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2. University of Michigan, TCAUP Structures II Slide 2/19 Architecture 324 Structures 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

3. University of Michigan, TCAUP Structures II Slide 3/19 Leonhard Euler (1707 – 1783) Euler Buckling (elastic buckling) –A = Cross sectional area (in 2 ) –E = Modulus of elasticity of the material (lb/in 2 ) –K = Stiffness (curvature mode) factor –L = Column length between pinned ends (in.) –r = radius of gyration (in.) Source: Emanuel Handmann (wikimedia commons)

4. University of Michigan, TCAUP Structures II Slide 4/19 Failure Modes Short Columns – fail by crushing (“compression blocks or piers” Engel) –f c = Actual compressive stress –A = Cross-sectional area of column (in 2 ) –P = Load on the column –F c = 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

5. 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 r x = 0.999 r y = 0.433

6. University of Michigan, TCAUP Structures II Slide 6/19 End Support Conditions K is a constant based on the end conditions l is the actual length l e is the effective length l e = 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.

7. University of Michigan, TCAUP Structures II Slide 7/19 Analysis of Wood Columns Data: Column – size, length Support conditions Material properties – F c, E Required: P crit for buckling and crushing 1.Calculate slenderness ratio; largest ratio governs. 2.Check slenderness against upper limit. 3.Calculate P crit for buckling using Euler’s equation: 4.Calculate P max for crushing: P max = F c A 5.Smaller of P crit or P max will fail first.

8. University of Michigan, TCAUP Structures II Slide 8/19 Example Problem : Analysis Data:section 3”x7” Full Dimension F c = 1000 psi E = 1,400,000 psi Find: P critical for buckling and crushing. Determine the mode of failure for the wood column.

9. University of Michigan, TCAUP Structures II Slide 9/19 Example Problem : Analysis (cont.) 1.Calculate slenderness ratios for each axis. The larger (more slender) controls. 2.Upper limits are usually given by codes.

10. 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.