6.13
Cadres : Résolution de problèmes I
Considérez une grue à flèche avec une charge externe suspendue à la poulie. Les dimensions des éléments de la grue sont indiquées dans la figure. Une…
Prenons l’exemple d’une potence avec une charge externe de la poulie.
Si les dimensions des éléments de la grue sont connues, quelles sont les forces de réaction au niveau des joints des axes, étant donné que les poulies sont sans frottement ?
Le système est une structure de cadre, composée d’un membre BD à deux forces et d’un membre ABC à plusieurs forces.
En considérant un schéma à corps libre et en appliquant les conditions d’équilibre des forces pour la section inférieure de la poulie, la tension dans le câble peut être obtenue.
En ce qui concerne le point C, la tension dans le câble vertical est dirigée vers le bas, tandis que pour le câble horizontal, elle est dirigée vers le joint A.
Dans le membre DB, la force FBD peut être exprimée à l’aide d’un triangle de pente.
La condition d’équilibre de moment à l’articulation A donne la force le long de BD.
En appliquant les conditions d’équilibre des forces, les forces de réaction horizontales et verticales à l’articulation A sont calculées.
Maintenant, si l’on considère le diagramme à corps libre de l’élément BD, les conditions d’équilibre des forces peuvent être appliquées à l’articulation D pour obtenir les forces de réaction horizontales et verticales à D.
Q1: What is the difference between a two-force member and a multi-force member in frame analysis?
A two-force member is a straight element subjected only to forces at its two ends, with forces equal in magnitude but opposite in direction, causing pure tension or compression. A multi-force member experiences more than two forces distributed along its length, including external loads, reaction forces at pin joints, and cable forces, resulting in complex stress distribution.
Q2: How do you determine cable tension in a jib crane pulley system?
For a frictionless pulley system, the load weight balances the tension in the cables. In the lower pulley section, each cable carries an upward tension equal to half the total load. For the upper pulley, the vertical cable tension remains the same as the lower section since it is part of the continuous cable system.
Q3: What equilibrium conditions are applied to find reaction forces at pin joints?
Force equilibrium conditions require that the sum of horizontal and vertical forces equals zero at each joint. Moment equilibrium conditions require that the sum of moments about a point equals zero. These conditions are applied systematically to free-body diagrams of each structural member to solve for unknown reaction forces.
Q4: How is the force in member BD resolved in a jib crane frame analysis?
The force FBD in member BD is resolved into horizontal and vertical components using a slope triangle, which relates the member's geometry to its force components. The moment equilibrium condition at joint A yields the magnitude of FBD. Substituting known dimensions and pulley radius values determines FBD as 50 kN in this example.
Q5: What are the typical reaction force values at pin joints in a loaded jib crane?
Reaction forces depend on load magnitude and structural geometry. In this jib crane example, the horizontal reaction force at joint A is 40 kN, while the vertical reaction force is -20 kN. At joint D, the horizontal reaction is -30 kN and the vertical reaction is 40 kN, determined by applying force equilibrium conditions.
Q6: Why is analyzing the lower pulley section important before analyzing the upper pulley?
The lower pulley section directly supports the external load, establishing the cable tension value. This tension value propagates through the continuous cable system to the upper pulley section. Determining lower pulley tension first provides known force values needed to solve for reaction forces at pin joints in the frame structure.
Q7: How does the free-body diagram method apply to frame problem solving?
Free-body diagrams isolate each structural member and show all forces acting on it, including external loads, cable tensions, and reaction forces at pin joints. Applying force and moment equilibrium conditions to these diagrams systematically yields unknown reaction forces. This method using method of joints problem solving principles allows solving complex frame structures step by step.