Macaulay’s method (the double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay’s technique is very convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads (u.d.l.) and uniformly varying loads (u.v.l.) over the span and a number of concentrated loads are conveniently handled using this technique.
Contents
- Method
- Example Simply supported beam with point load
- Boundary Conditions
- Maximum deflection
- Deflection at load application point
- Deflection at midpoint
- Special case of symmetrically applied load
- References
The first English language description of the method was by Macaulay. The actual approach appears to have been developed by Clebsch in 1862. Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression, to Timoshenko beams, to elastic foundations, and to problems in which the bending and shear stiffness changes discontinuously in a beam
Method
The starting point is the relation between bending moment and curvature from Euler-Bernoulli beam theory
Where
where the quantities
Ordinarily, when integrating
However, when integrating expressions containing Macaulay brackets, we have
with the difference between the two expressions being contained in the constant
Example: Simply supported beam with point load
An illustration of the Macaulay method considers a simply supported beam with a single eccentric concentrated load as shown in the adjacent figure. The first step is to find
Therefore,
Using the moment-curvature relation and the Euler-Bernoulli expression for the bending moment, we have
Integrating the above equation we get, for
At
For a point D in the region BC (
In Macaulay's approach we use the Macaulay bracket form of the above expression to represent the fact that a point load has been applied at location B, i.e.,
Therefore, the Euler-Bernoulli beam equation for this region has the form
Integrating the above equation, we get for
At
Comparing equations (iii) & (vii) and (iv) & (viii) we notice that due to continuity at point B,
The above argument holds true for any number/type of discontinuities in the equations for curvature, provided that in each case the equation retains the term for the subsequent region in the form
Reverting to the problem, we have
It is obvious that the first term only is to be considered for
Note that the constants are placed immediately after the first term to indicate that they go with the first term when
Boundary Conditions
As
or,
Hence,
Maximum deflection
For
or
Clearly
or,
Deflection at load application point
At
or
Deflection at midpoint
It is instructive to examine the ratio of
Therefore,
where
Special case of symmetrically applied load
When
and the maximum deflection is