Dip and Strike: Mastering Sloping Planes in Earthworks

Overview

In earthworks and geological engineering, we often deal with tilted planes—such as rock strata or sloping site excavations. To define these planes mathematically, surveyors use the concepts of Dip and Strike. Understanding these is essential for predicting the depth of rock layers or designing the intersection of different site slopes .

Theory

• Full Dip: The direction of maximum tilt on a plane .
• Strike Line: A level line on the plane, which is always at right angles ($90^\circ$) to the direction of full dip .
• Apparent Dip: Any grade on the plane measured in a direction other than full dip. An apparent dip is always less steep than the full dip .

Mathematical Principles

The fundamental relationship between full dip ($\theta$) and an apparent dip ($\theta_1$) measured at an angle $\phi$ to the full dip direction is: $\tan \theta_1 = \tan \theta \cos \phi$

Step-by-Step Example

Problem: On a stratum plane, an apparent dip of $1$ in $16$ bears $170^\circ$, and another apparent dip in the direction $194^\circ$ is $1$ in $11$. Calculate the direction and rate of full dip .

1. Set up equations: Let full dip direction be $\delta$. $\tan \text{Dip}_1 = \tan \text{Full Dip} \times \cos(170^\circ - \delta)$$\tan \text{Dip}_2 = \tan \text{Full Dip} \times \cos(194^\circ - \delta)$
2. Equate through Full Dip:$16 \cos(170^\circ - \delta) = 11 \cos(194^\circ - \delta)$ .
3. Solve for $\delta$: Using trigonometric expansion ($\cos(A-B) = \cos A \cos B + \sin A \sin B$): $\tan \delta = -3.6 / 6.5 \implies \delta = -29^\circ$ . Adjusting for direction: Since the grade is increasing from $1/16$ to $1/11$, the full dip must be beyond the second direction. Result: Full Dip Direction = $223^\circ$ .
4. Find Rate of Full Dip:$1/11 = (1/x) \cos(223^\circ - 194^\circ)$$x = 11 \cos 29^\circ = 9.6$ . Result: Rate of Full Dip = $1$ in $9.6$ .

Field Workflow

Practical Tips

• Strike Line Check: If you find the direction of full dip is $180^\circ$, then the strike lines (where the level is constant) must run exactly $090^\circ$ and $270^\circ$ .
• Reciprocal Ratios: In these formulas, it is often easier to work with gradients as decimals ($1/16 = 0.0625$) to avoid confusion .

Common Mistakes

• Sign Errors: Forgetting that the full dip must be the steepest possible line. If your calculated full dip is flatter than your observed apparent dip, your math is wrong .
• Angular Difference: The angle $\phi$ is the difference between the full dip direction and the apparent dip direction, not the bearing itself .

FAQ

Conclusion

Dip and strike calculations allow the surveyor to model complex 3D surfaces from just a few field observations. By mastering the $\tan \theta \cos \phi$ formula, you can provide engineers with critical data on subsurface rock layers and optimize the design of site slopes.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann.