Plane Beam Approximations: Euler Bernoulli Beam
- Reproduce the derivation of the equilibrium equation of the Euler Bernoulli beam.
- Describe the three basic assumptions for the equilibrium equation of the Euler Bernoulli beam.
- Identify the relationship between the load, displacement, bending moment, and shear force.
- Compute the bending moment, the shear force, the stress distribution, and the strain distribution in a beam by solving the differential equation of equilibrium.
There are three basic assumptions for an Euler Bernoulli beam that will be used to derive the equations. These are:
- Plane sections perpendicular to the neutral axis before deformation stay plane and perpendicular to the neutral axis after deformation (Figure 1).
- The deformations are small.
- The beam is linear elastic isotropic and Poisson’s ratio effects are ignored.
The beam is assumed to lie such that its long axis is aligned with . The deformation is controlled by a function which describes the deformation of the neutral axis (Figure 1). The first assumption allows the calculation of the small strain matrix at every point. The coordinates of an arbitrary point before deformation are given by:
Ignoring any effect due to Poisson’s ratio, the coordinates of the point after deformation (Figure 1) are given by:
The second assumption (small deformation) allows the approximations . Therefore, the displacement function after incorporating the small deformations assumption is given by:
We can drop the superscript since this applies to any arbitrary point. We also note that . Therefore, the displacement function has the form:
We remind the reader that is a function of only and so is its derivative. Therefore, The gradient of the displacement matrix is given by:
Therefore, the small strain matrix has the form:
In essence, the deformation assumptions result in all of the strain components being zero except for . Furthermore, we will look carefully at the variation of on a given cross section. For a given cross section of the beam we have X_1 is equal to constant. Therefore, the quantity is constant on the cross section and therefore is linear in the vertical direction (Figure 2). The value of is equal to zero at the neutral axis . The positive convention for and for lead to a positive strain below the neutral axis with the maximum positive value at the bottom fibre of the cross section. The positive convention also leads to a negative strain above the neutral axis with the maximum negative value at the top fibre of the cross section. Using the linear elastic constitutive relationship for the beam material and ignoring Poisson’s ratio lead to the same distribution for (Figure 2). Therefore, the normal stress component distribution on the cross section is given by:
The deformation assumption in beams allows considering only the deformation of the neutral axis. In particular, the deformation of the beam is controlled by the unknown function which is the vertical deformation of the neutral axis. We can also look at the forces acting on the cross section that is perpendicular to the neutral axis. These forces are termed: “Internal Forces” and can be obtained by integrating the various stress components acting on the cross section under consideration (Figure 3). For a beam in plane, only two internal forces are considered, the bending moment , and the shearing force . The positive convention for these internal forces is shown in Figure 3. can be obtained by integrating the stress component on the cross section:
The negative sign is introduced to keep the positive convention for causing compression on the top fibre of the beam. By substituting for using Equation 1, the moment can be calculated as:
where I is the moment of inertia of the beam calculated according to the formula:
It is possible to derive the equilibrium equations for the Euler Bernoulli beam by integrating the general equilibrium equations of a continuum. However, we will follow the classical method as described in many structural analysis books following the sign convention shown in Figure 3. First, two planes that are very close to each other (separated by a small distance and perpendicular to the are considered. Then, by assuming that and are smooth functions of , then the first approximation in a Taylor’s series can be utilized to describe the changes in the internal forces between those two planes (Figure 4). The loading on the beam is assumed to be transverse in the direction of upwards and given by with units of load per beam length. Assuming the sum of forces in the vertical direction (direction of ) is equal to zero yields:
Similarly, assuming the sum of moments around is equal to zero yields:
We can substitute for using Equation 2 so the equilibrium equation in terms of the displacement function and the loading becomes:
The solution for has the following form:
Four boundary conditions are needed to fully solve for and its derivatives. These boundary conditions can be given in terms of the displacement , the rotation , the bending moment or the shearing force . Note that if the beam can have a polynomial shape up to the third degree. For this reason, this formulation is sometimes termed: The cubic formulation.
Shear Stress in Euler Bernoulli Beam:
The small strain matrix obtained above for the Euler Bernoulli beam shows that the shear strains are equal to zero. However, this is an approximation that simplifies the beam model. While shear strains are directly proportional to shear stresses in linear elastic materials, for the Euler Bernoulli beam, shear stresses arise from the existence of the shearing force even though the shear strains are equal to zero. The shearing stresses associated with can be calculated by first assuming that the shear stress is constant across the beam width (along the direction). Then, by considering a small slice in the beam with width of the shear stress at the point represented by the black circle in Figure 5 can be calculated by considering the equilibrium of the shaded section shown in the figure. There are three horizontal forces acting on the shaded section. On the left we have caused by integrating on the left cross section of the shaded area. On the right we have caused by integrating on the right cross section of the shaded area. At the bottom we have caused by integrating on the bottom of the shaded area. Using Equation 3 that relates with , the three forces can be calculated as follows:
where is the width of the cross section (the width in the direction of ). Equilibrium dictates:
We will use the equality: . Therefore:
Since , , and are not functions of or , they can be brought outside the integral. Denote to be the moment of the area defined by:
A similar proof can be found in the Mechanics ebook here.
Discrepancies of The Beam Theory:
The Euler Bernoulli beam model is a simple approximation. While it is very successful because of it simplicity and practicality, it has a few discrepancies. One discrepancy is that the shear stress is not associated with any shear strains. The second discrepancy is that the calculated stresses, while they satisfy equilibrium in a holistic sense, they don’t necessarily satisfy the equilibrium equation of the continuum at a local sense. In particular, the equation of horizontal equilibrium of the continuum is:
Notice that is assumed to be zero and that we are using the undeformed coordinates since small deformations are assumed. Also, no body forces are assumed to be acting on the beam. By setting and substituting for and using Equation 3 and Equation 6 respectively, we get the following:
Assuming is constant along and since and are functions only of we get:
The left hand side is only equal zero if is constant (i.e., rectangular cross section). Otherwise, the left hand side calculation for any cross section other than a rectangle will give nonzero values, i.e., equilibrium is not necessarily satisfied!