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.

Structural Evaluation of Bamboo Bike Frames: Experimental and Numerical Analysis DOI: http://dx.doi.org/10.5772/intechopen.89858

#### Figure 11.

Bike-frame finite element model.


#### Table 1.

Cross sectional of the bike frame parts.

#### 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:

$$\text{Damage} = \left(\frac{\text{N}\_{applied}}{\text{N}\_{allow}}\right) \tag{4}$$

where Napplied is obtained from the dynamic simulation obtained from the finite element analysis of the bicycle traveling a given distance, at a given speed, over a road of given characteristics, whereas Nallowed is defined by the S-N curve. Where Napplied is in cycles per kilometer.

#### 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 λ ¼ 0:74 m, the frequency f ¼ 9:57 Hz and the angular velocity ω ¼ 9:57 Hz; the movement equation is:

$$y = 0.06 \sin 58.92t \tag{5}$$

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.

Figure 12. Road profile (distances in mm).

Structural Evaluation of Bamboo Bike Frames: Experimental and Numerical Analysis DOI: http://dx.doi.org/10.5772/intechopen.89858

Figure 13. Finite element model of the bike.

### 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).

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.

Figure 14. Typical load vs. displacement curve of bamboo specimen, three-point bending test.


### Table 2.

Longitudinal Young's module for bamboo.


#### Table 3.

Average of vertical (A and B) and horizontal (C) displacements.
