Deflection

below: a bending line  (blue), whose distance at a point x of the straight line (black) ₁ w local deflection ₁ is

As deflection of elongated items such as bars or rods , the offset between loaded and unloaded position is referred to, the at bending strain transverse to the longitudinal axis arises.

The deflection can be calculated with linear-elastic deformation with the help of the beam theory . The deflection is i. d. Usually the offset is indicated, which is shown in the determined bending line at one point . ${\displaystyle w_{1}}$ ${\displaystyle w(x)}$ ${\displaystyle x_{1}}$

Deflection of beams

The first theory of bending comes from Galileo  (1564–1642). It was further expanded by v. a. by Hooke's law  (1678) and in the 17th and 18th centuries by research by Jakob I Bernoulli , Leonhard Euler and Claude Navier .

Assuming that y and z are the main axes of inertia (y horizontal to the rear and z vertical) and that the curvature is in the y direction, i.e. H. the derivation of the angle of inclination w ' in the vertical xz image plane at the point x can be calculated as follows:  ${\displaystyle \kappa _{y}(x)}$ 

${\displaystyle \kappa _{y}(x)=-{\frac {\frac {\mathrm {d} ^{2}w(x)}{\mathrm {d} x^{2}}}{\left(1+\left({\frac {\mathrm {d} w(x)}{\mathrm {d} x}}\right)^{2}\right)^{1{,}5}}}\approx -{\frac {\mathrm {d} ^{2}w(x)}{\mathrm {d} x^{2}}}=-{w}''(x)}$,

applies:

${\displaystyle {\begin{aligned}w''(x)&={w^{b}}''(x)+{w^{e}}''(x)+{w^{s}}''(x)\\&=-\kappa _{y}^{b}(x)-\kappa _{y}^{e}(x)+{\frac {\mathrm {d} \gamma (x)}{\mathrm {d} x}}\end{aligned}}}$ 

With

• Curvature due to bending (assuming beam theory) ${\displaystyle \kappa _{y}^{b}(x)={\frac {M_{y}(x)}{EI_{yy}(x)}}}$
  •  Bending moment M _y transverse to the bar direction, at point  x
  • Bending stiffness ${\displaystyle EI_{yy}(x)=E\cdot I_{yy}(x)}$
    • Modulus of elasticity  E (a material parameter ) (in the inelastic (e.g. concrete ) or non-linear area (e.g. elastomer bearings ) this must be replaced with a suitable secant modulus )
    • Area moment of inertia  I of the cross-section of the beam (a purely geometric property)
• impressed curvature (e.g. due to temperature difference) ${\displaystyle \kappa _{y}^{e}(x)}$
• Shear deformation due to lateral force ${\displaystyle \gamma (x)={\frac {V(x)}{GA(x)}}}$ ${\displaystyle V}$
  • Shear stiffness ${\displaystyle GA(x)}$
    • Shear modulus ${\displaystyle G}$
    • Beam cross-sectional area in the yz plane. ${\displaystyle A}$

For the bending line of a sufficiently elastic, slender component with a constant cross-section, an often used approximation formula for the curvature for small helix angles w'≈0 under exclusive moment load is : ${\displaystyle V=0\Rightarrow \gamma =0}$

${\displaystyle \kappa _{y}(x)=-w''(x)\approx {\frac {M_{y}(x)}{E\cdot I_{y}}}}$

The deflection w actually sought  is obtained by integrating the curvature twice , taking into account the boundary and transition conditions (including: no deflection at the bearing points , i.e. ): ${\displaystyle w(x=0)=w(x=L)=0}$

${\displaystyle {\begin{aligned}\Rightarrow w(x)&=-\int \int \kappa _{y}(x)\,\mathrm {dx} \\&\approx -\int \int {\frac {M_{y}(x)}{E\cdot I_{y}}}\,\mathrm {dx} \end{aligned}}}$

Examples

Beam under medium load

1st example

If the force  F acts centrally (i.e. at half the bar length ) on a girder with constant cross-sectional properties on two columns, the bending moment and thus also the bar curvature is greatest in the bar center (explanation here ): ${\displaystyle {\frac {l}{2}}}$

${\displaystyle w_{\mathrm {max} }=w\left(x={\frac {l}{2}}\right)}$

For , neglecting the shear deformations (GA = ∞): ${\displaystyle 0\leq x\leq {\frac {l}{2}}}$

${\displaystyle M_{y}(x)={\frac {F}{2}}\cdot x}$

${\displaystyle \Rightarrow w''(x)=-{\frac {{\frac {F}{2}}\cdot x}{EI}}}$

with consideration of the boundary condition and the transition condition follows : ${\displaystyle w(x=0)=0}$ ${\displaystyle w'(x={\frac {l}{2}})=0}$

${\displaystyle w(x)=-{\frac {F\cdot x^{3}}{12EI}}+{\frac {F\cdot l^{2}\cdot x}{16EI}}}$

and thus:

${\displaystyle w_{\mathrm {max} }={\frac {F\cdot l^{3}}{48EI}}}$

2nd example

If a constant uniform load ( in N / m) acts on a beam on two columns with constant cross-sectional properties, the following applies, ignoring the shear deformations (GA = ∞): ${\displaystyle q_{0}}$

${\displaystyle w(x)={\frac {q_{0}}{12\cdot EI}}\left(-l\cdot x^{3}+{\frac {x^{4}}{2}}+{\frac {l^{3}\cdot x}{2}}\right)}$

This gives:

${\displaystyle w_{\mathrm {max} }=w\left(x={\frac {l}{2}}\right)={\frac {5\cdot l^{4}\cdot q_{0}}{384\cdot EI}}}$

Note:
In the case of a line load , the output equation is the 4th derivative of the bending line: ${\displaystyle q(x)}$

${\displaystyle w''''(x)={\frac {q(x)}{EI}}}$

This (with ) was integrated four times, whereby after the second integration the interrelationship between the bending line and the bending moment curve was found: ${\displaystyle q(x)=q_{0}}$

${\displaystyle w''(x)=-{\frac {M(x)}{EI}}={\frac {q_{0}\cdot x}{2\cdot EI}}\cdot (x-l)}$

Deflection of circular surfaces

→ Main article : Bending line #circular membrane

If the object extends over a large area, the calculation becomes quite complicated, but it can also be estimated for circular areas - e.g. for membranes (e.g. loudspeakers) or large lenses (e.g. telescope lenses ).

If the membrane has only a slight thickness  d , the bending moments follow a radial or tangential differential equation . However, the bending line of the circular membrane requires a composite differential formula, which is approximated for a transverse force  Q :

${\displaystyle {\frac {\mathrm {d} ^{3}w}{\mathrm {d} r^{3}}}+{\frac {1}{r}}{\frac {\mathrm {d} ^{2}w}{\mathrm {d} r^{2}}}-{\frac {1}{r^{2}}}{\frac {\mathrm {d} w}{\mathrm {d} r}}={\frac {Q}{D}}}$

With

• Moment of resistance ${\displaystyle D={\frac {E\cdot d^{3}}{12\cdot (1-\nu ^{2})}}}$
  • Poisson's number  ν of the material.

More complex cases

As long as an object is clearly reproducible and homogeneous , orthotropic and linearly elastic on a plane with cross-sectional properties / plate-producing properties, analytical mechanics also offers possible solutions for other regular shapes ( Airy's stress function ). Even cases with different materials can be approximately solved if their connection points are clearly defined mechanically, e.g. B. with an axial arrangement.

However, more complex forms are not strictly predictable. They are often examined by bending tests in the laboratory or mathematically and physically by breaking them down into network-like parts ( mainly finite element methods). For concrete there are sufficiently precise assumptions for building practice to be able to consider it as a smeared homogeneous material in the non-cracked area (which contains micro-cracks , but no macro-cracks).

literature

• Heinz Parkus : Mechanics of Solid Bodies , 2nd edition. Springer-Verlag, Vienna 1966, ISBN 3-211-80777-2
• Th. Dorfmüller, W. Hering, K. Stierstadt: Ludwig Bergmann - Clemens Schaefer textbook of experimental physics. Volume 1: Mechanics, Relativity, Heat. 11., rework. Edition, De Gruyter, Berlin 1998, ISBN 3-11-012870-5 .
• H. Mang, G Hofstetter: Strength theory. Springer Verlag, Vienna New York 2008 (3rd edition), ISBN 978-3-211-72453-8 , p. 176; 249.
• Karl-Eugen Kurrer : History of Structural Analysis. In Search of Balance , Ernst and Son, Berlin 2016, ISBN 978-3-433-03134-6 .

See also

• Tensile structure

Individual evidence

1. ↑ ^{a b} H. Mang, G Hofstetter: strength theory. Springer Verlag, Vienna New York 2008 (3rd edition), ISBN 978-3-211-72453-8 , p. 176; 249
2. ^ Pichler, Bernhard. Eberhardsteiner, Josef: Structural Analysis VO - LVA no.202.065 . Grafisches Zentrum at the Technical University of Vienna , TU Verlag ( 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. Vienna 2016 ISBN 9783903024175 Chapter 2.7.1 Transverse components and 10.2 Selected load links for the transverse components 
3. ↑ Tobias Renno: www.statik-lernen.de. Retrieved August 23, 2017 .