Column Analysis and Design
-
Upload
waqas-javaid -
Category
Education
-
view
897 -
download
4
description
Transcript of Column Analysis and Design
Unless otherwise noted, the content of this course material is licensed under a Creative Commons Attribution 3.0 License. http://creativecommons.org/licenses/by/3.0/
© 2009, Peter Von Buelow
You assume all responsibility for use and potential liability associated with any use of the material. Material contains copyrighted content, used in accordance with U.S. law. Copyright holders of content included in this material should contact [email protected] with any questions, corrections, or clarifications regarding the use of content. The Regents of the University of Michigan do not license the use of third party content posted to this site unless such a license is specifically granted in connection with particular content. Users of content are responsible for their compliance with applicable law. Mention of specific products in this material solely represents the opinion of the speaker and does not represent an endorsement by the University of Michigan. For more information about how to cite these materials visit https://open.umich.edu/education/about/terms-of-use.
Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. You should speak to your physician or make an appointment to be seen if you have questions or concerns about this information or your medical condition. Viewer discretion is advised: Material may contain medical images that may be disturbing to some viewers.
University of Michigan, TCAUP Structures II Slide 2/19
Architecture 324Structures II
Column Analysis and Design
• Failure Modes• End Conditions and Lateral Bracing• Analysis of Wood Columns• Design of Wood Columns• Analysis of Steel Columns• Design of Steel Columns
University of Michigan, TCAUP Structures II Slide 3/19
Leonhard Euler (1707 – 1783)
Euler Buckling (elastic buckling)
– A = Cross sectional area (in2)– E = Modulus of elasticity of the material (lb/in2)– K = Stiffness (curvature mode) factor– L = Column length between pinned ends (in.)– r = radius of gyration (in.)
crcr F
rKL
Ef
2
2
2
2
rKL
AEPcr
A
Ir
Source: Emanuel Handmann (wikimedia commons)
University of Michigan, TCAUP Structures II Slide 4/19
Failure Modes
• Short Columns – fail by crushing(“compression blocks or piers” Engel)
– fc = Actual compressive stress
– A = Cross-sectional area of column (in2)– P = Load on the column – Fc = Allowable compressive stress per codes
• Intermediate Columns – crush and buckle(“columns” Engel)
• Long Columns – fail by buckling(“long columns” Engel)
– E = Modulus of elasticity of the column material– K = Stiffness (curvature mode) factor– L = Column length between pinned ends (in.)– r = radius of gyration = (I/A)1/2
cc FA
Pf
crcr F
rKL
Ef
2
2
University of Michigan, TCAUP Structures II Slide 5/19
Slenderness Ratio
• Radius of Gyration: a geometric property of a cross section
– r = Radius of Gyration– I = Moment of Inertia– A = Cross-sectional Area
• Slenderness Ratios:
The larger ratio will govern.
Try to balance for efficiency
A
Ir 2ArI
x
x
r
L
y
y
r
L
rx = 0.999
ry = 0.433
University of Michigan, TCAUP Structures II Slide 6/19
End Support Conditions
K is a constant based on the end conditionsl is the actual length
le is the effective length
le = Kl
K= 0.5
K= 2.0
K= 0.7
K= 1.0
Both ends fixed.
One end pinned, one end fixed.
Both ends pinned.
One end free, one end fixed.
University of Michigan, TCAUP Structures II Slide 7/19
Analysis of Wood Columns
Data:• Column – size, length• Support conditions• Material properties – Fc , E
Required:• Pcrit for buckling and crushing
• Calculate slenderness ratio; largest ratio governs.
• Check slenderness against upper limit.• Calculate Pcrit for buckling using Euler’s
equation:• Calculate Pmax for crushing:
Pmax = Fc A
• Smaller of Pcrit or Pmax will fail first.
University of Michigan, TCAUP Structures II Slide 8/19
Example Problem : Analysis
Data: section 3”x7” Full DimensionFc = 1000 psiE = 1,400,000 psi
Find: Pcritical for buckling and crushing.Determine the mode of failure for the wood column.
University of Michigan, TCAUP Structures II Slide 9/19
Example Problem : Analysis (cont.)
1. Calculate slenderness ratiosfor each axis.
The larger (more slender) controls.
2. Upper limits are usually given by codes.
University of Michigan, TCAUP Structures II Slide 10/19
Example Problem : Analysis (cont.)
3. Calculate critical Euler buckling load.
4. Calculate crushing load.
5. Smaller of the two will fail first and control.
University of Michigan, TCAUP Structures II Slide 11/19
Analysis of Steel Columnsby Engel
Data:• Column – size, length• Support conditions• Material properties – Fy
• Applied load - Pactual
Required:• Pactual < Pallowable
• Calculate slenderness ratios.The largest ratio governs.
• Check slenderness ratio against upper limit of 200
• Use the controlling slenderness ratio to find the critical Euler buckling stress, fcr.
• Apply some Factor of Safety (like 3) to fcr.
• Determine yield stress limit, Fy.
• Fallowable is the lesser stress: (fcr / F.S.) or Fy
• Compute allowable capacity: Pallowable = Fallow A.
• Check column adequacy:Pactual < Pallowable
2
2
rKL
Efcr
University of Michigan, TCAUP Structures II Slide 12/19
Design of Steel Columnsby Engel
Data:• Column – length• Support conditions• Material properties – Fy
• Applied load - Pactual
Required:• Column – section
• Use the Euler equation to solve for Ar2 which is equal to I for both x and y axis.
• Enter the section tables and find the least weight section that satisfies BOTH Ix and Iy.
• Check the slenderness ratios are both < 200.
• Calculate the actual Euler stress fcr for the final section.
• Fallowable is the lesser stress: fcr / F.S. or Fy
• Compute allowable capacity: Pallowable = Fallow A.
..)(
2
2
SFE
lKPI xxx
..)(
2
2
SFE
lKPI yyy
University of Michigan, TCAUP Structures II Slide 13/19
Example Problem : Design
Select a steel section that can carry the given load.
University of Michigan, TCAUP Structures II Slide 14/19
Example Problem : Design (cont.)
University of Michigan, TCAUP Structures II Slide 15/19
Example Problem : Design (cont.)
• Determine the controlling slenderness (larger controls)
• Find the actual buckling stress, fcr
• Compare to allowable stress, Fallowable is lesser of : fcr/F.S. or Fy
• Determine safe allowable load, Pallowable = Fallowable A
University of Michigan, TCAUP Structures II Slide 16/19
Determining K factorsby AISC
Sidesway Inhibited:Braced frame1.0 > K > 0.5
Sidesway Uninhibited:Un-braced frameunstable > K > 1.0
If Ic/Lc is largeand Ig/Lg is smallThe connection is more pinned
If Ic/Lc is smalland Ig/Lg is largeThe connection is more fixed
Source: American Institute of Steel Construction, Manual of Steel Construction, AISC 1980
University of Michigan, TCAUP Structures II Slide 17/19
Steel Frame Construction
University of Michigan, TCAUP Structures II Slide 18/19
Analysis of Steel Columnsby AISC-ASD
Data:• Column – size, length• Support conditions• Material properties – Fy
• Applied load - Pactual
Required:• Pactual < Pallowable
• Calculate slenderness ratios. largest ratio governs.
• In AISC Table look up Fa for given slenderness ratio.
• Compute: Pallowable = Fa A.• Check column adequacy:
Pactual < Pallowable
Source: American Institute of Steel Construction, Manual of Steel Construction, AISC 1980
University of Michigan, TCAUP Structures II Slide 19/19
Design of Steel Columnswith AISC-ASD Tables
Data:• Column – length• Support conditions• Material properties – Fy
• Applied load - Pactual
Required:• Column Size
1. Enter table with height. 2. Read allowable load for each section to
find the smallest adequate size.3. Tables assume weak axis buckling. If
the strong axis controls the length must be divide by the ratio rx/ry
4. Values stop in table (black line) at slenderness limit, KL/r = 200
Source: American Institute of Steel Construction, Manual of Steel Construction, AISC 1980