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Diagramme de cisaillement et de moment fléchissant : Résolution de problème
Lors de l'analyse d'une poutre soutenant des charges concentrées et une charge répartie, il est essentiel de dessiner les diagrammes de cisaillement e…
Considérons une poutre supportant deux charges concentrées et une charge distribuée. Dessinez le diagramme des moments de cisaillement et de flexion de la poutre.
Tout d’abord, dessinez un schéma du faisceau et, à l’aide de l’équation d’équilibre, les forces de réaction sont obtenues.
Ensuite, divisez le faisceau en différentes sections et dessinez le schéma du corps libre de chaque section. En appliquant l’équation d’équilibre pour les sections, le cisaillement pour des sections individuelles peut être déterminé.
Le cisaillement reste constant entre les charges concentrées et les forces de réaction, tandis qu’il varie linéairement avec une pente constante dans la section de charge répartie.
L’aire sous la courbe de cisaillement entre deux points est égale à la variation du moment de flexion entre les deux mêmes points.
En considérant le moment de flexion zéro à l'extrémité de la poutre et en rappelant la relation entre la variation du moment de flexion et l'aire sous la courbe de cisaillement, le moment de flexion en chaque point est calculé.
Le diagramme du moment de flexion est dessiné en reliant les points connus par des lignes droites pour les régions à cisaillement constant et une ligne parabolique pour les régions à cisaillement linéaire.
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Q1: What is the first step in drawing shear and bending moment diagrams?
Start by drawing a free-body diagram of the entire beam, including all concentrated loads, distributed loads, and reaction forces at the supports. Apply equilibrium equations (sum of forces and moments equal zero) to determine the reaction forces. This foundation is essential before analyzing internal forces in different sections.
Q2: How does shear force behave in different sections of a loaded beam?
Shear force remains constant between concentrated loads and reaction forces, creating horizontal segments on the shear diagram. In sections with a distributed load, shear varies linearly with a constant slope. Understanding this relationship between the distributed load and shear helps predict diagram shape without calculating every point.
Q3: Why is the area under the shear curve important for finding bending moments?
The area under the shear curve between two points equals the change in bending moment between those same points. This relationship allows you to calculate bending moments at various locations by integrating shear values. Starting from a known bending moment (typically zero at a free end), you can determine moments throughout the beam.
Q4: How do you divide a beam into sections for analysis?
Divide the beam based on load distribution: create separate sections between concentrated loads, between concentrated loads and reaction forces, and within distributed load regions. Draw a free-body diagram for each section and apply equilibrium equations to calculate shear forces. This systematic approach ensures accurate internal force determination across the entire beam.
Q5: What shapes appear in a bending moment diagram?
Connect known bending moment points with straight lines in regions where shear is constant, and parabolic curves in regions where shear varies linearly. The diagram shape reflects the underlying shear distribution: constant shear produces linear moment segments, while linear shear produces parabolic segments, creating a visual representation of internal moment distribution.
Q6: Why is sectioning the beam necessary for solving shear and bending moment problems?
Sectioning isolates portions of the beam to reveal internal forces and moments at specific locations. By drawing free-body diagrams of each section and applying equilibrium equations, you can calculate shear and bending moment values that vary along the beam's length. This method transforms a complex problem into manageable steps for internal loadings structural members problem solving.
Q7: How do concentrated loads affect the shear diagram?
Concentrated loads create vertical jumps in the shear diagram at their points of application. The magnitude of each jump equals the load value, and the direction indicates whether the load acts upward or downward. Between concentrated loads, shear remains constant, producing horizontal segments that simplify diagram construction and interpretation.