1. Introduction and Structural Context

In structural and mechanical engineering, a beam is a slender structural member designed to support loads applied transverse (perpendicular) to its longitudinal centroidal axis. Unlike axial bar elements (which undergo tension or compression along their axes) and pinned truss elements (which transmit forces strictly along their member lines), beams undergo flexural deformation (bending). Bending is characterized by transverse displacement (deflection) and rotation of the cross-sections (slope).

To transition from 1D axial bar elements to 1D flexural beam elements, we must examine the nature of their strain fields:

$C^0$ vs. $C^1$ Continuity: The Mathematical Necessity

In Finite Element Analysis, the selection of shape functions is governed by the order of derivatives in the variational statement (weak form). A field is $C^1$ continuous if both the function itself ($v(x)$) and its first derivative (slope $\theta(x) = \frac{dv}{dx}$) are continuous across element boundaries.

If we were to use a simple linear or quadratic Lagrangian interpolation (which is only $C^0$ continuous), the slope would be discontinuous at the nodes. This discontinuity means that the second derivative $\frac{d^2v}{dx^2}$ would become a Dirac delta function (infinite) at the nodes, implying infinite strain energy and infinite bending moments, which is physically impossible.

To resolve this issue and maintain a valid finite element formulation, we must ensure that both deflection $v$ and slope $\theta$ are continuous at the nodes. Thus, we define the slope $\theta = \frac{dv}{dx}$ as an active nodal degree of freedom. Consequently, each node of a 1D beam element has 2 degrees of freedom:

For a two-node element, this yields a total of 4 degrees of freedom per element, requiring cubic polynomials for shape interpolation.

2. Kinematic Assumptions of Euler-Bernoulli Beam Theory

Euler-Bernoulli beam theory (often referred to as classical beam theory) is built upon several simplifying kinematic assumptions originally formulated by Leonhard Euler and Daniel Bernoulli. These assumptions allow us to reduce a three-dimensional elasticity problem to a simple one-dimensional governing differential equation:

3. Displacement, Strain, and Stress Fields

Let the beam be aligned along the coordinate $x$-axis, with the $y$-axis representing the vertical direction. Let $v(x)$ represent the vertical deflection of the neutral axis ($y = 0$).

According to the normality constraint, when the beam bends, a cross-section at coordinate $x$ rotates by an angle $\theta(x) = \frac{dv}{dx}$. As a result of this rotation, any point in the cross-section at a vertical distance $y$ from the neutral axis experiences a longitudinal axial displacement $u(x, y)$ given by:

The negative sign is crucial: for a positive slope (counter-clockwise rotation), points above the neutral axis ($y > 0$) move in the negative $x$-direction (compression), whereas points below the neutral axis ($y < 0$) move in the positive $x$-direction (tension).

The vertical displacement of any point in the beam is assumed to depend only on $x$:

From the displacement field, we can compute the normal strain $\epsilon_{xx}$ using the strain-displacement relation:

Using Hooke's Law, the normal bending stress $\sigma_{xx}$ is:

This demonstrates that the bending stress varies linearly through the depth of the beam, being zero at the neutral axis ($y=0$), reaching its maximum compressive value at the top fiber, and reaching its maximum tensile value at the bottom fiber.

4. Bending Moments, Shear Forces, and Governing Equations

The bending moment $M(x)$ at any section along the beam is the resultant couple of the internal normal stresses:

Substituting the bending stress expression into this integral:

The integral $\int_A y^2 \, dA$ is the second moment of area (commonly referred to as the area moment of inertia) $I$ about the neutral axis. Thus, we obtain the classical moment-curvature relation:

By applying static equilibrium to a differential beam element, we can relate the bending moment to the transverse shear force $V(x)$:

For a beam with a constant flexural rigidity $EI$:

Similarly, the transverse distributed load $q(x)$ (force per unit length, positive upwards) is the derivative of the shear force:

For constant $EI$, this yields the governing fourth-order ordinary differential equation (strong form) for beam deflection:

5. Physical Significance and Sign Conventions

To construct a consistent FEA formulation, we must define a clear sign convention. In standard structural mechanics, a sagging moment is considered positive (causing compression on the top fibers). However, in finite element analysis, it is more convenient to use a global coordinate sign convention:

The flexural rigidity $EI$ represents the beam's resistance to bending. It is the product of a material property ($E$, Young's Modulus) and a geometric property ($I$, Moment of Inertia). For a rectangular cross-section of width $b$ and height $h$, $I = \frac{bh^3}{12}$. This shows that the height of the beam has a cubic effect on bending resistance, explaining why structural members are oriented vertically (like I-beams) to maximize depth and minimize material volume.