7.9
Relación entre el cortante y el momento flector
Cuando una viga está sujeta a diversas cargas, como una carga distribuida, cargas concentradas y un momento de torción, experimenta tanto fuerzas cort…
Supongamos que una viga está sometida a una carga distribuida, dos cargas concentradas y un par de momentos. Establecer la relación entre el cizallamiento y el momento flector.
Considere una sección elemental de la viga y dibuje un diagrama de cuerpo libre de la sección.
Para que la sección esté en equilibrio, el momento que actúa en el lado derecho de la sección debe ser mayor en una cantidad pequeña y finita.
Una fuerza resultante de la carga distribuida se ejerce a una distancia fraccionaria desde el extremo derecho de la sección.
Usando la ecuación de equilibrio para el momento, se puede obtener una relación entre el momento y el corte.
Además, dividiéndolo por Δx y dejando que Δx se acerque a cero, se determina la pendiente del diagrama de momentos, que es equivalente al cortante.
Un momento flector máximo se produce en el punto donde la pendiente del momento y el cortante son cero.
Integrando la carga distribuida sobre la sección elemental, situada entre dos puntos arbitrarios, se obtiene una correlación entre el cambio en el momento flector y el área bajo el diagrama de cortante.
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Q1: How do you establish the relationship between shear and bending moment in a beam?
To establish the relationship, analyze an elemental section of the beam using a free-body diagram. Apply equilibrium equations for moments, noting that the moment on the right side should be higher by a small amount. By dividing the moment equation by Δx and letting Δx approach zero, you determine the slope of the moment diagram, which equals the shear force.
Q2: What does the slope of the moment diagram represent?
The slope of the moment diagram is equivalent to the shear force acting on the beam. This relationship is derived by dividing the moment equilibrium equation by the elemental length Δx and taking the limit as Δx approaches zero, establishing a direct mathematical connection between moment and shear.
Q3: Where does the maximum bending moment occur on a beam?
The maximum bending moment occurs at the point where both the slope of the moment diagram and the shear force are zero. This critical location represents a transition point where the shear changes sign, making it essential for structural design and analysis.
Q4: How does the area under the shear diagram relate to bending moment?
By integrating the distributed load over an elemental section between two arbitrary points, a correlation emerges: the change in bending moment equals the area under the shear diagram. This relationship provides a graphical method for determining moment variations along the beam's length.
Q5: Why must the moment on the right side of an elemental section be higher than on the left?
For the elemental section to remain in equilibrium, the moment on the right side must be higher by a small finite amount to balance the effects of shear forces and distributed loads acting on the section. This moment difference is essential for maintaining rotational equilibrium of the infinitesimal element.
Q6: What role does the distributed load play in the shear-moment relationship?
The distributed load exerts a resultant force at a fractional distance from the section's right end, creating the moment differential needed for equilibrium. When integrated over the elemental section, the distributed load determines how shear and moment vary along the beam, directly influencing the shape of both diagrams.
Q7: How do concentrated loads and couple moments affect the shear-moment relationship?
Concentrated loads and couple moments contribute to the overall loading condition on the beam, affecting both shear and bending moment distributions. These loads must be included in the free-body diagram analysis to accurately establish the relationship between shear and moment throughout the beam.