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.${\ displaystyle C}$

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. ${\ displaystyle {\ frac {\ mathrm {d} q} {\ mathrm {d} x}} = {\ textrm {const}}}$

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

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

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

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

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)