Euler-Bernoulli Beam Equation Derivation and Assumptions

Key Takeaways

• The design of beams is generally based on bending moments. 

• The Euler-Bernoulli beam equation is derived from four segments of beam theory: kinematics, constitutive, resultants, and equilibrium.

• The Euler-Bernoulli beam equation derivation assumptions should be met completely in order to obtain accurate results. 

The design of beams is generally based on bending moments

Beam theory is invariably used for the design and analysis of buildings and bridges (as well as bones of the human body!). In problems involving slender and long beams, beam theory is used for addressing strain-stress relations. Multiple beam theories are involved with various assumptions for solving problems related to beams. The accuracy of each beam theory is different.

The Euler-Bernoulli beam theory is one of the simplest yet most useful theories. Euler-Bernoulli beam theory derivation is based on assumptions that deal with in-plane (assumption 1) and out-of-plane displacements of the beams (assumptions 2 and 3). Let’s learn more about the Euler-Bernoulli beam theory.

Defining Beams

In engineering mechanics, the beam is a structural element that is subjected to force. Beams typically support loads that act perpendicular to their longitudinal axis. Beams are designed to withstand transverse loads. The beams are commonly seen extending between one or more supports.

The design of a beam is generally based on bending moments. The support given by the beam to the load is through bending. The beam is supported in different ways. Roller supports, fixed supports, and pin supports are some examples. Usually, beams are rectangular in cross-section. However, the cross-sectional shape can be any shape. 

Types of Beams

Simple beam - In simple support beams, the end supports are free to rotate as the beam undergoes loading. These types of beams have no resistance to bending moments.

Fixed beam - The end supports of fixed beams are immovable or fixed so that there is no rotation, vertical, or horizontal movement possible.

Cantilever beam - The cantilever beam permits vertical and horizontal movement at one end, which is not fixed like the other end.

Overhanging beam - The simple beam, when it extends beyond the supports at one or both ends, forms an overhanging beam.

Terminologies Used in Beam Theory

Neutral surface - A beam is the formation of many fibers aligned longitudinally. Under the action of downward transverse loads, the fibers near the top of the beam contract, whereas the fibers near the bottom of the beam extend. The plane where the fibers of the beam do not change their length is called the neutral surface.

Axis of the beam - The axis of the beam is the interaction of the neutral surface and the longitudinal symmetry.

Deflection curve - When the beam is loaded, the neutral axis or surface deforms into a curve, which is called a deflection curve.

Plane and bending - Consider a beam on which the x-axis of the conventional coordinate system is incited with the longitudinal axis of the beam. The transverse direction of the beam aligns with the y-axis – hence the longitudinal plane of symmetry in the beam structure lies along the x-y plane. The n-y plane forms the plane of bending.

The Longitudinal Plane of Symmetry in Beams

According to elementary beam theory, only bending is possible in beam structures. Beams are assumed to have a longitudinal plane of symmetry, which refers to the cross-sectional symmetry of the longitudinal plane. Supports and loading are symmetric to the longitudinal plane. With the above-mentioned symmetry, supports, and loading, there is no way for the beam to twist other than by bending.

The Assumptions of the Euler-Bernoulli Beam Equation 

The fundamental assumptions of the Euler-Bernoulli beam equation are:

1. The beam section is infinitely rigid in its own plane. According to this assumption, there is no deformation in the plane of the cross-section.
2. The cross-section of the beam remains plane to the deformed axis of the beam.
3. The cross-section of the beam remains normal to the deformed axis of the beam.

Euler-Bernoulli Beam Equation Derivation

The Euler-Bernoulli beam equation is derived from four segments of beam theory; the combination of the equations governing the four segments formulate the Euler-Bernoulli beam equation. The four distinct segments and their equations are:

1. Kinematics  

θ is the neutral surface rotation and w is the beam's displacement.

2. Constitutive

∈ is the direct strain and E is Young’s modulus of the beam.

3. Resultants

M is the resultant moment and V is the shear resultant.

4. Equilibrium

To find the relationship between out-of-the-plane displacement w to the pressure loading p, the above equations are combined in the order given below.

1. Combine the equilibrium equations and eliminate V.
2. Replace the moment resultant M using the first equation in the resultant’s segment.
3. Eliminate σ using the constitutive relation.
4. Use kinematics, replacing ∈ to get the Euler-Bernoulli beam equation in terms of the beam’s displacement w.

The Euler-Bernoulli beam equation:

I is the area moment of inertia of the beam’s cross-section.

The Euler-Bernoulli beam equation derivation assumptions should be met completely in order to obtain accurate results. Cadence’s suite of CFD tools can help you solve beam-related problems in solid mechanics.

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