Laterally Loaded Pile (Preview)

Theory of beam on elastic foundation can also be applied for the analysis of structures subjected to lateral loads. Here, BOF program was slightly modified in which the orientation of the beam is now perpendicular to X-axis, and the external forces - displacements can be set whether lead to positive or negative axis. Load on nodes profile also to be displayed for a quick verification on inputted pressure magnitudes at both elements ends. The program for solution of laterally loaded structure is named XLAT.

Case Example

Figure below shows an example of retaining wall with given geometry, soil properties, pile and anchor data. It also shows finite element (FE) model and numbering system.

Find:

• Magnitude of displacements, distribution of moments and shear forces along the wall.
• Forces and stress acting on anchor.

After the FE model (as the Figure) is established, now is to find lateral pressure behind the wall acting from ground surface to Point O using active pressure coefficient from Coulomb or Rankine. You can either input the lateral pressure to 1 node below the point O (referred to as either excavation point or dredge line).

Slight modifications were made in calculating fixed-end moments and forces on elements, that is to allow pressure profile varies linearly with depth (triangular distribution) into account. The description is shown below:

The code:

'member fixed-end forces due to pressure
For i = 1 To NM
    With Member(i)
    d = Abs(.Pres1 - .Pres2)
    sq = Application.Min(.Pres1, .Pres2)
    'rectangular part
    w = sq * .bm
        Pa(Idm(1, i), i) = -w * .Lh ^ 2 / 12 'moment
        Pa(Idm(2, i), i) = -w * .Lh / 2 'horizontal
        Pa(Idm(3, i), i) = w * .Lh ^ 2 / 12 'moment
        Pa(Idm(4, i), i) = -w * .Lh / 2 'horizontal
    'triangular part & summing
    w = d * .bm
        Pa(Idm(1, i), i) = Pa(Idm(1, i), i) + (-w * .Lh ^ 2 / 30) 'moment
        Pa(Idm(2, i), i) = Pa(Idm(2, i), i) + (-w * .Lh * 3 / 20) 'horizontal
        Pa(Idm(3, i), i) = Pa(Idm(3, i), i) + (w * .Lh ^ 2 / 20) 'moment
        Pa(Idm(4, i), i) = Pa(Idm(4, i), i) + (-w * .Lh * 7 / 20) 'horizontal
    End With
Next i
 
Equivalent load at node due to pressure on element:

'equivalent joint load due to pressure
For i = 1 To NM
    For j = 1 To 4
        For n = 1 To 4
            Peq(Idm(j, i), i) = -Pa(Idm(j, i), i)
        Next n
     Next j
Next i

'sum load = joint load + eq.load
For i = 1 To NP
    Ps(i) = Pj(i)
Next i

'load superposition
For i = 1 To NM
    For j = 1 To 4
        Ps(Idm(j, i)) = Ps(Idm(j, i)) + Peq(Idm(j, i), i)
    Next j
Next i

Notes: 

• The above code is the same as used in the previous programs to find the equivalent load on nodes. 
• Variable Idm states displacement index of elements ends. It has been described earlier in the previous chapters.

To generalize program where variation in the physical characteristics of the structure can be incorporated into calculation, thus the input of height of beam, t, in BOF is replaced by moment of inertia, I, in XLAT.

The following chart window (Excel 2003) will appear by clicking on Plot Geometry button. It shows finite element geometry and pressure profile that have been inputted into the program refer to the Figure above.

Modulus of subgrade reaction or spring constant of the soil (Ks) of sand layer in the front of the wall is assumed to increase with depth by the approach taken below:
ks     = As + Bs.Zn. ton/m3,
    = 1000 + 2000.Zn above GWL
    = 600 + 1100. Zn below GWL
where,
    As =  40(c.Nc+0.5g.B.Ng)
    Bs = 40(g.Nq)
    z = depth from ground surface
    Nc, Ng, Nq = bearing capacity factors
    n = 0.5, adopted exponent for non-linearity of Ks
 
Anchor rods are made of steel, with the specification shown in the Figure. The anchor rod spring per meter width of the wall is obtained by the formula:
k = AE/sl  
=  1400 ton/m.

Input-output form of XLAT is shown in the figure below:

The results of moment and shear force diagrams generated by the program are shown in the Figure below. For the wall design purpose, it can be taken the maximum positive bending moment (blue lines) that works below the anchor point, while negative moments (red lines) are located around the anchor and around the wall tip.

Application