Line load

Symbol for constant line load ,
i. d. Usually, however, a line load is not constant ( equal ) over the route

A line load is a length-related load (e.g. in Newtons per meter). The term can be found in technical mechanics , especially statics , but also in technical dynamics . ${\displaystyle q}$

The line load is generally a mathematical idealization

• a volume force (e.g. dead weight ) that is allocated to the member axis as effectively as possible, or
• a surface tension , which integrates over the area of ​​influence, becomes a load related to the length of the associated rod axis section.

A line load has the dimension force per length :

${\displaystyle q=-{\frac {\mathrm {d} V_{z}}{\mathrm {d} x}}}$

with the shear force . ${\displaystyle V_{z}}$

definition

A line load for a static system is calculated from the integral of the volume force densities over the cross section and the integral of the surface tensions over the surface: ${\displaystyle \mathbf {q} (x)}$

${\displaystyle \mathbf {q} ({x})=\int _{A}\mathbf {f} (\mathbf {x} )\,\mathrm {d} A+\int _{C}\mathbf {T} (\mathbf {n} ,\mathbf {x} )\,\mathrm {d} C}$

With

• the link load vector ${\displaystyle \mathbf {q} (x)=\left({\begin{matrix}n_{x}(x)\\q_{y}(x)\\q_{z}(x)\end{matrix}}\right)}$
  • of the component in -direction ${\displaystyle n_{x}(x)}$ ${\displaystyle x}$
  • of the component in -direction ${\displaystyle q_{y}(x)}$ ${\displaystyle y}$
  • of the component in -direction ${\displaystyle q_{z}(x)}$ ${\displaystyle z}$
• the volume force density , e.g. B. Own weights ${\displaystyle \mathbf {f} (\mathbf {x} )}$ ${\displaystyle \mathbf {f} (\mathbf {x} )=\rho (\mathbf {x} )\,\mathbf {g} (\mathbf {x} )}$
  • the density ${\displaystyle \rho (\mathbf {x} )}$
  • the gravitational field ${\displaystyle \mathbf {g} (\mathbf {x} )}$
• the cross-sectional area ${\displaystyle A}$
• the surface tension , e.g. B. Surface pressure due to a contact force with another continuum ${\displaystyle \mathbf {T} (\mathbf {n} ,\mathbf {x} )}$
  • the normal vector ${\displaystyle \mathbf {n} }$
• the circumference of the cross-section. ${\displaystyle C}$

General line loads

Line loads

various loads

A general line load is any function , for example a Fourier series . A line load as a Fourier series can replace individual loads or bending moments with any precision. ${\displaystyle q(x)}$

A triangular load is a line load that tends towards a value of at one end and increases with a constant (possibly negative) gradient to the other end. ${\displaystyle q(x)=0}$

A trapezoidal load is a line load, applies when: . They can be put together from a constant line load and a triangular load. ${\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} x}}={\textrm {const}}}$

Constant line load

A uniform line load is a line load which has a constant value over the respective member axis area:

${\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} x}}=0}$

Load model 71

A uniform load sometimes represents an unrealistic load pattern, but still serves a realistic dimensioning . An example of this is the load model 71 in railways , in which the wheel loads , which represent concentrated loads to a good approximation , are modeled as uniform loads . With this load model, the load envelopes of all common loads with the associated wheel spacings are mapped on the safe side as a good approximation .

Superposition

→ Main article : Superposition (physics)

q _ges = q ₁ ⊕q ₂ ⊕q ₃ ⊕… ⊕q _n

therein bedeutet means: "to be superimposed with".

In the theory of the first order or for linear problems it follows:

${\displaystyle \Rightarrow }$q _ges = ∑q _i

Bending theory

→ Main articles : bending line  and beam theory

From the equilibrium conditions with reference to the undeformed position, i.e. in the first order theory, it follows:

• ${\displaystyle {\frac {\mathrm {d} V_{z}(x)}{\mathrm {d} x}}=-q_{z}(x)}$
• ${\displaystyle {\frac {\mathrm {d} M_{y}(x)}{\mathrm {d} x}}=V_{z}(x)+m_{y}(x)}$
  • ${\displaystyle M_{y}(x)=\int \sigma _{xx}\cdot z\,\mathrm {d} A}$[kN * m] ... bending moment (stress resultant)
  • m (x) [kN * m / m] ... external moment per unit of length (moment load per length)

See also

• Meter load
• Weight per meter
• Bernoulli's assumptions
• Single span
• Structural analysis
• Area load

Individual evidence

1. ↑ In other words , a system in which in a Galilean reference system there are no annoyance forces.
2. ↑ ^{a b} 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. SS 2016 edition. TU Verlag, Vienna 2016, ISBN 978-3-903024-17-5 , angle of rotation method (520 pages, online - first edition: 2012). online ( 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.