# Study on the Style Design and Anchoring Mechanism of Enlarged Head Anchors

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis and the Style Design Principle of Enlarged Head Anchors

- ①
- Calculation of the pull-out bearing capacity of the anchor rod reinforcement:

- ②
- Calculation of the pull-out bearing capacity between grout and reinforcement in the anchor section of the bolt:

- ③
- Calculation of the pull-out capacity of the anchor under the condition of strength restriction of the surrounding soil:

## 3. Calculation of the Pull-Out Force of an Enlarged Head Anchor

#### 3.1. Derivation and Calculation of the Pull-Out Force Formula

- ①
- Cylinder-shaped:

- ②
- Frustum of a cone:

- ③
- Stepped shape:

- ④
- Semi-ellipsoid:

_{1}is derived as follows:

#### 3.2. Calculation and Comparison of the Pull-Out Force

- (1)
- The embedment depth of the anchor bolt in the soil is substantial;
- (2)
- The influence of the part above the enlarged head on the stress state of soil is very minimal and can be ignored;
- (3)
- When pressure is applied to the soil element in a certain direction, the soil element produces lateral pressure on the plane perpendicular to this direction. Let the lateral pressure coefficient be ξ, and the magnitude of ξ is the same in all directions.

## 4. Laboratory Model Test of the Enlarged Head Anchor

#### 4.1. Model Design

- (1)
- Similarity ratio calculation

^{a}, L

^{b}, γ

^{c}, E

^{d}, μ

^{e})

^{−2}], [P] = [F], [L] = [L], [γ] = [FL

^{−3}], [E] = [FL

^{−2}], and [μ] = [F

^{0}L

^{0}].

_{1}, π

_{2}, π

_{3}, and π

_{4}are the similarity criteria for each physical quantity.

_{L}=$\frac{1}{20}$, C

_{μ}= 1, and $C$

_{γ}= 1.

- (2)
- Model design and production
- (1)
- Design and fabrication of model box soilThe physical and mechanical parameters for the actual prototype soil mass are shown in Table 5. According to the calculation of the similarity ratio, C
_{σ}= 20, C_{γ}= 1, C_{E}= 2, and C_{μ}= 1. Compared with the prototype soil mass in engineering practice, the reduction parameters for the model’s soil mass are C and E. Considering practical feasibility, in the process of soil preparation, the mix ratio of the model soil was restructured and adjusted many times, and soil weight, cohesion, internal friction angle, and the elastic modulus of the soil were measured. The soil preparation closest to meeting the requirements of the similarity ratio was selected, and the final ratio was determined as sand–soil–water = 1:0.18:0.95. The physical and mechanical parameters for the designed model soil mass are shown in Table 6.Severe γ (kN/m ^{3})Cohesion C (kPa) Angle of Internal Friction φ (°) Poisson’s Ratio μ Modulus of Elasticity E (MPa) 18.5 20 17 0.35 42 Severe γ (kN/m ^{3})Cohesion C (kPa) Angle of Internal Friction φ (°) Poisson’s Ratio μ Modulus of Elasticity E (MPa) 18.5 2.2 17 0.35 5.5 - (2)
- Design and fabrication of the model box and anchor rod

#### 4.2. Test Method

- (1)
- Test soil is poured into the transparent box. When the filling height reaches a certain value, the enlarged head anchor composite is buried at the center of the transparent box, and soil continually fills the box until its height reaches a certain value; finally, the soil is leveled.
- (2)
- One end of the traction rope is connected to the round hole at one end of the anchor rod, and it is connected with the anchor rod. The other end is connected with the weight disc, and the middle section of the traction rope is erected on the upper end and the side surface of the external support frame by virtue of the cross beam and the pulley frame. The overall diagram of the device is shown in Figure 5.

- (3)
- According to a certain weight gradient, more weights are gradually added to the weight plate; when the anchor is pulled out uniformly, weights are no longer added. The total mass of the weight plate is obtained using a tensiometer.
- (4)
- The anchor rod is replaced, steps 1, 2, and 3 are repeated, and the total mass value for the weight plate and weight when different anchor rods are pulled is recorded.

#### 4.3. Results and Analysis

_{D}in the derivation formula uses a burial depth of 5 m, while the burial depth of the anchor rod in the test is 1 m.

## 5. Analysis of Factors Influencing the Anchoring Mechanism of an Enlarged Head Anchor

#### 5.1. Model Building and Parameter Selection

#### 5.2. Influencing Factors and Measurement Point Arrangement

- (1)
- When monitoring the anchor shaft force, the measurement points are arranged on the center axis of the inner anchorage section with the starting point coordinates (0, 14, 0) and the ending point coordinates (0, 14 + L, 0). The specific arrangement is shown in Figure 10a.
- (2)
- When monitoring the shear stress on the outer side of the anchor, the measurement point is arranged at the outer edge of the inner anchorage section, with the coordinates of the starting point (0, 14, r) and the coordinates of the ending point (0, 14 + L, R), as shown in Figure 10b.
- (3)
- To monitor the soil y-directional axial force and shear force more comprehensively, the measurement points are arranged mainly along two directions. For the first direction, the monitoring starting point coordinates are (0, 5, 0.5) and the end point coordinates are (0, 15, 0.5), hereinafter referred to as the horizontal direction; for the second direction, the monitoring starting point coordinates are (0.5, 15, 3) and the end point coordinates are (0.5, 15, 0), hereinafter referred to as the vertical direction. The specific arrangement is shown in Figure 10c.

#### 5.3. Results and Analysis

- (1)
- Influence of inner anchorage section rear section radius R

_{y}under the action of the enlarged head anchor for the radius of section R behind the inner anchorage section at 0.4, 0.45, and 0.5 m.

_{y}around the inner anchorage section increases and then decreases along the horizontal direction from the end of the inner anchorage section to the outer anchorage head, with the peak value appearing at the front of the inner anchorage section, and gradually increases along the vertical direction from the position far from the anchor rod to the position near the anchor rod, i.e., the phenomenon of high in the middle and low in the surroundings; furthermore, with the gradual increase in radius R of the section after the inner anchorage section, stress σ

_{y}around the anchor position generally decreases, while stress σ

_{y}around the anchor position increases, i.e., the stress distribution spreads outward. The analysis shows that with the increase in radius R of the rear section of the inner anchorage section, the pressure value for the soil at the center is gradually shared by the side of the inner anchorage section, the stress concentration phenomenon is weakened, soil stress σ

_{y}is spread from the anchor rod to the surrounding distribution, and the strength of the surrounding soil is more fully utilized, so the increase in radius R of the rear section of the inner anchorage section of the anchor rod improves the soil reinforcement effect.

- (2)
- Analysis of the effect of the rate of change in the anterior and posterior sections

_{y}stress under the action of the enlarged head anchor with different front and rear section change rates (front section radius at 0.2, 0.25, and 0.3 m).

_{y}applied to the soil gradually increases along the horizontal monitoring point, with the peak value appearing at the front end of the enlarged section, and soil stress σ

_{y}gradually increases along the vertical direction from the far anchor end to the near anchor end. Further, unlike the effect of the change in radius R of the section after the inner anchorage section, the change in the front and rear section change rate causes soil stress σ

_{y}to vary in the range L~1.3L from the axis direction of the front end in the inner anchorage section, i.e., the change in the front and rear section change rate has a greater effect on the force on the soil in front of the front end of the inner anchorage section. The analysis shows that under the condition that the change rate of the front and rear sections decreases (the radius of front section increases), the y-directional stress value for the soil body is gradually transferred to the direction of the outer anchor head, and the soil body is subjected to more reasonable axial stress, which has a positive effect on the reinforcement effect of the soil body.

- (3)
- Analysis of the effect of length L of the enlarged section

_{y}under the action of the enlarged head anchor for the inner anchorage section length L at 3, 4, and 5 m.

_{y}in the soil around the inner anchorage section gradually decreases, and there is a tendency for the peak position to shift toward the outer anchor head direction (the offset direction shown in the figure). Correspondingly, the longer the inner anchorage section is, the larger the distribution range in the compressive zone formed in the soil body along the horizontal direction, and the smaller the range in the tensile zone formed at the rear (the range gradually decreases, as shown in the figure), and the lower the peak maximum tensile stress. Combined with Figure 24, it can be seen that the increase in length L of the expanded section reduces the peak stress σ

_{y}in the near-anchored soil body.

## 6. Conclusions

- (1)
- Under the condition that the volume and length of the extended anchorage section are the same, the ultimate pulling capacity of the cylindrical enlarged head anchor is generally less than that of the circular, stepped, and semi-elliptic enlarged head anchor, and its value is about 0.2~0.5 times the latter three. Moreover, it increases with an increase in the anchoring section’s length. The ultimate uplift capacity of the circular table-shaped enlarged head anchor is obviously higher than that for the stepped, semi-elliptic and cylindrical enlarged head anchors. The ultimate uplift capacity of the circular table-shaped enlarged head anchor increases with the increase in anchorage length under the constant volume condition of the enlarged head anchor. The ultimate pulling capacity of a stepped enlarged head anchor is affected by its order and length of the anchoring section. With the increase in order n and anchoring section length L, the ultimate pulling capacity of the stepped enlarged head anchor increases. The ultimate pulling capacity of the semi-elliptic enlarged head anchor falls between the cylindrical and stepped enlarged head anchors, and its variation law is consistent with that for circular and stepped bolts. The longer the inner anchorage section, the larger the distribution range in the compression zone formed in the soil body, and the smaller the range in the tension zone formed in the rear section. The increase in the length of the inner anchoring section helps improve the reinforcement effect of the soil in front of the inner anchoring section; thus, the parameter plays an important role in the redistribution of soil that experiences the force.
- (2)
- The drawing capacity of the expanded head anchor is affected by the bearing capacity of the front and rear ends, the side bearing capacity, and the side friction resistance. For circular anchor bolts, stepped anchor bolts, and semi-ellipsoidal anchor bolts, with the increase in front section radius r, the lateral friction resistance in the inner anchoring section is gradually shared by the bearing force of the front end of the inner anchoring section; the front-end-bearing effect of the inner anchoring section is enhanced; the bolt’s pulling performance is enhanced. Therefore, the pull-out force of the circular rock bolt is the greatest, followed by that of the stepped rock bolt, and the pull-out force of the semi-elliptic rock bolt is the lowest. The increase in rear section R can provide greater lateral friction resistance and rear-end bearing capacity. Compared with cylindrical enlarged head anchors and circular, stepped, and semi-elliptic enlarged head anchors, although the front section is smaller, the rear section is larger. The bearing capacity of the front section decreases less than the side bearing capacity, and the rear bearing capacity increases; thus, the cylindrical bolt has the least pulling force. Compared with front radius r, back radius R has more influence on the drawing ability of the enlarged head anchor.
- (3)
- The numerical calculation and analysis model of the anchor rod with expanded horizontal pullout head was established by using the numerical calculation software FLAC 3D (Version 5.0.), and the effects of the parameters such as the diameter of the rear section of the inner anchorage section, the ratio of the radius of the front and rear sections, and the length of the inner anchorage section on the soil reinforcement effect, together with the force characteristics of the anchor rod itself, were analyzed in depth, and the significance of the effects of each parameter was compared. The results showed that increasing radius R of the rear section of the inner anchorage section helped to improve the reinforcement effect of the soil perpendicular to the distribution direction of the anchor rod; decreasing the change rate in the front and rear sections and increasing length L of the inner anchorage section helped to improve the reinforcement effect of the soil in front of the inner anchorage section of the anchor rod, and the latter was better than the former.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

- (1)
- The derivation of the formula for the back section radius of the inner anchorage section of the stepped enlarged head anchor rod.The volume of the stepped anchor rod is as follows:$$\begin{array}{c}V={\displaystyle \sum _{k=1}^{n}}\pi {{r}_{k}}^{2}h\\ =\left[{\left(\frac{{R}_{step}-\left(n-1\right)r}{n}\right)}^{2}+{\left(\frac{2{R}_{step}+\left(n-1\right)r}{n}\right)}^{2}+\dots +{\left(\frac{\left(n-1\right){R}_{step}-r}{n}\right)}^{2}+{{R}^{2}}_{step}\right]\pi \frac{L}{n}\end{array}$$The following is obtained:

- (2)
- The derivation of pulling force ${T}_{1}$ by the side friction of the stepped enlarged head anchor:$$\begin{array}{c}{T}_{1}=2\pi {\displaystyle \sum _{k=1}^{n}}\left({R}_{k}\frac{L}{n}\right){\tau}_{f}\\ =2\pi \frac{L}{n}{\tau}_{f}\left[\frac{{R}_{step}+\left(n-1\right)r}{n}+\frac{2{R}_{step}+\left(n-2\right)r}{n}+\dots +\frac{\left(n-1\right){R}_{step}+r}{n}+\frac{n{R}_{step}}{n}\right]\\ =2\pi \frac{L}{{n}^{2}}{\tau}_{f}\left[\left(1+2+3+\dots +n\right){R}_{step}+\left(1+2+3+\dots +\left(n-1\right)\right)r\right]\\ =2\pi \frac{L}{n}{\tau}_{f}\left(\frac{n+1}{2}{R}_{step}+\frac{n-1}{2}r\right)\\ =2\pi \frac{L}{n}\left(\frac{n+1}{2}{R}_{step}+\frac{n-1}{2}r\right)\frac{{f}_{mg}}{K}\psi \end{array}$$
- (3)
- The derivation of pulling force ${T}_{1}$ by the side friction of the semi-ellipsoid head anchor. We obtain the following:$$\begin{array}{c}z=z\left(x,y\right)=c\sqrt{1-\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}}\\ {T}_{1}={\tau}_{f}\underset{\sum}{\iint}f\left(x,y,z\right)dS\\ ={\tau}_{f}\underset{{D}_{\begin{array}{c}xy\\ \end{array}}}{\iint}f\left(x,y,z\left(x,y\right)\right)\sqrt{1+{z}_{x}^{2}+{z}_{y}^{2}}dxdy\\ ={\tau}_{f}\underset{{D}_{\begin{array}{c}xy\\ \end{array}}}{\iint}\mathit{cos}\gamma \sqrt{1+{z}_{x}^{2}+{z}_{y}^{2}}dxdy\\ ={\tau}_{f}\underset{{D}_{\begin{array}{c}xy\\ \end{array}}}{\iint}\sqrt{{z}_{x}^{2}+{z}_{y}^{2}}dxdy\end{array}$$

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**Figure 7.**Comparison between the mean value for measured tension and the theoretical value derived by checking calculations.

**Figure 10.**Diagram of the monitoring point layout location. (

**a**) Anchor shaft force monitoring point layout. (

**b**) Arrangement of shear stress monitoring points on the outer side of the anchor. (

**c**) Arrangement of monitoring points for soil y axial force and shear stress.

**Figure 11.**Soil stress σ

_{y}distribution under the influence of section radius R after different internal anchorage sections (horizontal monitoring).

**Figure 12.**Soil stress σ

_{y}distribution under the influence of section radius R after different internal anchorage sections (vertical monitoring).

**Figure 13.**Soil shear stress τ distribution under the influence of section radius R after different internal anchorage sections (horizontal monitoring).

**Figure 14.**Soil shear stress τ distribution under the influence of section radius R after different internal anchorage sections (vertical monitoring).

**Figure 15.**Axial y-direction stress diagram for the inner anchorage section under the influence of section radius R after different internal anchorage sections.

**Figure 16.**External shear stress diagram for the inner anchorage section under the influence of section radius R after different internal anchorage sections.

**Figure 17.**Soil stress σy distribution under the influence of different before and after section change rates (horizontal monitoring).

**Figure 18.**Soil stress σy distribution under the influence of different before and after section change rates (vertical monitoring).

**Figure 19.**Soil shear stress τ distribution under the influence of different before and after section change rates (horizontal monitoring).

**Figure 20.**Soil shear stress τ distribution under the influence of different before and after section change rates (vertical monitoring).

**Figure 21.**Axial y-direction stress diagram for the inner anchorage section under the influence of different before and after section change rates.

**Figure 22.**External shear stress diagram for the inner anchorage section under the influence of different before and after section change rates.

**Figure 23.**Soil stress σy distribution under the influence of different expanded section length L (horizontal monitoring).

**Figure 24.**Soil stress σy distribution under the influence of different expanded section length L (vertical monitoring).

**Figure 25.**Soil shear stress τ distribution under the influence of different expanded section length L (horizontal monitoring).

**Figure 26.**Soil shear stress τ distribution under the influence of different expanded section length L (vertical monitoring).

**Figure 27.**Axial y-direction stress diagram for the inner anchorage section under the influence of different expanded section length L.

**Figure 28.**External shear stress diagram for the inner anchorage section under the influence of different expanded section length L.

Parameter | ${\mathit{f}}_{\mathit{m}\mathit{g}}$ | K | $\mathit{\psi}$ |
---|---|---|---|

Value range and value | 0.10~0.15 | 2.0 | 1.3~1.6 |

Soil Type | $\mathit{\gamma}/\mathbf{kN}{\mathbf{m}}^{-3}$ | $\mathit{c}/\mathbf{kPa}$ | $\mathit{\phi}{/}^{\mathit{o}}$ |
---|---|---|---|

Sandy soil | 18 | 0 | 20 |

Length of the Internal Anchorage Section | ||||||
---|---|---|---|---|---|---|

4 m | 5 m | 6 m | 7 m | 8 m | ||

Theoretical derivation of pulling force value/kN | Cylinder-shaped | 33.32513 | 33.44759 | 33.57005 | 33.69251 | 33.81497 |

Frustum of a cone | 55.5145 | 55.51451 | 55.51452 | 55.51452 | 55.51452 | |

Second-order Stepped shape | 50.69843 | 50.86896 | 51.02736 | 51.17883 | 51.32597 | |

Third-order Stepped shape | 53.83549 | 53.89467 | 53.93413 | 53.96231 | 53.98345 | |

Semi-ellipsoid | 38.62513 | 38.94759 | 39.27005 | 39.27005 | 39.91497 |

Similarity Criterion | Similarity Constant |
---|---|

π_{1m} = π_{1p} | ${C}_{\sigma}$= 20 |

π_{2m} = π_{2p} | $C$_{γ} = 1 |

π_{3m} = π_{3p} | $C$_{E} = 20 |

π_{4m} = π_{4p} | C_{μ} = 1 |

**Table 7.**Measured values and theoretically calculated values for the pull-out resistance of bolts with different shapes.

Group Number | Cylinder-Shaped | Frustum of a Cone | Stepped Shape | Semi-Ellipsoid | ||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

Bolt size/cm | r = R = 2.5 L = 10 | r = 2.1 R = 2.9 L = 10 | r = 1.6 R = 3.4 L = 10 | ${r}_{1}$$=1\phantom{\rule{0ex}{0ex}}{r}_{2}$= 3.1 L = 10 | ${r}_{1}$$=1\phantom{\rule{0ex}{0ex}}{r}_{2}$$=2.1\phantom{\rule{0ex}{0ex}}{r}_{3}$= 3.3 L = 10 | a = 2.7 b = 3.5 c = 10 |

Check and derive the theoretical value/N | 70.809 | 96.382 | 99.386 | 90.461 | 94.461 | 78.037 |

$\mathrm{Volume}/\mathrm{c}{\mathrm{m}}^{3}$ | 196.25 | |||||

Measured value for pulling force/N (g = 9.8 N/kg) | 53.214 | 81.144 | 86.632 | 68.698 | 74.676 | 56.252 |

52.038 | 80.556 | 86.142 | 67.228 | 74.284 | 55.468 | |

52.822 | 80.360 | 84.476 | 65.954 | 73.500 | 55.370 | |

51.842 | 79.086 | 84.672 | 64.876 | 74.186 | 53.802 |

Severe γ (kN/m^{3}) | Poisson’s Ratio μ | Modulus of Elasticity E (MPa) |
---|---|---|

25 | 0.2 | 33,500 |

Name | Anchor Cable Modulus of Elasticity (GPa) | Adhesion Strength of Anchored Section/(MPa) | Friction Angle of Anchorage Section (°) | Anchorage Section Perimeter (m) | Anchor Cable Cross-Sectional Area (mm^{2}) | Yield Strength of Anchor Cable (kN) |
---|---|---|---|---|---|---|

Free section | 195 | 0 | 0 | 0.040 | 98 | 203 |

Anchorage section | 195 | 0.7 | 38 | 0040 | 98 | 203 |

Normal Stiffness (MPa/m) | Tangential Stiffness (MPa/m) | Shear Strength (kPa) | Tensile Strength (kPa) | Friction Angle (°) | Expansion Angle (°) | Cohesion (kPa) |
---|---|---|---|---|---|---|

20 | 10 | 3.45 | 1.15 | 20.0 | 20.0 | 5 |

Simulation Group Serial Number | Radius of Section Before Inner Anchorage Section (r/m) | Radius of Section After Inner Anchorage Section (R/m) | Length ofInternal Anchorage Section Segment (L/m) | $\mathit{t}\mathit{a}\mathit{n}\mathit{\theta}=\frac{\mathit{R}-\mathit{r}}{\mathit{L}}$ | Volume of Inner Anchorage Section (V/m ^{3}) |
---|---|---|---|---|---|

① | 0.2 | 0.4 | 3 | 0.2/3 | 1 V |

② | 0.2 | 0.45 | 3 | 0.25/3 | 1.4 V |

③ | 0.2 | 0.5 | 3 | 0.3/3 | 1.9 V |

④ | 0.25 | 0.4 | 3 | 0.15/3 | 2.3 V |

⑤ | 0.3 | 0.4 | 3 | 0.1/3 | 2.7 V |

⑥ | 0.2 | 0.4 | 4 | 0.2/4 | 2 V |

⑦ | 0.2 | 0.4 | 5 | 0.2/5 | 2.2 V |

⑧ | 0.2 | 0.4 | 6 | 0.2/6 | 2.8 V |

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## Share and Cite

**MDPI and ACS Style**

Zhang, S.; Wang, Y.; Li, C.; Wang, C.; Zhang, G.; Sun, S.
Study on the Style Design and Anchoring Mechanism of Enlarged Head Anchors. *Sustainability* **2023**, *15*, 8645.
https://doi.org/10.3390/su15118645

**AMA Style**

Zhang S, Wang Y, Li C, Wang C, Zhang G, Sun S.
Study on the Style Design and Anchoring Mechanism of Enlarged Head Anchors. *Sustainability*. 2023; 15(11):8645.
https://doi.org/10.3390/su15118645

**Chicago/Turabian Style**

Zhang, Sifeng, Yushuai Wang, Chao Li, Changwei Wang, Guojian Zhang, and Shengzhi Sun.
2023. "Study on the Style Design and Anchoring Mechanism of Enlarged Head Anchors" *Sustainability* 15, no. 11: 8645.
https://doi.org/10.3390/su15118645