6.8
Méthode des sections : Résolution de problèmes I
Considérez une structure de fermes de toit symétrique, composée de membres verticaux, diagonaux et horizontaux. La longueur de chaque membre horizonta…
Envisagez une structure symétrique en treillis de toit comprenant les éléments verticaux, diagonaux et horizontaux.
Un schéma à corps libre est dessiné pour analyser les forces sur les éléments DC et HC à l’aide de la méthode de section.
Ici, les charges et les longueurs des éléments horizontaux et verticaux sont les paramètres connus.
Tout d’abord, en additionnant les moments autour du point A, on calcule la force de réaction en E.
De plus, en utilisant la condition d’équilibre de force pour le treillis, la force de réaction en A est déterminée.
La symétrie de la poutre garantit que les deux forces de réaction sont égales.
Maintenant, une coupe est faite le long d’un plan coupant les membres DC, HC et HG, et un diagramme de corps libre de la plus petite section est dessiné.
En prenant la somme des moments autour de H, on obtient la force le long de DC. Le signe positif indique la force de traction.
La force le long de HC est résolue en ses composantes sinus et cosinus, et la trigonométrie est utilisée pour estimer l’angle thêta.
La condition d’équilibre du moment en E donne la force sur HC, le signe négatif indiquant la force de compression.
Q1: How do you calculate reaction forces in a symmetrical roof truss using the method of sections?
For a symmetrical roof truss, apply moment equilibrium about a support point, substituting known loads and distances into the moment equation. This yields the reaction force at one support. Then use vertical force equilibrium to find the reaction at the other support. Due to symmetry, both reaction forces are equal. In the example, both reaction forces equal 4 kN.
Q2: What does a positive force value indicate when analyzing truss members?
A positive force value indicates a tensile force, meaning the member is being pulled or stretched. Conversely, a negative force value indicates a compressive force, where the member is being pushed or compressed. In the roof truss example, member DC experiences a tensile force of 3 kN, while member CH experiences a compressive force of -1.41 kN.
Q3: How does the method of sections isolate forces on specific truss members?
The method of sections involves making a cut through a truss along a plane that intersects the members you want to analyze. A free-body diagram is drawn for the smaller section created by this cut. Moment and force equilibrium equations are then applied to this isolated section to calculate internal forces on the cut members without analyzing the entire truss.
Q4: Why is moment equilibrium applied at specific points when using the method of sections?
Moment equilibrium is applied at points where unknown forces intersect or align, eliminating those unknowns from the equation. For example, taking moments about point H eliminates the force at H from the calculation, allowing you to solve directly for the force on member DC. This strategic point selection simplifies the analysis and reduces the number of simultaneous equations needed.
Q5: What role does trigonometry play in analyzing diagonal truss members?
Trigonometry is used to resolve diagonal member forces into horizontal and vertical components and to determine the angle between members and the horizontal axis. In the roof truss example, trigonometry revealed that member CH makes a 45-degree angle with the horizontal. This angle information is essential for applying force equilibrium conditions and calculating accurate member forces.
Q6: How does truss symmetry simplify the analysis process?
Truss symmetry ensures that reaction forces and member forces are equal on both sides of the centerline, reducing calculation work. In a symmetrical roof truss, once you calculate the reaction force at one support, you immediately know the reaction force at the opposite support is identical. This property eliminates redundant calculations and provides a quick verification check for your results.
Q7: What information must be known before applying the method of sections to a truss?
You must know the loads acting at each joint, the lengths of all members, and the geometry of the truss structure. These known parameters allow you to set up moment and force equilibrium equations. Additionally, you need to identify which members you want to analyze and determine an appropriate cutting plane that intersects those members without cutting through more than three unknowns.