In structural engineering, mechanical stresses or section stresses are physical magnitudes with units of force over area used in the calculation of prismatic parts such as beams or columns and also in the calculation of plates and sheets.
Definition
The internal forces on a plane cross section of a structural element are defined as a set of forces and moments statically equivalent to the distribution of internal stresses over the area of that section.
Thus, for example, the stresses on a plane cross section Σ of a beam is equal to the integral of the stresses t over that plane area. Normally a distinction is made between the forces perpendicular to the section of the beam (or thickness of the plate or sheet) and the tangents to the section of the beam (or surface of the plate or sheet):
Section forces in beams and columns
Contenido
Para un prisma mecánico o elemento unidimensional los esfuerzos se designan como:.
Dado un sistema de ejes ortogonales, en que el eje X coincide con el eje baricéntrico de un elemento unidimensional con sección transversal uniforme, los anteriores esfuerzos son las resultantes de las tensiones sobre cada sección transversal:.
Es común también denominar esfuerzos a:.
Donde es el alabeo seccional de la sección transversal.
Cada uno de estos esfuerzos van asociados a cierto tipo de tensión:.
Practical calculation of efforts in prisms
Let's consider the mechanical beam or prism seen in the first figure and assume that it is linked to the rest of the structure in an isostatic way. We will also assume that active external forces act on this prism in the plane of its barycentric axis (or straight line that joins the barycenters of all the straight cross sections of the prism).
The first step is to divide the rigid into two smaller blocks. Blocks 1 and 2 of the figure are determined.
Total efforts
Introduction
In structural engineering, mechanical stresses or section stresses are physical magnitudes with units of force over area used in the calculation of prismatic parts such as beams or columns and also in the calculation of plates and sheets.
Definition
The internal forces on a plane cross section of a structural element are defined as a set of forces and moments statically equivalent to the distribution of internal stresses over the area of that section.
Thus, for example, the stresses on a plane cross section Σ of a beam is equal to the integral of the stresses t over that plane area. Normally a distinction is made between the forces perpendicular to the section of the beam (or thickness of the plate or sheet) and the tangents to the section of the beam (or surface of the plate or sheet):
Section forces in beams and columns
Contenido
Para un prisma mecánico o elemento unidimensional los esfuerzos se designan como:.
Dado un sistema de ejes ortogonales, en que el eje X coincide con el eje baricéntrico de un elemento unidimensional con sección transversal uniforme, los anteriores esfuerzos son las resultantes de las tensiones sobre cada sección transversal:.
Es común también denominar esfuerzos a:.
Donde es el alabeo seccional de la sección transversal.
Cada uno de estos esfuerzos van asociados a cierto tipo de tensión:.
Practical calculation of efforts in prisms
Let's consider the mechanical beam or prism seen in the first figure and assume that it is linked to the rest of the structure in an isostatic way. We will also assume that active external forces act on this prism in the plane of its barycentric axis (or straight line that joins the barycenters of all the straight cross sections of the prism).
Next we will study block 1, where 2 external reactive forces appear acting (P and P). As you can see, this block is now not isostatically linked, so in order for it to remain in balance, there must be forces that balance it. These forces are also reactive forces and correspond to the action of block 2 on block 1. The reactive forces of block 2 on block 1 can be reduced to a force and a moment acting on the barycenter of the straight section A. In fact these forces and moments are the resultant force and the resultant moment of the stress distribution over the straight area A.
As we are dealing with the special case of active external forces acting on the plane of the barycentric axis, the moment and force to which the reactive forces of block 2 on block 1 are reduced must be a force contained in said plane and a moment perpendicular to the same plane.
We will call the force R of block 2 on the block and immediately call it M. The force R can be decomposed into a vertical and a horizontal component in the plane in which it is contained. We will call R the force decomposed vertically and R the force decomposed horizontally. Summarizing, we have the system of forces in equilibrium that is formed by:.
The reactive forces R, R and the moment M are known as internal forces. And they respectively represent the normal stress (N = R), the shear stress (Q = R) and the "Bending (engineering)" bending moment (M = M).
Calculation of tensions in prisms
In prismatic pieces subjected to compound bending (not deflected and without torsion), the calculation of the stresses is simple if the internal forces are known, for a symmetrical piece in which the center of gravity is aligned with the center of shear and with a total depth small enough compared to the length of the prismatic piece, so that the Navier-Bernouilli theory can be applied, the tension tensor of a beam is given as a function of the internal forces by:.
Where the normal (σ) and tangential (τ) stresses can be determined 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 and shear turn out to be:
Where is the coefficient that relates the maximum shear stress and the average shear stress of the section. A frequently used criterion for metal beams is to verify that the following condition is verified in all sections:
Being the ultimate stress or admissible stress normally defined in terms of the elastic limit of the material. For prismatic pieces susceptible to buckling, the previous calculation does not lead to a safe design, since in that case the normal stress likely to develop in the piece is underestimated.
Stresses in plates and sheets
En un elemento bidimensional, parametrizado por dos coordenadas α y β, el número de esfuerzos que deben considerarse es mayor que en elementos unidimensionales:.
Calculation of stresses in plates
In a sheet fundamentally subjected to bending in which shear deformation and membrane stress are neglected, it is called Love-Kirchhof sheet, the internal forces are characterized by two bending moments according to two mutually perpendicular directions and a torsional force. These efforts are directly related to the vertical arrow w(x, y) at each point by:.
Where:.
The stresses on a plate are directly calculable from the previous stresses:
The first step is to divide the rigid into two smaller blocks. Blocks 1 and 2 of the figure are determined.
Next we will study block 1, where 2 external reactive forces appear acting (P and P). As you can see, this block is now not isostatically linked, so in order for it to remain in balance, there must be forces that balance it. These forces are also reactive forces and correspond to the action of block 2 on block 1. The reactive forces of block 2 on block 1 can be reduced to a force and a moment acting on the barycenter of the straight section A. In fact these forces and moments are the resultant force and the resultant moment of the stress distribution over the straight area A.
As we are dealing with the special case of active external forces acting on the plane of the barycentric axis, the moment and force to which the reactive forces of block 2 on block 1 are reduced must be a force contained in said plane and a moment perpendicular to the same plane.
We will call the force R of block 2 on the block and immediately call it M. The force R can be decomposed into a vertical and a horizontal component in the plane in which it is contained. We will call R the force decomposed vertically and R the force decomposed horizontally. Summarizing, we have the system of forces in equilibrium that is formed by:.
The reactive forces R, R and the moment M are known as internal forces. And they respectively represent the normal stress (N = R), the shear stress (Q = R) and the "Bending (engineering)" bending moment (M = M).
Calculation of tensions in prisms
In prismatic pieces subjected to compound bending (not deflected and without torsion), the calculation of the stresses is simple if the internal forces are known, for a symmetrical piece in which the center of gravity is aligned with the center of shear and with a total depth small enough compared to the length of the prismatic piece, so that the Navier-Bernouilli theory can be applied, the tension tensor of a beam is given as a function of the internal forces by:.
Where the normal (σ) and tangential (τ) stresses can be determined 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 and shear turn out to be:
Where is the coefficient that relates the maximum shear stress and the average shear stress of the section. A frequently used criterion for metal beams is to verify that the following condition is verified in all sections:
Being the ultimate stress or admissible stress normally defined in terms of the elastic limit of the material. For prismatic pieces susceptible to buckling, the previous calculation does not lead to a safe design, since in that case the normal stress likely to develop in the piece is underestimated.
Stresses in plates and sheets
En un elemento bidimensional, parametrizado por dos coordenadas α y β, el número de esfuerzos que deben considerarse es mayor que en elementos unidimensionales:.
Calculation of stresses in plates
In a sheet fundamentally subjected to bending in which shear deformation and membrane stress are neglected, it is called Love-Kirchhof sheet, the internal forces are characterized by two bending moments according to two mutually perpendicular directions and a torsional force. These efforts are directly related to the vertical arrow w(x, y) at each point by:.
Where:.
The stresses on a plate are directly calculable from the previous stresses: