12.10
Structure indéterminée
Les structures indéterminées désignent des structures dont les forces internes et les réactions ne peuvent pas être déterminées uniquement à l'aide de…
Dans une structure indéterminée, les équations d’équilibre statique ne peuvent pas déterminer suffisamment les forces et les réactions internes sur celle-ci.
Prenons l’exemple d’une table oscillante avec quatre pieds cylindriques, chacun ayant une section transversale de 1cm2. La longueur de trois pattes est de 2 m, tandis que la quatrième patte est plus longue de 0,50 mm. Lorsqu’une masse de 300 kg est placée, les pieds sont comprimés et la table est de niveau et ne vacille plus. Si le module de Young des pieds en bois est de 1,3 x 1010 N/m2, déterminez l'intensité des forces agissant sur les pieds.
En rappelant l'équation du module de Young, une relation entre la jambe allongée et les jambes plus courtes peut être établie.
En équilibrant toutes les forces verticales agissant sur le système, la force sur la jambe allongée peut être obtenue.
En comparant les équations de la jambe allongée et en substituant les valeurs, la force sur les jambes les plus courtes peut être déterminée.
En utilisant l’équation de la force et en substituant les valeurs, la force agissant sur la jambe allongée peut être obtenue.
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Q1: What makes a structure indeterminate?
An indeterminate structure has more unknown forces and reaction forces than static equilibrium equations available to solve them. Standard equilibrium equations alone cannot determine all internal forces and reactions. These structures require additional equations based on material properties and deformation to find all unknowns, making them common in engineering for efficient, complex designs.
Q2: How do you solve forces in an indeterminate structure?
Solving indeterminate structures requires supplementary equations beyond static equilibrium. The force method, displacement method, and flexibility method are common approaches. These techniques use material properties like Young's modulus and deformation relationships to create additional equations. For example, relating compression of different legs using Young's modulus allows you to solve for individual forces on each support.
Q3: Why are suspension bridges indeterminate structures?
Suspension bridges are indeterminate because they have more unknown forces than static equilibrium equations can resolve. Cables suspended between towers experience tension while towers experience compression, creating complex load distributions. Advanced analysis techniques are required to determine how forces distribute through the cables, towers, and bridge deck under various loading conditions.
Q4: What role does Young's modulus play in analyzing indeterminate structures?
Young's modulus quantifies material stiffness and relates stress to strain in structural members. In indeterminate structures, Young's modulus enables you to establish relationships between deformations of different components. By comparing compression or elongation of structural elements using their Young's modulus values, you create additional equations needed to solve for unknown forces that static equilibrium alone cannot determine.
Q5: How do cantilever bridges exemplify indeterminate structures?
Cantilever bridges consist of anchored supports with a central span supported by cantilever arms. The cantilever arms experience bending stress while supports experience compressive stress, creating multiple unknown forces. This complex stress distribution makes cantilever bridges indeterminate, requiring advanced analysis to determine how loads distribute through the structure and ensuring safe, efficient design.
Q6: Why are multistory buildings considered indeterminate structures?
Multistory buildings have loads from each floor distributed to columns and beams in complex ways that exceed the number of available equilibrium equations. The distribution of forces through interconnected structural members creates more unknowns than static equilibrium can resolve. Advanced analysis techniques are necessary to determine how each floor's weight distributes through the support system.
Q7: What advantages do indeterminate structures offer in engineering design?
Indeterminate structures enable efficient, cost-effective designs by distributing loads across multiple load paths, allowing them to carry large loads with minimal material. Their redundancy improves safety and stability. However, this efficiency requires more advanced analysis calculations and techniques to determine internal forces and reactions, making design more complex but ultimately producing stronger, more economical structures.