Cylindrical Water Tank as a Beam on an Elastic Foundation Finite Element

wafflecanadianMechanics

Jul 18, 2012 (5 years and 28 days ago)

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Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 1 of 7 
 
Cylindrical Water Tank as a Beam on an Elastic Foundation Finite Element 
Introduction 
The ring bending moments and ring transverse shear forces (per unit length)  in cylindrical shells with 
axisymmetric loadings can be modeled as a beam on an elastic foundation.  The hoop strength of the shell 
provides the elastic foundation effect, which can be either compressive or tensile.  The equivalent foundation 
stiffness is ݇ ൌ ܧݐ ܴ


 for a shell mean radius, R, thickness, t, and elastic modulus E.  The beam has a width of 
unity, in the circumferential direction.  The effect of Poisson ratio increases the moment of inertial of the 
section to ܫ ൌ ݐ

12ሺ1 െݒ



.  The induced hoop force (per unit length) in the shell is ܰ ൌ െ݌ ܴ, where the 
equivalent pressure from the foundation is ݌ ൌ ݇ݒ for a transverse (radial) deflection of v.  To illustrate the 
fifth‐degree beam on an elastic foundation to this class of problem, consider the Hetenyi example of a 
cylindrical water tank.  His semi‐infinite beam solution is compared to the three‐element solution. 
Cylindrical Water Tank 
 
     
 
 
Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 2 of 7 
 
 
 
Three‐element approximation 
The deflection, foundation pressure (proportional to N), moment and shear force plots are shown next.  The plots are 
followed by the listing of the input and output results.  Note the remarks section at the top of the output file. 
 
Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 3 of 7 
 
 
 
Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 4 of 7 
 
 
 
Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 5 of 7 
 
>> addpath /net/course-a/mech517/public_html/Matlab_Plots
>> L3_Quintic_BoEF(0)
NOTE: no point load data supplied. Reset flag?

(Echo of msh_remarks.tmp)
=================== Begin Application Remarks ============================
Numerical check of Hetenyi Fig 80 cyl water tank
k_f = E*t/R^2 = 458.1 psi, I = 1*t^3/[12*(1-v^2)] = 243.91 in^4
E=41.5e6 psi, v= 0.25, R=360 inch, t=14 inch
Fluid = 0.0361 lb/in^3, w = 1*fluid*y, 0 <= y <= H=312 inch

A 3 element model:
w=11.3 W=7.51 w=3.75 w=0
1 2 3 4 5 6 7
Fixed *----*----*----*----*----*----* top
Bottom (1) (2) (3)

At node 1: M_theory = 14,000 in-lb/in, M_fea = 13,950 in-lb/in
Q_theory = 564 lb/in, Q_fea = 563 lb/in
At 88 inch: M_theory = 3,400 in-lb/in, M_fea = 3,527 in-lb/in
Hoop force: N = -k_f*v*R = - p_f*R = 2,550 lb/in, N_fea = 2,456

While the differences are tiny, the jumps in the shear plot (at points
of no load discontinuity) shows are finer mesh should be used.

Nodal dof: deflection and slope
Element type = 1 (a beam, everywhere)
Element connection: three nodes per element
Element properties 6 (columns in msh_properties.tmp):
modulus (E), inertia (I), line load w1 w2, foundation (k_f), density (Rho)
here k_f = 0 (no foundation), Rho = any value (not used in statics)
==================== End Application Remarks ============================

Read 7 nodes.
(Echo of file msh_bc_xyz.tmp)
bc_flags, 1 coordinates
11 0
00 52
00 104
00 156
00 208
00 260
00 312

Note: expecting 2 displacement BC values.


Read 3 elements with (ignored) type & 3 nodes each.
(Echo of file msh_typ_nodes.tmp)
1 1 2 3
1 3 4 5
1 5 6 7


Read 2 EBC data sets with Node, DOF, Value.
(Echo of file msh_ebc.tmp)
1 1 0
1 2 0


Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 6 of 7 
 

(Echoing file msh_properties.tmp)
6 property values per element
4.25e+06 243.91 11.26 7.51 459.1 0.
4.25e+06 243.91 7.51 3.75 459.1 0.
4.25e+06 243.91 3.75 0.0 459.1 0.


WARNING: BoEF moment & shear inaccurate for crude meshes


Properties for element 1
Elastity modulus (N/m^2) = 4.25e+06
Moment of inertia (m^4) = 243.91
Line Load (N/m) = [ 11.26 7.51 ]
Foundation stiffness (N/m^2) = 459.1
Mass per unit length (km/m) = 0

Properties for element 2
Elastity modulus (N/m^2) = 4.25e+06
Moment of inertia (m^4) = 243.91
Line Load (N/m) = [ 7.51 3.75 ]
Foundation stiffness (N/m^2) = 459.1
Mass per unit length (km/m) = 0

Properties for element 3
Elastity modulus (N/m^2) = 4.25e+06
Moment of inertia (m^4) = 243.91
Line Load (N/m) = [ 3.75 0 ]
Foundation stiffness (N/m^2) = 459.1
Mass per unit length (km/m) = 0

Resultant input sources:
Node, DOF, Resultant input sources
1 1 255.6
1 2 1836.66
2 1 520.555
2 2 -772.571
3 1 364.438
3 2 -386.801
4 1 312.277
4 2 -774.632
5 1 182.047
5 2 -386.801
6 1 104
6 2 -772.571
7 1 17.6429
7 2 -193.143
Totals = 1.0e+03 * [1.7566 -1.4499]

Calculated Displacements
Node, Y_displacement, Z_rotation at 7 nodes
1 0 0
2 0.00854383 0.000196124
3 0.0146638 3.36936e-05
4 0.0133002 -6.89518e-05
5 0.00891241 -9.1255e-05
6 0.00423243 -8.78324e-05
7 -0.000268683 -8.60696e-05

Rice University, Advanced Mechanics of Materials 
J.E. Akin, Page 7 of 7 
 

Max foundation pressure value is 0.123352
Min foundation pressure value is -6.8237
Created BoEF_pressure.png

Max deflection value is 0.0148632
Min deflection value is -0.000268683
Created Beam_deflections.png

Max slope value is 0.00021249
Min slope value is -9.1321e-05
Created Beam_slope.png

Max moment value is 13929.3
Min moment value is -3527.11
Created Beam_moments.png

Max shear value is 43.1949
Min shear value is -562.608
Created Beam_shears.png

Reactions at Displacement BCs
Node, DOF, Reaction Value
1 1 -563.082
1 2 -13948.6
Totals = 1.0e+04 * [-0.0563 -1.3949]

Individual Reaction Summaries:
Elem, F_1, M_1, F_2, M_2, F_3, M_3
1, -563.082, -13948.6, 0.0, 0.0, -34.8132, -3073.19
2, 34.8132, 3073.19, 0.0, 0.0, -8.6469, -36.2939
3, 8.6469, 36.2939, 0.0, 0.0, 0.0, 0.0

>> quit