Beam Columns I by vRaGRt

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									Beam-Columns
            Members Under Combined Forces


Most beams and columns are subjected to some degree of both bending and axial load
                 e.g. Statically Indeterminate Structures
             A                                                      B
    P1




             C                                                      D
     P2




             E                                                      F
     Interaction Formulas for Combined Forces



           Load Effects  1.0
               Resistance
                        e.g. LRFD
If more than one resistance is involved consider interaction

          iQi      iQi 
                              1.0
         Rn  LS 1  Rn  LS 2
               Basis for Interaction Formulas


  Tension/Compression & Single Axis Bending
                    Pu     Mu
                                  1.0
                    c Pn  b M n

       Tension/Compression & Biaxial Bending

              Pu    M ux     M uy     
                                      1.0
             c Pn  b M nx b M ny
                   
                                       
                                       

Quite conservative when compared to actual ultimate strengths
especially for wide flange shapes with bending about minor axis
         AISC Interaction Formula – CHAPTER H


 AISC Curve


Pr 8  M rx M ry 
                   1.0 for Pr  0.2
         
Pc 9  M cx M cy 
                           Pc


 Pr  M rx M ry   1.0 for Pr  0.2
         
2 Pc  M cx M cy 
                           Pc


  r = required strength

  c = available strength
Pr 8  M rx M ry 
                   1.0 for Pr  0.2
         
Pc 9  M cx M cy 
                           Pc

 Pr  M rx M ry   1.0 for Pr  0.2
         
2 Pc  M cx M cy 
                           Pc



REQUIRED                CAPACITY
   Pr                     Pc
   Mrx                    Mcx
      Mry                    Mcy
Pr 8  M rx M ry 
                  1.0 for Pr  0.2
          
Pc 9  M cx M cy 
                           Pc



 Pr  M rx M ry             Pr
                1.0 for     0.2
2 Pc  M cx M cy 
                           Pc
                 Axial Capacity Pc


   Pn  Fcr Ag
            QFy
                        KL           E
       0.658 Fe 
                   QF if      4.71
       
                 y     r          QFy
                  
      
      
Fcr                    or Fe  0.44QFy
      
      
      
      
          0.877Fe otherwise
                          Axial Capacity Pc


Fe:    Elastic Buckling Stress corresponding to the controlling mode of
           failure (flexural, torsional or flexural torsional)

      Theory of Elastic Stability (Timoshenko & Gere 1961)

Flexural Buckling   Torsional Buckling Flexural Torsional   Flexural Torsional
                    2-axis of symmetry Buckling             Buckling
                                       1 axis of symmetry   No axis of symmetry


        2E          AISC Eqtn           AISC Eqtn            AISC Eqtn
Fe 
     KL / r 2      E4-4                E4-5                 E4-6
                               Effective Length Factor

Free to rotate and translate          Fixed on top       Free to rotate




   Fixed on bottom                  Fixed on bottom      Fixed on bottom

           2 EA                             2 EA                 2 EA
    Pcr                             Pcr 
          2L 
              r
                2
                                           
                                           0.5L
                                                 r
                                                   2
                                                     
                                                           Pcr 
                                                                  
                                                                 0.7 L
                                                                         2
                                                                           
                                                                       r
                     Effective Length of Columns


                         Ic Lc
                Ig Lg            Ig Lg                   Assumptions
                     A                                   • All columns under
                         Ic Lc                             consideration reach buckling
                                                           Simultaneously
                     B
                                                         • All joints are rigid

                                                         • Consider members lying in the
                                                           plane of buckling
                Define:

       I
                                                         • All members have constant A

GA   
            c   Lc
                             GB          
                                           I   c   Lc
       I   g   Lg                         I
                                            g       Lg
            Effective Length of Columns


Use alignment charts (Structural Stability Research Council SSRC)


       LRFD Commentary Figure C-C2.2 p 16.1-241,242



                  Connections to foundations
                  (a) Hinge
                       G is infinite - Use G=10
                  (b) Fixed
                        G=0 - Use G=1.0
                Axial Capacity Pc


LRFD
  Pc  c Pn

 c  resistance factor for compressio n  0.90

 c Pn  design compressiv e strength
                Axial Capacity Pc


ASD
       Pn
  Pc 
       c


 c  safety factor for compressio n  1.67

 Pn c  allowable compressiv e strength
Pr 8  M rx M ry 
                  1.0 for Pr  0.2
          
Pc 9  M cx M cy 
                           Pc



 Pr  M rx M ry 
                  1.0 for Pr  0.2
          
2 Pc  M cx M cy 
                           Pc
               Moment Capacity Mcx or Mcy


       M p for Lb  Lp
      
                             Lb  Lp 
M n  Cb  M p  M p  M r            M p for Lp  Lb  Lr
                            Lr  Lp 
                                       
       F S  M for L  L
       cr x       p     r      b


                             M r  0.7 Fy S x
                                                                        2
                                   Cb E2
                                                        Jc  Lb 
                            Fcr             1  0.078         
                                  Lb rts 2
                                                       S x ho  rts 
                                                               

                              REMEMBER TO CHECK FOR NON-
                                   COMPACT SHAPES
            Moment Capacity Mcx or Mcy


        REMEMBER TO ACCOUNT FOR LOCAL
           BUCKLING IF APPROPRIATE



       M p for    p
      
                            p 
M n   M p  M p  M r             M p for  p    r
                         r   p 
                                     
       F S  M for   
       cr x      p       r
      Moment Capacity Mcx or Mcy



  LRFD                        ASD

M c  b M n                    Mn
                           Mc 
                                b
b  0.90                   b  1.67
              Demand



Pr 8  M rx M ry 
                  1.0 for Pr  0.2
          
Pc 9  M cx M cy 
                           Pc



 Pr  M rx M ry 
                  1.0 for Pr  0.2
          
2 Pc  M cx M cy 
                           Pc
           Axial Demand Pr



LRFD                          ASD


Pr  Pu                      Pr  Pa
factored                     service
              Demand



Pr 8  M rx M ry 
                  1.0 for Pr  0.2
          
Pc 9  M cx M cy 
                           Pc



 Pr  M rx M ry 
                  1.0 for Pr  0.2
          
2 Pc  M cx M cy 
                           Pc
    Second Order Effects & Moment Amplification

              P                                   P


W

                                                   y

                                             M




                               ymax @ x=L/2 = d



    Mmax @ x=L/2 = Mo  Pd  wL2/8 + Pd
                           additional moment causes additional
                                        deflection
 Second Order Effects & Moment Amplification


Consider
           P                                   P
                                           D

    H                           H




               Mmax = Mo  PD
                  additional moment causes additional
                               deflection
Second Order Effects & Moment Amplification


• Total Deflection cannot be Found Directly
• Additional Moment Because of Deformed Shape

     • First Order Analysis
        • Undeformed Shape - No secondary moments


     • Second Order Analysis (P-d and P-D)
        • Calculates Total deflections and secondary moments
         • Iterative numerical techniques
         • Not practical for manual calculations
         • Implemented with computer programs
           Design Codes


          AISC Permits


    Second Order Analysis

                  or

Moment Amplification Method
Compute moments from 1st order analysis
   Multiply by amplification factor
Derivation of Moment Amplification


                       x 
            yo  e sin 
                       L
        Derivation of Moment Amplification


Moment Curvature                                     x 
                                          yo  e sin 
   d2y     M                                         L
      2
        
   dx      EI
                                                P
  M  P  yo  y                          M

 d2y     P        x   
           e sin  y 
 dx 2
         EI        L   

 d2y P          Pe   x
    2
          y   sin        2nd order nonhomogeneous DE
 dx     EI      EI    L
           Derivation of Moment Amplification


  d2y P          Pe   x
     2
           y   sin
  dx     EI      EI    L

Boundary Conditions
     @x 0      y0

     @x  L     y0

Solution
                 x
     y  B sin
                  L
             Derivation of Moment Amplification

                           x
Substitute     y  B sin        in DE
                           L
       2        x
                 P       x    Pe     x
      2 B sin    B sin        sin
      L       L EI        L    EI      L
 Solve for B

       Pe
     
 B    EI   e       e
                            
                                 e
    P    2
               EI  EI
                2    2
                              Pe
        2 1     2    2
                         1       1
    EI L       PL   PL        P
        Derivation of Moment Amplification


Deflected Shape


             x      e       x
    y  B sin               sin L 
              L  Pe P   1
           1            x        1      
        P P   1 e sin L    P P   1 yo
        e                    e           
         Derivation of Moment Amplification


Moment
                            x              x 
M  P  yo  y     P e sin  
                                      e
                                              sin L 
                             L  Pe P   1       

                            x       1      
                    P e sin                
                             L  1  P P e 

                        Mo(x)     Amplification
                                    Factor
Braced vs. Unbraced Frames




             M r  B1M nt  B2 M lt
                       Eq. C2-1a

            M r  required moment strength
                M u for LRFD
                M a for ASD
        Braced vs. Unbraced Frames


M r  B1M nt  B2 M lt          Eq. C2-1a

Mnt = Maximum 1st order moment assuming no
      sidesway occurs
Mlt = Maximum 1st order moment caused by sidesway

B1 = Amplification factor for moments in member
     with no sidesway

B2 = Amplification factor for moments in member
     resulting from sidesway
Braced Frames




                M  P  yo  y 


                       1     
             Mo 
                  1  Pe P 
                              
Braced Frames
                      Braced Frames


          Cm
B1                 1
     1  aPr Pe1 
                                 AISC Equation C2 - 2

    Pr = required axial compressive strength
      = Pu for LRFD
      = Pa for ASD

    Pr has a contribution from the PD effect and is given by

             Pr  Pnt  B2 Plt
                       Braced Frames


          Cm
B1                 1
     1  aPr Pe1 
                              AISC Equation C2 - 2

    a = 1 for LRFD
      = 1.6 for ASD


               2 EI
      Pe1 
              K1L2
                 Braced Frames


Cm coefficient accounts for the shape of the moment
     diagram
                Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

                             M1 
                             M  AISC C2 - 4
              Cm  0.6  0.4    
                             2
              M1: Absolute smallest End Moment
              M2: Absolute largest End Moment
               Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

                      aPr 
                      P  AISC CommentaryC2 - 2
           Cm  1       
                      e1 

                      2d o EI
                         2
                                 -1
                      MoL
                AISC CommentaryTable C - C2.1


        COSERVATIVELY                 Cm= 1

								
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