12.10
Struttura indeterminata
Le strutture indeterminate sono strutture in cui le forze interne e le reazioni non possono essere determinate utilizzando solo le equazioni di equili…
In una struttura indeterminata, le equazioni di equilibrio statiche non possono determinare sufficientemente le forze interne e le reazioni su di essa.
Considera un tavolo oscillante con quattro gambe cilindriche, ciascuna con una sezione trasversale di 1 cm2. La lunghezza di tre gambe è di 2 m, mentre la quarta gamba è più lunga di 0,50 mm. Quando viene posizionata una massa di 300 kg, le gambe vengono compresse e il tavolo è in piano e non oscilla più. Se il modulo di Young delle gambe di legno è 1,3 x 1010 N/m2, determinare l'entità delle forze che agiscono sulle gambe.
Ricordando l'equazione del modulo di Young, si può stabilire una relazione tra la gamba allungata e le gambe più corte.
Bilanciando tutte le forze verticali agenti sul sistema, si può ottenere la forza sulla gamba allungata.
Confrontando le equazioni della gamba allungata e sostituendo i valori, è possibile determinare la forza sulle gambe più corte.
Utilizzando l'equazione della forza e sostituendo i valori, è possibile ottenere la forza che agisce sulla gamba allungata.
View the full transcript and gain access to JoVE Core videos
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.