17.6
Contrainte : Conditions générales de chargement
Pour saisir la complexité des conditions réelles dans lesquelles plusieurs charges sont appliquées simultanément à une structure, on peut visualiser u…
Considérons un corps soumis à de multiples charges. Lorsqu’on dissèque un point à l’aide d’un plan parallèle au plan xy, on observe que les forces de cisaillement agissent sur une petite zone entourant le point.
Considérons une partie du corps soumise aux charges précédemment indiquées, la force normale agissant le long de l’axe z
Les forces de cisaillement n’ont pas de direction bien définie sur un plan de section, elles sont donc résolues en deux forces composantes parallèles aux axes x et y.
L’amplitude de chaque force est ensuite divisée par l’aire pour obtenir les composantes de contrainte.
Le premier indice des contraintes indique qu’elles s’exercent sur la surface perpendiculaire à l’axe z. Le deuxième indice des contraintes de cisaillement identifie la direction.
Lorsque l’autre côté du plan vertical est analysé, il produit les mêmes composantes de contrainte mais dans des directions opposées.
En faisant passer des sections par le point parallèlement aux plans yz et zx, des composants de contrainte supplémentaires peuvent être définis.
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Q1: What are stress components and how are they determined at a point in a loaded body?
Stress components are determined by passing a sectional plane through a point in a body and analyzing forces acting on that section. Normal forces and shearing forces are resolved into components parallel to coordinate axes. Each force component is then divided by the area to obtain stress values. This process yields three distinct stress components at any point when sections are passed parallel to different planes.
Q2: What do the subscripts in stress notation indicate?
In stress notation, the first subscript indicates the plane on which the stress acts—specifically, the direction of the surface normal. The second subscript identifies the direction of the stress component itself. For example, τzy indicates shearing stress acting on a surface perpendicular to the z-axis, directed along the y-axis. This notation system clarifies both location and orientation of each stress component.
Q3: How do shearing forces differ from normal forces in stress analysis?
Normal forces act perpendicular to a sectional plane, typically along one axis. Shearing forces, however, lack a single well-defined direction on the sectional plane and must be resolved into two component forces parallel to the coordinate axes. This resolution allows shearing stresses to be calculated by dividing each component by the section area, yielding multiple shearing stress components at a point.
Q4: Why do opposite sides of a sectional plane show stress components in opposite directions?
When analyzing opposite sides of a vertical plane through a point, Newton's third law applies: action and reaction forces are equal and opposite. Therefore, the stress components calculated on one side of the plane are identical in magnitude but opposite in direction on the other side. This symmetry reflects the equilibrium of forces within the body under load.
Q5: How many stress components exist at a point in a three-dimensional loaded body?
At any point in a three-dimensional body, nine stress components exist: three normal stresses (σx, σy, σz) and six shearing stresses (τxy, τyx, τxz, τzx, τyz, τzy). These components are determined by passing sectional planes parallel to the xy, yz, and zx planes through the point. Together, they fully describe the stress state at that location under general loading conditions.
Q6: What do positive and negative signs indicate in stress component values?
The signs of stress components indicate the type of loading: positive values represent tension, where the material is being pulled apart, while negative values represent compression, where the material is being pushed together. This sign convention applies to both normal and shearing stress components, allowing engineers to quickly identify whether a region experiences tensile or compressive loading.
Q7: How does analyzing multiple sectional planes help understand stress under complex loading?
Analyzing sections parallel to different coordinate planes reveals the complete stress state at a point. By examining sections parallel to xy, yz, and zx planes, all nine stress components become defined. This comprehensive approach unravels the complex interplay of forces and stresses under intricate loading conditions, enabling accurate prediction of material behavior and structural failure.