Macaulay brackets are a notation used to describe the

ramp function The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative inputs, output equals input for non-negative inputs". The term "ramp" can also be used for ...

\ = \begin 0, & x < 0 \\ x, & x \ge 0. \end

A popular alternative transcription uses angle brackets, ''viz.''

\langle x \rangle

.Lecture 12: Beam Deflections by Discontinuity Functions.
Introduction to Aerospace Structures. Department of Aerospace Engineering Sciences, University of Colorado at Boulder Another commonly used notation is

x

₊ or

(x)

₊ for the positive part of

x

, which avoids conflicts with

\

for set notation.

In engineering

Macaulay's notation is commonly used in the static analysis of bending moments of a beam. This is useful because shear forces applied on a member render the shear and moment diagram discontinuous. Macaulay's notation also provides an easy way of integrating these discontinuous curves to give bending moments, angular deflection, and so on. For engineering purposes, angle brackets are often used to denote the use of Macaulay's method. :

\langle x-a\rangle^n = \begin 0, & x < a \\ (x-a)^n, & x \ge a. \end

(n \ge 0)

The above example simply states that the function takes the value

(x-a)^n

for all ''x'' values larger than ''a''. With this, all the forces acting on a beam can be added, with their respective points of action being the value of ''a''. A particular case is the

unit step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Different ...

, :

\langle x-a\rangle^0 \equiv \^0 = \begin 0, & x < a \\ 1, & x > a. \end

References

Mathematical analysis {{mathanalysis-stub

In engineering

See also

References