Abstract
This chapter presents the analytical description of thin, or so-called shear-rigid, beam members according to the Euler–Bernoulli theory. Based on the three basic equations of continuum mechanics, i.e., the kinematics relationship, the constitutive law, and the equilibrium equation, the partial differential equations, which describe the physical problem, are presented. All equations are introduced for single plane bending in the x-y plane as well as the x-z plane. Analytical solutions of the partial differential equation are given for simple cases. In addition, this chapter treats the case of unsymmetrical bending, a superposition of tensile and bending modes as well as the shear stress distribution for solid section and thin-walled beams.
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Notes
- 1.
A historical analysis of the development of the classical beam theory and the contribution of different scientists can be found in [7].
- 2.
More precisely, this is the neutral fibre or the bending line.
- 3.
Consequently, the width b and the height h of a, for example, rectangular cross section remain the same.
- 4.
The sum of all points with \(\sigma = 0\) along the beam axis is called the neutral fiber..
- 5.
Note that according to the assumptions of the Euler–Bernoulli beam the lengths 01 and \(0'1'\) remain unchanged.
- 6.
However, this formulation works well in the case of rod elements since stress and strain are constant over the cross section, i.e. \(\sigma _x=\sigma _x(x)\) and \(\varepsilon _x=\varepsilon _x(x)\).
- 7.
A similar stress resultant can be stated for the shear stress based on the shear force: \(Q_y(x)=\iint \tau _{xy}(x,y)\,\text {d}A\).
- 8.
The curvature is then called a generalized strain.
- 9.
A positive cut face is defined by the surface normal on the cut plane which has the same orientation as the positive x-axis. It should be regarded that the surface normal is always directed outward.
- 10.
If the axis is grasped with the right hand in a way so that the spread out thumb points in the direction of the positive axis, the bent fingers then show the direction of the positive rotational direction.
- 11.
In the general case, the unit of the elastic foundation modulus is force per unit area per unit length, i.e. \(\tfrac{\text {N}}{\text {m}^2}/\text {m}=\tfrac{\text {N}}{\text {m}^3}\).
- 12.
Note that according to the assumptions of the Euler–Bernoulli beam the lengths 01 and \(0'1'\) remain unchanged.
- 13.
However, this formulation works well in the case of rod elements since stress and strain are constant over the cross section, i.e. \(\sigma _x=\sigma _x(x)\) and \(\varepsilon _x=\varepsilon _x(x)\).
- 14.
A similar stress resultant can be stated for the shear stress based on the shear force: \(Q_z(x)=\iint \tau _{xz}(x,z)\,\text {d}A\).
- 15.
The curvature is then called a generalized strain.
- 16.
A positive cut face is defined by the surface normal on the cut plane which has the same orientation as the positive x-axis. It should be regarded that the surface normal is always directed outward.
- 17.
If the axis is grasped with the right hand in a way so that the spread out thumb points in the direction of the positive axis, the bent fingers then show the direction of the positive rotational direction.
- 18.
In the general case, the unit of the elastic foundation modulus is force per unit area per unit length, i.e. \(\tfrac{\text {N}}{\text {m}^2}/\text {m}=\tfrac{\text {N}}{\text {m}^3}\).
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Öchsner, A. (2021). Euler–Bernoulli Beam Theory. In: Classical Beam Theories of Structural Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-76035-9_2
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