Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying capacity and deflection of beams.
When external forces are applied to a beam, internal shear forces and bending moments develop causing bending and curvature. Euler–Bernoulli beam theory states that the shear force at any point on a beam is the cumulative sum of the loads applied along the length of the beam up to that point. Similarly, the bending moment at any point is the sum of the shear forces along the beam up to that point. Additionally, the theory states that the deflection at any point on the beam is the fourth integral of the applied loads up to that point, and depends on flexural rigidity. Through the use of calculus, and boundary conditions describing the beam's curvature at its supports, the theory provides a mathematical model to predict the structural behavior of beams.
Euler–Bernoulli beam theory is limited to small deflections of a beam that is subjected to lateral loads only, causing elastic bending. Beam theories were later refined in the 20th century to account for the effects of shear deformation and rotatory inertia; Euler–Bernoulli beam theory is now considered a case of Timoshenko–Ehrenfest beam theory that neglects these effects.
It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.
Additional models have been developed, such as plate theory, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.
History
Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Leonardo lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made.
Jacob Bernoulli postulated that the curvature at any point on a deflected beam is proportional to its bending moment at that point, first published in Histoire de l'Académie royale des sciences in 1705. He also formulated an approach for calculating the deflection of a cantilevered beam with a concentrated load at its free end, however he erroneously assumed a beam's curvature is due to its cross-section rotating about an axis that intersects the extreme compression fiber.
Leonhard Euler built on Jacob Bernoulli's work, and applied differential calculus to describe the behavior of elastic curves and published Methodus inveniendi lineas curvas in 1744.
Daniel Bernoulli furthered theories and formulated the differential equation of motion of a vibrating beam.
Static beam equation
The Euler–Bernoulli equation describes the relationship between the beam's deflection and the applied load:The curve describes the deflection of the beam in the direction at some position (recall that the beam is modeled as a one-dimensional object). is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of , , or other variables. is the elastic modulus and is the second moment of area of the beam's cross section. must be calculated with respect to the axis which is perpendicular to the applied loading.For an Euler–Bernoulli beam not under any axial loading this axis is called the neutral axis. Explicitly, for a beam whose axis is oriented along with a loading along , the beam's cross section is in the plane, and the relevant second moment of area is
It can be shown from equilibrium considerations that the centroid of the cross section must be at .
Often, the product (known as the flexural rigidity) is a constant, so that
This equation, describing the deflection of a uniform, static beam, is used widely in engineering practice. Tabulated expressions for the deflection for common beam configurations can be found in engineering handbooks. For more complicated situations, the deflection can be determined by solving the Euler–Bernoulli equation using techniques such as "direct integration", "Macaulay's method", "moment area method", "conjugate beam method", "the principle of virtual work", "Castigliano's method", "flexibility method", "slope deflection method", "moment distribution method", or "direct stiffness method".
Sign conventions are defined here since different conventions can be found in the literature. In this article, a right-handed coordinate system is used with the axis to the right, the axis pointing upwards, and the axis pointing into the figure. The sign of the bending moment is taken as positive when the torque vector associated with the bending moment on the right hand side of the section is in the positive direction, that is, a positive value of produces compressive stress at the bottom surface. With this choice of bending moment sign convention, in order to have , it is necessary that the shear force acting on the right side of the section be positive in the direction so as to achieve static equilibrium of moments. If the loading intensity is taken positive in the positive direction, then is necessary for force equilibrium.
Successive derivatives of the deflection have important physical meanings: is the slope of the beam, which is the anti-clockwise angle of rotation about the -axis in the limit of small displacements;
is the bending moment in the beam; and
is the shear force in the beam.
The stresses in a beam can be calculated from the above expressions after the deflection due to a given load has been determined.
Derivation of the bending equation
Because of the fundamental importance of the bending moment equation in engineering, we will provide a short derivation. We change to polar coordinates. The length of the neutral axis in the figure is The length of a fiber with a radial distance below the neutral axis is Therefore, the strain of this fiber is
The stress of this fiber is where is the elastic modulus in accordance with Hooke's law. The differential force vector, resulting from this stress, is given by
This is the differential force vector exerted on the right hand side of the section shown in the figure. We know that it is in the direction since the figure clearly shows that the fibers in the lower half are in tension. is the differential element of area at the location of the fiber. The differential bending moment vector, associated with is given by
This expression is valid for the fibers in the lower half of the beam. The expression for the fibers in the upper half of the beam will be similar except that the moment arm vector will be in the positive direction and the force vector will be in the direction since the upper fibers are in compression. But the resulting bending moment vector will still be in the direction since Therefore, we integrate over the entire cross section of the beam and get for the bending moment vector exerted on the right cross section of the beam the expression
where is the second moment of area. From calculus, we know that when is small, as it is for an Euler–Bernoulli beam, we can make the approximation , where is the radius of curvature. Therefore,
This vector equation can be separated in the bending unit vector definition ( is oriented as ), and in the bending equation:
Dynamic beam equation
The dynamic beam equation is the Euler–Lagrange equation for the following action:
The first term represents the kinetic energy where is the mass per unit length, the second term represents the potential energy due to internal forces (when considered with a negative sign), and the third term represents the potential energy due to the external load . The Euler–Lagrange equation is used to determine the function that minimizes the functional . For a dynamic Euler–Bernoulli beam, the Euler–Lagrange equation is Derivation of Euler–Lagrange equation for beamsSince the Lagrangian is
the corresponding Euler–Lagrange equation is
Now,
Plugging into the Euler–Lagrange equation gives
or,
which is the governing equation for the dynamics of an Euler–Bernoulli beam. When the beam is homogeneous, and are independent of , and the beam equation is simpler:
Free vibration In the absence of a transverse load, , we have the free vibration equation. This equation can be solved using a Fourier decomposition of the displacement into the sum of harmonic vibrations of the form
where is the frequency of vibration. Then, for each value of frequency, we can solve an ordinary differential equation
The general solution of the above equation is
where are constants. These constants are unique for a given set of boundary conditions. However, the solution for the displacement is not unique and depends on the frequency. These solutions are typically written as
The quantities are called the natural frequencies of the beam. Each of the displacement solutions is called a mode, and the shape of the displacement curve is called a mode shape.
Example: Cantilevered beam
The boundary conditions for a cantilevered beam of length (fixed at ) are
If we apply these conditions, non-trivial solutions are found to exist only if
This nonlinear equation can be solved numerically. The first four roots are , , , and .
The corresponding natural frequencies of vibration are
The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:
The unknown constant (actually constants as there is one for each ), , which in general is complex, is determined by the initial conditions at on the velocity and displacements of the beam. Typically a value of is used when plotting mode shapes. Solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency , i.e., the beam can resonate. The natural frequencies of a beam therefore correspond to the frequencies at which resonance can occur.
Example: free–free (unsupported) beam
A free–free beam is a beam without any supports. The boundary conditions for a free–free beam of length extending from to are given by:
If we apply these conditions, non-trivial solutions are found to exist only if
This nonlinear equation can be solved numerically. The first four roots are , , , and .
The corresponding natural frequencies of vibration are:
The boundary conditions can also be used to determine the mode shapes from the solution for the displacement:
As with the cantilevered beam, the unknown constants are determined by the initial conditions at on the velocity and displacements of the beam. Also, solutions to the undamped forced problem have unbounded displacements when the driving frequency matches a natural frequency .
Example: hinged-hinged beam
The boundary conditions of a hinged-hinged beam of length (fixed at and ) are
This implies solutions exist for
Setting enforces this condition. Rearranging for natural frequency gives
Stress Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler–Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.