Buried and embankment conduits (Marston–Spangler concepts): For external load per unit length, which proportionality statements are correct for the following cases? • A pipe laid on or projecting above undisturbed ground and then covered with embankment (projecting/embankment condition) • A flexible pipe buried in a narrow trench with thoroughly compacted side fills • A rigid pipe buried in a narrow trench with thoroughly compacted side fills Assume customary soil-structure interaction behavior used in municipal pipeline design.

Introduction / Context:
External soil loads on buried or embankment pipes are classically estimated using Marston–Spangler theory. The proportionality of load to geometric parameters depends on the installation condition (embankment vs. trench) and on whether the pipe behaves flexibly (soil carries part of the load) or rigidly (pipe carries most of the load).

Given Data / Assumptions:

• Standard earth load conditions for municipal pipelines.
• Narrow trench with well-compacted side fills where stated.
• Projecting (embankment) condition when the pipe is above undisturbed ground and covered.
• Proportionality trends, not exact numerical coefficients.

Concept / Approach:

Marston’s formulas express earth load per unit length as W = C * γ * (geometric term). For embankment conditions, the governing geometric term scales with the square of the outside width (for circular pipe, the external diameter). In narrow trenches, load trends depend on trench width and the relative flexibility/rigidity of the pipe–soil system, yielding proportionalities to width*diameter (flexible) or to width squared (rigid).

Step-by-Step Solution:

Verification / Alternative check:

These are the well-known proportionality forms presented in buried pipeline design texts and codes for preliminary comparisons, prior to inserting site-specific coefficients and depth terms for final design.

Why Other Options Are Wrong:

• “None of the above” conflicts with established Marston–Spangler proportionalities for the stated conditions.

Common Pitfalls:

• Applying trench formulas to embankment cases (and vice versa).
• Ignoring flexibility factor; flexible vs. rigid behavior changes the geometric dependence.

Final Answer:

All the above.