Deformations and stresses in the beams

If the components of the strain tensor are calculated from these displacements, we arrive at:

From these deformations, the stresses can be obtained using the Lamé-Hooke equations, assuming:.

where E is the longitudinal modulus of elasticity, or Young's modulus, and G the transverse modulus of elasticity. It is clear that the Euler-Bernoulli theory is incapable of approximating the tangential deformation energy, for this purpose we must resort to the Timoshenko theory, in which:

Internal forces in beams

From the previous results and the equivalence equations, the normal stress, the shear stress and the bending moment to which a section of a beam subjected to simple bending in the Euler-Bernouilli theory is subjected can be simply obtained.

where: A area of ​​the cross section, I the moment of inertia according to the axis with respect to which the bending occurs. The last of these equations is precisely the elastic curve equation, one of the basic equations of beam theory that relates internal forces to the vertical displacement field.

Equilibrium equations

The equilibrium equations for a beam are the application of the equations of statics to a beam span in equilibrium. The forces that intervene on the section would be the external load applied on the beam and the shear forces acting on the extreme sections that delimit the section. If the section is in equilibrium, this implies that the sum of vertical forces must be zero, and also the sum of moments of force at the neutral fiber must be zero in the tangent direction to the neutral fiber. These two conditions can only be met if the variation of shear stress and bending moment are related to the vertical load per unit length by:.

Calculation of stresses in beams

The calculation of stresses in beams generally requires knowing the variation of internal forces and from them applying the appropriate formula depending on whether the beam is subjected to bending, torsion, normal stress or shear stress. The tension tensor of a beam is given as a function of the internal forces by:.

where the stresses can be determined, approximately, from the internal forces. If a system of main axes of inertia on the beam is considered, considered as a mechanical prism, the stresses associated with extension, flexion, shear and torsion turn out to be:

where:.

The maximum normal and tangential stresses on any cross section of the beam can be calculated from the first () and third () principal stress:.

In metallic beams, the failure criterion is frequently used if at some point the Von Mises equivalent stress exceeds a certain ultimate stress defined from the elastic limit. In that case, the failure criterion can be written as:

Deformations and stresses in the beams

If the components of the strain tensor are calculated from these displacements, we arrive at:

From these deformations, the stresses can be obtained using the Lamé-Hooke equations, assuming:.

where E is the longitudinal modulus of elasticity, or Young's modulus, and G the transverse modulus of elasticity. It is clear that the Euler-Bernoulli theory is incapable of approximating the tangential deformation energy, for this purpose we must resort to the Timoshenko theory, in which:

Internal forces in beams

From the previous results and the equivalence equations, the normal stress, the shear stress and the bending moment to which a section of a beam subjected to simple bending in the Euler-Bernouilli theory is subjected can be simply obtained.

where: A area of ​​the cross section, I the moment of inertia according to the axis with respect to which the bending occurs. The last of these equations is precisely the elastic curve equation, one of the basic equations of beam theory that relates internal forces to the vertical displacement field.

Equilibrium equations

The equilibrium equations for a beam are the application of the equations of statics to a beam span in equilibrium. The forces that intervene on the section would be the external load applied on the beam and the shear forces acting on the extreme sections that delimit the section. If the section is in equilibrium, this implies that the sum of vertical forces must be zero, and also the sum of moments of force at the neutral fiber must be zero in the tangent direction to the neutral fiber. These two conditions can only be met if the variation of shear stress and bending moment are related to the vertical load per unit length by:.

Calculation of stresses in beams

The calculation of stresses in beams generally requires knowing the variation of internal forces and from them applying the appropriate formula depending on whether the beam is subjected to bending, torsion, normal stress or shear stress. The tension tensor of a beam is given as a function of the internal forces by:.

where the stresses can be determined, approximately, from the internal forces. If a system of main axes of inertia on the beam is considered, considered as a mechanical prism, the stresses associated with extension, flexion, shear and torsion turn out to be:

where:.

The maximum normal and tangential stresses on any cross section of the beam can be calculated from the first () and third () principal stress:.

In metallic beams, the failure criterion is frequently used if at some point the Von Mises equivalent stress exceeds a certain ultimate stress defined from the elastic limit. In that case, the failure criterion can be written as: