Cantilever Beam with Applied Moment at Tip — Shell Elements

Problem Statement

Note

To view this project in FLAC3D, use the menu command Help ► Examples…. Choose “Structure/Shell/Cantilever” and select “Cantilever.f3prj” to load. The main data files used are shown at the end of this example. The remaining data files can be found in the project.

A cantilever beam is subjected to an applied moment at its tip, as described in Cantilever Beam with Applied Moment at Tip — Beam Elements. Here we replicate this problem using shell elements to compute the large-strain y-direction deflection at the beam tip.

The FLAC3D model simulates a beam of 10 m length and 1 m depth. We utilize a cross-diagonal mesh pattern to ensure symmetric response, and we utilize the DKT-CST Hybrid Shell Element. We set the Young’s modulus and Poisson’s ratio equal to 200 GPa and 0, respectively. We set the shell thickness equal to 0.133887 m to produce a second moment of inertia, \(I\), equal to 200 × 10^-6 m⁴. The left end is fully fixed in all six degrees of freedom. The total applied moment at the tip is 5 MN-m, one-half of which is assigned to each of the two nodes at the tip.

The final structural configuration is shown in Figure 1. The y-direction deflection at the beam tip equals 5.471 m, which is within 0.1% of the analytical solution.

Figure 1: Final structural configuration.

Reference

Cook, R. D., D. S. Malkus and M. E. Plesha. Concepts and Applications of Finite Element Analysis, Third Edition. New York: John Wiley & Sons Inc. (1989).

Data File

Cantilever.f3dat

model new
model title "Cantilever beam (using shell elements), applied tip moment (large-strain)"
; Create shells and assign properties
struct shell create by-quad (0,0,0) (10,0,0) (10,0,1) (0,0,1) size (10,1) ...
                    cross-diagonal element-type=dkt-csth
struct shell property isotropic=(2e11, 0.0) thick=0.133887
; Boundary conditions
struct node fix velocity rotation range position-x=0 ; fully fix left end
struct node fix velocity-z rotation-x rotation-y  ; restrict non-beam deformation modes
struct node apply moment=(0,0,2.5e6) range position-x=10
; Solve model
model largestrain on
model solve ratio-local 1e-6
model save 'Cantilever'