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GTStrudl Modeling and Analysis Using 2-D and 3-D Finite Elements

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GTStrudl Modeling and Analysis Using 2-D and 3-D Finite Elements Powered By Docstoc
					            GTStrudl Training
                    …
     Nonlinear Geometric Analysis
                    of
                Structures
                    …
Some Practical Fundamentals and Insights

              Michael H. Swanger

              Georgia Tech CASE Center
                     June, 2011
                               Topics
•     Lite Overview of Basic Concepts
      -     Equilibrium Formulation
      -     Element Nodal Forces
      -     Element Implementation Behavior Assumptions
      -     Tangent Stiffness

•     Simple Basic behavior Examples
      - Simply-supported beam under axial load, imperfect geometry
      - Shallow truss arch: snap-through behavior
      - Shallow arch toggle: SBHQ6 model, snap-through behavior
      - Slender cantilever shear wall under axial load -- in-plane
             SBHQ plate behavior
      - The P-δ Question!


•     Additional Examples
June 22-25, 2011          GTSUG, 2011, Delray Beach,FL         2
            Overview of Basic Concepts
                        Equilibrium Formulation

                   The Principle of Virtual Work :


                     (u )  (u ) dV  P u  0

                    u T   BT (u ) (u ) dV  P   u  0
                                                 


                        BT (u ) (u ) dV  P  0

June 22-25, 2011             GTSUG, 2011, Delray Beach,FL     3
                 Overview of Basic Concepts
                             Equilibrium Formulation

The Element Equation of Equilibrium :


 BT (u ) (u ) dV  P  0


BT (u )  BL  BNL (u )
           T    T


                                        1  ui u j  u1 u1 u2 u2 u3 u3   
 (u )  D{ L   NL }            ij                                      
                                        2  x j        x x x x x x   
                                                    xi  j j                   
                                                                j   j   j   j  




 {BL  BNL (u )}D{ L   NL }dV  P  0
    T    T




    June 22-25, 2011             GTSUG, 2011, Delray Beach,FL                  4
              Overview of Basic Concepts
                           Element Nodal Forces

The Equation of Element Equilibrium -- Element Nodal Forces :


 {BL D L  BL D NL  BNL (u ) D L  BNL (u ) D NL } dV  P
    T         T          T               T




       L  [ BL ]{u},       NL  [GNL (u )]{u}


 {[BL DBL ]{u} 
     T



               [ BL DGNL (u )]{u}  [ BNL (u ) DBL ]{u} 
                  T                    T


                                             [ BNL (u ) DGNL (u )]{u}} dV  P
                                                T




 June 22-25, 2011               GTSUG, 2011, Delray Beach,FL                    5
              Overview of Basic Concepts
        Element Implementation Behavior Assumptions

             Assumptions related to the scope of nonlinear geometric
             behavior are introduced into the definition of strain and
             the equilibrium equation:

                    Example: Frame Member Strain and Equilibrium

     u    u     u
                    2
                       1  u  u
                                22
                                       u
                                        2    u y   uz 2 
                                             2       2

 x  x  y 2y  z 2z   x  y 2y  z 2z   
                            x                          
     x    x     x   2      x     x   x   x  
                                                              
                                              0                 P and M are coupled                   
                                                                                                      
                                                       Modified UAxial and U Transverse are uncoupled 
      {[BL DBL ]{u} 
          T                                                     θ                                     
                                                                 Torsionand U Transverse are uncoupled

                        [ BL DGNL (u )]{u}  [ BNL (u ) DBL ]{u} 
                           T                    T
                                                                               0


                                                          [ BNL (u ) DGNL (u )]{u}} dV  P
                                                             T


 June 22-25, 2011                     GTSUG, 2011, Delray Beach,FL                                         6
              Overview of Basic Concepts
       Element Implementation Behavior Assumptions
                   Summary of GTSTRUDL NLG Behavior Assumptions

    1. Plane and Space Frame

          −    Small strains; σ = Eε remains valid
          −    Internal rotations and curvatures are small; θ ≈ sinθ
          −    Member chord rotations are small
          −    P and M are coupled
          −    Uaxial and UTransverse are uncoupled
          −    θTorsion and UTransverse are uncoupled
          −    Other member effects are not affected by member displacement
          −    Member loads are not affected by member displacement

    2. Plane and Space Truss

          − Small strains; σ = Eε remains valid
          − No assumptions limiting magnitude of displacements
June 22-25, 2011                GTSUG, 2011, Delray Beach,FL             7
             Overview of Basic Concepts
       Element Implementation Behavior Assumptions
                   Summary of GTSTRUDL NLG Behavior Assumptions

3. SBHQ and SBHT Plate Elements

     − Small strains; σ = Dε remains valid
     − BPH + PSH + 2nd order membrane effects
           Internal rotations and curvatures are small
           Uin-plane and UTransverse are coupled in 2nd order membrane effects
           BPH and 2nd order membrane effects are uncoupled
     − Element loads are not affected by element displacements

4. The IPCABLE Element

     − Small strains; σ = Eε remains valid
     − No assumptions limiting magnitude of displacements
     − Regarding NLG, 2-node version and the truss are the same

June 22-25, 2011                GTSUG, 2011, Delray Beach,FL                8
              Overview of Basic Concepts
                           The Tangent Stiffness Matrix

          Incremental Equation of Element Equilibrium:

           B(TL  NL ) ( L  NL ) dV  P  0


          d    B T
                   ( L  NL )          
                             ( L  NL ) dV u  P, where d 
                                                                
                                                               u

           dB(TL  NL ) ( L  NL ) dV  B(TL  NL ) d ( L  NL ) dV  u  P
                                                                    


                    K  Ku  u  P;
                                                        KT u  P
June 22-25, 2011                  GTSUG, 2011, Delray Beach,FL                     9
             Overview of Basic Concepts
                     The Tangent Stiffness Matrix

              P

                                                KT = [Kσ + Ku]
                    B σdV
                     T


                                     a          b
           Pi+1
                                                  2
                                      1
                                                                  u2=u1+u2
             Pi
                              u1         u2

                         ui    u1=ui+u1                       ui+1       u
June 22-25, 2011                GTSUG, 2011, Delray Beach,FL                  10
       Simple Basic behavior Examples


 •     Simply-supported beam under axial load, imperfect geometry

 •     Shallow truss arch: snap-through behavior

 •     Shallow arch toggle: SBHQ6 model, snap-through behavior

 •     Slender cantilever shear wall under axial load -- in-plane SBHQ
       plate behavior

 •     The P-δ Question!




June 22-25, 2011           GTSUG, 2011, Delray Beach,FL             11
       Simple Basic Behavior Examples
    Simply-supported beam under axial load, imperfect geometry




                                                                                  P

                                                    20 @ 1 ft



           Imperfection: Yimp = -0.01sin(πx/L) ft

                                                 E = 10,000 ksi
                                Plane Frame:   Ax = 55.68 in2,  Iz = 100.00 in4




June 22-25, 2011                    GTSUG, 2011, Delray Beach,FL                      12
       Simple Basic Behavior Examples
    Simply-supported beam under axial load, imperfect geometry




                                      Pe = 171.2 kips




June 22-25, 2011        GTSUG, 2011, Delray Beach,FL             13
           Simple Basic Behavior Examples
        Simply-supported beam under axial load, imperfect geometry

                                Push-over Analysis Procedure

UNITS KIPS
LOAD 1
JOINT LOADS
  21 FORCE X -1000.0     $ Load P

NONLINEAR EFFECTS                        Load P
  GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
  INCREMENTAL LOAD 1
  MAX NUMBER OF LOAD INCR 200
  MAX NUMBER OF TRIALS 20
  MAX NUMBER OF CYCLES 100
  LOADING RATE     0.005    $ f1
  CONVERGENCE RATE     0.8
  CONVERGENCE TOLERANCE COLLAPSE 0.0001
  CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
                                              1
PERFORM PUSHOVER ANALYSIS
                                                          f1P




                                                                      Displacement




    June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL             14
           Simple Basic Behavior Examples
        Simply-supported beam under axial load, imperfect geometry

                               Push-over Analysis Procedure

UNITS KIPS
LOAD 1
JOINT LOADS
  21 FORCE X -1000.0
                                       Load P
NONLINEAR EFFECTS
  GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
  INCREMENTAL LOAD 1
  MAX NUMBER OF LOAD INCR 200
  MAX NUMBER OF TRIALS 20
  MAX NUMBER OF CYCLES 100
  LOADING RATE     0.005    $ f1                2
  CONVERGENCE RATE     0.8
  CONVERGENCE TOLERANCE COLLAPSE 0.0001
  CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END
PERFORM PUSHOVER ANALYSIS                       1                (2f1)P

                                                         f1P




                                                                          Displacement



    June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL                15
           Simple Basic Behavior Examples
        Simply-supported beam under axial load, imperfect geometry

                               Push-over Analysis Procedure

UNITS KIPS
LOAD 1
JOINT LOADS
  21 FORCE X -1000.0
                                     Load P
NONLINEAR EFFECTS
  GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA                     3
  INCREMENTAL LOAD 1
  MAX NUMBER OF LOAD INCR 200
  MAX NUMBER OF TRIALS 20
  MAX NUMBER OF CYCLES 100
  LOADING RATE     0.005    $ f1           2
  CONVERGENCE RATE     0.8
  CONVERGENCE TOLERANCE COLLAPSE 0.0001
  CONVERGENCE TOLERANCE DISPLACEMENT 0.001                               (3f1)P
END
                                           1                    (2f1)P
PERFORM PUSHOVER ANALYSIS
                                                        f1P




                                                                                  Displacement




    June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL                         16
           Simple Basic Behavior Examples
        Simply-supported beam under axial load, imperfect geometry

                               Push-over Analysis Procedure

UNITS KIPS
LOAD 1
JOINT LOADS
  21 FORCE X -1000.0
                                     Load P
NONLINEAR EFFECTS
  GEOMETRY MEMBERS EXISTING

PUSHOVER ANALYSIS DATA
                                           3
  INCREMENTAL LOAD 1
  MAX NUMBER OF LOAD INCR 200
                                           4
  MAX NUMBER OF TRIALS 20
  MAX NUMBER OF CYCLES 100
  LOADING RATE     0.005    $ f1           2
  CONVERGENCE RATE     0.8  $ r
  CONVERGENCE TOLERANCE COLLAPSE 0.0001
  CONVERGENCE TOLERANCE DISPLACEMENT 0.001
END                                                                      (3f1)P
                                                                                  (2f1 + rf1)P
PERFORM PUSHOVER ANALYSIS
                                              1                 (2f1)P

                                                        f1P




                                                                                             Displacement


    June 22-25, 2011                    GTSUG, 2011, Delray Beach,FL                                    17
       Simple Basic Behavior Examples
                   Shallow truss arch: snap-through behavior



                                              P



                                                                  u

                             L
                                                                       3 in
                                                                 3-u
                                 L’
                         θ




                                         2 @ 100 in




                                       E = 29,000 ksi
                                 Plane Truss: Ax = 1.0 in2




June 22-25, 2011                  GTSUG, 2011, Delray Beach,FL                18
       Simple Basic Behavior Examples
                   Shallow truss arch: snap-through behavior




June 22-25, 2011              GTSUG, 2011, Delray Beach,FL     19
       Simple Basic Behavior Examples
      Shallow arch toggle: SBHQ6 model, snap-through behavior

                                         P
                                                    Θz = 0
                        A

                                                                0.36 67 in
                            A


                                  2 @ 1 2.94 3 in                Fixed (typ)


                            E = 1 .030 000 0E+07 lb s/in 2
                             ν = 0.0
              Y

                                                    0.24 3 in
                                     0.75 3 in
                   X
                                   Section A-A




                                SBHQ6 Arch Leg, 20 x 4




June 22-25, 2011        GTSUG, 2011, Delray Beach,FL                           20
       Simple Basic Behavior Examples
      Shallow arch toggle: SBHQ6 model, snap-through behavior




                                               Note: Pbuck = 152.4 lbs (linear buckling load)




June 22-25, 2011        GTSUG, 2011, Delray Beach,FL                                            21
        Simple Basic Behavior Examples
Slender cantilever shear wall under axial load -- in-plane SBHQ plate behavior
                                                                  P

                                             0.01 kips




                         Mesh =   2X50
                    Material =    concrete
                        POISSON   = 0.0                               2 ft
                      Thickness    = 4 in
                                                         100 ft




 June 22-25, 2011                   GTSUG, 2011, Delray Beach,FL             22
           Simple Basic Behavior Examples
Slender cantilever shear wall under axial load -- in-plane SBH plate behavior




                                                    Pbuck (FE) = 41.95 kips

                                                    (Pe (SF) = 28.42 kips)




    June 22-25, 2011         GTSUG, 2011, Delray Beach,FL                     23
                          The P-δ Question
                         Does GTSTRUDL Include P-δ?

                   E = 10,000 ksi,   Plane Frame:   Ax = 55.68 in2,   Iz = 100.0 in4




                                            No Mid Span Nodes




                                            1 Mid Span Node




June 22-25, 2011                     GTSUG, 2011, Delray Beach,FL                      24
                   The P-δ Question




June 22-25, 2011      GTSUG, 2011, Delray Beach,FL   25
                   The P-δ Question

                          Mtot = M0 + Pδmid




June 22-25, 2011      GTSUG, 2011, Delray Beach,FL   26

				
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