The dependent pressures method models an embedded retaining wall as a beam supported by elasto-plastic springs, in which the pressure acting at any point is derived from the displacement calculated at that point rather than assumed in advance. It is the basis of most dedicated excavation support software, and it occupies the ground between hand calculation using limiting earth pressures and full finite element analysis of the soil continuum.<\/p>\n\n\n\n
The name describes the essential idea. The pressure on the wall depends on how far the wall has moved. A wall that has not moved carries at-rest pressure. As it deflects away from the retained soil, pressure falls toward the active limit; as it moves into the soil, pressure rises toward the passive limit. Neither limit can be exceeded, because the soil cannot deliver more or less than those states represent.<\/p>\n\n\n\n
At-rest state \u2192 wall deflects \u2192 pressure migrates toward Ka or Kp \u2192 bounded at the limiting value<\/p>\n\n\n\n
A classical hand calculation assumes active pressure behind the wall and passive pressure in front, and solves for equilibrium. It is transparent, fast, and adequate for a cantilever wall in uniform ground.<\/p>\n\n\n\n
Its difficulty is that it prescribes the answer to the question the analysis is being asked. Active pressure exists only where the wall has moved enough to produce it. Full passive resistance exists only where the wall has moved far more. A propped wall restrained at several levels does not achieve either state uniformly: it may be near active at mid-span, near at-rest at a stiff prop, and well short of passive at the toe.<\/p>\n\n\n\n
Assuming a distribution therefore predetermines the bending moments, the support forces, and the displacements \u2014 the three results the analysis exists to produce.<\/p>\n\n\n\n
A limiting-pressure calculation tells you what the wall must resist if the soil reaches failure. It does not tell you whether the soil reaches failure.<\/strong><\/p>\n\n\n\n The dependent pressures method removes that assumption. It does not remove all assumptions, and it introduces one of its own, which is the subject of the following section.<\/p>\n\n\n\n The wall is divided into finite elements along its length. Nodes are placed first at every topological point \u2014 the wall head and toe, each anchor and strut level, each excavation level, and every point at which the section properties change \u2014 and the remaining nodes are distributed so that elements are of approximately equal size.<\/p>\n\n\n\n Each element is assigned a horizontal modulus of subgrade reaction, and the soil is represented as an independent Winkler spring at each node. The stiffness of that spring is the parameter that governs how much pressure develops per unit of wall movement, and its determination by the Schmitt or Chadeisson method is discussed in the guide to the modulus of subgrade reaction<\/a>.<\/p>\n\n\n\n The structure is first loaded with at-rest pressure and solved by the matrix-displacement method. The resulting pressure at each node is then compared with the active and passive limits<\/a> for that location. Where a limit has been violated, the spring at that node is removed \u2014 its stiffness set to zero \u2014 and the wall is loaded with the limiting pressure instead. The system is solved again.<\/p>\n\n\n\n Load with at-rest pressure \u2192 solve \u2192 check limits \u2192 set kh to zero where exceeded, apply the limit \u2192 solve again \u2192 repeat until equilibrium<\/p>\n\n\n\n The iteration continues until the horizontal pressures on the wall are in equilibrium with no limit violated. The result is a pressure distribution that is neither at-rest, nor active, nor passive, but a combination determined by the calculated deflected shape.<\/p>\n\n\n\n The analysis proceeds stage by stage, and plastic deformation accumulated in earlier stages is carried forward. This is what allows support elements to be introduced into a wall that has already deformed.<\/p>\n\n\n\n A prestressed anchor is modelled as a force in the stage in which it is stressed, because it is installed into an already-deflected wall. In subsequent stages it acts as a spring carrying that force, extending as the wall continues to move. Struts and unstressed anchors are springs throughout.<\/p>\n\n\n\n The consequence is that construction stages must reflect the actual execution sequence. A model that jumps from ground level to final excavation with all supports present produces a wall that never deflected, springs that never yielded, and forces that do not correspond to anything that will be built.<\/p>\n\n\n\n The dependent pressures method captures the mechanisms that govern most excavation support design:<\/p>\n\n\n\nHow the Method Works<\/h2>\n\n\n\n
Discretisation<\/h3>\n\n\n\n
Soil Representation<\/h3>\n\n\n\n
Solution<\/h3>\n\n\n\n
Construction Stages<\/h3>\n\n\n\n
What the Method Represents Well<\/h2>\n\n\n\n