Euler-Bernoulli beam theory concerns itself with the bending of beams (e.g. the metal beams used in large buildings), subject to certain simplifying assumptions, which are generally valid for beams that are narrow, i.e. longitudinally much larger than transversely.
Consider a beam of length , placed upright on the plane, above the origin. If we pull the top of this beam in the postive -direction, we assume that it bends uniformly, i.e. with constant radius of curvature . We also assume that the bending is shear-free: if we treat the beam as a bundle of elastic strings, then there is no friction between them.
The central string has its length unchanged (i.e. still ), while an arbitrary non-central string is extended or compressed to . The Cauchy strain tensor element is then:
Because the bending is uniform, the central string is an arc with radius and central angle , where . The non-central string has , where is geometrically shown to be , with being the -coordinate of that string at the beam’s base. So:
Where is the elastic modulus of the material. By Hooke’s inverse law, the other nonzero strain components are as follows, where is Poisson’s ratio:
For completeness, we turn the strain tensor into a full displacement field :
By integrating the above strains , we get the components of :
Where , and are integration constants, which we find by demanding that the off-diagonal strains are zero. Starting with :
Here, only may depend on , and only may depend on . This equation thus tell us:
Where is an unknown integration constant. Moving on to :
Only may contain , so its -derivative must be a constant, so . Therefore:
Where is an unknown integration constant. Finally, we put everything in :
Only the first term contains , so to satisfy this equation, we must set . The remaining terms then tell us that . Therefore:
Inserting this into the components , and then yields the full displacement field.
In any case, the beam experiences a bending torque with an -component given by:
Where is the cross-section. Th above integral is known as the area moment, and is typically abbreviated by . This brings us to the Euler-Bernoulli law:
The product is called the flexural rigidity, i.e. the beam’s “stiffness”. For a small deformation, i.e. a large radius of curvature , the law can be approximated by:
A beam that is very thin in the transverse directions ( and in this case), can be approximated as a single string or rod . Each infinitesimal piece of the rod exerts forces and on the next piece, and is feels external forces-per-length and , e.g. gravity. In order to have equilibrium, the total force must be zero:
Rearranging these relations yields these equations for the internal forces and :
Meanwhile, the rod also feels a torque with -component , where equilibrium entails:
This can be rearranged to get a differential equation for , namely:
If and are small, the last term can be dropped. These equations are widely applicable, but there is one especially important application, so much so that it is usually what is meant by “Euler-Bernoulli law”: the shape of a laterally loaded rod.
Consider a beam along the -axis, carrying a lateral load , e.g. its own weight or more. Assuming there is no other load and , the above equations become:
Which we can simply substitute into each other, eventually leading to:
This is often referred to as the Euler-Bernoulli law as well. Typically the flexural rigidity is a constant in , in which case we can reduce the equation to:
Which is clearly solved by a fourth-order polynomial, given some boundary conditions.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.