Macaulay’s method (the double integration method) is a technique used in

structural analysis Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...

to determine the

deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflectio ...

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 Macaulay, Macauley, MacAulay, or McAulay may refer to: Name Surname *Macaulay (surname), an English-language surname with multiple etymological origins (also includes surnames ''Macauley'', ''MacAulay'' and ''McAulay''). People Surname *Thomas B ...

.W. H. Macaulay, "A note on the deflection of beams", Messenger of Mathematics, 48 (1919), 129.
/ref> The actual approach appears to have been developed by

Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...

in 1862.J. T. Weissenburger, ‘Integration of discontinuous expressions arising in beam theory’, AIAA Journal, 2(1) (1964), 106–108. Macaulay's method has been generalized for Euler-Bernoulli beams with axial compression,

W. H. Wittrick Professor William Henry Wittrick FAA FRS (1922–1986) was a British engineer and academic who spent 20 years in Australia, including 10 years as Professor of Aeronautical Engineering at the University of Sydney (1956–1964). He was Professor of ...

, "A generalization of Macaulay’s method with applications in structural mechanics", AIAA Journal, 3(2) (1965), 326–330. to Timoshenko beams,A. Yavari, S. Sarkani and J. N. Reddy, ‘On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory’, International Journal of Solids and Structures, 38(46–7) (2001), 8389–8406. to elastic foundations,A. Yavari, S. Sarkani and J. N. Reddy, ‘Generalised solutions of beams with jump discontinuities on elastic foundations’, Archive of Applied Mechanics, 71(9) (2001), 625–639. and to problems in which the bending and shear stiffness changes discontinuously in a beam.Stephen, N. G., (2002), "Macaulay's method for a Timoshenko beam", Int. J. Mech. Engg. Education, 35(4), pp. 286-292.

Method

The starting point is the relation from Euler-Bernoulli beam theory :

\pm EI\dfrac = M

Where

w

is the deflection and

M

is the bending moment. This equationThe sign on the left hand side of the equation depends on the convention that is used. For the rest of this article we will assume that the sign convention is such that a positive sign is appropriate. is simpler than the fourth-order beam equation and can be integrated twice to find

w

if the value of

M

as a function of

x

is known. For general loadings,

M

can be expressed in the form :

M = M_1(x) + P_1\langle x - a_1\rangle + P_2\langle x - a_2\rangle + P_3\langle x - a_3\rangle + \dots

where the quantities

P_i\langle x - a_i\rangle

represent the bending moments due to point loads and the quantity

\langle x - a_i\rangle

is a

Macaulay bracket Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.''

\langle x \rangle

.cfrac - ax\right+ C However, when integrating expressions containing Macaulay brackets, we have :

\int P\langle x-a \rangle~dx = P\cfrac + C_m

with the difference between the two expressions being contained in the constant

C_m

. 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 In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...

s and

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. Informally speaking, a step function is a piecewise constant function having on ...

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

M

. The reactions at the supports A and C are determined from the balance of forces and moments as :

R_A + R_C = P,~~ L R_C = P a

Therefore,

R_A = Pb/L

and the bending moment at a point D between A and B (

0 < x < a

) is given by :

M = R_A x = Pbx/L

Using the moment-curvature relation and the Euler-Bernoulli expression for the bending moment, we have :

EI\dfrac = \dfrac

Integrating the above equation we get, for

0 < x < a

, :

\begin
    EI\dfrac &= \dfrac +C_1 & &\quad\mathrm\\
    EI w &= \dfrac + C_1 x + C_2    & &\quad\mathrm
  \end

x=a_

\begin
    EI\dfrac(a_) &= \dfrac +C_1 & &\quad\mathrm \\
    EI w(a_) &= \dfrac + C_1 a + C_2    & &\quad\mathrm
  \end

For a point D in the region BC (

a < x < L

), the bending moment is :

M = R_A x - P(x-a) = Pbx/L - P(x-a)

In Macaulay's approach we use the

Macaulay bracket Macaulay brackets are a notation used to describe the ramp function :\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end A popular alternative transcription uses angle brackets, ''viz.''

\langle x \rangle

.bending moment In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending mome ...

and hence for the

curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...

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

\langle x-a\rangle ^n, \langle x-b\rangle ^n, \langle x-c\rangle ^n

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 to the problem, we have :

EI\dfrac = \dfrac - P\langle x-a \rangle

It is obvious that the first term only is to be considered for

x < a

and both the terms for

x > a

and the solution is :

- \cfrac \end

Note that the constants are placed immediately after the first term to indicate that they go with the first term when

x < a

and with both the terms when

x > a

. The Macaulay brackets help as a reminder that the quantity on the right is zero when considering points with

x < a

Boundary Conditions

w = 0

x = 0

C2 = 0

. Also, as

w = 0

x = L

, :

- \cfrac = 0

or, :

C_1 = -\cfrac(L^2-b^2) ~.

Hence, :

- \cfrac \end

Maximum deflection

For

w

to be maximum,

dw/dx = 0

. Assuming that this happens for

x < a

we have :

\dfrac -\cfrac(L^2-b^2) = 0

or :

x = \pm \cfrac

Clearly

x < 0

cannot be a solution. Therefore, the maximum deflection is given by :

-\cfrac

or, :

w_ = -\dfrac~.

Deflection at load application point

x = a

, i.e., at point B, the deflection is :

EI w_B = \dfrac -\cfrac(L^2-b^2) = \frac(a^2+b^2-L^2)

or :

w_B = -\cfrac

Deflection at midpoint

It is instructive to examine the ratio of

w_/w(L/2)

. At

x = L/2

EI w(L/2) = \dfrac -\cfrac(L^2-b^2) = -\frac\left frac -b^2\right

Therefore, :

\frac = \frac 
    = \frac
    = \frac

where

k = B/L

and for

a < b; 0 < k < 0.5

. 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

a = b = L/2

, for

w

to be maximum :

x = \cfrac = \frac

and the maximum deflection is :

w_ = -\dfrac = -\frac = w(L/2)~.