Macaulay's method
Encyclopedia
Macaulay’s 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.
The first English language description of the method was by Macaulay. The actual approach appears to have been developed by Clebsh 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
and curvature
from Euler-Bernoulli beam theory
This equation is simpler than the fourth-order beam equation and can be integrated twice to find
if the value of 
as a function of 
is known. For general loadings, 
can be expressed in the form
where the quantities
represent the bending moments due to point loads and the quantity 
is a Macaulay bracket defined as
Ordinarily, when integrating
we get
However, when integrating expressions containing Macaulay brackets, we have
with the difference between the two expressions being contained in the constant
. Using these integration rules makes the calculation of the deflection of Euler-Bernoulli beams simple in situations where there are multiple point loads and point moments. The Macaulay method predates more sophisticated concepts such as Dirac delta function
s and step function
s but achieves the same outcomes for beam problems.
. The reactions at the supports A and C are determined from the balance of forces and moments as
Therefore
and the bending moment at a point D between A and B (
) is given by
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 (
), the bending moment is
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,
and 
. The above observation implies that for the two regions considered, though the equation for bending moment
and hence for the curvature
are different, the constants of integration got during successive integration of the equation for curvature for the two regions are the same.
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
etc.
It should be remembered that for any x, giving the quantities within the brackets, as in the above case, -ve should be neglected, and the calculations should be made considering only the quantities which give +ve sign for the terms within the brackets.
Reverting back to the problem, we have
It is obvious that the first term only is to be considered for
and both the terms for 
and the solution is
Note that the constants are placed immediately after the first term to indicate that they go with the first term when
and with both the terms when 
. The Macaulay brackets help as a reminder that the quantity on the right is zero when considering points with 
.
at 
, 
. Also, as 
at 
,
or,
Hence,
to be maximum, 
. Assuming that this happens for 
we have
or
Clearly
cannot be a solution. Therefore, the maximum deflection is given by
or,
, i.e., at point B, the deflection is
or
. At 
Therefore,
where
and for 
. Even when the load is as near as 0.05L from the support, the error in estimating the deflection is only 2.6%. Hence in most of the cases the estimation of maximum deflection may be made fairly accurately with reasonable margin of error by working out deflection at the centre.
, for 
to be maximum
and the maximum deflection is
Structural analysis
Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and...
to determine the deflection
Deflection
Deflection or deflexion may refer to:* Deflection , the displacement of a structural element under load* Deflection , a technique of shooting ahead of a moving target so that the target and projectile will collide...
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.
The first English language description of the method was by Macaulay. The actual approach appears to have been developed by Clebsh in 1862. Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression, to Timoshenko beams
Timoshenko beam theory
The Timoshenko beam theory was developed by Ukrainian-born scientist Stephen Timoshenko in the beginning of the 20th century. The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or...
, to elastic foundations, and to problems in which the bending and shear stiffness changes discontinuously in a beam
Method
The starting point for Maucaulay's method is the relation between bending momentBending Moment
A bending moment exists in a structural element when a moment is applied to the element so that the element bends. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres , or pound-foot or foot-pound...
and curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
from Euler-Bernoulli beam theory
This equation is simpler than the fourth-order beam equation and can be integrated twice to find
if the value of as a function of is known. For general loadings, can be expressed in the formwhere the quantities
represent the bending moments due to point loads and the quantity is a Macaulay bracket defined asOrdinarily, when integrating
we getHowever, when integrating expressions containing Macaulay brackets, we have
with the difference between the two expressions being contained in the constant
. Using these integration rules makes the calculation of the deflection of Euler-Bernoulli beams simple in situations where there are multiple point loads and point moments. The Macaulay method predates more sophisticated concepts such as Dirac delta functionDirac delta function
The Dirac delta function, or δ function, is a generalized function depending on a real parameter such that it is zero for all values of the parameter except when the parameter is zero, and its integral over the parameter from −∞ to ∞ is equal to one. It was introduced by theoretical...
s and step function
Step function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
s but achieves the same outcomes for beam problems.
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. The reactions at the supports A and C are determined from the balance of forces and moments asTherefore
and the bending moment at a point D between A and B () is given byUsing 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 (
), the bending moment isIn 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,
and . The above observation implies that for the two regions considered, though the equation for bending momentBending Moment
A bending moment exists in a structural element when a moment is applied to the element so that the element bends. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres , or pound-foot or foot-pound...
and hence for the curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
are different, the constants of integration got during successive integration of the equation for curvature for the two regions are the same.
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
etc.It should be remembered that for any x, giving the quantities within the brackets, as in the above case, -ve should be neglected, and the calculations should be made considering only the quantities which give +ve sign for the terms within the brackets.
Reverting back to the problem, we have
It is obvious that the first term only is to be considered for
and both the terms for and the solution isNote that the constants are placed immediately after the first term to indicate that they go with the first term when
and with both the terms when . The Macaulay brackets help as a reminder that the quantity on the right is zero when considering points with .Boundary Conditions
As at , . Also, as at ,or,
Hence,
Maximum deflection
For to be maximum, . Assuming that this happens for we haveor
Clearly
cannot be a solution. Therefore, the maximum deflection is given byor,
Deflection at load application point
At, i.e., at point B, the deflection isor
Deflection at midpoint
It is instructive to examine the ratio of. At Therefore,
where
and for . Even when the load is as near as 0.05L from the support, the error in estimating the deflection is only 2.6%. Hence in most of the cases the estimation of maximum deflection may be made fairly accurately with reasonable margin of error by working out deflection at the centre.Special case of symmetrically applied load
When, for to be maximumand the maximum deflection is