6.13
Telai: Risoluzione dei problemi I
Considera una gru a cavalletto con un carico esterno sospeso dalla puleggia. Le dimensioni dei membri della gru sono mostrate nella figura. Per determ…
Si consideri una gru a bandiera con un carico esterno dalla puleggia.
Se le dimensioni degli elementi della gru sono note, quali sono le forze di reazione ai giunti dei perni, considerando che le pulegge sono prive di attrito?
Il sistema è una struttura a telaio, costituita da un membro BD a due forze e da un membro ABC a più forze.
Considerando un diagramma a corpo libero e applicando le condizioni di equilibrio della forza per la sezione inferiore della puleggia, si può ottenere la tensione nel cavo.
Considerando il punto C, la tensione nel cavo verticale è diretta verso il basso, mentre per il cavo orizzontale è diretta verso il giunto A.
Nell'asta DB, la forza FBD può essere espressa utilizzando un triangolo di pendenza.
La condizione di equilibrio del momento al giunto A fornisce la forza lungo BD.
Applicando le condizioni di equilibrio delle forze, vengono calcolate le forze di reazione orizzontali e verticali al giunto A.
Ora, considerando il diagramma di corpo libero per l'asta BD, le condizioni di equilibrio delle forze possono essere applicate al giunto D per ottenere le forze di reazione orizzontali e verticali in 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.