Axially loaded columns Effective length Axial Load Eccentric load

Effective Length of columns 1 Effective length (le) depends on degree of ‘Fixity’ at the supports. Fixed Ends, le = 0.7 L L

Effective Length of columns 2 Effective length (le) depends on degree of ‘Fixity’ at the supports. One end fixed, other pinned le = 0.85 L L

Effective Length of columns 3 Effective length (le) depends on degree of ‘Fixity’ at the supports. Pinned Ends le = L L

Effective Length of columns 4 Effective length (le) depends on degree of ‘Fixity’ at the supports. One end fixed, other end free (sway in one direction) le = 1.5 L L

Effective Length of columns 5 Effective length (le) depends on degree of ‘Fixity’ at the supports. One end fixed, other end free (no restraints) le = 2.0 L L

 Slenderness Ratio = = Sr = = Sr = le b OK for square section effective length of column least width of column = Sr Better to use le r = effective length of column Least radius of gyration = Sr where I A = r  I, being the 2nd Moment of area about C. of G. of cross section A, the cross sectional area

Slenderness Ratio Note for long columns, Ultimate buckling load reduces with increasing slenderness Buckling failure – excess deflection Buckling load << shear failure load for short columns, Failure due to shear failure

Column subjected to Lateral & Axial loads Direct or axial load causes direct stress, of same intensity right across section W A = fd M Z = fb Bending stress causes stress to vary across section Non axial load causes a combination of the two. (eccentric load)

Effect of load eccentricity W Effect of load not being central is to produce bending in the column With compression under the load And tension under other side Value of Bending Moment produced is Load x Eccentricity ( M = W x e) Deflected shape

Column subjected to Lateral & Axial loads “e” is the load eccentricity, i.e. Distance from the Neutral Axis or Centroid to the load e W Direct stress (compression) = W/A + Bending stress (tension & compression = M/Z + - Combined stress = W/A + M/Z or W/A - M/Z + When this becomes zero i.e. W/A = M/Z , then further eccentricity would causes TENSILE STRESS

Middle third rule for no tension For square or rectangular sections, load may be eccentric from one axis by no more than 1/6 of width unless tension is permissible on one face (eg Brickwork cannot resist tension ---must always be under compression.) Tension just starts when W/A = M/Z Prove the middle third rule by considering a rectangular section with I = bd3/12, y = d/2, A = bxd , M = W x e. Hint: Make e the subject of the equation .

Load Eccentricity about both axis + - - Myy /A Myy /A - - Mxx /A Mxx /A + + W/A X + W/A - Mxx /A - Myy /A + Myy /A Y Y e e Column loaded with double eccentric force W. Produces bending about both axis Myy & Mxx Which need to be combined in each corner E.g. At this corner stress is: W/A -Mxx /A + Myy /A X + W/A - Mxx /A + Myy /A W/A + Mxx /A

3D ! Stress block x Y x + + - + + - W/A Mxx /A - Mxx /A - Myy /A