Das Soil Mechanics Ch10c

Vertical Strip Load in Foundation Design

Introduction to Vertical Strip Load

• The session continues with Chapter 10, focusing on various loading types, specifically the vertical strip load.

• This load is described as infinite in the direction perpendicular to the page, emphasizing its rectangular shape.

Real-World Applications

• An example of a real-world application includes roadway loading, which simulates traffic loads on shallow foundations.

• The concept of rectangular strip loading is introduced as essential for understanding how loads are distributed over structures like roadways.

Formula and Calculation Insights

• A complex formula (referred to as Equation 10.9) is presented for calculating stress due to strip loading, where Q represents load per unit area.

• Key variables include:

• X: Distance from the center of the strip load to the point of interest.

• B: Total width of the loading; often half B is used for calculations due to symmetry.

• Z: Depth below ground surface.

Important Calculation Considerations

• It’s crucial to use radians instead of degrees when calculating arc tangent values in this context.

• The formula applies only when X exceeds half B; if X is less than or equal to half B, adjustments must be made by adding π to certain terms.

Simplified Methods for Stress Calculation

• To avoid calculation errors with complex formulas, simpler methods such as graphical or tabular approaches are recommended.

• Table 10.4 provides variations of Δσz/Q based on two key ratios: 2/B and 2X/B , facilitating easier calculations.

Influence Zones and Graphical Representation

• Figure 10.6 illustrates contours showing Δσz/Q beneath a strip load, indicating that only half B needs consideration due to symmetry.

• The depth range considered extends from zero up to six times B while horizontal influence remains within two times B on either side.

Understanding Load Types in Structural Engineering

Influence Zone of Cube Load

• The influence zone of a cube load can be determined, particularly when assessing the impact on vertical structures like roadways. It is noted that this impact extends approximately 6B vertically.

• To enhance calculations and estimations regarding the influence zone, a fun grade method is introduced to provide more detailed information.

Triangle Loading Concept

• Triangle loading is discussed as it relates to roadway shoulders and areas outside these shoulders, emphasizing its triangular shape.

• Understanding triangle loading is crucial for analyzing embankment loads, which will be elaborated upon later.

Superposition Method for Embankment Load Analysis

• The superposition method allows for the consideration of two loading types together when analyzing embankment loads. Key variables include depth (Z), horizontal distance (X), and maximum magnitude (Q).

• Two angles are essential for calculations: Delta (the angle between a line from the triangle's corner to vertical) and Alpha (the angle between two lines).

Calculation Methods for Triangle Loading

• A table method is presented as an alternative to formulas for calculating triangle loading, reducing potential calculation errors.

• It's important to reference Table 10.5 specifically when discussing triangle loads to avoid confusion with other data sets.

Transitioning to Embankment Loading

• The discussion transitions into embankment loading, which resembles a trapezoid that can be divided into triangular and rectangular components.

• Key parameters include widths B1 and B2, maximum magnitude q0, and angles Alpha One and Alpha Two; Equation 10.21 should be used for calculations related to this type of load.

Simplifying Calculations with I2 Index

• An easier formula (Equation 10.24) combines several parameters into one index (I2), simplifying calculations related to embankment loading.

• A graphical method using curves based on B1 over Z and B2 over Z helps estimate I2 values effectively.

Example Application in Embankment Calculation

• An example involving a full trapezoidal embankment illustrates key dimensions: left side at 40m, right side at 40m, middle part at 5m height with soil density at 17.5 kN/m³.

• Focus shifts towards calculating point A2 within the context of this example; specific details about its depth are provided as part of the instructional process.

Calculating DAT Sigma Z at Point A2

Understanding the Parameters

• The distance from the center line to point A2 is 11.5 meters, which is crucial for calculations.

• To find DAT Sigma Z at point A2, the trapezoid shape needs to be divided into smaller sections for easier calculation.

Superposition Method

• The first part of the calculation involves a triangle that overlaps with point A2, which has a width of 5 meters. This triangle's dimensions are essential for further calculations.

• Interpolation can be used to determine q0; in this case, it is calculated as 2.5 based on previous dimensions (5 in the middle and 14 on the side).

Calculating Remaining Areas

• The height of the small triangle is confirmed to be 2.5 meters, while other parts form an irregular shape that requires additional methods for calculation.

• By adding another triangle to create a standard embankment shape, numerical or graphical methods can be employed to calculate DAT Sigma Z from this new configuration.

Final Calculation Steps

• The final formula combines different components: Delta Sigma Z1 plus Delta Sigma Z2 minus Delta Sigma Z3 will yield the total DAT Sigma Z at point A2. This highlights the importance of understanding superposition in these calculations.

Loading Types and Their Calculations

Uniformly Loaded Circular Area

• Introduction of uniformly loaded circular areas where stress distribution is uniform from the center outward; specific formulas are provided for calculating stress under various conditions including depth and radius parameters.

Stress Calculation Beyond Center

• For points outside the center of a circular area, factors related to spatial parameters must be considered using tables found in textbooks for accurate calculations involving Q magnitude and distances involved.

Rectangularly Loaded Area Insights

Characteristics of Rectangular Loading

• Rectangular loading differs significantly from strip loading; visualizing it as a blanket over ground helps understand its impact on stress distribution beneath it at depth z.

Specific Calculation Requirements

• Stress can only be accurately calculated under one corner of this rectangular area using complex integration formulas; simplicity lies in utilizing combined indices like I3 for practical applications despite lengthy equations involved in deriving results under such loads.

How to Calculate I3 for Shallow Foundation Design

Understanding the I3 Calculation Formula

• The formula for calculating I3 is complex, involving multiple variables (M and N). Here, M represents B over Z, while N is L over Z. These parameters are crucial in shallow foundation design.

• The vertical axis of the figure used for calculations represents I3, while the horizontal axis denotes M. The curve itself corresponds to N, emphasizing the importance of understanding how to interpret this figure.

• To calculate stress under a rectangularly loaded area, it’s essential to focus on specific points—particularly at corners. This highlights a unique requirement in stress calculation.

Calculating Stress Within Rectangular Areas

• When needing to calculate stress at a point inside a rectangular area (not at the corner), one can divide the rectangle into four smaller rectangles. This method simplifies calculations significantly.

• After dividing into four sections, you must compute Delta Sigma for each section (Z1 through Z4). Summing these values will yield Delta Sigma at that internal point.

• The total stress calculation involves adding together all four parts: Delta Sigma Z1 + Delta Sigma Z2 + Delta Sigma Z3 + Delta Sigma Z4 results in your final value for that internal point.

Importance of Practice and Homework

• Understanding these concepts is vital for shallow foundation design; practical application through examples enhances learning outcomes.

• Although today's lecture focused on logic rather than numerical examples, students are encouraged to refer to their textbooks where various loading types are illustrated with examples.

• For homework assignments (10.4, 17, and 19), using Excel is recommended due to its ability to manage complex calculations and help debug errors effectively as numerical methods become increasingly intricate.