# 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

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
deflection
Second Order Effects & Moment Amplification

Consider
P                                   P
D

H                           H

Mmax = Mo  PD
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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