How to use the Legrange method to solve optimisation cantilever beam?

In summary, the minimum weight cantilever beam consisting of three steps that meets the condition of strength is determined by beam diameters d1, d2, and d3, length x1 of the end shoulder, coordinate x2 that determines the length of the middle shoulder, and the method of Lagrange Multiplier.
  • #1
lak91
4
0
esign a cantilever beam of minimum weight (volume) consisting of
three steps that meets condition of strength. Given parameters are: total length of the beam
L, force F at the end of the beam, and allowable stresses [sigma].

Parametres to be determined:
Beam Diameters d1 d2 d3
Length x1 of the end shoulder
Coordinate x2 that determines the length of the middle shoulder

How to solve using Legrange!

Conditions of strength
σ A = Ma/Za = (32 x F x L) / (pi x d1 ^3)
σ B= MB/ZB = (32 x F x L) / (pi x d2 ^3)
σ C = Mc/Zc = (32 x F x L) / (pi x d3 ^3)

where za = (pi x d1^3)/32 zB = (pi x d2^3)/32 zC = (pi x d3^3)/32 section modulus


How would I use the legrange method to derive expressions for d1,d2,d3 and x2 minimum eight (volume) of the beam for the given L,F, σ
 
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  • #2
A, σB and σC? The Lagrange method involves setting up an objective function that must be minimized. The objective function is: F = (π/32) x [d1^3 + d2^3 + d3^3] - w [x1 + (L - x2) + (L - x1 - x2)]where w is the weight of the beam per unit length. The constraints are: σ A = Ma/Za = (32 x F x L) / (pi x d1 ^3)σ B= MB/ZB = (32 x F x L) / (pi x d2 ^3)σ C = Mc/Zc = (32 x F x L) / (pi x d3 ^3)The Lagrange multipliers then need to be determined using the method of Lagrange Multiplier. This involves solving the following system of equations: F = (π/32) x [d1^3 + d2^3 + d3^3] - w [x1 + (L - x2) + (L - x1 - x2)] + λ1 [σA - (32 x F x L) / (pi x d1^3)] + λ2 [σB - (32 x F x L) / (pi x d2^3)] + λ3 [σC - (32 x F x L) / (pi x d3^3)]0 = λ1 + λ2 + λ3Once the Lagrange multipliers have been determined, the optimal values of d1, d2, d3 and x2 can then be calculated by substituting the values of the Lagrange multipliers back into the objective function.
 

1. What is the Legrange method?

The Legrange method is a mathematical technique used to solve optimization problems, which involve finding the maximum or minimum value of a function subject to certain constraints. It involves using the method of Lagrange multipliers to convert a constrained optimization problem into an unconstrained one.

2. How does the Legrange method specifically apply to solving optimization problems for cantilever beams?

The Legrange method can be applied to solving optimization problems for cantilever beams by using the beam's governing equation, which is a fourth-order differential equation, as the objective function. The constraints, such as the beam's length and load, can be incorporated into the method using Lagrange multipliers to find the optimal solution for the beam's dimensions.

3. What are the advantages of using the Legrange method for solving optimization problems for cantilever beams?

The Legrange method offers several advantages for solving optimization problems for cantilever beams. It allows for the incorporation of multiple constraints, including non-linear ones, into the optimization problem. It also provides a systematic approach to finding the optimal solution and can handle complex objective functions. Additionally, the method can be applied to a wide range of optimization problems, making it a versatile tool for engineers and scientists.

4. Are there any limitations to using the Legrange method for solving optimization problems for cantilever beams?

As with any mathematical method, there are some limitations to using the Legrange method for solving optimization problems for cantilever beams. One limitation is that it may not always provide the global optimum solution, as it relies on finding the critical points of the objective function. There may also be cases where the method is unable to find a solution or produces incorrect results. In these situations, alternative methods may need to be used.

5. Can the Legrange method be applied to other types of optimization problems?

Yes, the Legrange method can be applied to a wide range of optimization problems, not just for cantilever beams. It is commonly used in engineering, economics, physics, and other fields to solve optimization problems with multiple constraints. However, the specific approach and equations used may vary depending on the problem at hand. It is always important to carefully consider the problem and determine if the Legrange method is the most appropriate approach.

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