基础是结构的基本要素，通过将结构的总载荷传递和分配到地面来提供整体稳定性. 浅基础, 例如矩形或方形隔离基础, are the preferred type of foundation due to the simplicity of their construction and overall cost compared to deep foundations. Estimating the base pressure dramatically affects the design and sizing of the footing. 通常, the utility ratio between the allowable bearing capacity of the soil and the governing base pressure under the footing is the basis of the initial size of the footing. 一旦设置了初始基础尺寸, 进一步的安全性和稳定性设计检查, 如单向和双向剪切, 弯曲能力, 和开发长度检查, are checked depending on which design code is used.
When a footing is subjected to a bi-axial bending (中号X, 中号与), it is assumed that the axial load (P) is acting on an eccentricity coordinate (ËX, Ë与) where there is a tendency to rotate from the center. The interaction between the soil and footing mainly depends on the footing dimension and the resultant eccentricity of the applied loads. 取决于产生的偏心的位置, 基础压力导致基础完全或部分受压. 在实践中, 建议设计一个完全压缩的基础. Partial compression or loss of contact between the soil and footing should not be neglected, but most designers avoid this scenario due to its calculation complexity. 当产生的偏心率位于紧缩内或 C 区下方时，基础处于完全压缩状态. C区外的偏心使基础部分受压. 数字 1 在矩形基础上显示不同的指定区域.
This article shall focus on calculating corner pressures under different zone classifications based on Bellos & 痕迹 (2017) 和SS. Ray’s (1995) studies.
矩形基础的区域分类
The zone classifications of a rectangular footing are derived from multiple studies by different authors to develop a practical approach to estimating the distribution of soil pressure under expected loading conditions. 如图 1, 有五个不同的地区 (A-E 区) depending on the location of resultant eccentricity. Each zone corresponds to a different loading, 基础压力分布, and deformation. C区, 也称为内核, is the main core. It is the ideal region to design a footing, resulting in full compression on the footing. 该区域的尺寸相当于 1/6 其各自的基础长度.
数字 1: 矩形基础的区域分类
次核心是椭圆区 (以图中虚线为界 1) 其长短半轴等于 1/3 其各自的基础长度. 该区域覆盖整个区域 B & C and some parts of zones D & Ë. 次要核心导致基础部分压缩. 保持次要区域内的偏心距以获得可接受的基础设计是一种很好的做法.
超出次要区域的偏心是高双轴载荷的结果. 它覆盖了整个 A 区和 D 区的其余部分 & Ë. 建议避免在这些区域设计基础，因为它会有倾覆的风险. 因此, it is advisable to redesign the footing dimensions for this loading type.
下面列举求解各区域分类中的角压力的解析公式.
C区 (主要核心, Full compression zone)
如上所述, this is the most preferred case for designing footings since it is capable of setting the whole base of the footing into compression, 如图 2. This case is represented by small eccentricity within the kern or no eccentricity. 数字 2 shows the eccentricity within the kern with its maximum pressure at corners P3 & P4 and minimum pressure at corners P1 & P2.
数字 2: 偏心 (-ËX, -Ë与) at Zone C & full compression area
最大 & minimum corner pressures (Bellos & 痕迹, 2017):
| Corner pressures based on eccentricity | ||||
|---|---|---|---|---|
| P1 | P2 | P3 | P4 | |
| +ËX, +Ë与 | P最高 | P最高 | P分 | P分 |
| +ËX, -Ë与 | P最高 | P最高 | P分 | P分 |
| -ËX, -Ë与 | P分 | P分 | P最高 | P最高 |
| -ËX, +Ë与 | P分 | P分 | P最高 | P最高 |
A区 (Triangular compression zone)
This case corresponds to four rectangular areas in every corner of the footing. It usually occurs with large bi-axial eccentricity, imposing a high triangular compressive area in one of the corners, as shown by the shaded region in Figure 3. The remaining corners lose contact with the soil. 因此, this case is not advisable for design.
数字 3: 偏心 (-ËX, -Ë与) at Zone A & triangular compression area around P3
最大压力 (Bellos & 痕迹, 2017):
| Corner pressures based on eccentricity | ||||
|---|---|---|---|---|
| P1 | P2 | P3 | P4 | |
| ËX(+), Ë与(+) | P最高 | 0 | 0 | 0 |
| ËX(+), Ë与(-) | 0 | P最高 | 0 | 0 |
| ËX(-), Ë与(-) | 0 | 0 | P最高 | 0 |
| ËX(-), Ë与(+) | 0 | 0 | 0 | P最高 |
D区 (Trapezoidal compression zone)
Zone D also corresponds to large eccentricities in the areas attached in the x-direction of the footing, 如图 4. The eccentricity in the z-direction (Ë与) is much greater than in the x-direction (ËX). 在这种情况下, two corners of the footing lose contact with soil and produce a trapezoidal compressive area. Compared to zone A, which is entirely outside the secondary zone, a portion of zone D is still covered by the secondary zone.
数字 4: 偏心 (-ËX, -Ë与) at Zone D & trapezoidal compression area around P3
最大 & minimum corner pressures (Bellos & 痕迹, 2017):
Vertical heights of the trapezoidal compressive area (Bellos & 痕迹, 2017):
| Corner pressures based on eccentricity | ||||
|---|---|---|---|---|
| P1 | P2 | P3 | P4 | |
| ËX(+), Ë与(+) | P最高 | 0 | 0 | P分 |
| ËX(+), Ë与(-) | 0 | P最高 | P分 | 0 |
| ËX(-), Ë与(-) | 0 | P分 | P最高 | 0 |
| ËX(-), Ë与(+) | P分 | 0 | 0 | P最高 |
E区 (Trapezoidal compression zone)
Similar to zone D, this case also produces a trapezoidal compressive area but is caused by a large eccentricity in the x-direction(ËX).
数字 5: 偏心 (-ËX, -Ë与) at Zone E & trapezoidal compression area around P3
最大 & minimum corner pressures (Bellos & 痕迹, 2017):
Horizontal bases of the trapezoidal compressive area (Bellos & 痕迹, 2017):
| Corner pressures based on eccentricity | ||||
|---|---|---|---|---|
| P1 | P2 | P3 | P4 | |
| ËX(+), Ë与(+) | P最高 | P分 | 0 | 0 |
| ËX(+), Ë与(-) | P分 | P最高 | 0 | 0 |
| ËX(-), Ë与(-) | 0 | 0 | P最高 | P分 |
| ËX(-), Ë与(+) | 0 | 0 | P分 | P最高 |
B区 (Pentagonal compression zone)
This case occurs when the applied loads on the footings generate a moderate eccentricity within the secondary zone. The areas covered by zone B are bounded by two curved sides and one flat base around the exteriors of zone C. 在这种情况下, a pentagonal compressive area is produced, and only a corner of the footing loses contact with the soil. 然而, the solutions provided below are slightly complex and require numerical solving methods for the corner pressures and the x & y intercepts of the compressive area.
Corner pressures (Bellos & 痕迹, 2017):
Pentagonal sides of the compressive area (Bellos & 痕迹, 2017):
| Corner pressures based on eccentricity | ||||
|---|---|---|---|---|
| P1 | P2 | P3 | P4 | |
| ËX(+), Ë与(+) | P最高 | Pq | 0 | Pp |
| ËX(+), Ë与(-) | Pp | P最高 | Pq | 0 |
| ËX(-), Ë与(-) | 0 | Pp | P最高 | Pq |
| ËX(-), Ë与(+) | Pq | 0 | Pp | P最高 |
或者, a more direct solution by S.S. 射线 (1995) can be used for the corner pressures and intercepts of the pentagonal compressive zone. The equations are given below:
Corner pressures (S.S. 射线, 1995):
Pentagonal sides of the compressive area (S.S. 射线, 1995):
SkyCiv的 Foundation Design Module is capable of solving the base pressures of a rectangular concrete footing. Additional design checks in accordance with different design codes (ACI 318-14, Australian standard 2009 & 2018, 和欧洲规范) are also available.
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参考资料:
- Bellos, J。, 痕迹, ñ. (2017). Complete Analytical Solution for Linear Soil Pressure Distribution under Rigid Rectangular Spread Footing.
- 计算极限承载能力, 计算极限承载能力. (2007). 计算极限承载能力 (7第版). 计算极限承载能力
- Rawat, S., et. al. (2020). Isolated Rectangular Footings under Biaxial Bending: A Critical Appraisal and Simplified Analysis Methodology.
- 射线, S.S. (1995). 钢筋混凝土. Blackwell Science
