,

    , , ,

    .

    131.262 (solid elements), ,

    .

    36 CPU RAM ,

    (131.262 solid elements)

    37 CPU RAM ,

    (30 )

  • & ()

    - 41 -

    ,

    (solid elements),

    :

    ()

    . ,

    , (pre-processing),

    .

    H ( )

    (, , , .).

    (dofs),

    .

    ,

    (shear-locking, membrane-locking phenomena),

    (shell elements)

    , .

    .

    , .

    , (post-

    processing),

    , , , , ..

    .

    6.4

    , Euler

    Bernoulli Saint-Venant Timoshenko Saint-Venant (

    1212),

    ( 2020)

    (131.262 ,

    - solid elements 85.317 , ,

    -solid elements).

  • 6o

    - 42 -

    :

    ,

    ,

    Euler Bernoulli Timoshenko.

    ,

    Euler Bernoulli Timoshenko,

    ,

    ,

    ( 21 35).

    .

    (max),

    Euler Bernoulli Timoshenko 37%

    ,

    11%

    .

    , ,

    ,

    , .

  • - 43 -

    . (2013), , , Timoshenko, Euler Bernoulli, , , , , 2013.

    Beer, G., Smith, I. and Duenser, Ch. (2008). The Boundary Element Method with Programming For Engineers and Scientists. Springer Wien New York.

    Chang, P. and Hijazi, H. (1989). General Analysis of Asymmetric Thin-Walled Members. Thin-Walled Structures, 7, 159-186.

    Chang, S.T. and Zheng, F.Z., (1987). Negative Shear Lag in Cantilever Box Girder with Constant Depth. Journal of Structural Engineering, 113(1), 20-35.

    Dezi, L. and Mentrasti, L. (1985). Nonuniform Bending-Stress Distribution (Shear Lag). Journal of Structural Engineering, 111(12), 2675-2690.

    Dikaros, I.C. and Sapountzakis, E.J. (2013). Nonuniform Shear Warping Analysis of Composite Beams of Arbitrary Cross Section using the Boundary Element Method. Civil-Comp Press, Proceedings of the 14th International Conference on Civil, Structural and Environmental Engineering Computing.

    Dong, S.B., arbas, S. and Taciroglu, E. (2013). On Principal Shear Axes for Correction Factors in Timoshenko Beam Theory. International Journal of Solids and Structures, 50, 1681-1688.

    El Fatmi, R. (2007). Non-uniform Warping Including the Effects of Torsion and Shear Forces. Part-II: Analytical and Numerical Applications. International Journal of Solids and Structures, 44, 5930-5952.

    El Fatmi, R. (2007a). Non-uniform Warping Including the Effects of Torsion and Shear Forces. Part-I: A General Beam Theory. International Journal of Solids and Structures, 44, 5912-5929.

    El Fatmi, R. (2007b). Non-uniform Warping Including the Effects of Torsion and Shear Forces. Part-II: Analytical and Numerical Applications. International Journal of Solids and Structures, 44, 5930-5952.

    El Fatmi, R. and Ghazouani, N. (2011). Higher Order Composite Beam Theory built on Saint-Venants Solution. Part-I: Theoretical Developments. Composite Structures, 93, 557-566.

    Eurocode 3 (2004): Design of Steel Structures Part 1.5: Plated Structural Elements, European Committee for Standardization, prEN 1993-1-5.

  • - 44 -

    Eurocode 4 (2004): Design of Composite Steel and Concrete Structures Part 1.1: General Rules and Rules for Buildings, European Committee for Standardization, prEN 1994-1-1.

    Eurocode 4 (2004): Design of Composite Steel and Concrete Structures Part 2: General Rules and Rules for Bridges, European Committee for Standardization, prEN 1994-2.

    FEMAP for Windows (2008). Finite element modeling and post-processing software. Help System Index, Version 10.

    Ferradi, M.K., Cespedes, X. and Arquier, M. (2013). A higher Order Beam Finite Element with Warping Eigenmodes. Engineering Structures, 46, 748-762.

    Gara, F., Ranzi, G. and Leoni, G. (2011). Simplified Method of Analysis Accounting for Shear-lag Effects in Composite Bridge Decks. Journal of Constructional Steel Research, 67, 1684-1697.

    Genoese, A., Genoese, A., Bilotta, A. and Garcea, G. (2013). A Mixed Beam Model with Non-Uniform Warpings Derived from the Saint Vennt Rod. Computers and Structures, 121, 87-98.

    Ghazouani, N. and El Fatmi, R. (2010). Extension of the non-uniform warping theory to an orthotropic composite beam. Comptes Rendus Mecanique, 338, 704-711.

    Ghazouani, N. and El Fatmi, R. (2011). Higher Order Composite Beam Theory built on Saint-Venants Solution. Part-II: Built-in Effects Influence on the Behavior of End-Loaded Cantilever Beams. Composite Structures, 93, 567-581.

    Gupta, P.K., Singh, K.K. and Mishra, A. (2010). Parametric Study on Behaviour of Box-Girder Bridges Using Finite Element Method, Technical Note. Asian Journal of Civil Engineering (Building and Housing), 11(1), 135-148.

    Hjelmstad, K.D. (1987). Warping Effects in Transverse Bending of Thin-Walled Beams. Journal of Engineering Mechanics, 113(6), 907-924.

    Ie, C.A. and Kosmatka, J.B. (1992). On the Analysis of Prismatic Beams Using First-Order Warping Functions. International Journal of Solids and Structures, 29(7), 879-891.

    Katsikadelis, J.T. (2002). The Analog Equation Method. A Boundary only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies. Theoretical and Applied Mechanics, 27, 13-38.

    Katsikadelis, J.T. (2002a). Boundary Elements: Theory and Applications, Elsevier. Amsterdam-London.

    Katsikadelis, J.T. (2002b). The Analog Equation Method. A Boundary only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies. Theoretical and Applied Mechanics, 27, 13-38.

  • - 45 -

    Katsikadelis, J.T. and Sapountzakis, E.J. (2002). A realistic estimation of the effective breadth of ribbed plates. International Journal of Solids and Structures, 39, 897-910.

    Koo, K.K. and Cheung, Y.K. (1989). Mixed Variational Formulation for Thin-Walled Beams with Shear Lag. Journal of Engineering Mechanics, 15(10), 2271-2286.

    Koo, K.K. and Wu, X.S. (1992). Shear Lag Analysis for Thin-Walled Members by Displacement Method. Thin-Walled Structures, 13, 337-354.

    Laudiero, F. and Savoia, M. (1990). Shear Strain Effects in Flexure and Torsion of Thin-Wailed Beams with Open or Closed Cross-Section. Thin-Walled Structures, 10, 87-119.

    Le Corvec, V. and Filippou, F.C. (2011). Enhanced 3D Fiber Beam-Column Element with Warping Displacements. Proc. of the 3rd International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering COMPDYN.

    Luo, Q.Z. and Li, Q.S. (2000). Shear Lag of Thin-Walled Curved Box Girder Bridges. Journal of Engineering Mechanics, 126(10), 1111-1114.

    Luo, Q.Z., Tang, J. and Li, Q.S. (2003). Shear Lag Analysis of Beam-Columns. Engineering Structures, 25, 1131-1138.

    Lutz, E., Ye, W. and Mukherjee, S. (1998). Elimination of Rigid Body Modes from Discretized Boundary Integral Equations. International Journal of Solids and Structures, 35(33), 4427-4436.

    Malcolm, D.J. and Redwood, R.G. (1970). Shear lag in stiffened box-girders. J. Struct. Div. ASCE, 96(ST7), 1403-15.

    Moffatt, K.R. and Dowling, P.J. (1975). Shear lag in steel box-girder bridges. Struct. Engineer, 53, 439-48.

    Mokos, V.G. and Sapountzakis, E.J. (2011). Secondary Torsional Moment Deformation Effect by BEM. International Journal of Mechanical Sciences, 53, 897-909.

    Murn, J., Kuti, V. (2008). An effective finite element for torsion of constant cross- sections including warping with secondary torsion moment deformation effect. Engineering Structures, 30(10), 2716-23.

    Muskhelishvili, N.I. (1963). Some Basic Problems of the Mathematical Theory of Elasticity. P. Noordhoff Ltd.

    Park, S.W., Fuji, D. and Fujitani, Y. (1997). A Finite Element Analysis of Discontinuous Thin-Walled Beams Considering Nonuniform Shear Warping Deformation. Computers and Structures, 65(1), 17-27.

    Proki, A. (2002). A New Finite Element for Analysis of Shear Lag. Computers and Structures, 80, 1011-1024.

  • - 46 -

    Razaqpur, A.G. and Li, H.G. (1991). A Finite Element with Exact Shape Functions for Shear Lag Analysis in Multi-Cell Box Girders. Computers and Structures, 39(1), 155-163.

    Reissner, E. (1946). Analysis of shear lag in box beams by the principle of minimum potential energy. Q. Appl. Math., 41, 268-78.

    Sapountzakis, E.J. and Dikaros, I.C (2015), Advanced 3D beam element of arbitrary composite cross section including generalized warping effects, International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.4849

    Sapountzakis, E.J. and Katsikadelis, J.T. (2000). Analysis of plates reinforced with beams. Computational Mechanics, 26, 66-74.

    Sapountzakis, E.J. and Mokos, V.G. (2003). Warping Shear Stresses in Nonuniform Torsion of Composite Bars by BEM. Computational Methods in Applied Mechanics and Engineering, 192, 4337-4353.

    Siemens PLM Software Inc. (2008), NX Nastran Users Guide.

    Tahan, N., Pavlovi, M.N. and Kotsovos, M.D. (1997). Shear-Lag Revisited: The Use of Single Fourier Series for Determining the Effective Breadth in Plated Structures. Computers and Structures, 63(4), 759-167.

    Tesar, A. (1996). Shear Lag in the Behavior of Thin-Walled Box Bridges. Computers and Structures, 59, 607-612.

    Tsipiras, V.J. and Sapountzakis, E.J. (2012). Secondary Torsional Moment Deformation Effect in Inelastic Nonuniform Torsion of Bars of Doubly Symmetric Cross Section by BEM. International Journal of Non-linear Mechanics, 47, 68-84.

    Vieira, R.F., Virtuoso, F.B.E. and Pereira, E.B.R. (2013). A Higher Order Thin-Walled Beam Model Including Warping and Shear Modes. International Journal of Mechanical Sciences, 66, 67-82.

    Vlasov, V. (1963), Thin-walled elastic beams. Israel Program for Scientific Translations, Jerusalem.

    Wu, Y., Lai, Y., Zhang, X. and Zhu, Y. (2004). A Finite Beam Element for Analyzing Shear Lag and Shear Deformation Effects in Composite-Laminated Box Girders. Computers and Structures, 82, 763-771.

    Wu, Y., Liu, S., Zhu, Y. and Lai, Y. (2003). Matrix Analysis of Shear Lag and Shear Deformation Effects in Thin-Walled Box Beams. Journal of Engineering Mechanics, 129(8), 994-950.

    Zhou, S.J. (2010). Finite Beam Element Considering Shear-Lag Effect in Box Girder. Journal of Engineering Mechanics, 136(9), 1115-1122.

    1o - i i 1.1 1.2 1.3 1.3.1 ( ):/

    1.3.2 ( Hooke ):1.3.3 ( ) :/

    1.3.4

    2o - 2.1 2.2 Euler - Bernoulli2.2.1 Euler Bernoulli () ( )( xxyy, zz yyzz 0 ). , ( i,i=) , , ( yy zz0 ). ( Ox) , ( , = ,,), ( xyxz0 , =, =) ( yz0 ) .

    2.2.2 Euler Bernoulli2.2.3 Euler Bernoulli2.2.4 2.2.5

    2.3 Timoshenko2.3.1 Timoshenko2.3.2 Timoshenko2.3.3 Timoshenko2.3.4 2.3.5

    3o - 3.1 (Saint-Venant)3.1.1 () ( )( xxyy, zz yyzz 0 ). , ( i,i=) , , ( yy zz0 ). ( yz0 ) . ( Ox) . . , (), ( xx0 ) ( xx 0 ). ( x

    x = ).

    3.1.2 3.1.3 3.1.4 3.1.5

    3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5

    4o - 4.1 - () ( )( xxyy, zz yyzz 0 ). , ( i,i=) , , ( yy zz0 ). ( yz0 ) . () . , (), .

    4.2 ( S)

    ,

    ( S)

    ,,

    ( CZ)

    ,

    ( C)

    ,

    4.3 4.4

    5o - 5.1 5.1.1 .5.1.2 ./

    5.1.3 5.1.4 ,

    5.2

    6o 6.1 6.1.1 /

    6.1.2 RHS 500X300X206.1.3 ////

    // //

    ()//////

    6.2 6.2.1 /

    6.2.2 /

    6.2.3 ///

    // //

    ()////////

    6.3 () . , , (pre-processing), . H ( ) (, , , .). (dofs), . , (shear-locking, membrane-locking phenomena), (shell elements) , . . , . , (post-processing), , , , , .. .

    6.4 , , Euler Bernoulli Timoshenko. , Euler Bernoulli Timoshenko, , , ( 21 35). . (max), Euler Bernoulli Timoshenko 37% , 11% .