Statics: When the reactions depend on the displacements

In summary, the conversation discusses problems where there is a dependency between the reactions at the supports and the displacements due to deformations, which cannot be solved using traditional statics and resistance of materials tools. This type of problem is known as statically undetermined and can be solved by finding the relation between internal axial forces and deformation. One book that covers this topic is Roark's Formulas for Stress and Strain, and techniques for analyzing these problems can be found in textbooks on Advanced Mechanics of Materials. The specific case of a beam under combined axial and transverse loading can be solved using the displacement formula from Roark's book. However, there is still a need for further research and insight on this topic.
  • #36
T1m0 said:
Juanda, you can find a fairly thorough explanation of the pressure vessel formulas in

https://pkel015.connect.amazon.auck...lasticity_Applications_03_Presure_Vessels.pdf

I am not aware of any analytical solutions to the pressure vessel problems for the large displacement case. This may be because most materials used in pressure vessels will experience yielding before the displacements get large. I suppose a really soft material might deform significantly before yielding. However, these materials tend to have more complicated stress-strain laws than Hooke's law.

In general, there are few analytical solutions to large displacement problems. The usual (and easiest) way to attack these problems is to use structural finite element analysis software (e.g., ANSYS, ABAQUS, etc). Although the full versions of these programs are very expensive, there are student versions that are free or very inexpensive.

That was a much lighter read. I appreciate it. It is similar to what's shown in Mechanics of Materials by Barry J. Goodno and James M. Gere.
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I wouldn't know how to apply what we discussed in this thread to it though. You mention how those solutions are only for small displacements. However, the equations we used for the beam are also related to small displacements. It's just that by studying equilibrium in the deformed state it's possible to find a more accurate solution which can be critical in some cases when that increment in accuracy reveals that the stress is greater than initially calculated.
 
<h2>What is the basic principle of statics when reactions depend on displacements?</h2><p>In statics, when reactions depend on displacements, the equilibrium conditions must account for the fact that the support reactions are not constant but vary with the displacements of the structure. This typically involves solving a system of equations that includes both the equilibrium equations and the compatibility conditions, which relate the displacements to the reactions.</p><h2>How do you formulate the equilibrium equations in such problems?</h2><p>The equilibrium equations are formulated by ensuring that the sum of all forces and moments acting on the structure equals zero. However, in cases where reactions depend on displacements, these equations will include terms that represent the relationship between the displacements and the reactions. This often requires the use of stiffness or flexibility matrices to describe the system's response.</p><h2>What role do compatibility conditions play in these problems?</h2><p>Compatibility conditions are crucial because they ensure that the displacements and deformations of the structure are consistent with the physical constraints and boundary conditions. These conditions are used to relate the displacements to the reactions and must be satisfied along with the equilibrium equations to accurately describe the behavior of the structure.</p><h2>How do you solve the system of equations that arise in these problems?</h2><p>Solving the system of equations typically involves using numerical methods, such as the finite element method (FEM), to handle the complexity of the relationships between displacements and reactions. The system of equations can be large and non-linear, requiring iterative techniques and computational tools to find a solution that satisfies both the equilibrium and compatibility conditions.</p><h2>Can you provide an example of a structure where reactions depend on displacements?</h2><p>A common example is a beam on an elastic foundation, where the support reactions are a function of the beam's displacements. The foundation provides a reaction force that depends on the local displacement of the beam, leading to a coupled system of equations that must be solved to determine the beam's deflection and the corresponding reactions.</p>

FAQ: Statics: When the reactions depend on the displacements

What is the basic principle of statics when reactions depend on displacements?

In statics, when reactions depend on displacements, the equilibrium conditions must account for the fact that the support reactions are not constant but vary with the displacements of the structure. This typically involves solving a system of equations that includes both the equilibrium equations and the compatibility conditions, which relate the displacements to the reactions.

How do you formulate the equilibrium equations in such problems?

The equilibrium equations are formulated by ensuring that the sum of all forces and moments acting on the structure equals zero. However, in cases where reactions depend on displacements, these equations will include terms that represent the relationship between the displacements and the reactions. This often requires the use of stiffness or flexibility matrices to describe the system's response.

What role do compatibility conditions play in these problems?

Compatibility conditions are crucial because they ensure that the displacements and deformations of the structure are consistent with the physical constraints and boundary conditions. These conditions are used to relate the displacements to the reactions and must be satisfied along with the equilibrium equations to accurately describe the behavior of the structure.

How do you solve the system of equations that arise in these problems?

Solving the system of equations typically involves using numerical methods, such as the finite element method (FEM), to handle the complexity of the relationships between displacements and reactions. The system of equations can be large and non-linear, requiring iterative techniques and computational tools to find a solution that satisfies both the equilibrium and compatibility conditions.

Can you provide an example of a structure where reactions depend on displacements?

A common example is a beam on an elastic foundation, where the support reactions are a function of the beam's displacements. The foundation provides a reaction force that depends on the local displacement of the beam, leading to a coupled system of equations that must be solved to determine the beam's deflection and the corresponding reactions.

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