IGCSE Design & Technology: Structures and Forces

Spaghetti Warren Truss Bridge Design

Complete Engineering Analysis for 2 kg Load Capacity

Concept Definitions and Fundamental Theory

What is a Truss?

A truss is a structural framework composed of triangulated elements (members) connected at joints (nodes). Trusses are designed to carry loads primarily through axial forces - either tension or compression - rather than bending moments.

Key Principles:

• Members are assumed to be connected by frictionless pins
• Loads are applied only at joints
• Members carry only axial loads (no bending)
• The structure is statically determinate

Warren Truss Characteristics

The Warren truss is a type of truss characterized by:

• Angled struts forming a series of equilateral or isosceles triangles
• No vertical members (except at supports)
• Alternating tension and compression in diagonal members
• Efficient load distribution through triangular geometry

Advantages:

• Fewer members than other truss types
• Good strength-to-weight ratio
• Uniform load distribution
• Suitable for moderate spans

Fundamental Structural Formulas

Static Equilibrium (Method of Joints)

For each joint in the truss:

• $\Sigma F_x = 0$ (Sum of horizontal forces = 0)
• $\Sigma F_y = 0$ (Sum of vertical forces = 0)
• $\Sigma M = 0$ (Sum of moments about any point = 0)

Stress and Strain Relationships

Stress: $\sigma = F/A$

where: $\sigma = \text{stress (Pa)}$, $F = \text{force (N)}$, $A = \text{cross-sectional area (m}^2)$

Strain: $\epsilon = \Delta L/L_0$

where: $\epsilon = \text{strain (unitless)}$, $\Delta L = \text{change in length (m)}$, $L_0 = \text{original length (m)}$

Hooke's Law: $\sigma = E \times \epsilon$

where: $E = \text{elastic modulus (Pa)}$

Buckling Analysis (Euler's Formula)

For compression members:

Critical Buckling Load: $$P_{cr} = \frac{\pi^2 \times E \times I}{(K \times L)^2}$$

where:

• $E = \text{elastic modulus (Pa)}$
• $I = \text{second moment of area (m}^4)$
• $K = \text{effective length factor (1.0 for pinned-pinned)}$
• $L = \text{member length (m)}$

Slenderness Ratio: $\lambda = KL/r$

where: $r = \text{radius of gyration} = \sqrt{(I/A)}$

Deflection Formulas

For simply supported beam under point load:

Maximum Deflection: $$\delta_{max} = \frac{P \times L^3}{48 \times E \times I}$$

For trusses (approximate):

$$\delta_{max} = \frac{P \times L^3}{192 \times E \times I_{\text{equivalent}}}$$

where: $$I_{\text{equivalent}} = \frac{A_{\text{chord}} \times h^2}{2}$$

h = truss height

Safety Factor

Factor of Safety (FS) = $\sigma_{\text{ultimate}}/\sigma_{\text{working}}$

Design Stress: $\sigma_{\text{allowable}} = \sigma_{\text{ultimate}}/\text{FS}$

Geometric Relationships for Warren Truss

Diagonal Length: $L_d = \sqrt{P^2 + h^2}$

where: $P = \text{panel length}$, $h = \text{truss height}$

Diagonal Angle: $\theta = \arctan(h/P)$

Load Types and Combinations

Dead Load (DL)

• Self-weight of the structure
• Permanent loads
• For spaghetti bridge: weight of pasta members

Live Load (LL)

• Applied loads during use
• Variable loads
• For bridge testing: the test weight

Load Combinations

• Service Load = DL + LL
• Design Load = (DL + LL) $\times$ Safety Factor

Material Properties Definitions

Tensile Strength

Maximum stress a material can withstand while being stretched before failure.

Compressive Strength

Maximum stress a material can withstand while being compressed before failure.

Elastic Modulus (E)

Measure of material stiffness - ratio of stress to strain in elastic region.

Second Moment of Area (I)

• For circular cross-section: $I = \pi \times d^4/64$
• For hollow circular: $I = \pi \times (D^4 - d^4)/64$

Where d = diameter, D = outer diameter

Design Specifications

• Bridge Type: Warren Truss
• Span Length: 600 mm
• Bridge Width: 100 mm
• Design Load: 2 kg (19.6 N)
• Safety Factor: 2.5
• Material: Standard spaghetti pasta

Step 1: Material Properties and Assumptions

Spaghetti Properties (Based on Testing Data)

• Diameter: 2.0 mm
• Cross-sectional Area: $A = \pi \times (1.0)^2 = 3.14 \text{ mm}^2$
• Tensile Strength: $\sigma_t = 45 \text{ MPa}$
• Compressive Strength: $\sigma_c = 35 \text{ MPa}$
• Elastic Modulus: $E = 12 \text{ GPa} = 12,000 \text{ MPa}$
• Density: $\rho = 1.5 \text{ g/cm}^3$

Design Parameters

• Bridge Length: $L = 600 \text{ mm}$
• Bridge Height: $H = 120 \text{ mm}$ (L/5 ratio for optimal efficiency)
• Panel Length: $P = 100 \text{ mm}$ (6 panels)
• Number of Trusses: 2 (parallel trusses connected by cross-bracing)

Step 2: Load Analysis

Load Calculations

Dead Load (estimated bridge weight):

• Spaghetti mass $\approx 80 \text{ strands} \times 0.6 \text{ m} \times 1.5 \text{ g/cm}^3 \times \pi \times (0.1)^2 \text{ cm}^2$
• Dead Load $\approx 0.25 \text{ kg} = 2.45 \text{ N}$

Live Load:

• Applied load = $2 \text{ kg} = 19.6 \text{ N}$

Total Service Load = $2.45 + 19.6 = 22.05 \text{ N}$

Design Load (with safety factor):

Design Load = $22.05 \times 2.5 = 55.1 \text{ N}$

Load Distribution

• Load applied at center of bridge (300 mm from each support)
• Load distributed equally between two trusses: $27.6 \text{ N}$ per truss

Step 3: Structural Analysis - Warren Truss

Geometry Setup

• Top Chord Nodes: T1, T2, T3, T4, T5, T6, T7 (7 nodes)
• Bottom Chord Nodes: B1, B2, B3, B4, B5, B6, B7 (7 nodes)
• Panel Points: 6 panels of 100 mm each
• Diagonal Angle: $\theta = \arctan(120/100) = 50.2^\circ$

Member Forces Using Method of Joints

Support Reactions:

• Left Support (vertical): $R_1 = 27.6 \text{ N} \uparrow$
• Right Support (vertical): $R_2 = 27.6 \text{ N} \uparrow$

Member Force Analysis (per truss):

• Bottom Chord Members (Tension):
  • B1-B2: $F = 27.6 \times 500/120 = 115.0 \text{ N}$ (Tension)
  • B2-B3: $F = 27.6 \times 400/120 = 92.0 \text{ N}$ (Tension)
  • B3-B4: $F = 27.6 \times 300/120 = 69.0 \text{ N}$ (Tension)
  • B4-B5: $F = 69.0 \text{ N}$ (Tension)
  • B5-B6: $F = 92.0 \text{ N}$ (Tension)
  • B6-B7: $F = 115.0 \text{ N}$ (Tension)
• Top Chord Members (Compression):
  • T1-T2: $F = 27.6 \times 500/120 = 115.0 \text{ N}$ (Compression)
  • T2-T3: $F = 92.0 \text{ N}$ (Compression)
  • T3-T4: $F = 69.0 \text{ N}$ (Compression)
  • T4-T5: $F = 69.0 \text{ N}$ (Compression)
  • T5-T6: $F = 92.0 \text{ N}$ (Compression)
  • T6-T7: $F = 115.0 \text{ N}$ (Compression)
• Diagonal Members:
  • Diagonal length = $\sqrt{100^2 + 120^2} = 156.2 \text{ mm}$
  • T1-B2: $F = 27.6/\sin(50.2^\circ) = 35.9 \text{ N}$ (Compression)
  • B1-T2: $F = 35.9 \text{ N}$ (Tension)
  • T2-B3: $F = 35.9 \text{ N}$ (Compression)
  • B2-T3: $F = 35.9 \text{ N}$ (Tension)
  • [Pattern continues for remaining diagonals]

Step 4: Member Design

Critical Member Analysis

Most Critical Member: Bottom chord B1-B2 (115.0 N tension)

Tension Member Design:

• Required stress capacity: $\sigma_{\text{required}} = F/A = 115.0 \text{ N} / 3.14 \text{ mm}^2$
• $\sigma_{\text{required}} = 36.6 \text{ MPa}$
• Available strength: $\sigma_{\text{allowable}} = 45 \text{ MPa} / 1.5 = 30 \text{ MPa}$ (Factor of safety = 1.5 on material strength)
• Since $36.6 > 30 \text{ MPa}$, we need multiple strands:
• Number of strands = $36.6 / 30 = 1.22 \approx 2$ strands minimum

Compression Member Design (Top Chord):

Critical buckling load for single strand:

$$P_{cr} = \frac{\pi^2 \times E \times I}{L^2}$$

Second moment of area: $I = \pi \times d^4/64 = \pi \times (2.0)^4/64 = 0.785 \text{ mm}^4$

For $L = 100 \text{ mm}$:

$$P_{cr} = \frac{\pi^2 \times 12,000 \times 0.785}{(100)^2} = 93.3 \text{ N}$$

Required load capacity = $115.0 \text{ N}$

Since $115.0 > 93.3 \text{ N}$, we need multiple strands:

Number of strands = $115.0 / 93.3 = 1.23 \approx 2$ strands minimum

Step 5: Final Member Design Summary

Bottom Chord (Tension Members):

• B1-B2, B6-B7: 2 spaghetti strands each
• B2-B3, B5-B6: 2 spaghetti strands each
• B3-B4, B4-B5: 2 spaghetti strands each

Top Chord (Compression Members):

• T1-T2, T6-T7: 2 spaghetti strands each
• T2-T3, T5-T6: 2 spaghetti strands each
• T3-T4, T4-T5: 2 spaghetti strands each

Diagonal Members:

• All diagonals: 1 spaghetti strand each (Force = 35.9 N < 93.3 N buckling capacity)

Step 6: Deflection Analysis

Maximum Deflection Calculation:

For warren truss under point load at center:

$$\delta_{\text{max}} = \frac{P \times L^3}{192 \times E \times I_{\text{equivalent}}}$$

Equivalent moment of inertia for truss:

$$I_{\text{equivalent}} = \frac{A_{\text{chord}} \times h^2}{2}$$

where h = truss height = 120 mm

$$I_{\text{equivalent}} = \frac{2 \times 3.14 \times 120^2}{2} = 45,216 \text{ mm}^4$$

$$\delta_{\text{max}} = \frac{27.6 \times 600^3}{192 \times 12,000 \times 45,216}$$

$$\delta_{\text{max}} = \frac{5,943,360}{104,186,880} = 0.057 \text{ mm}$$

Allowable deflection = $L/250 = 600/250 = 2.4 \text{ mm}$

Since $0.057 < 2.4 \text{ mm}$, Deflection is acceptable.

Step 7: Connection Design

Joint Analysis:

• Critical Joint: Center bottom joint (highest force concentration)
• Connection Method: Epoxy adhesive bonds
• Joint Reinforcement: Small cardboard gusset plates

Adhesive Joint Capacity:

• Shear strength of epoxy $\approx 20 \text{ MPa}$
• Required bond area per connection: $A_{\text{bond}} = F_{\text{max}} / \sigma_{\text{shear}} = 115.0 \text{ N} / 20 \text{ MPa} = 5.75 \text{ mm}^2$
• Minimum bond length = $5.75 / (\pi \times 2.0) = 0.91 \text{ mm}$
• Recommended bond length = 10 mm (safety factor > 10)

Step 8: Material Quantity Calculation

Total Spaghetti Required:

• Bottom Chord: 6 members $\times 100 \text{ mm} \times 2 \text{ strands} = 1,200 \text{ mm}$
• Top Chord: 6 members $\times 100 \text{ mm} \times 2 \text{ strands} = 1,200 \text{ mm}$
• Diagonals: 12 members $\times 156.2 \text{ mm} \times 1 \text{ strand} = 1,874 \text{ mm}$
• Cross Bracing (between trusses): 8 cross members $\times 100 \text{ mm} \times 1 \text{ strand} = 800 \text{ mm}$
• Total Length Required: $5,074 \text{ mm} = 5.07 \text{ m}$

Number of Standard Spaghetti (250mm long): 21 pieces minimum

Recommended Purchase: 30 pieces (includes waste factor)

Step 9: Construction Guidelines

Assembly Sequence:

• Cut all members to required lengths
• Create assembly jig with precise node locations
• Assemble bottom chord first
• Add vertical supports temporarily
• Install diagonal members
• Install top chord
• Add cross-bracing between trusses
• Apply reinforcing gussets at critical joints
• Allow adhesive to cure for 24 hours before testing

Quality Control Checks:

• Verify all joint alignments
• Check member lengths $\pm 1 \text{ mm}$
• Ensure no twisted or bent members
• Test joint strength on sample connections

Step 10: Expected Performance

Load Capacity Analysis:

• Theoretical Capacity: $55.1 \text{ N}$ (with SF = 2.5)
• Expected Test Capacity: $60-70 \text{ N}$
• Target Load: $19.6 \text{ N}$ (2 kg)
• Performance Margin: 3.0x - 3.5x above required load

Weight Estimate:

• Total Bridge Weight: $\approx 250-300$ grams
• Strength-to-Weight Ratio: $65-78 \text{ N/kg}$

Conclusion

This warren truss design provides a robust solution for supporting 2 kg with significant safety margin. The dual-strand approach for chord members ensures adequate capacity while maintaining reasonable material usage. The design balances structural efficiency with practical construction considerations for a spaghetti bridge competition.