Truss Optimizer

• Truss Optimizer
  • Problem Definition
  • Cost Model
    • Simple Cost Function
  • Constraint Model
    • Failure Modes (TODO)
      • Euler Buckling Load Estimation
  • Computing Problem
    • Input Specification
    • Output Specification
  • Appendix: Material Properties
    • Balsa wood Properties
    • Dowel Pin Properties

Problem Definition

$PV$ is the largest load to weight ratio. The main objective is to maximize $PV$. Since there are way too many factors to incorporate to simulate an actual load-to-weight ratio, a simplified truss and cost/weight model is required. The goal of this report is to find well-defined cost function (i.e. weight function) for any truss, and use the existing optimization algorithm to maximize the truss' load to cost ratio.

Cost Function is now a synonym for Weight Function.

Cost Model

Consider a truss, $T=(\mathcal J, \mathcal M)$ where $\mathcal J$ and $\mathcal M$ are the set of joints and members. Let $\mathcal J = {j_1, j_2,\dots,j_r}$, $\mathcal M={m_1, m_2,\dots,m_k}$ where $r,k\in\mathbb Z^+$. Let $L(m_1)$ denotes the length of member $m_1$, and let $F(m_1)$ denotes the internal force of $m_1$.

Simple Cost Function

Since a member can have variable thickness, i.e. more internal force in a member means thicker the member should be. I therefore assume that $F(m_i) \propto$ Volume/Weight of $m_i$.

Define function $C(T)$ as the cost function of the truss:

$$ C(T) = \sum_{i=1}^r F(M_i)\times L(M_i) $$

This cost function is easy to implement, with the downside being it completely ignores the extra weight iccured by $\mathcal J$.

Constraint Model

• Truss should be $40\pm 1$ cm long and $10\pm 0.5$ cm tall truss that can handle the largest load-to-weight ratio.
• The truss should be loaded at the middle of the bridge.
• A minimum load-to-mass ratio of 100 is required, this is the lower threshold requirement.

Failure Modes (TODO)

• Member in tension: Rupture and Member Tearing
• Member in compression: Buckling and Bearing Stress
• Pin Shear (bending, don't worry about it here)

Euler Buckling Load Estimation

This is an over estimation

$$ P_{cr} = \frac{\pi^2EI}{(KL)^2} $$

Where $P_{cr}$ is Euler's critical load, $E$ is modulus of elasticity of column material, $I$ is minimum area moment of inertia of the cross section of the column, $L$ is unsupported length of column, $K$ is column effective length factor.

Computing Problem

Input Specification

Truss configuration should be entered as a textfile. To run the application, enter truss-optimizer some_truss.txt in the command prompt.

The first line consists of 3 inputs $P$, $Q$, and $R$. $P$ is the number of joints, $Q$ is the number of members, and $R$ is the number of loaded members.

The next $P$ lines are the $x, y$ coordinate of joints, each line contains 2 numbers and 2 characters 'T'/'F' (separated by space). If the first character is T, it means its a free joint. if the second character is T, it means it's the highest point that should be used to maintain the dimension of the truss bridge. Note that each joints entered are indexed from $0$ to $(P-1)$.

The next $Q$ lines are the member connections of joints, each line contains 2 numbers, and each number represents the index of a joint entered above.

The next $R$ lines consists of 2 numbers each, the joint number and the vertical load. Note that the vertical load in the up direction is positive and int he down direction is negative.

Example:

3 3 3
0 0 F F
20 10 T T
40 0 F F
0 1
1 2
0 2
0 1
1 -2
2 1

Output Specification

Output the most optimized truss configuration to optimized.txt in the following format:

• First Line: Score: $SCORE where $SCORE is Cost divided by the Force applied.
• Next $P$ lines: ($X, $Y) representing the coordinate of each joint.

On the console, print whatever.

Appendix: Material Properties

Balsa wood Properties

$$\rho = 123\pm 23\ kg/m^3$$ $$E=3.51\pm 1.05\ GPa$$ $$\sigma_{ut} =7.7 \pm 2.5\ MPa$$ $$\tau_{ut}=1.6\ MPa$$

Dowel Pin Properties

$$\rho = 650\ kg/m^3$$ $$E=17\ GPa$$ $$\sigma_{ut} =171\ MPa$$ $$M_{max} = 368\ N.mm$$ $$\tau_{ut} = 23\ MPa$$