# Beam theory

The beam theory describes the behavior of beams under load . It is a branch of technical mechanics . In particular, the elastic bending of a beam is investigated with the help of strength theory and elasticity theory , which is why one speaks of the bending theory of the beam .

It is developed and applied in the engineering sciences civil engineering and mechanical engineering .

In addition to the bending moment , the load variables are also longitudinal and transverse forces and torsional moments . The bend is also dependent on the geometry of the beam ( cross-section , possibly variable over the length) and its mounting as well as the elasticity of the beam material. Strength values ​​of the material determine the transition to plastic bending and bending fracture .

The bar theory has gradually been refined over time. The bending process was modeled better and better, but the handling of the theory became more complex. In most applications, sufficiently accurate results are calculated with the classical bending theory ( first order theory ).

## Main features

### Approximation steps

Generally one differentiates

First order beam theory
A beam element is considered on the undeformed beam, forces and moments are balanced. In most cases the result agrees reasonably with reality. (In general, if the buckling load is below 10% of the ideal buckling compressive load and the maximum angle of rotation does not exceed 0.1, and the pre-deformations are negligible.)
linearized beam theory of the second order
A beam element is considered on the deformed beam and the mathematical model is linearized. It is used for stability problems and for large deflections at angles of inclination of up to approx. 20 °.
Third order beam theory
A beam element is considered on the deformed beam and the mathematical model is not linearized. It is required in special cases, with very large deflections and angles of inclination over approx. 20 °.

In the second-order beam theory, non-linear terms can also be taken into account, depending on the literature, so the boundary between second-order theory and third-order theory is fluid.

### Classic assumptions: the Bernoulli assumptions

The content of the Bernoulli assumptions is:

1. The beam is slim : its length is much greater than its cross-sectional dimensions .
2. Beam cross-sections that were perpendicular to the beam axis before the deformation are also perpendicular to the deformed beam axis after the deformation.
3. Cross-sections remain even after deformation.
4. The bending deformations are small compared to the length of the beam (maximum in the size of the cross-sectional dimensions).
5. The beam is made of isotropic material and follows Hooke's law .

When these requirements are met (approximately), one speaks of an Euler-Bernoulli bar . However, these are model assumptions that are only more or less precisely fulfilled in real bars. In the real world, there is no bar that exactly matches this model.

The assumptions 2. u. 3. step i. A. never on loaded beams, but if the assumptions / consequences 2. u. 3. are permitted in an approximation, is z. E.g. a beam, which is dealt with under the heading Timoshenko beam .

If the load is only in the longitudinal direction, a bar can fail in the first order bar theory due to a strength criterion (due to normal force and bending); In the second order bar theory, this strength criterion must be fulfilled in the deformed position and it also enables a statement to be made about the risk of stability failure due to lateral buckling ( buckling bar ). Furthermore, one generally has to rule out failure due to shear force with Bernoulli beams.

Furthermore, beams often have constant cross-sectional properties (constant cross-section, modulus of elasticity, ...) over their entire length, as these are often easier to handle in terms of production technology and arithmetic.

### First order theory: statics

The classical theory essentially coincides with the theory of the first order, whereby equilibrium conditions are used on the cross-sectional areas of the undeformed beam, whose flatness is assumed by the theory.

#### Static determinacy

Statically determined beam

With statically determined supported beams, the support forces and internal forces can be determined from the equilibrium conditions. In the case of statically overdetermined beams, in addition to the equilibrium conditions, compatibility conditions must also be met in order to be able to determine the support forces and internal forces. In the simplest case, a bar is calculated using the bending line equation , a linear inhomogeneous differential equation. It creates a connection between the deflection (in the direction) and the transverse loads (line load , individual load across the beam, individual moment, ...) as a function of the coordinate along the beam axis. ${\ displaystyle w}$${\ displaystyle z}$${\ displaystyle q}$${\ displaystyle x}$

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

#### Bending stiffness

The bending stiffness indicates how large the bending moment is in relation to the curvature. For homogeneous cross-sections, it results from the product of the elasticity module of the material and the geometrical moment of inertia of the given cross-section. The latter is calculated as ${\ displaystyle EI}$ ${\ displaystyle E}$ ${\ displaystyle I}$

${\ displaystyle I_ {y} = \ int z ^ {2} {\ rm {d}} A = \ iint z ^ {2} {\ rm {d}} y \, {\ rm {d}} z \ quad}$where and are the orthogonal coordinates measured away from the center of gravity.${\ displaystyle y}$${\ displaystyle z}$

For a bar with a rectangular cross-section (in - or - direction) is ${\ displaystyle b \ cdot h}$${\ displaystyle y}$${\ displaystyle z}$

${\ displaystyle I_ {y} (x) = \ int _ {- h (x) / 2} ^ {h (x) / 2} \ int _ {- b (x) / 2} ^ {b (x) / 2} z ^ {2} {\ rm {d}} y \, {\ rm {d}} z = {\ left (h (x) \ right) ^ {3} \ cdot b (x) \ over 12}}$.

Boundary and transition conditions result from the type of support and consist of kinematic boundary conditions and dynamic boundary conditions (relating to forces and moments).

What is relevant for the dynamic boundary conditions is the relationship between the deflection and the cutting loads, namely

${\ displaystyle M (x) = - EI (x) \, w '' (x)}$

Shear force:

${\ displaystyle Q (x) = - (EI (x) \, w '' (x)) '}$

#### Bending stress

The bending moment is made up of bending stresses; these are stresses acting in the axial direction with a distribution of normal stresses that varies across the bar:

In the simplest case, the beam theory is based on Bernoulli's theory , which assumes that the cross-sections remain planar, in combination with a linear elastic material behavior. This simplification leads to the formula:

${\ displaystyle \ sigma _ {B} (x, y, z) = - {\ frac {M_ {z} (x) \ cdot I_ {y} (x) + M_ {y} (x) \ cdot I_ { yz} (x)} {I_ {y} (x) \ cdot I_ {z} (x) - (I_ {yz} (x)) ^ {2}}} \ cdot y + {\ frac {M_ {y} (x) \ cdot I_ {z} (x) + M_ {z} (x) \ cdot I_ {yz} (x)} {I_ {y} (x) \ cdot I_ {z} (x) - (I_ {yz} (x)) ^ {2}}} \ cdot z}$

If the deviation moment I yz is equal to zero, the stress component due to bending follows:

${\ displaystyle \ sigma _ {B} (x, y, z) = {\ frac {M_ {y} (x)} {I_ {y} (x)}} z - {\ frac {M_ {z} ( x)} {I_ {z} (x)}} y}$

This is the geometrical moment of inertia of the cross-section around the axis around which the bending moment rotates. The characteristic value at the maximum (at the outermost fiber of the cross-section) is also called the section modulus . A well-known result follows from this: the carrying capacity of a beam is proportional to . ${\ displaystyle I}$${\ displaystyle I / z}$${\ displaystyle z}$ ${\ displaystyle W}$${\ displaystyle I / h \ propto b \ cdot h ^ {2}}$

Bending (greatly exaggerated) of an evenly loaded beam for different support positions; blue: storage in the Bessel points

In the case of asymmetrical cross-sections, the coordinate system must be rotated in the direction of the main axes of inertia so that the bending can be calculated separately in both directions. Example: if an L-profile is loaded from above, it generally also bends in direct proportion to the side. A beam bends only in the direction of any of the main axes of inertia, exclusively in the direction of the load.

How much a beam bends also depends very much on the position of the supports; with uniform loading = const, the Bessel points are obtained from the differential equation as the optimal bearing positions . ${\ displaystyle q (x)}$

The bending stress in particular describes the force that acts on the cross section (e.g. a beam) that is loaded perpendicular to its direction of expansion.

The normal stress in the beam cross-section is:

${\ displaystyle \ sigma (x, y, z) = {\ frac {N (x)} {A (x)}} - {\ frac {M_ {z} (x) \ cdot I_ {y} (x) + M_ {y} (x) \ cdot I_ {yz} (x)} {I_ {y} (x) \ cdot I_ {z} (x) - (I_ {yz} (x)) ^ {2}} } \ cdot y + {\ frac {M_ {y} (x) \ cdot I_ {z} (x) + M_ {z} (x) \ cdot I_ {yz} (x)} {I_ {y} (x) \ cdot I_ {z} (x) - (I_ {yz} (x)) ^ {2}}} \ cdot z}$

If the moments of deviation are zero and one has a simple bend in the z-direction without a normal force:

${\ displaystyle \ sigma (x, z) \, = {\ frac {M_ {y} (x)} {I_ {y} (x)}} \ cdot z}$

If the moment M y is positive, tensile stresses for > 0 and compressive stresses for <0 occur in the case of pure bending stress caused by M y . The greatest amount of stress occurs in the case of pure bending stress caused by M y in the outermost fiber . ${\ displaystyle z}$${\ displaystyle z}$${\ displaystyle | z | _ {\ mathrm {max}}}$

The section modulus is a pure cross-sectional value and indicates the ratio of the applied moment to the associated stress in the "critical" fiber ${\ displaystyle W}$${\ displaystyle \ sigma \,}$

${\ displaystyle W_ {u} (x) = {\ frac {I_ {y} (x)} {z _ {\ mathrm {max}} (x)}} \ ,, \ quad W_ {o} (x) = {\ frac {I_ {y} (x)} {z _ {\ mathrm {min}} (x)}}}$

Describes the area moment of inertia . The following results for the maximum bending stress: ${\ displaystyle I}$

${\ displaystyle | \ sigma | _ {\ mathrm {max}} (x) = {\ frac {| M (x) |} {| W (x) | _ {\ mathrm {min}}}}}$

The greater the amount of section modulus, the smaller the amount of bending stress in the edge fiber.

Beam cutouts, bent under bending moment loads M

When a beam is bent, its longitudinal fibers on the tension side are stretched (in the front in the adjacent picture, left part) and those on the pressure side are compressed (in the back in the adjacent picture, left part). Tensile stresses arise in the stretched fibers and compressive stresses in the compressed ones. The stress curve from the outer maximum tensile stresses to the inner maximum compressive stresses is i. d. Typically non-linear, but linear distribution is a common assumption.

With a relatively small bend and no normal force, the neutral (tension-free) fiber is in the middle of the height of the beam. The tensile and compressive stresses in a cross-sectional area are of the same magnitude, provided there is no normal force.

#### Bend line of the beam

The deflection (deflection) of     the beam in its place     can be described with the following linear differential equation : ${\ displaystyle w}$${\ displaystyle x}$

${\ displaystyle w '' (x) = - {M_ {y} (x) \ over EI_ {y}} \.}$

It depends on the load caused by the bending moment     , the area moment   of inertia of the beam cross-section and the modulus   of elasticity of the beam material (index    : bending around the transverse axis ). With the first integration, the inclination of   the bending line follows   from its curvature    : ${\ displaystyle M_ {y} (x)}$  ${\ displaystyle I_ {y}}$  ${\ displaystyle E}$${\ displaystyle _ {y}}$${\ displaystyle y}$${\ displaystyle w '}$${\ displaystyle w ''}$

${\ displaystyle w '(x) = - {{\ int _ {0} ^ {x} M_ {y} (\ xi) \, \ mathrm {d} \ xi + C_ {1}} \ over {EI_ { y}}} \.}$

In the second integration, the inclination of the bending line results in its deflection : ${\ displaystyle w}$

${\ displaystyle w (x) = - {{\ int _ {0} ^ {x} (\ int _ {0} ^ {x} M_ {y} (\ xi) \, \ mathrm {d} \ xi + C_ {1}) \, \ mathrm {d} \ xi + C_ {2}} \ over {EI_ {y}}} \.}$
Beam on 2 supports, central force load (blue: bending line)${\ displaystyle P}$

In the example of a beam resting at both ends with a central point load (adjacent picture), the curve of the bending moment has a kink. In this case, integration is usually carried out separately for the left and right beam parts. The combination of the two results to form a continuously running bending line results from the fact that both their inclination and their deflection are the same for both parts. In the example there is symmetry (in the bending line and moment line). The integration z. B. the differential equation for the left half is sufficient. This half can also be seen as a cantilever  beam clamped in the middle and loaded with the force (through the support) at the other end   . ${\ displaystyle P / 2}$

For     : ${\ displaystyle x \ leq L / 2}$

${\ displaystyle M (x) = Px / 2 \,}$
${\ displaystyle w '(x) = - {Px ^ {2} / 4 + C_ {1} \ over EI_ {y}} \,}$                                          at     the slope     is zero →     ${\ displaystyle x = L / 2}$${\ displaystyle w '}$${\ displaystyle C_ {1} = - PL ^ {2} / 16 \,}$
${\ displaystyle w (x) = - {Px ^ {3} / 12-P \ cdot L ^ {2} x / 16 + C_ {2} \ over EI_ {y}} \,}$         wherein     the deflection is   equal to zero →     ${\ displaystyle x = 0}$${\ displaystyle w}$${\ displaystyle C_ {2} = 0 \,}$
${\ displaystyle w (x) = {P (L ^ {2} x / 16-x ^ {3} / 12) \ over EI_ {y}}}$, at     the deflection is the     same  ${\ displaystyle x = L / 2}$${\ displaystyle w}$${\ displaystyle {P \ cdot L ^ {3} \ over 48 \ cdot EI_ {y}} \.}$

### First order theory: dynamics

So far only the statics have been dealt with. The beam dynamics, for example to calculate beam vibrations, is based on the equation

${\ displaystyle (EI (x) \, w '' (x, t)) '' + b \, {\ dot {w}} (x, t) + m \, {\ ddot {w}} (x , t) = q (x, t)}$

The problem here depends not only on the location , but also on the time . There are two additional parameters of the beam, namely the mass distribution and the structural damping . If the component vibrates under water, it also includes the hydrodynamic mass, and a linearized form of the hydrodynamic damping can be included in, see Morison's equation . ${\ displaystyle x}$${\ displaystyle t}$${\ displaystyle m}$${\ displaystyle b}$${\ displaystyle m}$${\ displaystyle b}$

### Second order theory: buckling bar

While so far the forces and moments were approximately balanced on the undeformed component, in the case of buckling bars it is necessary to consider a beam element in the deformed state. Buckling bar calculations are based on the equation

${\ displaystyle (EI (x) \, w '' (x)) '' + (N \, w '(x))' = q (x)}$

in the simplest case with . In addition, there is the compressive force acting axially in the buckling bar , which, depending on the boundary conditions , must not exceed the buckling load so that the bar does not buckle. ${\ displaystyle q = 0}$${\ displaystyle N}$

Differential relationships

In the shear-soft beam theory Ⅱ. Order there are the following differential equations for the transverse components under the Bernoulli assumptions :

• ${\ displaystyle {\ frac {\ mathrm {d} R (x)} {\ mathrm {d} x}} = - q (x)}$
• ${\ displaystyle {\ frac {\ mathrm {d} M (x)} {\ mathrm {d} x}} = R (x) -N ^ {II} (x) \ cdot \ left [{\ frac {\ mathrm {d} w_ {v}} {\ mathrm {d} x}} + {\ frac {\ mathrm {d} w} {\ mathrm {d} x}} \ right] + m (x)}$
• ${\ displaystyle {\ frac {\ mathrm {d} \ varphi (x)} {\ mathrm {d} x}} = - \ left [{\ frac {M (x)} {E \ cdot I (x)} } + \ kappa ^ {e} (x) \ right]}$
• ${\ displaystyle {\ frac {\ mathrm {d} w (x)} {\ mathrm {d} x}} = \ varphi (x) + {\ frac {V (x)} {G {\ tilde {A} } (x)}}}$

With

• the running coordinate  along the beam axis${\ displaystyle x}$
• the modulus of elasticity ${\ displaystyle E}$
• the shear modulus  (term does not appear in the differential equations in rigid theory)${\ displaystyle G}$
• the area moment of inertia ${\ displaystyle I (x)}$
• ${\ displaystyle R (x)}$the transverse force ( applies in the first order theory )${\ displaystyle R (x) = V (x)}$
• ${\ displaystyle V (x)}$the shear force
• ${\ displaystyle N ^ {II} (x)}$the normal force according to theory theory Ⅱ. Order (in first order theory this term does not appear in the differential equation)
• ${\ displaystyle q (x)}$ the uniform load (transverse load per unit length)
• ${\ displaystyle M (x)}$the bending moment
• ${\ displaystyle m (x)}$ the insertion torque (bending load per unit length)
• ${\ displaystyle \ varphi (x)}$ the twist
• ${\ displaystyle \ kappa ^ {e} (x)}$ the impressed curvature
• ${\ displaystyle w (x)}$ load due to deflection
• ${\ displaystyle w_ {v} (x)}$ pre-deformation due to deflection
• ${\ displaystyle {\ tilde {A}} (x)}$ the shear area (term does not appear in the rigid theory).

### Third order theory

The third order theory also covers large deformations ; the simplifications of the second order theory no longer apply here.

One application where third order beam theory is necessary is e.g. B. the laying of offshore pipelines from a watercraft in great water depths, shown here only as a flat static case.

A very long pipe string hangs down from the vehicle to the sea floor, is curved like a rope, but rigid . The nonlinear differential equation is here:

${\ displaystyle EI \, \ varphi '' (s) -H \, \ sin \ varphi (s) + (ws-V) \ cos \ varphi (s) = 0}$

With

• the coordinate ( arc length along the pipeline)${\ displaystyle s}$
• the angle of inclination , which is related to the horizontal coordinate and the vertical coordinate as follows:${\ displaystyle \ varphi}$${\ displaystyle x (s)}$${\ displaystyle z (s)}$
${\ displaystyle \ sin \ varphi (s) = {\ frac {\ partial z (s)} {\ partial s}}}$
${\ displaystyle \ cos \ varphi (s) = {\ frac {\ partial x (s)} {\ partial s}}}$
• ${\ displaystyle H}$is the constant horizontal component of the cutting force (horizontal pull) along the pipeline ; H is influenced by how hard the vehicle pulls on the pipeline with its anchors and the tensioner so that it does not sag and break; the tensioner is a device made of two caterpillar chains that clamps the pipeline on board and keeps it under tensile load
• the weight per length , minus buoyancy${\ displaystyle w}$
• a mathematical parameter , which can be as small ground reaction force can imagine.${\ displaystyle V}$

## history

After previous, mainly intellectual experiments by Leonardo da Vinci , the bar theory was founded by Galileo Galilei . With the work of Claude Louis Marie Henri Navier a preliminary, classical beam theory called conclusion was reached.

"Fathers" of the classical bending theory from Leonardo da Vinci to Navier:

• Leonardo da Vinci (1452–1519) - Tensile tests on wires
• Galileo Galilei (1564–1642) - Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti alla mecanica ei movimenti locali : tensile strength of marble columns, ropes and wires (first day), considerations on the breaking strength of beams (second day)
• Edme Mariotte (1620–1684) - Linear distribution of fiber strains over cross-section, neutral fiber at half the height of the double-symmetrical beam cross-section
• Robert Hooke (1635–1703) - Proportionality between strain and tension ( Hooke's law )
• Isaac Newton (1643–1727) - balance of forces, calculus
• Gottfried Wilhelm Leibniz (1646–1716) - infinitesimal calculus, moment of resistance
• Jakob I Bernoulli (1655–1705) - assumptions that simplify the theory: a flat cross-sectional area perpendicular to the beam axis is also flat after bending and perpendicular to the beam axis
• Leonhard Euler (1707–1783) - first attempt to treat a statically indeterminate system (four-legged table), investigation of the buckling of bars (second order theory)
• Charles Augustin de Coulomb (1736–1806) - first representation of the bar, vault and earth pressure theory connected by the calculus; Structural engineering becomes "scientific subject"
• Johann Albert Eytelwein (1764–1848) - solution of statically indeterminate systems: continuous beams
• Claude Louis Marie Henri Navier (1785–1836) - his works represent the “constitutional phase of structural engineering”; In his technical bending theory he brings together the mathematical-mechanical analysis of the elastic line (Bernoulli, Euler) and the primarily engineering-oriented beam statics;
• Georg Rebhann (1824–1892) - gave formulas in 1856 for verifying bending stresses in single-symmetrical cross-sections.

## Individual evidence

1. ^ Fritz Stüssi : Structural engineering I. 4th edition. 1971, ISBN 3-7643-0374-3 , from p. 173
2. ^ Pichler, Bernhard. Eberhardsteiner, Josef: Structural Analysis VO - LVA no.202.065 . Vienna 2016, ISBN 978-3-903024-17-5 , 2.7.1 transverse components and 10.2 selected load links for transverse components ( TU Verlag [accessed on December 10, 2016]). 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.
3. Oliver Romberg, Nikolaus Hinrichs: Don't panic about mechanics! - Success and fun in the classic "loser subject" of engineering studies . In: Study without panic . 8th, revised edition. tape 4 . Vieweg + Teubner Verlag, 2011, ISBN 978-3-8348-1489-0 (349 pp., Springer.com [PDF] first edition: 1999).
4. B. Kauschinger, St. Ihlenfeldt: 6. Kinematics. (No longer available online.) Archived from the original on December 27, 2016 ; accessed on December 27, 2016 .
5. Jürgen Fröschl, Florian Achatz, Steffen Rödling, Matthias Decker: Innovative component test concept for crankshafts . In: MTZ-Motortechnische Zeitschrift . tape 71 , no. 9 . Springer, 2010, p. 614-619 ( springer.com ).
6. a b Herbert Mang , G Hofstetter: Strength theory. Ed .: Springer Verlag. (3. Edition). Vienna / New York 2008, ISBN 978-3-211-72453-8 , 6.4 "Normal stresses", p. 156 (487 pp., Springer.com ).
7. see main article: bending line
8. It is possible to integrate Heaviside functions over the entire bar
9. Due to the symmetry of the bending line, the antimetry of the twisting line follows, since there is no impressed change in angle (at this point), the angle is constant and thus the left-hand limit value is equal to the right-hand limit value , which follows from these two formulas and from this equation follows that is${\ displaystyle \ textstyle w '(x = {\ frac {l} {2}} ^ {-}) = - w' (x = {\ frac {l} {2}} ^ {+})}$${\ displaystyle \ textstyle w '(x = {\ frac {l} {2}} ^ {-})}$${\ displaystyle \ textstyle w '(x = {\ frac {l} {2}} ^ {+})}$${\ displaystyle \ textstyle w '(x = {\ frac {l} {2}} ^ {+}) = - w' (x = {\ frac {l} {2}} ^ {+})}$${\ displaystyle \ textstyle w '(x = {\ frac {l} {2}} ^ {+}) = 0}$
10. a b c d Bernhard Pichler: 202.068 structural analysis 2 . WS2013 edition. Vienna 2013, VO_06_ThIIO_Uebertragungsbeektiven ( online platform of the Vienna University of Technology ).
11. a b c Pichler, Bernhard. Eberhardsteiner, Josef: Structural Analysis VO LVA no.202.065 . Ed .: TU Verlag. SS2016 edition. TU Verlag, Vienna 2016, ISBN 978-3-903024-17-5 , linear bar theory of planar bar structures (520 pages, Grafisches Zentrum at the Technical University of Vienna [accessed on January 12, 2017]). Grafisches Zentrum at the Technical University of Vienna ( 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.
12. IFBS: 8.3 From Galileo's bending theory to sandwich theory see: The great mathematicians intervene ( Memento from March 15, 2016 in the Internet Archive )