Consequences of Inextensibility of an in-plane line in Euler-Bernoulli Beam Theory

Question

The Euler-Bernoulli beam theory is based on the hypothesis that the beam deforms such that a straight line normal to the beam axis before deformation remains

1. inextensible
2. straight
3. normal to the beam axis

after deformation.

I am trying to find out the consequences of these assumption number 1 and 2 for displacement field. In the following discussion, $x_1$ is the axial direction and $x_2$, and $x_3$ are in-plane directions. Similarly, $u_1$ is the displacement along axial direction, and $u_2$, and $u_3$ are in-plane displacements. The coordinate system is placed at the centroid of the cross-section of the beam.

Assumption 1: A straight line normal to the beam axis before deformation remains inextensible after deformation.

Consider a straight line $\overline{AB}$ in the cross-section of the beam as shown in Fig. 1.. The length of the beam before deformation is given by

$$l_{AB}=\sqrt{(x_{2A} - x_{2B})^2 + (x_{3A} - x_{3B})^2}$$

where, $x_{ij}$ represent the ith-coordinate of jth point. For example, $x_{3B}$ is displacement along $x_3$ direction of point B.

Similarly, the length of line $\overline{AB}$ after deformation is given by

$$l'_{AB}=\sqrt{(x_{2A} + u_{2A} - x_{2B} - u_{2B})^2 + (x_{3A} + u_{3A} - x_{3B} - u_{3B})^2}$$

Since the length of line $\overline{AB}$ cannot change, i.e., $l_{AB} = l'_{AB}$, therefore,

$$\sqrt{(x_{2A} - x_{2B})^2 + (x_{3A} - x_{3B})^2} = \sqrt{(x_{2A} + u_{2A} - x_{2B} - u_{2B})^2 + (x_{3A} + u_{3A} - x_{3B} - u_{3B})^2}$$

which is possible if and only if $u_{2A} = u_{2B}$ and $u_{3A} = u_{3B}$.

Since, points A and B are arbitrary points on the cross-section, therefore, we can conclude that the $u_2$ and $u_3$ displacement will be same for all points on the cross-section of the beam. In other words, $u_2$ and $u_3$ are functions of $x_1$ only.

$$u_2(x_1, x_2, x_3) = u_2(x_1)$$ $$u_3(x_1, x_2, x_3) = u_3(x_1)$$

Assumption 2: A straight line normal to the beam axis before deformation remains straight after deformation.

This is possible if and only if the line translate as rigid-body along axis $x_1$ and rotate as a rigid-body about axes $x_2$ and $x_3$.

Rigid-body translation: If the line moves as a rigid-body along $x_1$, its displacement along $x_1$ is given by $\overline{u_1}(x_1)$.

Rigid-body rotation about $x_2$: If the line rotates as a rigid-body about $x_2$, the displacement of any point on the line along $x_1$ is given by $x_3 \phi_2(x_1)$, where $\phi_2(x_1)$ is the angle of rotation about $x_2$.

Rigid-body rotation about $x_3$: If the line rotates as a rigid-body about $x_3$, the displacement of any point on the line along $x_1$ is given by $-x_2 \phi_3(x_1)$, where $\phi_3(x_1)$ is the angle of rotation about $x_3$.

So, the total displacement along axis $x_1$ is given by $$u(x_1, x_2, x_3) = \overline{u_1}(x_1) + x_3 \phi_2(x_1) - x_2 \phi(x_1)$$

Owing to the reasoning mentioned above for both assumptions, are the conclusions made sensible and correct?

Answer 1

You are misunderstanding what the assumptions are.

The only lines which are assumed to remain straight and normal to the axis (but not inextensible) are those perpendicular to the plane of neutral bending of the beam.

If the beam bends in two planes, since the theory assumes small displacements and linear behaviour you can combine the deflections in each principal plane by superposition.

The deformed shape of the section changes because of Poisson's ratio for the material. Since the beam has tensile axial strain on one side of the neutral axis and compressive axial strain on the other side, it should be fairly obvious that Poisson's ratio will deform the cross section into some sort of trapezium shape. In fact the top and bottom faces end up curved, as in Figure 7.9 on page 12 here.

The real assumption of E-B beam theory is ignoring the strain energy caused by shear strain deformations (the statements about straight lines used to derive the equations are just consequences of that assumption) and therefore the deformation of the cross section is irrelevant for the theory.

E-B beam theory doesn't answer the question "what shape is the deformed cross section of the beam" because it that is irrelevant to the bending along the length of the beam if the assumptions made by the theory are valid, and for short and deep beams they are not valid.

In a beam theory that does include shear effects, the cross section can deform, it does not remain perpendicular to the neutral plane of the beam, and in fact it does not even remain plane.