Cross sectional of the bike frame parts.

## Abstract

Construction of bicycles with bamboo frames has become an alternative to improve the quality of life of some communities, be friendly with the environment and be ecologically sustainable. However, the production of bike frames is made in an artisanal way and there are few antecedents that have proven their reliability. This work presents the evaluation and simulation of the mechanical behavior of bike frames made in bamboo. Three-points bending tests were performed using bamboo bars with similar dimensions to bike frames, and an equivalent elasticity modulus was determined and used as the input datum of a finite element model. A linear model material and beam elements were used to model the bike frame. Tests were performed using bike frames of bamboo applying loads greater than 7000 N, and the displacements were measured. The experimental displacements were used to calibrate the model, which consisted of modifying the rigidity of the connections until the displacements of the model fit near to 90%. The calibrated model was used for a fatigue simulation in order to predict the lifespan of the bike frame. Some technical values of bamboo bike frames were obtained so that these will allow them to define the technical characteristics of the product and guarantee their operating conditions.

### Keywords

- bicycle
- bike frame
- bamboo
- fatigue
- three-point bending tests
- finite element analysis

## 1. Introduction

Bicycles offer a cost-effective transportation alternative primarily for low-income communities [1]. Additionally, bikes have other advantages, such as zero greenhouse gas emissions, low-cost maintenance, quick displacement in high traffic zones, and physical fitness promotion for the users.

Conventional bike frames, using materials, such as steel [2], aluminum [2, 3], carbon fiber [4, 5], and titanium, have been studied via numerical model analyses and experimental tests. The numerical studies generally use the finite element method, and the tests generally obtain the static load carrying capacity; in this direction, the researches’ focus has been oriented to improving the relationship between the weight and the strength. However, the new research trends are focused on the replacement of the bike frame material using a low-cost alternative, like environmentally friendly materials, low weight and very attractive esthetically [6]. In this direction, the bamboo can become a good alternative.

There are more than 1000 species of bamboo around the world of which 70 are abundant in South America and Asia [7]. Bamboo is a natural fiber species that belongs to grass Poaceae family and subfamily Bambusoideae and grows in diverse types of climate. Compared to other trees, the bamboo has significant low density, high strength, and stiffness [8], most high growth rate (30–100 cm per day). Also, the bamboo plays an important environmental role by preventing ground erosion and landslides in mountainous zones and retaining significant amounts of water that restore ground conditions where it grows [9].

Currently, some companies and foundations have been building a bike frame using bamboo [10, 11, 12]. Locally, the bike frames are being manufactured using bamboo and their joints with a composite that uses an epoxy matrix reinforced with natural fibers (fique). However, the composite material joints are highly dependent on the geometric and material characteristics; for this reason, the testing of the actual bamboo bike frame is imperative. Some information can be found in the literature about the experimental strength of bamboo bike frame, although some analyses have been made using the finite element method [2, 13, 14, 15]. Once experimental data has been obtained, sensitivity and fatigue studies can be performed in order for the useful life of the bamboo bikes to be assessed.

In this direction, this work pretends to estimate the maximum allowable distance traveled by bikes made using bamboo frames. The general structural performance of the bike frames was evaluated under static and dynamic loads using experimental tests and the finite element method. Also, the mechanical strength of the joints was evaluated.

## 2. Methods and materials

Several steps were defined in order to perform the research (Figure 1). The first step was to determine the experimental properties of bamboo using a three-point bending test, measuring the displacement of some points of the bike under to external load and experimental modal analysis. The next step was to perform a static analysis of the bamboo bike frame using the finite element method. Using the experimental displacements, the finite element model was calibrated (step 3). The model calibrated was validated using the natural frequencies obtained experimentally (step 4). Finally, a fatigue analysis using the finite element method was performed using S-N curves reported and an estimation of the maximum distance that the bike can travel under the load conditions defined.

### 2.1 Geometry

The bamboo bike frames were made of bamboo tubes joined with a composite material, a resin as a matrix reinforced with fique (Figure 2).

A bamboo frame size M was used to perform the experiments and finite element analysis (Figure 3).

### 2.2 Experimental tests

#### 2.2.1 Three-point bending tests

Three-point bending tests were performed in order to characterize the structural behavior of bamboo. The load was applied perpendicularly at midspan of a simply supported beam according to ASTM D790-17 standards [16]; from this test Young’s modulus was obtained. Bending tests were performed using a universal test machine UTS 200.3. Wood blocks were placed on load application points in order to avoid the load concentration effects (Figure 4).

Sixteen bamboo specimens with two sets of radii were tested, eight samples with 15.5 mm of outer radius and 10.5 of inner radius and another eight samples with 10.5 mm of outer radius and 10.25 mm of inner radius, approximately. A monotonically increasing load was applied to bamboo specimens until fracture and load versus displacement curves were obtained. The Young’s modulus was calculated as explained below.

The *Y* displacement of the midspan of the specimen is:

Knowing the applied load (*P)* and the displacement (*Y)*, the Young’s modulus (E) must be obtained as:

where *L* is the distance between supports and *I* the inertia of the cross section. The inertia can be calculated as:

where *Douter *is the outer diameter and *Dinner *is the inner diameter.

#### 2.2.2 Experimental displacements

A universal test machine UTS 200.3 was used to obtain the experimental displacements of the A, B, and C points (Figures 5 and 6). Three load ramps were applied (2000, 3500, and 6500 N) on the seat tube (Figure 7) of the bike frame, and the displacements were measured. The experimental displacements of the bike frame were used to calibrate the numerical model.

#### 2.2.3 Experimental natural frequencies

To validate the finite analysis model of the bike, experimental modal analysis of the frame was performed to obtain their natural frequencies and compare it to the results of the finite element model.

The experimental frequencies were obtained via a mobile application VibSensor [17], loaded on cell phones. The software provides the natural frequency on the system and its direction as responses to impulse loads are induced on the frame by tapping the frame several times at different locations with different intensities. The planes were defined by the phone (Z direction is normal to the screen of the phone).

In order to prove the accuracy of the mobile application, a simple model was used. The simple model consisted of a steel plate. The plate was fixed on both ends and the cell phone placed on the mid-length; then a load was applied to excite the plate, and their experimental natural frequencies were obtained. The first experimental natural frequencies were 19 Hz (Figure 8) and 19.95 Hz from the finite element model (Figure 9). In this direction, the accuracy of the mobile application was near to 5%. For this reason, the results obtained from the mobile application can be considered acceptable, and it can be used to the bamboo bike frame.

A setup was designed to obtain the natural frequencies of the bamboo bike frame (Figure 10). Cell phones were placed on three tubes: seat stay, chain stay, and down tube.

The bike frame was supported simulating the real conditions: the rear dropouts were fixed along *x-* and *y*-axes, and the front dropouts were fixed along *y*-axis and free along x-axis. The bike frame was excited using a load, and the cell phones registered the accelerations from which modal frequencies were processed, via a mobile application VibSensor.

### 2.3 Finite element model

The finite element analysis was performed using Abaqus 6.14-3 [18]. A static load of 3500 N and a moment of 350,000 N-mm were applied to the frame at the saddle. The boundary conditions were defined as the rear dropouts were fixed along x- and y-axes, and the front dropouts were fixed along y-axis and free along x-axis (Figure 11). The elastic modulus was taken from the three-point bending tests.

The geometry was represented using straight bars and beam elements. A standard mesh was used in the model with an approximate global size of 60 mm per element. The cross section of each beam elements is shown in Table 1.

Name part | Outer radius (mm) | Thickness (mm) |
---|---|---|

Down tube | 15.5 | 5.o |

Top tube | 15.5 | 5.0 |

Seat tube | 15.5 | 5.0 |

Seat stay | 10.25 | 3.5 |

Chain stay | 10.25 | 3.5 |

Thicker seat joint | 25.0 | 10.5 |

Thinner seat joint | 13.0 | 3.25 |

Headset joint | 23.0 | 7.5 |

Thicker bottom bracket joint | 25.0 | 10.5 |

Thinner bottom bracket joint | 12.2 | 1.95 |

Dropouts joints | 10.5 | 0.25 |

#### 2.3.1 Model calibration

A SimFlow was created using iSight [19] and Abaqus software [18] in order to fit the displacement of the finite element model (A, B, and C points) with experimental values. The elastic module of the joints was considered as input, and the displacements of points A, B, and C were considered as output data.

#### 2.3.2 Model validation

The finite element model was validated comparing the experimental natural frequencies obtained in the 2.2.3 item with the modal analysis performed using Workbench/Ansys 19.0 software [20].

#### 2.3.3 Fatigue analysis

Due to the bicycle moving on the irregular road, this induces loads dependent on the time on the bike frame. The fatigue strength is generally represented using alternative stress versus the number of cycles diagram, called the S-N curve.

The structural performance of the bike is important to corroborate the quality of products and to assign warranties. Some standards were found as ASTM F2711-08 [21], F2043-13 [22] and EN 14766 [23]. The standards to evaluate the fatigue performance of bicycle frames require a test setup where the frame is positioned at its normal attitude with the rear dropouts is free to rotate but without translation, while the front axle is free to translate and rotate. In this way, the whole frame is free to bend as it is the case when used on a road.

Fatigue analysis was performed using Workbench/Ansys software [20]. A bar is used to simulate the seat-stem, and it is inserted at 70 mm of distance from the top of the seat tube. The load at the bar simulates the weight of the rider (Figure 11). Then 50,000 test cycles load between 0 and 1200 N are applied vertically downward using a 25 Hz of frequency. From a practical standpoint, this cyclic load regime seems arbitrary and not related to any particular road the bicycle may be traveling.

Due to the impossibility of performing experiments to obtain the S-N curve of the bamboo, the S-N curve reported by Song et al. [24] was used to perform the finite element analysis.

Applying the Palmgren-Miner’s [25] ratio to the bamboo bike frame, it was possible to determine their life in years:

where

#### 2.3.4 Dynamic analysis

A dynamic analysis of the bike frame was performed. A displacement vs. time function was applied (Figure 12) at the front and rear dropout nodes and spring elements used to simulate the tires and fork stiffness (Figure 13).

The displacement prescription was obtained from the technical specification of the speed reducer, 6 cm of height and 37 cm of width (Figure 12); also we used a bicycle speed of 25 km/h.

Using the road profile and knowing that the wavelength

The spring constant (Figure 13) was calculated using the vertical deflection of the tire of the bicycle measured at the laboratory under applying a weight. As results were obtained, K = 15.5 N/mm for the rear tire and K = 56.93 N/mm for the front tire. The model was restricted to all translation degrees on dropouts and fixed the displacement in the *Z*-axis on the fork.

## 3. Results

### 3.1 Three-point bending tests

The Young’s module was calculated using Eq. (2), and the displacements obtained from three-point bending tests (Figure 14). The average values were 9420.8 MPa for the large diameter bamboo and 12610.3 MPa for the small diameter (Table 2).

Thicker bamboo | Thinner bamboo | |
---|---|---|

Average [MPa] | 9420.8 | 12610.3 |

Standard deviation [MPa] | 459.4 | 1668.9 |

Coefficient of variation (%) | 4.9 | 13.3 |

Minimum [MPa] | 8589.3 | 10015.0 |

Maximum [MPa] | 10551.2 | 14764.8 |

Additionally, the coefficient of variation for a thicker bamboo was good (4.9) considering the variability of the bamboo as a biological material; instead, for thinner bamboo, the variability was 2.7 times higher (Table 2).

### 3.2 Experimental displacements

The maximum displacements were obtained on C point followed by B point and A point (Table 3). The behavior of the displacement versus load was lineal, to all points.

Load | Points displacement (mm) | ||
---|---|---|---|

A | B | C | |

2000 N | −0.06 | −2.19 | 5.09 |

3500 N | −2.89 | −4.14 | 9.36 |

6000 N | −5.17 | −7.43 | 16.34 |

### 3.3 Finite element model

#### 3.3.1 Model calibration

To calibrate the finite element model, a load of 3500 N and a moment of 35,000 N-mm were used. The target of the model was to fit the displacement of A point near to the experimental displacement, −2.89 mm along *y-*direction (Table 3). The range of Young’s modulus used to the joints was 6000–200,000 MPa. The solution converged for a range of Young’s modulus between 6000 and 9880 MPa (Table 4).

Zone | E (MPa) |
---|---|

Bottom bracket | 8061 |

Dropouts joint | 6000 |

Headset | 9880 |

Seat joint | 6000 |

#### 3.3.2 Model validation

Young’s modulus obtained from the calibration process (Table 4) was used to define the rigidity of the joints. The model was validated using the experimental natural frequency (Figure 15), and the first modal shape of the bike frame was out of the plane (*Z*) with a natural frequency of 38.0 Hz (Figure 15). For the finite element model, the first natural frequency was 35.0 Hz (Figure 16). The difference between the experimental result and the finite element model was 7.9%.

This difference may be due to the difficulty of exactly modeling the supports and material properties, specifically at the fork and the joints. These results show that the model represents satisfactorily the real bike frame. Then, these results can be used to fatigue analysis using the finite element method.

### 3.4 Dynamic response of the finite element model

Using the finite element model calibrated and validated, a dynamic response analysis was performed. The maximum reaction force at dropout joint zone was 114 N (Figure 17), the maximum stress was 114.6 MPa at the dropout, and the maximum displacement was 70.6 mm at the seat joint (Figure 18, Table 5). The results show that the maximum stress occurs at dropout joints; hereby, this joint requires special attention.

Control points | Maximum stress (MPa) | Maximum displacement (mm) |
---|---|---|

Seat joint | 15.6 | 70.6 |

Bottom bracket | 34.3 | 59.8 |

Fork | 19.6 | 34.0 |

Dropout | 114.6 | 0.0 |

From on the dynamic simulation of bike frame and assuming that the bike travel at 25 km/h, with a rider of 100 kg, and a road with contiguous bumps of 6 cm in height and 34 cm in length. The mechanical response at the intersection between the seat stay and the rear dropout was 114 MPa. Using a rate of application of 1351 cycles/km, 90 km is the distance a rider rides per week. Then, for a year the distance is 4680 km, and assuming a life of the bike of 10 years, the bike works:

Using the bamboo fatigue S-N curve reported by Song [24] and the stress profile obtained from the dynamic analysis (Figure 19), a fatigue analysis was performed. The life obtained was 1.7 × 10^{9} cycles, the damage ratio, N_{applied}/N_{allowed} = 0.037; this indicates that the bike frame would last 26 times longer than its intended use; then the fatigue life of the frame will be more than 100 years, for these work conditions.

## 4. Discussion and conclusions

The relatively large scatter obtained for the elastic moduli can be explained by the non-exactly replicated nature of the bamboo material from plant to plant [25]. The differences of Young’s moduli between thicker and thinner diameter bamboo specimens may be explained by the differences between the compaction of the bamboo structure or the relation between the thickness and diameter of the bamboo samples, depending on the zone of the stem where the specimens were extracted.

In general, bamboo is thicker at the top than at the base of the culms [25]. Comparing the structures for the different diameters, the thinner diameter bamboo (Figure 20) has a structure more compact than the thicker diameter bamboo (Figure 21) and consequently higher Young’s modulus [26].

The vascular bundle is a small longitudinal interstice of the bamboo stem (Figures 20 and 21). It affects directly the mechanical properties of the specimen due to these pores that act as stress concentrations. Kanzawa et al. [27] proposed to measure the maximum width and length of the vascular bundles (Figure 22). In this work, an average of 0.45 mm length and 0.38 mm width for thicker bamboo and 0.19 mm length and 0.14 mm width for thinner bamboo. Because the gap in the thinner bamboo is smaller than the thicker bamboo, the Young’s modulus in the thinner is going to be higher making it more rigid.

In addition, a dynamic simulation of the bamboo frame was performed to obtain the acting forces at the bike frame and thus the stresses, at the most critical joint entering the rear dropout. With this information, the generic specimen representative of this joint was prepared to generate additional fatigue data to evaluate the useful life of the frame in future works (Figure 23).

Finally, a methodology was proposed to evaluate the fatigue life of the bamboo bike frame from the experimental data reported for bamboo samples. However, the results should be taken as an approximation because the fatigue life of the bamboo bike frame has a high dependency on their joints. In this direction and in future works, it is necessary to perform fatigue experiments using the whole bike frame or at least to test the joints separated. This can be explained as there are substantial differences between the fatigue life of base material and the fatigue life of the final product. Also, some technical values of bamboo bike frames were obtained, so that these will allow them to define the technical characteristics of the product and guarantee their operating conditions.

## Acknowledgments

The author appreciates the support of the Universidad Autónoma de Occidente (Cali-Colombia).

## Conflict of interest

The author declares that there is no conflict of interest regarding the publication of this chapter.