Influence line

Influence lines and on the bending moment or on the shear force at the interface of the single-span girder under a right-left-shifting load at the location ${\displaystyle \eta _{M}}$ ${\displaystyle \eta _{Q}}$ ${\displaystyle M}$ ${\displaystyle Q}$ ${\displaystyle s}$ ${\displaystyle F_{z}}$ ${\displaystyle b}$

An influence line is static , in particular in the structural analysis , a mathematically by the influence function recordable (see below) curve. The line of influence of a force or displacement variable is the line that clearly shows the influence of a load on a structure (force, moment, angle change, relative displacement) acting at the variable location on a static state variable (cutting reaction, position reaction, displacement, twist) occurring at a fixed location , normalized form shows. It serves (among other things) in the case of several possible load cases for the efficient determination of the load that leads to a maximum (or minimum) internal size ( N , V , M ) for a certain position x _{measurement point.} An influence line does not only state whether a load is favorable ( relieving) or unfavorable (burdensome), but also how great the quantitative influence of a load at a point x _{load is} on the internal size to be searched for at point x _measurement point .

With the help of influence lines z. B. show how a vehicle rolling over a bridge constantly changes or influences the reaction force of a bridge bearing or the cutting reactions at any point on the bridge beam . The illustration shows the influence lines and for the influence of a right-left sliding load on Belastungsort on the cutting reactions bending moment and shear force at the point of a stored beam statically determined. In statically determined systems, the lines of influence for force values ​​are piece-wise linear functions over the length of the bar ${\displaystyle \eta _{M}}$ ${\displaystyle \eta _{Q}}$ ${\displaystyle F_{z}}$ ${\displaystyle b}$ ${\displaystyle s}$

Comparison with state lines

Status and influence lines . Internal shear force in a single-span girder loaded with an external shear force, shown above the bs -plane. The lines b = const are state lines, the lines s = const are influence lines (multiplied by the load force) . ${\displaystyle Q}$

→ Main article : State line

Most static structures in building construction (e.g., as pedestrians, partitions) with migratory loads charged, this will be in the static calculation , however, often pragmatic reasons stationary surface load (z. B. crowds, intermediate wall charge) and individual loads (e.g. B. Tressor / bookshelf, local wind pressure according to Eurocode) shown at the most unfavorable place in terms of statics. For each stationary load case defined condition lines (also internal force lines) the course of a load (condition) along a component. State lines have only the abscissa characterizing a variable location in common with influence lines:

• Influence line: Dependency of a static quantity at a certain location (e.g. impact of two beams) on the variable location of the stressing quantity;
• Condition line: Internal force profile along a component as a result of a load that is considered to be stationary for the load case (car at a specific point).

In principle, working with influence lines does not require any knowledge that goes beyond that required for determining the component condition (internal parameters) with a given unchanged load.

With the help of an influence line, the influence on both a fixed force and a displacement can be represented.

Table 1: Comparison of state line and influence line

	State lines	Lines of influence
Place of loading:	stationary: (previously defined, constant for the LF )	variable point of application: (where the line is applied)
Place of exposure:	variable: (where the line is applied)	stationary: (certain point)

Like a state line, an influence line is a function depending on the bar axis coordinate x, but influence lines differ significantly from state lines. While condition lines only apply to a specific load case (LC), influence lines can be evaluated universally for any loads. Influence lines are usually used when you want to dimension something at a certain point. An example of this would be an assembly joint or a support reaction .

The influence function

The influence function , which the influence line formulates analytically or numerically, is equal to the state variable occurring at the fixed (intersection) location , divided by the load at the location . The variable load location and the fixed location of the examined state variable are the variable or the parameter of the influence line. Its ordinate has the dimension of the fixed state variable divided by the dimension of the portable load . ${\displaystyle \eta _{Z}(b,s=const)}$ ${\displaystyle s}$ ${\displaystyle Z(b,s=const)}$ ${\displaystyle B}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle s}$ ${\displaystyle Z}$ ${\displaystyle Z}$ ${\displaystyle B}$

Since is proportional , when deriving the influence function - instead of assuming an arbitrary load value that is ultimately divided by - a unit load can be used, which makes division superfluous. The equivalent definition applies accordingly: The influencing function is equal to the state variable occurring at the fixed location as a result of a unit load acting at the variable location . Since it is not physically correct to set a load equal to one, the load is sometimes also used in units (e.g. 1k N ). ${\displaystyle Z(b,s=const)}$ ${\displaystyle B}$ ${\displaystyle \eta _{Z}(b,s=const)={\frac {Z(b,s=const)}{B}}}$ ${\displaystyle B}$ ${\displaystyle B=1}$ ${\displaystyle \eta _{Z}(b,s)}$ ${\displaystyle s}$ ${\displaystyle Z(b,s=const)}$ ${\displaystyle b}$

Influence lines prove to be particularly advantageous when the state variable is to be determined for a group of forces. If z. B. the moment of reaction at the point under the load of the forces and with or is to be determined, is obtained with the superposition theorem . ${\displaystyle M}$ ${\displaystyle s}$ ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$ ${\displaystyle b_{1}}$ ${\displaystyle b_{2}}$ ${\displaystyle M(x=s)=F_{1}\eta _{M}(b_{1},s)+F_{2}\eta _{M}(b_{2},s)}$

A line load can also be recorded with the same influencing function. The force contribution along the path element generates the moment contribution . The moment of the total line load is equal to the integral of over the length of the line load. ${\displaystyle q(x)}$ ${\displaystyle q(x){\text{d}}x}$ ${\displaystyle {\text{d}}x}$ ${\displaystyle s}$ ${\displaystyle {\text{d}}M=q(x){\text{d}}x\cdot \eta _{M}(x,s)}$ ${\displaystyle {\text{d}}M}$

The state line explained in the introduction (delimiting the line of influence) is the graph of the function if the point of attack of the load is treated as a fixed parameter and the interface as a variable. The drawing on the right illustrates the relationship between the state and influence lines for the transverse force (thick lines) using the example of the girder shown above (designations as there). ${\displaystyle Z(b,s)}$ ${\displaystyle b}$ ${\displaystyle s}$ ${\displaystyle Q}$

calculation

Theorem of Betti

2 loads

System 1: first the left, then the right place is loaded

System 2: first the right, then the left place is loaded

2 loads with respective deflections

Reciprocity theorems are an essential basis for both the lines of influence of displacement variables and the lines of influence of internal variables in statically indeterminate systems .

In the case of purely linear elastic material behavior, deformations are not dependent on the load history, but only dependent on the current load . Since all the work done in the case of purely elastic material behavior can be recovered in terms of deformation energies, the work done must be independent of the order in which the loads were applied.

Let us consider system 1 , as shown in the picture on the right: First F is applied to position i and then P to position j, here with linear elasticity the work done by the external forces is:

${\displaystyle A_{1}^{(a)}=F^{(i)}\cdot \left({\frac {\delta _{ii}}{2}}+\delta _{ij}\right)+P^{(j)}\cdot \left({\frac {\delta _{jj}}{2}}\right)}$

In system 2 , the load P is first applied at point j and then the load F is applied to the left, as shown in the figure on the right. Assuming the linear elasticity, the work of the external forces is thus:

${\displaystyle A_{2}^{(a)}=F^{(i)}\cdot \left({\frac {\delta _{ii}}{2}}\right)+P^{(j)}\cdot \left(\delta _{ji}+{\frac {\delta _{jj}}{2}}\right)}$

Since the work done is independent of the load curve, i.e. not dependent on the route, it follows:

${\displaystyle A_{1}^{(a)}=A_{2}^{(a)}}$

hence Betti's theorem follows :

${\displaystyle F^{(i)}\cdot \delta _{ij}=P^{(j)}\cdot \delta _{ji}}$

Betti's theorem applies to any loads, i.e. F and P just do not stand for forces, as in this case, but apply to all force quantities, such as moments. It should be noted here that the path variables δ are then not only for displacements, but also for path variables of any kind, including rotations.The following formulations follow:

${\displaystyle F^{(i)}\cdot u_{ij}=F^{(j)}\cdot u_{ji}}$

${\displaystyle F^{(i)}\cdot u_{ij}=M^{(j)}\cdot \varphi _{ji}}$

${\displaystyle M^{(i)}\cdot \varphi _{ij}=M^{(j)}\cdot \varphi _{ji}}$

Note that the first index of the path size describes the location of the path size and the second index describes the cause. The path quantity δ _ij must be the energetically conjugate work partner of the force quantity P ^{(j)} .

If one specializes Betti's theorem for F = 1 and P = 1, Maxwell's theorem follows :

${\displaystyle \delta _{ij}=\delta _{ji}}$

Betti's theorem for lines of influence

If we want to know the displacement at the measurement point x _{i due} to a load at a loading point x _j , it is possible, with the help of Maxwell's theorem, instead of placing the load at the loading point x _j, to place this load on the measurement point x _i and the To determine the displacement at the loading point x _j , since this is identical to the displacement sought.

This analogy is used for the construction of lines of influence of displacements by placing an imaginary load F = 1 exclusively at the design point x _i with which the deflections of the system are determined. Since these deflections δ _ij at the point x _{i due} to a unit load at the point x _j are identical to the deflection δ _ji at the point x _{j due} to a unit load at the point x _i , it follows that the deflections of the influence line for the deflection at the point x _i .

Influence lines for internal forces in statically determined systems

Influence line for the normal force in point b

In statically determined systems , the influence lines consist of piecewise linear functions. The construction is done by applying an energetically conjugated (virtual) path variable. This is to be chosen so that the force variable does negative work on it.

annotation

1. ↑ In comparison to kinematics , path sizes in statics are very small. Often it is only a matter of displacements caused by elastic deformation of the components.

Individual evidence

1. ↑ E. Pestel: Technical Mechanics Part 1 Statics, BI University Pocketbooks Volume 205, 1969, Section Inner Forces
2. ↑ ^{a b} Karl-Heinrich Dubbel, Jörg Feldhusen (Hrsg.): Dubbel - paperback for mechanical engineering . 23rd, updated and expanded edition. Springer, Berlin / Heidelberg 2011, ISBN 978-3-642-17305-9 , doi : 10.1007 / 978-3-642-38891-0 .
3. ^ Fritz Stüssi: Baustatik I , Birkhäuser Verlag , 1971, page 86
4. ^ Lecture notes at the Ruhr University Bochum, Section 7.2 Characteristics of lines of influence
5. ^ Fritz Stüssi: Baustatik I , Birkhäuser Verlag , 1971, page 94
6. ↑ COSMiQ Die Wissenscomunity: What is an influence line? ( Memento of the original from April 26, 2017 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. : "The lines of influence of statically determined structures are always straight lines, whereas in statically indeterminate structures they resemble bending lines." 
7. ↑ ^{a b} payload (construction) # high-rise buildings
8. ↑ University of Siegen : Lecture Manuscript Structural Analysis, Chapter 7 Influence Lines (PDF) 7.2 Definition of Influence Lines ( Memento of the original from April 26, 2017 in the Internet Archive ) Info: The archive link has been inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. : "In principle, all methods that have already been learned to determine a condition line can be used to determine the influence lines." 
9. ↑ ^{a b c d e} Konstantin Meskouris and Erwin Hake: Statics of the rod support structures: Introduction to structural engineering . Springer-Verlag, 2009, ISBN 978-3-540-88993-9 , ISSN  0937-7433 , doi : 10.1007 / 978-3-540-88993-9 ( springer.com ).
10. ↑ Ruhr University Bochum - Chair for Statics and Dynamics: Influence Lines. (PDF) Ruhr University Bochum - Chair for Statics and Dynamics, November 21, 2011, pp. 79–87 , accessed on April 16, 2017 . 
11. ↑ University of Siegen - Chair of Structural Analysis: Influence Lines (calculation information). (PDF) (No longer available online.) University of Siegen - Chair of Structural Analysis, April 8, 2008, archived from the original on July 17, 2016 ; accessed on April 16, 2017 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
12. ↑ When a force is applied, the influencing function assigned to a bearing force or an internal force (an internal moment) has the dimension one (length); see. Siegfried Wetzel: The influence line, a tool for strength and deformation studies in structural engineering
13. ^ Karl-Eugen Kurrer: History of structural engineering . John Wiley & Sons, Berlin 2002, ISBN 3-433-01641-0 , pp. 462 (539 p., Limited preview in Google Book search).
14. ^ Karl-Eugen Kurrer: History of structural analysis: In search of balance . 2nd, greatly expanded edition. John Wiley & Sons, Berlin 2016, ISBN 978-3-433-03134-6 , pp. 518, 523, 948, 962, 1015, 1161 (1188 p., Limited preview in Google Book search).
15. ↑ Dietmar Gross, Thomas Seelig: fracture mechanics: with an introduction to micromechanics . 6th, expanded edition. Springer-Verlag, Berlin Heidelberg 2016, ISBN 978-3-662-46737-4 , p. 32 , doi : 10.1007 / 978-3-662-46737-4 (370 p., Springer.com ; limited preview in the Google book search).
16. ↑ ^{a b c d e f g h} Bernhard Pichler, Josef Eberhardsteiner: Structural Analysis VO - LVA-Nr . 202.065 . Ed .: E202 Institute for Mechanics of Materials and Structures - Faculty of Civil Engineering, TU Vienna - 1040 Vienna, Karlsplatz 13/202. SS 2017 edition. TU Verlag, Vienna 2017, ISBN 978-3-903024-41-0 , 12.6 Influence lines in 12 force determination in Part II statically determinate structures , S. 183–193 (516 pages, tuverlag.at ). Structural Analysis VO - LVA-Nr. 202.065 ( Memento of the original from March 13, 2016 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. * Bernhard Pichler, Josef Eberhardsteiner: Structural Analysis VO - LVA No. 202.065 . Ed .: E202 Institute for Mechanics of Materials and Structures - Faculty of Civil Engineering, TU Vienna - 1040 Vienna, Karlsplatz 13/202. SS 2017 edition. TU Verlag, Vienna 2017, ISBN 978-3-903024-41-0 , 23 reciprocity theorems as a basis for lines of influence , p.  
     423–443 (516 pages, tuverlag.at ). Structural Analysis VO - LVA-Nr. 202.065 ( Memento of the original from March 13, 2016 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice.  
17. ↑ ^{a b c d e} Dieter Dinkler: Lines of influence for path sizes . In: Basics of structural engineering . Springer, 2016, ISBN 978-3-658-13850-9 , pp. 172-178 , doi : 10.1007 / 978-3-658-13850-9_12 ( springer.com ).
18. ↑ Dieter Dinkler: Lines of influence of statically indeterminate systems . In: Basics of structural engineering . Springer, 2014, ISBN 978-3-658-13850-9 , pp. 269-278 , doi : 10.1007 / 978-3-658-05172-3_19 ( springer.com ).