Bending moment

circular bending of a rod as a result of a constant bending moment over its length

A moment that can bend a slim ( rod , beam , shaft , etc.) or thin component ( plate , etc.) is referred to as a bending moment . ${\ displaystyle M}$

Bending moment in beam theory

Cantilever : tensile and compressive stress in a cross-section near the clamping point
(cut out for illustration)
when subjected to a bending moment (generated by force F at the free end)

The behavior of a slim component or a beam under load is the subject of beam theory . In particular, its behavior under a bending moment that loads it is examined with the aid of strength and elasticity . Instead of the beam theory, we therefore often speak of the bending theory of the beam , or in a narrower sense .

With the help of the theoretical individual disciplines of strength theory and elasticity theory , the bending stresses resulting from the loading bending moment inside the beam and the external elastic deformation (e.g. deflection ) of the beam are calculated and compared with the respective permissible values. The bending stresses should be smaller than the material values ​​permissible for elastic deformation ( proof of strength against plastic deformation or breakage). In some applications there is an additional limitation in the form of a permissible (elastic) deflection. This should not be exceeded by the calculated value.

The total bending stress in a cross-sectional area of the beam is proportional to the bending moment at this point. In the cross-section, it runs from maximum pressure on the inner edge (concave bend) via zero in the neutral zone to maximum tensile stress on the outer edge (convex bend). The proof of strength is i. d. Usually carried out with the maximum tensile stress (the compressive stress that can be tolerated by a beam material is usually the greater).

The bending of the beam is represented by its curvature , which is also proportional to the bending moment acting there at each cross-sectional point. To make a statement about z. B. a permissible deflection is used by the bending line determined from the curvature that is variable over the length of the bar .

Examples of bending moment curves on the beam

Clamped beam ( cantilever beam ) with a force P at the free end

Cantilever beam, single force at the free end

A cantilever beam clamped on one side is loaded by a force at the free end at a distance (see adjacent figure). The bending moment curve is ${\ displaystyle L}$${\ displaystyle P}$

${\ displaystyle M (x) = P \ cdot (Lx)}$.

At the initiation point ( ) of the force, it is zero. It rises linearly to its maximum value up to the clamping point ( ) . ${\ displaystyle x = L}$${\ displaystyle x = 0}$${\ displaystyle M = P \ cdot L}$

Beams supported at the ends, individual force in between

Bending moment curve M (x) over bars on two bearings, single force F: max. Bending. at the place of F (e.g. at l / 2)

To calculate the internal moments, the component is mentally cut through at the point of interest , and those moments are considered which act on a section at its intersection . The bending moment at a point is therefore the sum of all torques that are caused by forces on one side of the interface . ${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$

In the beam with a single load supported at its ends (see adjacent figure), the left-hand section is subject to a clockwise torque (briefly called a moment in technical mechanics ), which can be described with the help of the contact force F L on the left-hand bearing . The torque increases linearly from zero on the support to the maximum value at the point of load F. On the right of this, a counterclockwise torque that increases linearly from zero to the same maximum value on the right support comes from the load F, so that the sum of moments from Maximum value at the load point decreases linearly to zero at the right end.

${\ displaystyle M (x) = {\ begin {cases} {\ frac {F} {2}} \ cdot x & {\ text {(left of center)}} x <{\ frac {l} {2}} \\ {\ frac {F} {2}} \ cdot (lx) & {\ text {(right of center)}} x> {\ frac {l} {2}} \ end {cases}}}$

Special case of central load: The at maximum bending moment has the value ${\ displaystyle x = l / 2}$

${\ displaystyle M _ {\ mathrm {max}} = {\ frac {F \ cdot l} {4}}}$

Bending moment and bending line

Course of a bending moment on a beam with a central force F, shown here as a point load P, with the maximum bending moment M at l / 2 including the transverse force course Q and the bending line w

The elastic deformation caused by the bending moment load is described by the bending line . For a bar of constant cross-section, the following approximation equation applies to its curvature : ${\ displaystyle w (x)}$${\ displaystyle w '' (x)}$

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

With

• the curvature (variable x in the direction of the bar)${\ displaystyle w '' (x)}$
• the modulus of elasticity (a material property )${\ displaystyle E}$
• the axial geometrical moment of inertia (a geometric quantity of the constant cross-section of the beam; index y : bending around the y-axis perpendicular to the x-axis)${\ displaystyle I_ {y} = constant}$

The curvature is proportional to the bending moment , which z. B. can be seen in the bending line shown on the left : bending moment u, curvature in the middle of the beam maximum and zero at the ends (radius of curvature minimal or infinitely large = straight beam end) ${\ displaystyle w ''}$${\ displaystyle M_ {y}}$${\ displaystyle w (x)}$

The deflection of the bending line is determined by integrating the curve twice . ${\ displaystyle w (x)}$${\ displaystyle w '' (x)}$

Bending moment and bending stress

The bending stresses to be determined for the strength verification in a beam cross-section are proportional to the bending moment acting there , as indicated in the following approximation equation for a beam with a constant cross-section: ${\ displaystyle \ sigma _ {x} (x, z)}$${\ displaystyle M_ {y} (x)}$

${\ displaystyle \ sigma (x, z) = {\ frac {M_ {y} (x)} {I_ {y}}} \ cdot z}$     (Variable in the direction of the bar , variable in the direction of the bar height).${\ displaystyle x}$${\ displaystyle z}$

The proportionality with the distance from the neutral beam layer indicates that the bending stress is greatest in the edge layers. The bending stress prevailing there is: ${\ displaystyle z}$

${\ displaystyle \ sigma _ {\ mathrm {max}} (x) \, = {\ frac {M_ {y} (x)} {W_ {y}}}}$   with       ( resistance moment in the beam cross-section to bending about the y-axis).${\ displaystyle W_ {y} = {\ frac {I_ {y}} {z _ {\ text {Rand}}}}}$

Individual evidence

1. So-called "pure bend" (see here ), which rarely occurs. Most of the time, there is a “transverse force bending”: a force multiplied by a part of the length of the beam acts across the beam as a lever arm.
2. The sign is ignored. Compressive and tensile stress are both the result of a bending moment.
3. Alfred Böge (Ed.): Manual mechanical engineering: Basics and applications of mechanical engineering . 20th edition. Springer DE, 2011 ( limited preview in the Google book search).
4. The observation leading from right to left leads to the same result with the help of the right reaction force F R via a left-turning moment.