Design Contents |

Preliminary Matters

Design Methodology

..brain storming

..evaluation matrix

..QFD

Statistical Considerations

..variability in materials

..variability in dimensions

..variability in loading

..preferred sizes

Tolerances

Design Factor

Introduction to Failure

Failure Theories

Application of von Mises

..criterion in 2 D

Stress Concentration

..and notch sensitivity

Failure Under Combined Loading

..combined bending and torsion

Failure Under Cyclic Loading

..fatigue

..fracture mechanics

Instability - Buckling

Concentrically Loaded Strut

..slender columns

..Euler formula

..effective length

..short and intermediate columns

Eccentrically Loaded Strut

.. theory

Shock Loading

..deflection

..stress

FAILURE OF AN ECCENTRICALLY LOADED STRUT - THE SECANT FORMULA |

**1. Introduction**

In practice struts will either be designed to withstand a compressive load applied at a specified eccentricity,
or else for a nominally concentric load which will inevitably be applied slightly non-concentrically due to manufacturing
and/or assembly tolerances.

A concentrically loaded strut remains straight under an increasing axial compression until the critical load is reached when it
suffers sudden collapse. A strut subject to an increasing eccentric load starts to bend immediately and the
deflection will increase as the load increases. In this situation failure is reached when the maximum stress
or the maximum deflection exceeds that allowable, rather than the sudden buckling collapse of a concentrically loaded strut.

This section deals with the eccentrically loaded strut (the case of a concentrically load strut has been covered in a previous section,
See this link).

Assuming the strut is subjected to a compressive load 'P' at a distance 'e' (the eccentricity) from the axis of the strut, this
system can be represented by the equivalent system consisting of a concentric load 'P' and moments at each end of M = Pe. This will
cause some bending of the beam, see diagrams below.

From the free body diagram of the portion of the strut AQ, the bending moment at Q is given by:

At end A, x = 0 and y = 0 and substituting in eqn. (b) gives B = e

Putting x = L and y = 0 and substituting in (b) gives:

y_{max} = e {[sin^{2}(pL/2) + cos^{2}(pL/2)]/cos(pL/2) - 1}

y_{max} = e {sec(pL/2) - 1} . . . (e)

Replacing p with (P/EI)^{0.5} and substituting in (e):

From (f): y_{max} theoretically becomes infinite when [P/EI]^{0.5}L/2 = 3.14159/2 . . . (g)

While the deflection does not become infinite, it will be unacceptably large and P should not be allowed to reach the critical value which satisfies eqn. (g). Solving eqn. (g) for P gives:

The maximum stress sigma_{max} occurs in the section of the strut where the bending moment is maximum, ie
in the transverse section through the midpoint for the above configuration and can be obtained by adding the normal
stresses due to the axial force and the bending momentexerted on that section.

From the free body diagram on the right (above):

Substituting this value into (h) and noting that I = Ak^{2} gives:

Link to applet to carry out calculations

David J Grieve Revised: 7th October 2013.

**Contact the Author:**

Please contact me for comments and / or corrections or to purchase the book, at: davejgrieve@aol.com