A PLANE TRUSS STRUCTURE

Delft University Masterclass, discusses a science application for optimizing the topology, size, and shape of plane trusses using genetic algorithms and finite element analysis to minimize structural mass while adhering to stress and displacement constraints. About members, nodes and joints.

The encoding technique used in this study involves representing the truss structure as a chromosome in the genetic algorithm (GA). 

1. **Parameterization**:

   - The truss structure is parameterized with a variable number of spans, each span having adjustable size and several parameters.

   - Each span contains 8 parameters: vertical position of the bottom node (VPi), length of the vertical bar (VLi), sectional areas of the vertical bar (AVi), bottom bar (ABi), top bar (ATi), diagonal conformation (Di), and sectional areas of the diagonal bars going down (AD1i) and up (AD2i).

   - The (n+1)th span only contains 3 parameters for the last vertical bar.

2. **Chromosome Representation**

   - Each truss design (individual) is represented as a chromosome, which is a vector of genes.

   - Each gene corresponds to one parameter of the truss structure.

   - The chromosome length is fixed and calculated as (n*8 + 3), where n is the number of spans.

3. **Gene Arrangement**

   - Genes are arranged in the chromosome in a left-to-right order corresponding to the structure parameters.

   - This arrangement helps maintain related genes close to each other, improving the algorithm’s performance.

4. **Feasible Value Intervals**

   - Each gene is bounded within an interval of feasible values.

   - For example, VP can range from [0, H-Lmin], VL from [Lmin, H], D from (0, 3], and sectional areas from [Amin, Amax].

5. **Mixed Integer-Real Representation**

   - MATLAB works with real variables, but the diagonal conformation parameter (D) needs to be an integer.

   - D is represented as a real value in the interval (0-3] and rounded up during decoding to get integer values {1, 2, 3}.

6. **Flexibility in Optimization**

   - The technique allows fixing certain parameters to specific values while optimizing the rest.

   - This is achieved by supplying an array with the span number, parameter number, and desired value, which is used during the coding and decoding process.

This encoding technique allows the simultaneous optimization of the topology, size, and shape of the truss structure, providing a flexible and efficient approach to structural optimization using genetic algorithms.

The document discusses a science application for optimizing the topology, size, and shape of plane trusses using genetic algorithms and finite element analysis to minimize structural mass while adhering to stress and displacement constraints.

The fitness value of a truss is determined by calculating its total mass and evaluating how well it conforms to the maximum stress and maximum allowable displacement constraints. The stresses and displacements are evaluated using the finite element method (FEA). 

If the truss violates any constraints, a penalty function is applied to the fitness value. The initial fitness is the mass of the truss, and the penalty function increases the fitness value based on the extent of the constraint violations, making the fitness value higher (worse) for nonconforming trusses.

Part I -16-  Chap. 3

FINITE ELEMENT METHOD FOR A TRUSS STRUCTURE

Basically, the structure of a finite element program is independent of the complexity of the

specific elements. The calculational procedure of a program for a simple plane truss structure

is the same as the calculational procedure of a program for the more complex plate and shell

elements. In this chapter the procedure will be introduced with the consideration of a simple

plane truss structure.

3.1 A plane truss structure

The structure to be analyzed consists of a series of plane truss members connected to each

other along a straight line and loaded by a uniformly distributed loading f(x) 2 F₁ F2

ول دی 2 f(x)→ figure 3.1

The structure has been subdivided into elements, the loading has been concentrated at the joints

We subdivide the structure into three elements and four joints. The distributed loading will

be replaced by statically equivalent forces at the nodes 1, 2, 3 and 4. The force at node 4 does

not influence the structure and will be neglected.

The forces F1, F2, and F3, applied in the positive x - direction, are positive. These external

forces yield a reaction force R4. The reaction force, also an external force of the structure, is

positive with the positive action direction. With positive forces F1, F2, F3, the reaction force

R4 is certainly negative. The joints are displaced by the quantities u₁, u₂, u3 and u

(although u₁ = 0) in the direction of the applied forces. These joint displacements are called

the degrees of freedom (d.o.f.).

The penalty function handles constraint violations by increasing the fitness value of a truss based on the extent of the violation. 

1. **Initial Fitness**

The initial fitness of an individual truss is its total mass, which needs to be minimized.

2. **Penalty Application**

If the truss violates stress or displacement constraints, a penalty is added to the initial fitness value. The penalty function is designed to be parabolic, meaning it increases more rapidly as the violation becomes more severe.

3. **Penalty Calculation**

The penalty function is defined as: - PF(V) = OF + IP * (V/Va - 1)^2

   - Where PF is the penalized fitness, OF is the original fitness (mass), IP is the initial penalty at the point of stress limit, V is the actual maximum stress or displacement, and Va is the allowable stress or displacement.

4. **Constraint Violation Handling**

   - For small violations, the penalty is relatively low, allowing potentially good designs with minor violations to remain in the gene pool.

   - For larger violations, the penalty increases significantly, discouraging the selection of highly nonconforming designs.

This approach ensures that the optimization process favors designs that meet the constraints while still exploring potentially promising solutions that are close to the limits.

The score of the individuals is considered 1/Sqr(rank), making the distance between the fitness values larger for top individuals and increasingly smaller for the less fit ones. The algorithm uses elitism in order to preserve the best individuals. This means that a portion of them automatically survives unaltered to the next generation. 

Generally their used an elite count of 2, meaning the 2 fittest individuals are guaranteed to be found in the next generation, where it’s possible to be replaced by even fitter children. In order to exploit the good solutions found so far, a portion of the children for the new generation are generated using two-point crossover. 

The principle behind this operation is illustrated in Fig. 5a. For each two selected parents, two integer values between 1 and size of chromosome are randomly selected. Using these two values, the algorithm recombines the genes from the two parents to form two children, as shown in the picture. The mutation operator is crucial for the algorithm to explore new portions of the design domain and not fall into the trap of local minima. Reference [13].

Part I -17- Chap. 3

3.2 Members, nodes and joints. For a better understanding of the f.e.m. it is useful to subdivide the structure into nodes and elements. The nodes are separate from the member elements and independent particles of the structure. The four nodes are labelled by numbers, the elements by the letters a, b and c.

O 2. (b) 2 Nodes and elements are independent structural components figure 3.2 For each element we introduce a local numbering for the connection points between the member and the nodes.

1 Start figure 3.3 an element e with local start point 1 and end point 22 end. External loading should be applied at the joint only. An external distributed loading applied to a member, must be replaced by concentrated joint loads which are statically equivalent to the distributed loading. At the connection between node and element, the node and the element apply loads to each other. 

For the global structure, these loads are internal forces

Delft University Masterclasses

A Plane truss Structure

The Individual Study Parts

On the Degrees of Freedom

These element or member forces are positive with the positive x - direction. The reaction forces at the nodes do act, of course, in opposite directions for optimizing the topology, size, and shape of plane trusses using genetic algorithms and finite element analysis to minimize structural mass while adhering to stress and displacement constraints.