Föppl clamp

The Föppl bracket is a simplistic notation introduced by August Föppl , especially for mechanics . It is also called the Föppl symbol . It is sometimes referred to as the Macaulay bracket after the British mathematician William Herrick Macaulay .

definition

The Föppl bracket is not a mathematical notation, it was adopted by engineers for use in technical mechanics .

${\displaystyle \langle x-a\rangle ^{n}={\begin{cases}0&\mathrm {f{\ddot {u}}r} ~x<a\\(x-a)^{n}&\mathrm {f{\ddot {u}}r} ~x>a\end{cases}}}$

This expression means that the bracket for x values ​​is less than 0, and for values ​​greater than the value of a normal bracket . Please note that the Föppl bracket is not defined for. For considerations on this point, other forms of description (e.g. the balance on the differential element) are necessary; however, in most cases such considerations are not necessary. ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle (x-a)^{n}}$ ${\displaystyle x=a}$

Specifically describes:

${\displaystyle \langle x-a\rangle ^{0}={\begin{cases}0&\mathrm {f{\ddot {u}}r} ~x<a\\1&\mathrm {f{\ddot {u}}r} ~x>a\end{cases}}}$

Thus, jumps, z. B. in a force curve by multiplying the bracket with the force (see example).

The derivative and antiderivative are also defined:

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\langle x-a\rangle ^{n}=n\langle x-a\rangle ^{n-1}}$

${\displaystyle \int \langle x-a\rangle ^{n}\mathrm {d} x={\frac {1}{n+1}}\langle x-a\rangle ^{n+1}+C}$

With differentiation and integration, the bracket symbol can be viewed like a round bracket.

use

The Föppl bracket allows the force and moment curves to be shown in a short form on bending beams and beams. Without this representation, a case distinction would have to be made for every attacking force and every attacking moment.

The exponents of a Föppl bracket are to be selected according to the force or torque curve. Examples: The surface loading q (x) is constant: n = 0; a force or a moment acts: n = 0; the surface loading q (x) is linear: n = 1; the surface loading q (x) is quadratic: n = 2; the surface loading q (x) is cubic: n = 3 etc.

When calculating the transverse force Q (x) by integrating z. B. with a linear surface loading q (x) with n = 1, the exponent n = 2 for Q (x) and, through further integration, the exponent n = 3 for the bending moment M (x).

example

A beam of length l is statically supported at its end points A and D. It is loaded in point B by the force F and in point C by the moment M.

The following applies to the relationship between load and internal forces:

${\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} x}}=-q\qquad {\frac {\mathrm {d} M}{\mathrm {d} x}}=Q}$

The course of the shear forces (in the z-direction) follows the formula:

• with Föppl clamp:

${\displaystyle Q(x)=-F_{Az}+F\langle x-f\rangle ^{0}}$

Explanation: The course of the transverse force corresponds to the left of point f of the negative reaction force F _Az , since the Föppl bracket is defined as zero for x <f. To the right of point f, the term assumes the value 1, which means that the load F flows into the transverse force curve through a jump.

• without Föppl clamp:

${\displaystyle Q(x)={\begin{cases}-F_{Az}&\mathrm {f{\ddot {u}}r} ~0<x<f\\-F_{Az}+F&\mathrm {f{\ddot {u}}r} ~f<x<l\end{cases}}}$

The bending moment curve (around the y-axis) follows the formula:

• with Föppl clamp:

${\displaystyle M_{y}(x)=-F_{Az}x+F\langle x-f\rangle ^{1}+M\langle x-m\rangle ^{0}}$

• without Föppl clamp:

${\displaystyle M(x)={\begin{cases}-F_{Az}x&\mathrm {f{\ddot {u}}r} ~0<x<f\\-F_{Az}x+F(x-f)&\mathrm {f{\ddot {u}}r} ~f<x<m\\-F_{Az}x+F(x-f)+M&\mathrm {f{\ddot {u}}r} ~m<x<l\end{cases}}}$

See also

• Spring clip
• Heaviside function

Individual evidence

1. ^ D. Gross, W. Hauger, J. Schröder, W. Wall: Technical Mechanics 1 - Statics. 12th edition. Springer-Verlag, 2013, pp. 198ff.