UPDATED MESH ANALYSIS OF A CANTILEVER
UPDATED MESH ANALYSIS OF A CANTILEVER
The range of problems with known solutions involving large displacement effects that
may be used to test the large displacement options in PLAXIS is very limited. The large
displacement elastic bending of a cantilever beam, however, is one problem which is well
suited as a large displacement benchmark problem since a known analytical solution
exists, see Mattiasson (1981).
L
u
w
F
θ0
Figure 1 True deformation of elastic cantilever
Geometry non-linearity is of major importance in problems involving slender structural
members like beams, plates and shells. Indeed, phenomena like buckling and bulging
cannot be described without considering geometry changes. Soil bodies, however, are
far from slender and consequently, most finite element formulations tacitly disregard
changes in geometry. This also applies to conventional PLAXIS calculations. Users
should check such results by considering the truly deformed mesh. In most practical
cases this will indicate very little change of geometry. In some particular cases, however,
it may be significant.
For special problems of extreme large deformation an Updated mesh analysis is needed.
For this reason PLAXIS involves a special module. For details on the implementation the
reader is referred the PhD thesis by Van Langen (1991). This module was programmed
using the Updated Lagrangian formulation as described by McMeeking & Rice (1975).
Used version:
•
PLAXIS 2D - Version 2011
Input: The analysis relates to the calculation of the horizontal and vertical tip
displacement for the cantilever beam shown in Figure 1. Two meshes are used in the
PLAXIS analysis: One composed of triangular (soil) elements with a thickness of 0.01 m
and one composed of beam elements with zero thickness. The geometry of the projects
is given in the Figure 2.
Materials: The material properties are:
Beam (plate) properties:
EA = 104 kN/m2
EI = 0.0835 kN/m2 /m
Soil properties (representing cantilever):
Linear elastic
E = 106 kN/m2
ν = 0.0
Meshing: The Medium option is selected for the Element distribution of the Global
coarseness. The resulting mesh is shown in Figure 3.
PLAXIS 2012 | Validation & Verification
1
VALIDATION & VERIFICATION
0.01 m
1.0 m
1.0 m
1.0 m
1.0 m
a. Soil element
b. Beam element
Figure 2 Geometry of the projects
b. Beam element
a. Soil element
Figure 3 Finite element mesh
Calculations: In the Initial phase zero initial stresses are generated by using the K0
procedure with Σ -Mweight equal to zero. A new calculation phase is introduced (Phase
1) and the Calculation type is set to Plastic analysis. The Reset displacements to zero is
selected and the Tolerated error for Iterative procedure is set to 0.001. In this phase the
soil clusters are deactivated and the beam (or the soil cluster) representing the cantilever
is activated. The Updated mesh option is selected in the Advanced general settings
window. The Additional step number is defined as 1500 and 1000 for the soil element
and beam element respectively.
Output: The resulting deformed mesh is given in Figure 4. The maximum reached
values of deformation ( |u| ) are 1.023 m and 1.019 m for soil element and beam element
respectively.
Verification: The computed load-displacement curves are plotted in Figure 6 below.
The numerical results of both the soil elements and the beam elements are clearly in
2
Validation & Verification | PLAXIS 2012
UPDATED MESH ANALYSIS OF A CANTILEVER
a. Soil element
b. Beam element
Figure 4 Deformed mesh
close agreement with the analytical solution.
4.0
Normalised load (FL2 /EI )
3.5
3.0
2.5
2.0
1.5
1.0
PLAXIS 2D Soil elements
PLAXIS 2D Beam elements
Analytical
Linear
0.5
0.0
0.00
0.15
0.30
0.45
Vertical displacement (w / L)
0.60
0.75
Figure 5 Variation of normalized load with vertical displacement
PLAXIS 2012 | Validation & Verification
3
VALIDATION & VERIFICATION
4.0
3.5
Normalised load (FL2 /EI )
3.0
2.5
2.0
1.5
1.0
PLAXIS 2D Soil elements
PLAXIS 2D Beam elements
Mattiasson K. (1981)
0.5
0.0
0.00
0.05
0.10
0.15
0.20
0.25
Horizontal displacement (u / L)
0.30
0.35
Figure 6 Variation of normalized load with horizontal displacement
REFERENCES
[1] Mattiasson, K. (1981). Numerical results from large deflection beam and frame
problems analyzed by means of elliptic integrals. Int. J. Numer. Methods Eng., 17,
145–153.
[2] McMeeking, R.M., Rice, J.R. (1975). Finite-element formulations for problems of
large elastic-plastic deformation. Int. J. Solids Struct., 11, 606–616.
[3] Van Langen, H. (1991). Numerical Analysis of Soil-Structure Interaction. Phd thesis,
Delft University of Technology. Plaxis users can request copies.
4
Validation & Verification | PLAXIS 2012