Static equivalence

From static equivalence is called in the technical mechanics when one of two power systems (voltages, forces, torques) static are effectively equivalent. The determination of a static equivalent force group that is as simple as possible is called reduction . A typical example of a reduction, a common application of static equivalence, is the formation of a resultant of a uniform load or the formation of stress resultants of stresses in the cross section.

Force system

A force system , also called a force group , consists of n≥0 force variables (e.g. forces, moments, volume weights ).

Resultant of a force system

→ Main article : Resulting force

The resultant is determined by vector addition : with
${\displaystyle \mathbf {R} :=\left[\sum _{i}^{n}\mathbf {F} _{i}\right]+\left[\sum _{i}^{m}\int _{x}\mathbf {q} _{i}(x)\mathrm {d} x\right]+\left[\sum _{i}^{o}\iint _{A}{\boldsymbol {\sigma }}_{i}(x,y)\mathrm {d} x\mathrm {d} y\right]+\left[\sum _{i}^{p}\iiint _{V}{\boldsymbol {\gamma }}_{i}(x,y,z)\mathrm {d} x\mathrm {d} y\mathrm {d} z\right]\quad {\textrm {mit}}\quad {n,m,o,p}\in \mathbb {N} _{0}}$

• ${\displaystyle \mathbf {R} }$ the resultant
• ${\displaystyle \mathbf {F} _{i}}$ Single force i
• ${\displaystyle \mathbf {q} _{i}(x)}$the line load i
• ${\displaystyle {\boldsymbol {\sigma }}_{i}(x,y)}$the traction vector i
• ${\displaystyle {\boldsymbol {\gamma }}_{i}(x,y,z)}$the volume force i

Moment of a force system

→ Main article : Moment (engineering mechanics)

The moment around a point O is defined as: with
${\displaystyle \mathbf {M} _{O}:=\left[\sum _{i}^{n}{\overrightarrow {OX}}\times \mathbf {F} _{i}\right]+\left[\sum _{i}^{m}\int _{x}{\overrightarrow {OX}}(x)\times \mathbf {q} _{i}(x)\mathrm {d} x\right]+\left[\sum _{i}^{o}\iint _{A}{\overrightarrow {OX}}(x,y)\times {\boldsymbol {\sigma }}_{i}(x,y)\mathrm {d} x\mathrm {d} y\right]+\left[\sum _{i}^{p}\iiint _{V}{\overrightarrow {OX}}(x,y,z)\times {\boldsymbol {\gamma }}_{i}(x,y,z)\mathrm {d} x\mathrm {d} y\mathrm {d} z\right]\quad {\textrm {mit}}\quad {n,m,o,p}\in \mathbb {N} _{0}}$

• ${\displaystyle \mathbf {M} _{O}}$ the moment around point O
• ${\displaystyle {\overrightarrow {OX}}}$the distance from point O to point of application X of the respective force size

definition

Static equivalence between internal forces (blue) and stresses (red)

Static equivalence exists when one force system is equivalent to another force system. If two force systems are equivalent, both force systems have the same (also the same sign) resultant and the same moment. A resultant formation is generally only useful for a statically determined rigid body . In the case of a deformable body as well as a multi-body system, forces must generally not be shifted along the line of action. If two systems are statically equivalent, it means that the two force groups can be exchanged with one another. A common application is a reduction (e.g. when forming a resultant).

Two statically equivalent force groups have an equally large and equally directed (same line of action) resultant. Statically equivalent forces must act on the same statically determined rigid body. Two typical examples of static equivalence:

• A uniform load or several partial loads that are combined to form a resultant.
• A stress distribution over the cross-section that is summarized statically equivalent in the internal forces.

In the case of equivalence considerations, in the case of statically determined rigid bodies, forces can be shifted along their line of action as in an equilibrium consideration.

Difference to balance

With static equivalence , the resulting two force systems have the same orientation , but when two force systems are in equilibrium , the resultants have the same amount and the same line of action, but the resultants of the two forces are oriented in opposite directions .

Static equivalence

1. ${\displaystyle \sum _{i=1}^{n_{F}}{\vec {F}}_{i}=\sum _{j=1}^{m_{F}}{\vec {F}}_{j}}$
2. ${\displaystyle \sum _{i=1}^{n_{M}}{\vec {M}}_{i}=\sum _{j=1}^{m_{M}}{\vec {M}}_{j}}$

balance

1. ${\displaystyle \sum _{i=1}^{n_{F}}{\vec {F}}_{i}+\sum _{j=1}^{m_{F}}{\vec {F}}_{j}=0}$
2. ${\displaystyle \sum _{i=1}^{n_{M}}{\vec {M}}_{i}+\sum _{j=1}^{m_{M}}{\vec {M}}_{j}=0}$

Further meaning

The term static equivalency is also used to represent a dynamic process quasi-statically equivalent.

Individual evidence

1. ↑ ^{a b c} Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and statics . Springer-Verlag, 2015, ISBN 978-3-658-10046-9 , II. 6 Equivalence and reduction of force groups , p. 85 ff ., doi : 10.1007 / 978-3-658-10047-6 ( preview in Google book search [accessed on December 5, 2019]).
2. ↑ Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and static . Springer-Verlag, 2015, ISBN 978-3-658-10046-9 ( preview in Google book search [accessed December 7, 2019]).
3. ↑ Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and static . Springer-Verlag, 2015, ISBN 978-3-658-10046-9 ( preview in Google book search [accessed December 7, 2019]).
4. ↑ Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and static . Springer-Verlag, 2015, ISBN 978-3-658-10046-9 ( preview in Google book search [accessed December 7, 2019]).
5. ↑ Manfred Braun: Technische Mechanik I - winter semester 2003/2004. In: stephanholzmann.de. 2003, accessed December 8, 2019 . 
6. ^ Hans H. Müller-Slany: Stereo-Statik . In: Tasks and solution methodology Technical Mechanics: Using strategy to systematically develop solutions . Springer Fachmedien Wiesbaden, Wiesbaden 2018, ISBN 978-3-658-22419-6 , pp. 10, 11 , doi : 10.1007 / 978-3-658-22420-2_2 .
7. ↑ Gerhard Silber, Florian Steinwender: Component calculation and optimization with the FEM . 2005, ISBN 978-3-519-00425-7 , doi : 10.1007 / 978-3-322-80048-0 .
8. ↑ Shear force bending of prismatic beams . In: Introduction to technical mechanics: strength theory (=  Springer textbook ). Springer, Berlin / Heidelberg 2008, ISBN 978-3-540-37890-7 , 5.2 Beams with thin-walled open cross-sections , p. 129 , doi : 10.1007 / 978-3-540-37892-1_6 .
9. ↑ spatial statics . In: Introduction to technical mechanics: statics (=  Springer textbook ). Springer, Berlin / Heidelberg 2005, ISBN 978-3-540-23194-3 , pp. 83-90 , doi : 10.1007 / 3-540-26939-8_7 .
10. ↑ D. Ringer, D. Harries: Material flow modeling and CO2 neutrality - a contradiction? In: Chemical Engineer Technology . tape 80 , no. 9 , September 1, 2008, ISSN  1522-2640 , p. 1386-1387 , doi : 10.1002 / cite.200750560 .