Pandeo. Carga critica de Euler. Explicación de forma intuitiva y sencilla

Summary

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Takeaways

• 😀 The phenomenon of buckling in slender bars and columns is explained as an instability in elements subjected to compression.
• 😀 A steel tie or plate, like a drawing ruler, resists tension but collapses under compression at much lower loads due to buckling.
• 😀 Buckling occurs when slender compressed elements experience large transversal displacements perpendicular to the compression direction.
• 😀 Buckling is an unstable phenomenon where deformation leads to greater moments, which in turn cause more deformation, leading to collapse.
• 😀 The critical load for buckling, also known as Euler's critical load, is calculated using the Euler's equation, which takes into account the material's modulus of elasticity and the moment of inertia of the section.
• 😀 The moment of inertia plays a key role in resisting buckling—higher values of inertia help the bar withstand buckling better.
• 😀 Euler's equation considers the minimum moment of inertia, which is always calculated with respect to the weakest axis of the section.
• 😀 The length of the bar (or its effective buckling length) is another critical factor in the Euler equation—shorter bars resist buckling better.
• 😀 The equation includes a factor (Alpha) that depends on the type of supports (e.g., pinned, fixed), affecting the buckling resistance.
• 😀 For real bars, safety factors must be applied since the Euler equation assumes perfectly straight elements and homogenous material, which is rarely the case in practice.

Q & A

What is buckling and why does it happen in slender bars?

Buckling is an instability phenomenon that occurs in slender columns or bars when subjected to compression. It happens because the bar deforms transversely under compression, and as it deforms more, it generates additional moments, which cause the deformation to increase, eventually leading to collapse.

How does buckling differ from the failure due to tension?

In tension, a bar or plate can withstand forces up to its yield strength without collapsing. However, when subjected to compression, the bar may fail due to buckling with much lower loads, because of the instability caused by lateral displacement.

What is the Euler's critical load equation, and what does it calculate?

Euler's critical load equation calculates the load at which a slender bar will buckle. The equation is given by: P_cr = (π² * E * I_min) / (L_p)², where E is the material's modulus of elasticity, I_min is the minimum moment of inertia of the bar's section, and L_p is the buckling length.

Why is the minimum moment of inertia important in Euler's equation?

The minimum moment of inertia is crucial because it determines how resistant a bar is to buckling. A bar with a higher moment of inertia will resist buckling more effectively than a bar with a lower one.

What is the role of the factor α in Euler's equation?

The factor α represents the boundary condition of the bar and affects the length of buckling (L_p). It depends on how the bar is supported. For example, in a bar with both ends pinned, α = 1; for a bar with one end fixed and the other pinned, α = √2; and for a fixed-pinned bar, α = 2.

How does the modulus of elasticity (E) affect the buckling behavior of a material?

The modulus of elasticity (E) determines the material's stiffness. A higher modulus means the material can resist buckling more effectively, as it is less likely to deform under compression.

What is the critical load for a bar with the following properties: pinned-pinned support, section of 15 mm by 2 mm, length of 500 mm, modulus of elasticity 65,000 MPa, and a safety factor of 2?

The critical load is calculated using Euler's equation. After calculating the moment of inertia, substituting the values, and applying the appropriate factor α (for pinned-pinned supports, α = 1), the critical load comes out to be 51 N. After applying the safety factor, the admissible load is 25.5 N.

What does the safety factor do in the calculation of critical load?

The safety factor is used to ensure that the material can handle unexpected conditions. It is a margin of safety applied to the critical load, reducing the maximum allowable load to prevent failure due to unforeseen factors.

When is the Euler's equation valid, and are there any limitations?

Euler's equation is valid only for perfectly straight elements with a load collinear to the axis of the element, made of isotropic and homogeneous material. It assumes that the material has a constant modulus of elasticity, which isn't true once the material reaches its yield point. It also doesn't account for short bars that may fail by crushing rather than buckling.

What happens if a bar is too short to buckle?

For very short bars, they may fail by crushing due to compressive stress exceeding the material's yield strength before any buckling occurs. This is why, in addition to buckling calculations, a compression check is also necessary to ensure that the material does not exceed its elastic limit.