Analysis of Geometrically Nonlinear Structures
by Robert Levy & William R. Spillers
Chapman & Hall, 1994
CHAPTER 6
NONLINEAR ANALYSIS OF MEMBRANES
6.1 Introduction
This chapter completes the sequence of discussions of discrete systems. It is included here for several reasons. In terms of applications, if you are going to discuss fabric structures you need access to a membrane finite element. The plane stress/strain element which follows is probably the most simple finite element which can be discussed. Since the functional dependence of the element upon the node coordinates is so simple, it is an easy matter to apply perturbation methods to it.
The subsequent chapters of this book discuss continuous rather than discrete systems. This membrane finite element is in these terms something of a hybrid. Membranes are, of course, continuous in the sense that the are properly described by their field equations. The most direct way to come up with a nonlinear membrane finite element would be to first of all derive the appropriate field equations and then formulate some discrete solution of them such as the finite element solution. In the material below an alternative approach is taken. That is, the discretization of the finite element method for the linear membrane problem is accepted as a starting point and then perturbation methods applied to it. Certainly the procedure is simple.
The work below proceeds as follows. First, Zienkiewicz's plane stress/strain finite element is used to derive its corresponding geometrically nonlinear finite element. This is a two-dimensional result. It is then a simple matter to construct the three-dimensional finite element using the rotation matrix. An application is included with the computer program which is discussed subsequently.
6.2 The Geometric Stiffness Matrix of the Plane Stress Triangular Finite Element
For the purpose of this book, the most simple finite element,
the triangular, plane stress (constant stress) element described
by Zienkiewicz (1977) is used. In the element, the node equilibrium
equations appear in the local coordinate system as
(6.1)
Here are the usual stresses of plane elasticity and the b's and c's are the coefficients that Zienkiewicz gives as typically
(6.2)
For completeness the plane stress elastic stiffness matrix for the system is,
(6.3)
where
(6.4)
Here E and are Young's modulus and Poisson's ratio and is the surface area of the triangular finite element facet.
As in the previous chapters the geometric stiffness matrix is derived from the nodal equilibrium equations. The gradient of Eq. 6.1 has a particularly simple form. It comprises the geometric stiffness matrix having 9 submatrices (instead of 4 in two noded elements like trusses and beams) arranged in an antisymmetric manner as
(6.5)
where,
(6.6)
The gradient in Eq. 6.2 has been taken with respect to the node coordinates while holding down the stresses fixed. Eqs. 6.1-6.6 give and explicitly for plane elasticity problems. An additional step is required to deal with three dimensional membranes.
6.3 Three-Dimensional Membranes
A plane stress problem defines a two-dimensional (x, y) displacement field while a three-dimensional membrane in general defines a three-dimensional (x,y,z,) displacement field. For linear elastic problems, a membrane has no out-of-plane stiffness so that the 3-D element stiffness can be constructed by simply "rearranging" terms from Eq. 6.4 and Eq. 6.6 in a 3-D array i.e. expanding the 22 matrices to 33 by inserting a zero third row and a zero third column. That is not true in the nonlinear case. Just as the string (truss bar) has a "geometric" stiffness component normal to its axes, the nonlinear membrane has an out-of-plane "geometric" stiffness. That effect will be described below using the relationship from mechanics (that was introduced in Chapter 5) which gives the change of a vector force which is subjected to a small rotation as,
(6.7)
This approach is possible since within small strain theory an out-of-plane displacement produces no strain in a membrane.
Figure 6.1 A Triangular Finite Element in its Local Coordinate System
First of all, it should be clear that small, in-plane (z component) rotations are included within the geometric stiffness matrix shown in Eq. 6.5. It is the effect of out-of-plane rotations (x, y, components) which must be added to the existing formulation to complete the 3-dimensional model. Given the x, y, z displacement components of the nodes of a typical finite element (Figure 6.1) the out-of-plane rotation components are
(6.8)
(6.9)
Figure 6.2 Out-of-Plane Rotations
These rotations are obtained from Figure 6.2 as
where
and
It only remains to construct a matrix representation of the incremental forces Eq. (6.7), produced by these rotations acting upon the element nodal forces of Eq. 6.1. The cross product of this equation is rewritten as,
(6.10)
When performed the product gives the three components of the vector for any given one node as
(6.11)
But since the rotation components of Eqs. 6.8, 6.9 are linear in displacements, the matrix can be written as
(6.12)
or
The required contribution to the geometric stiffness matrix, is then simply
where F* is a 93 matrix obtained by stacking the 33 matrix of Eq 6.11 and setting for each node. Each entry of is an element that comes from Eq. 6.1. It yields
(6.13)
where,
for r =i, j, m,
Definitions of stress and strain are more complicated than the case of the rod. But even without recourse to the equations of nonlinear elasticity, it is possible to remove rigid body effects (move to the deformed local coordinate system) prior to computing strains. In the case of the truss again this provides an "exact" solution; for the case of the membrane, this technique can be used to achieve results with small strains but large rotations.
6.4 Computer Programs
6.4.1 Program P12-FEMPS.FOR. This program is one of the most simple finite element analysis programs. It is the constant stress finite element program carefully described in Zienkiewicz's book and is in fact quite similar to the other programs described in thise book. In terms of loads and coordinates this program is identical to the earlier programs; it also includes the same solver used in these earlier programs. The coefficients b (here B) and c (here CZ) are those of Eq. 6.2.
6.4.2 Program P13-MEMBR.FOR. This program is a step along the way toward a computer program for fabric structures or membranes. It is a linear analysis program which does not take into account changes in geometry. The elements themselves only have in-plane stiffness so that you end up with a structure which can only transmit axial forces and in-plane shear which is of course the membrane shell.
It is well known that to construct a membrane shell finite element program it is simply necessary to start with a plane stress finite element and rotate the element until it has the proper geometric orientation in space. That is what is done here and also the reason that much of the code which follows is taken from program P12.
This program has the structure of the other analysis programs. It uses the following subroutines:
ABC. This subroutine computes the Zienkiewicz coefficients "b" and "c" and the surface area A.
COMPKT. This subroutine collects terms from a global array and places them in a local array.
PCOORD sets up so-called "plane coordinates" and determines the rotation matrix from the element coordinates.
PLSTR sets the member stiffness for a plane stress/strain element.
FORCES computes the member stresses/strains given the member displacements.
TRANS rotates a vector from global to local coordinates.
RASM rotates the member stiffness to global coordinates.
6.4.3 Program P14-MEMNL.FOR. This program is simply program P13 with geometrical nonlinearities added. Truss bars are also included as an added capability since they would be needed for a typical fabric structure application. New subroutines in this program include:
MGSTIF generates the geometric stiffness matrix.
UPDATE calculates the new member forces once rigid body rotations have been removed.
This program uses the same input as program P13.
6.5 Examples
6.5.1 Example 6.1. This example is a "deep beam", that is a beam in which the depth is not small in comparison to its span. In this case, the problem data is generated by a simple computer program FEMPSDAT.FOR. This program generates a rectangular finite element grid which is subsequently divided into triangles. The variables in this program include: NX = number of grid lines in the x direction; NY = number of grid lines in the y direction; T = beam thickness; H = beam height and AL = beam length.
It this case the program places a unit load at the end of the beam. The easiest way to run these programs is to compile both the data generation program, FEMPSDAT.FOR,and the finite element analysis program, FEMPS.FOR, and then use the pipeline command to concatenate their execution FEMPSDAT|FEMPS>FEMPS.OUT. The beam is 30 inches long and 28 inches deep and 1 inch thick. Young's modulus is 29,000,000 psi and Poison's ratio is .32. Figure 6.3 shows the structural analysis model.
The program reads joint coordinates (R), loads (P), member incidence nodes (NP), and thickness (S). It computes joint displacements (P), strains (AZ) and stresses,(AI). The output file is listed below. Assessment of the results as representative of the phenomenon is left as an exercise to the reader.
Output file FEMPS.OUT
COORDINATES LOADS
X Y PX PY
1 .30000000D+02 .00000000D+00 .00000000D+00 -.10000000D+01
2 .30000000D+02 .93333330D+01 .00000000D+00 .00000000D+00
3 .30000000D+02 .18666670D+02 .00000000D+00 .00000000D+00
4 .30000000D+02 .28000000D+02 .00000000D+00 .00000000D+00
5 .22500000D+02 .00000000D+00 .00000000D+00 .00000000D+00
6 .22500000D+02 .93333330D+01 .00000000D+00 .00000000D+00
7 .22500000D+02 .18666670D+02 .00000000D+00 .00000000D+00
8 .22500000D+02 .28000000D+02 .00000000D+00 .00000000D+00
9 .15000000D+02 .00000000D+00 .00000000D+00 .00000000D+00
Figure 6.3 The Plane Stress Cantilever Beam
10 .15000000D+02 .93333330D+01 .00000000D+00 .00000000D+00
11 .15000000D+02 .18666670D+02 .00000000D+00 .00000000D+00
12 .15000000D+02 .28000000D+02 .00000000D+00 .00000000D+00
13 .75000000D+01 .00000000D+00 .00000000D+00 .00000000D+00
14 .75000000D+01 .93333330D+01 .00000000D+00 .00000000D+00
15 .75000000D+01 .18666670D+02 .00000000D+00 .00000000D+00
16 .75000000D+01 .28000000D+02 .00000000D+00 .00000000D+00
17 .00000000D+00 .00000000D+00 .00000000D+00 .00000000D+00
18 .00000000D+00 .93333330D+01 .00000000D+00 .00000000D+00
19 .00000000D+00 .18666670D+02 .00000000D+00 .00000000D+00
20 .00000000D+00 .28000000D+02 .00000000D+00 .00000000D+00
ELEMENT ELEMENT NODES THICKNESS
1 5 1 2 .10000000E+01
2 5 2 6 .10000000E+01
3 6 2 3 .10000000E+01
4 6 3 7 .10000000E+01
5 7 3 4 .10000000E+01
6 7 4 8 .10000000E+01
7 9 5 6 .10000000E+01
8 9 6 10 .10000000E+01
9 10 6 7 .10000000E+01
10 10 7 11 .10000000E+01
11 11 7 8 .10000000E+01
12 11 8 12 .10000000E+01
13 13 9 10 .10000000E+01
14 13 10 14 .10000000E+01
15 14 10 11 .10000000E+01
16 14 11 15 .10000000E+01
17 15 11 12 .10000000E+01
18 15 12 16 .10000000E+01
19 17 13 14 .10000000E+01
20 17 14 18 .10000000E+01
21 18 14 15 .10000000E+01
22 18 15 19 .10000000E+01
23 19 15 16 .10000000E+01
24 19 16 20 .10000000E+01
DISPLACEMENTS
X Y
1 -.12561651D-06 -.29653664D-06
2 -.24175047D-07 -.23195584D-06
3 .25915595D-07 -.20460964D-06
4 .79456403D-07 -.19862350D-06
5 -.96539144D-07 -.16937614D-06
6 -.20086171D-07 -.15520392D-06
7 .24816674D-07 -.14673469D-06
8 .75709248D-07 -.14552769D-06
9 -.72706518D-07 -.95010275D-07
10 -.15657549D-07 -.86800959D-07
11 .19421555D-07 -.84794679D-07
12 .61803372D-07 -.87771887D-07
13 -.41962270D-07 -.40817802D-07
14 -.80423984D-08 -.32318339D-07
15 .10651986D-07 -.32806949D-07
16 .35986177D-07 -.37874193D-07
STRAINS STRESSES
EX EY GA SX SY TAU
1 -.38770D-08 .69194D-08 -.60860D-08 -.53722D-01 .18347D+00 -.66854D-01
2 -.54518D-09 .15185D-08 -.20422D-08 -.19152D-02 .43422D-01 -.22433D-01
3 -.54518D-09 .29299D-08 -.48667D-08 .12678D-01 .89025D-01 -.53460D-01
4 .14652D-09 .90742D-09 -.29056D-08 .14115D-01 .30832D-01 -.31918D-01
5 .14652D-09 .64137D-09 -.19801D-08 .11365D-01 .22237D-01 -.21752D-01
6 .49962D-09 .12932D-09 -.16267D-08 .17479D-01 .93436D-02 -.17869D-01
7 -.31777D-08 .15185D-08 -.17241D-08 -.86967D-01 .16206D-01 -.18939D-01
8 -.59048D-09 .87957D-09 -.30080D-08 -.99840D-02 .22313D-01 -.33042D-01
9 -.59048D-09 .90742D-09 -.43094D-08 -.96960D-02 .23212D-01 -.47338D-01
10 .71935D-09 .21496D-09 -.45002D-08 .25463D-01 .14382D-01 -.49434D-01
11 .71935D-09 .12932D-09 -.28059D-08 .24578D-01 .11615D-01 -.30822D-01
12 .18541D-08 -.31899D-09 -.31599D-08 .56606D-01 .88632D-02 -.34711D-01
13 -.40992D-08 .87957D-09 -.11133D-08 -.12335D+00 -.13963D-01 -.12229D-01
14 -.10154D-08 .91066D-09 -.36301D-08 -.23389D-01 .18924D-01 -.39876D-01
15 -.10154D-08 .21496D-09 -.35059D-08 -.30582D-01 -.35525D-02 -.38512D-01
16 .11693D-08 -.52351D-10 -.49287D-08 .37236D-01 .10397D-01 -.54141D-01
17 .11693D-08 -.31899D-09 -.23908D-08 .34479D-01 .17828D-02 -.26262D-01
18 .34423D-08 -.54292D-09 -.39386D-08 .10560D+00 .18048D-01 -.43265D-01
19 -.55950D-08 .91066D-09 -.18081D-08 -.17135D+00 -.28423D-01 -.19862D-01
20 -.10723D-08 .00000D+00 -.43091D-08 -.34645D-01 -.11086D-01 -.47335D-01
21 -.10723D-08 -.52351D-10 -.23061D-08 -.35186D-01 -.12778D-01 -.25333D-01
22 .14203D-08 .00000D+00 -.43743D-08 .45886D-01 .14684D-01 -.48051D-01
23 .14203D-08 -.54292D-09 -.16599D-08 .40273D-01 -.28572D-02 -.18234D-01
24 .47982D-08 .00000D+00 -.50499D-08 .15502D+00 .49607D-01 -.55472D-01
Stop - Program terminated.
6.5.2 Example 6.2. This example analyses a spherical cap
for membrane stresses . Figure 6.4 shows a plan view of the analysis
model. Again, the data is generated using a computer program (MEMBRNDAT.FOR)
with the following variables: RAD = shell radius; THICK = shell
thickness; PHI = shell central angle; NSEG = number of sectors
taken around the shell and NRING = number of levels into which
the shell is subdivided. Again, both the data generation program
and the analysis program must first be compiled. At run time they
can be concatenated using the pipeline command MEMBRNDAT|MEMBRN>MEMBRN.OUT.
The cap has a radius of 4.76 inches, a central angle of 10.9 degrees and a thickness of 0.01576 inches. Young's modulus and Poisson's ratio are 29,000,000 psi and 0.32 respectively. The output for this example is listed below.
Figure 6.4 The Spherical Membrane Shell
Output file MEMBRN.OUT
NUMBER OF BARS 0
NUMBER OF NODES 25
NUMBER OF SUPPORTS 8
NUMBER OF CONSTRAINTS 0
NUMBER OF ITERATIONS 1
NUMBER OF FINITE ELEMENTS 40
COORDINATES LOADS
X Y Z PX PY PZ
1 .00000D+00 .00000D+00 .47600D+01 .00000D+00 .00000D+00 -.10000D+01
2 .30165D+00 .00000D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
3 .21330D+00 .21330D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
4 .38236D-06 .30165D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
5 -.21330D+00 .21330D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
6 -.30165D+00 .76473D-06 .47504D+01 .00000D+00 .00000D+00 .00000D+00
7 -.21330D+00 -.21330D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
8 -.11471D-05 -.30165D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
9 .21330D+00 -.21330D+00 .47504D+01 .00000D+00 .00000D+00 .00000D+00
10 .55625D+00 .23041D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
11 .23041D+00 .55625D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
12 -.23041D+00 .55625D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
13 -.55625D+00 .23041D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
14 -.55625D+00 -.23040D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
15 -.23041D+00 -.55625D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
16 .23040D+00 -.55625D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
17 .55625D+00 -.23041D+00 .47218D+01 .00000D+00 .00000D+00 .00000D+00
18 .90009D+00 .00000D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
19 .63646D+00 .63646D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
20 .11410D-05 .90009D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
21 -.63646D+00 .63646D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
22 -.90009D+00 .22819D-05 .46741D+01 .00000D+00 .00000D+00 .00000D+00
23 -.63646D+00 -.63646D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
24 -.34229D-05 -.90009D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
25 .63646D+00 -.63647D+00 .46741D+01 .00000D+00 .00000D+00 .00000D+00
FINITE ELEMENT DATA
NO. NODES THICKNESS
I J M
1 1 2 3 .1576E-01
2 1 3 4 .1576E-01
3 1 4 5 .1576E-01
4 1 5 6 .1576E-01
5 1 6 7 .1576E-01
6 1 7 8 .1576E-01
7 1 8 9 .1576E-01
9 10 11 3 .1576E-01
10 2 10 3 .1576E-01
11 11 12 4 .1576E-01
12 3 11 4 .1576E-01
13 12 13 5 .1576E-01
14 4 12 5 .1576E-01
15 13 14 6 .1576E-01
16 5 13 6 .1576E-01
17 14 15 7 .1576E-01
18 6 14 7 .1576E-01
19 15 16 8 .1576E-01
20 7 15 8 .1576E-01
21 16 17 9 .1576E-01
22 8 16 9 .1576E-01
23 17 10 2 .1576E-01
24 9 17 2 .1576E-01
25 18 19 10 .1576E-01
26 17 18 10 .1576E-01
27 19 20 11 .1576E-01
28 10 19 11 .1576E-01
29 20 21 12 .1576E-01
30 11 20 12 .1576E-01
31 21 22 13 .1576E-01
32 12 21 13 .1576E-01
33 22 23 14 .1576E-01
34 13 22 14 .1576E-01
35 23 24 15 .1576E-01
36 14 23 15 .1576E-01
37 24 25 16 .1576E-01
38 15 24 16 .1576E-01
39 25 18 17 .1576E-01
40 16 25 17 .1576E-01
DISPLACEMENTS
X Y Z
1 -.28711713D-10 .17348089D-10 -.23013107D-02
2 .26981282D-04 -.58449789D-11 .12403418D-03
3 .19078607D-04 .19078568D-04 .12403315D-03
4 .28488593D-10 .26981224D-04 .12403335D-03
5 -.19078584D-04 .19078653D-04 .12403344D-03
6 -.26981239D-04 .74573549D-10 .12403344D-03
7 -.19078664D-04 -.19078536D-04 .12403328D-03
8 -.85858794D-10 -.26981189D-04 .12403299D-03
9 .19078543D-04 -.19078664D-04 .12403294D-03
10 .82616315D-05 .34220653D-05 .34118350D-04
11 .34220896D-05 .82616316D-05 .34118449D-04
12 -.34220756D-05 .82616545D-05 .34118522D-04
13 -.82616462D-05 .34221082D-05 .34118544D-04
14 -.82616561D-05 -.34220602D-05 .34118502D-04
15 -.34221060D-05 -.82616211D-05 .34118430D-04
16 .34220644D-05 -.82616785D-05 .34118600D-04
17 .82616260D-05 -.34221213D-05 .34118451D-04
MEMBER ANALYSIS
STRAINS STRESSES
EX EY GA SX SY TAU
1 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42271E+02 -.81153E+03
2 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42271E+02 -.81153E+03
3 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42271E+02 -.81153E+03
4 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42271E+02 -.81153E+03
5 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42271E+02 -.81153E+03
6 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42270E+02 -.81153E+03
7 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42270E+02 -.81153E+03
8 -.16540E-03 .45773E-04 -.21100E-03 -.16667E+04 -.42277E+02 -.81153E+03
9 .14852E-04 -.33342E-04 -.30056E-10 .53294E+02 -.31744E+03 -.11560E-03
10 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
11 .14852E-04 -.33342E-04 -.30224E-11 .53294E+02 -.31744E+03 -.11625E-04
12 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
13 .14852E-04 -.33342E-04 .18448E-11 .53294E+02 -.31744E+03 .70953E-05
14 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
15 .14852E-04 -.33342E-04 -.56942E-11 .53294E+02 -.31744E+03 -.21901E-04
16 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
17 .14852E-04 -.33342E-04 .13423E-11 .53294E+02 -.31744E+03 .51628E-05
18 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
19 .14852E-04 -.33342E-04 -.28137E-11 .53294E+02 -.31744E+03 -.10822E-04
20 -.11794E-04 .76648E-04 .71993E-04 .12309E+03 .80341E+03 .27689E+03
21 .14852E-04 -.33342E-04 .16695E-09 .53293E+02 -.31744E+03 .64210E-03
22 -.11794E-04 .76648E-04 .71992E-04 .12309E+03 .80341E+03 .27689E+03
23 .14852E-04 -.33342E-04 .22670E-09 .53294E+02 -.31743E+03 .87192E-03
24 -.11794E-04 .76648E-04 .71992E-04 .12308E+03 .80341E+03 .27689E+03
25 .00000E+00 -.77662E-05 -.44590E-10 -.25603E+02 -.85343E+02 -.17150E-03
26 -.24579E-05 .72260E-05 .22979E-04 -.31876E+01 .71304E+02 .88383E+02
27 .00000E+00 -.77662E-05 -.24496E-11 -.25603E+02 -.85343E+02 -.94214E-05
28 -.24580E-05 .72261E-05 .22979E-04 -.31882E+01 .71305E+02 .88382E+02
29 .00000E+00 -.77662E-05 .61971E-11 -.25603E+02 -.85343E+02 .23835E-04
30 -.24580E-05 .72262E-05 .22979E-04 -.31883E+01 .71305E+02 .88382E+02
31 .00000E+00 -.77662E-05 -.33468E-11 -.25603E+02 -.85343E+02 -.12872E-04
32 -.24580E-05 .72262E-05 .22979E-04 -.31883E+01 .71305E+02 .88382E+02
33 .00000E+00 -.77662E-05 -.84355E-11 -.25603E+02 -.85343E+02 -.32444E-04
34 -.24580E-05 .72261E-05 .22979E-04 -.31883E+01 .71305E+02 .88382E+02
35 .00000E+00 -.77662E-05 .12291E-10 -.25603E+02 -.85343E+02 .47275E-04
36 -.24580E-05 .72262E-05 .22979E-04 -.31882E+01 .71305E+02 .88382E+02
37 .00000E+00 -.77662E-05 .19779E-10 -.25603E+02 -.85343E+02 .76073E-04
38 -.24580E-05 .72262E-05 .22979E-04 -.31883E+01 .71305E+02 .88382E+02
39 .00000E+00 -.77661E-05 -.88205E-10 -.25603E+02 -.85342E+02 -.33925E-03
40 -.24580E-05 .72262E-05 .22979E-04 -.31884E+01 .71305E+02 .88382E+02
Stop - Program terminated.
6.5.3 Example 6.3. The transversely loaded flat cable-net of Example 3.2 is transformed to a flat stretched membrane ending a sequence of examples that started with Biot's prestressed 2-bar truss. The symmetric analysis model of Figure 3.3 is turned to a symmetric finite element analysis model. (See Figure 6.5.) The thickness of the membrane is taken as the cross-sectional area of a single cable divided by one span i.e. . Similarly a uniform stress of is applied all around the membrane. Young's modulus and Poisson's ratio are taken as 30,000,000 psi and 0.3 respectively. Input file MEMNL.DAT is invoked by MEMNL.FOR and the results are partially listed in MEMNL.OUT. Actually only the echo of the input file and the computations of the last iteration of the last load step are listed. It is important to mention here that LINEAR INCREMENTAL ANALYSIS lists the strains and stresses that are computed from the usual plane nodal displacements/plane strain relations of the plane stress triangular element i.e. nodal displacements are transformed to local coordinates and the z-coordinate discarded. INCREMENTAL NONLINEAR ANALYSIS, however, lists strains and stresses which are computed after rigid body rotations have been removed. (See problem 6.2.) UPDATED PRESTRESS simply adds the stresses from the current increment to those of the previous increment.
It might be of interest to observe that the central deflection here 6.62 inches whereas the central deflection of the cable net is 7.11 inches. But then again this is to be expected because of the shear in the membrane that tends to stiffen it.
Figure 6.5 The Transversely Loaded Stretched Membrane
Output file MEMNL.OUT
NUMBER OF BARS 0
NUMBER OF NODES 25
NUMBER OF SUPPORTS 16
NUMBER OF CONSTRAINTS 0
NUMBER OF ITERATIONS 10
NUMBER OF FINITE ELEMENTS 32
NUMBER OF LOAD STEPS 5
COORDINATES LOADS
X Y Z PX PY PZ1 -.60000E+02 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
2 .00000E+00 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
3 .60000E+02 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
4 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
5 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00 -.10000E+05
6 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
7 -.60000E+02 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
8 .00000E+00 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
9 .60000E+02 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
10 -.12000E+03 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
11 -.60000E+02 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
12 .00000E+00 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
13 .60000E+02 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
14 .12000E+03 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
15 -.12000E+03 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
16 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
17 -.12000E+03 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
18 -.12000E+03 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
19 -.60000E+02 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
20 .00000E+00 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
21 .60000E+02 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
22 .12000E+03 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
23 .12000E+03 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
24 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
25 .12000E+03 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
FINITE ELEMENT DATA
NO. NODES THICKNESS PRESTRESS
I J M SIGX SIGY TAU
1 15 1 10 .41600E-02 .80000E+05 .80000E+05 .00000E+00
2 11 10 1 .41600E-02 .80000E+05 .80000E+05 .00000E+00
3 1 2 11 .41600E-02 .80000E+05 .80000E+05 .00000E+00
4 12 11 2 .41600E-02 .80000E+05 .80000E+05 .00000E+00
5 13 12 2 .41600E-02 .80000E+05 .80000E+05 .00000E+00
6 2 3 13 .41600E-02 .80000E+05 .80000E+05 .00000E+00
7 14 13 3 .41600E-02 .80000E+05 .80000E+05 .00000E+00
8 3 23 14 .41600E-02 .80000E+05 .80000E+05 .00000E+00
9 16 4 15 .41600E-02 .80000E+05 .80000E+05 .00000E+00
10 1 15 4 .41600E-02 .80000E+05 .80000E+05 .00000E+00
11 4 5 1 .41600E-02 .80000E+05 .80000E+05 .00000E+00
12 2 1 5 .41600E-02 .80000E+05 .80000E+05 .00000E+00
13 3 2 5 .41600E-02 .80000E+05 .80000E+05 .00000E+00
14 5 6 3 .41600E-02 .80000E+05 .80000E+05 .00000E+00
15 23 3 6 .41600E-02 .80000E+05 .80000E+05 .00000E+00
16 6 24 23 .41600E-02 .80000E+05 .80000E+05 .00000E+00
17 4 16 17 .41600E-02 .80000E+05 .80000E+05 .00000E+00
18 17 7 4 .41600E-02 .80000E+05 .80000E+05 .00000E+00
19 5 4 7 .41600E-02 .80000E+05 .80000E+05 .00000E+00
20 7 8 5 .41600E-02 .80000E+05 .80000E+05 .00000E+00
21 8 9 5 .41600E-02 .80000E+05 .80000E+05 .00000E+00
22 6 5 9 .41600E-02 .80000E+05 .80000E+05 .00000E+00
23 9 25 6 .41600E-02 .80000E+05 .80000E+05 .00000E+00
24 24 6 25 .41600E-02 .80000E+05 .80000E+05 .00000E+00
25 7 17 18 .41600E-02 .80000E+05 .80000E+05 .00000E+00
26 18 19 7 .41600E-02 .80000E+05 .80000E+05 .00000E+00
27 19 20 8 .41600E-02 .80000E+05 .80000E+05 .00000E+00
28 8 7 19 .41600E-02 .80000E+05 .80000E+05 .00000E+00
29 20 21 8 .41600E-02 .80000E+05 .80000E+05 .00000E+00
30 9 8 21 .41600E-02 .80000E+05 .80000E+05 .00000E+00
31 21 22 9 .41600E-02 .80000E+05 .80000E+05 .00000E+00
32 25 9 22 .41600E-02 .80000E+05 .80000E+05 .00000E+00
****ITERATION NUMBER 10
LOAD STEP 5
COORDINATES LOADS
X Y Z PX PY PZ
1 -.59985E+02 .59985E+02 -.14309E+01 .00000E+00 .00000E+00 .00000E+00
2 -.25607E-15 .59983E+02 -.26046E+01 .00000E+00 .00000E+00 .00000E+00
3 .59985E+02 .59985E+02 -.14309E+01 .00000E+00 .00000E+00 .00000E+00
4 -.59983E+02 -.82747E-15 -.26045E+01 .00000E+00 .00000E+00 .00000E+00
5 .39495E-16 .36615E-16 -.66418E+01 .00000E+00 .00000E+00 -.10000E+05
6 .59983E+02 .82668E-15 -.26045E+01 .00000E+00 .00000E+00 .00000E+00
7 -.59985E+02 -.59985E+02 -.14309E+01 .00000E+00 .00000E+00 .00000E+00
8 .27357E-15 -.59983E+02 -.26046E+01 .00000E+00 .00000E+00 .00000E+00
9 .59985E+02 -.59985E+02 -.14309E+01 .00000E+00 .00000E+00 .00000E+00
10 -.12000E+03 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
11 -.60000E+02 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
12 .00000E+00 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
13 .60000E+02 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
14 .12000E+03 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
15 -.12000E+03 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
16 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
17 -.12000E+03 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
18 -.12000E+03 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
19 -.60000E+02 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
20 .00000E+00 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
21 .60000E+02 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
22 .12000E+03 -.12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00
23 .12000E+03 .60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
24 .12000E+03 .00000E+00 .00000E+00 .00000E+00 .00000E+00 .00000E+00
25 .12000E+03 -.60000E+02 .00000E+00 .00000E+00 .00000E+00 .00000E+00
ERROR = .12389175D-02
DISPLACEMENTS
X Y Z
1 .52016588D-09 -.10951429D-08 -.46234717D-09
2 .11365291D-15 -.21571404D-08 -.16859462D-09
3 -.52016577D-09 -.10951434D-08 -.46234508D-09
4 -.29205406D-09 .72583523D-15 .28383451D-09
5 -.12545721D-16 -.14985675D-16 .45354019D-09
6 .29205406D-09 -.72999134D-15 .28383420D-09
7 .52016577D-09 .10951434D-08 -.46234496D-09
8 -.12116067D-15 .21571404D-08 -.16859486D-09
9 -.52016589D-09 .10951429D-08 -.46234720D-09
MEMBER ANALYSIS
MEMBRANE FINITE ELEMENTS
ELEMENT 1
NODES 15 1 10
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
1 .88504E-11 -.44298E-14 -.18238E-10 .29173E-03 .87385E-04 -.21044E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
1 .88505E-11 -.43817E-14 -.18239E-10 .29173E-03 .87388E-04 -.21044E-03
UPDATED PRESTRESS
SX SY TAU
.9737767E+05 .8521242E+05 -.2801543E+04
ELEMENT 2
NODES 11 10 1
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
2 .00000E+00 .18421E-10 -.86649E-11 .18219E-03 .60729E-03 -.99979E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
2 .00000E+00 .18421E-10 -.86649E-11 .18219E-03 .60729E-03 -.99979E-04
UPDATED PRESTRESS
SX SY TAU
.8521326E+05 .9737754E+05 -.2802885E+04
ELEMENT 3
NODES 1 2 11
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
3 -.87633E-11 .18403E-10 -.26405E-10 -.10689E-03 .52004E-03 -.30467E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
3 -.87633E-11 .18403E-10 -.26405E-10 -.10689E-03 .52004E-03 -.30467E-03
UPDATED PRESTRESS
SX SY TAU
.8351018E+05 .9685909E+05 -.8657109E+04
ELEMENT 4
NODES 12 11 2
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
4 .00000E+00 .35996E-10 -.18919E-17 .35601E-03 .11867E-02 -.21830E-10
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
4 .00000E+00 .35996E-10 -.18919E-17 .35601E-03 .11867E-02 -.21830E-10
UPDATED PRESTRESS
SX SY TAU
.9211395E+05 .1203798E+06 .2734976E-10
ELEMENT 5
NODES 13 12 2
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
5 .00000E+00 .35996E-10 -.18919E-17 .35601E-03 .11867E-02 -.21830E-10
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
5 .00000E+00 .35996E-10 .00000E+00 .35601E-03 .11867E-02 .00000E+00
UPDATED PRESTRESS
SX SY TAU
.9211395E+05 .1203798E+06 .0000000E+00
ELEMENT 6
NODES 2 3 13
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
6 -.87633E-11 .18403E-10 .26405E-10 -.10689E-03 .52004E-03 .30467E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
6 -.87633E-11 .18403E-10 .26405E-10 -.10689E-03 .52004E-03 .30467E-03
UPDATED PRESTRESS
SX SY TAU
.8351018E+05 .9685909E+05 .8657109E+04
ELEMENT 7
NODES 14 13 3
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
7 .00000E+00 .18421E-10 .86649E-11 .18219E-03 .60729E-03 .99979E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
7 .00000E+00 .18421E-10 .86649E-11 .18219E-03 .60729E-03 .99979E-04
UPDATED PRESTRESS
SX SY TAU
.8521326E+05 .9737754E+05 .2802885E+04
ELEMENT 8
NODES 3 23 14
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
8 .88504E-11 -.44298E-14 .18238E-10 .29173E-03 .87385E-04 .21044E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
8 .88505E-11 -.43817E-14 .18238E-10 .29173E-03 .87388E-04 .21044E-03
UPDATED PRESTRESS
SX SY TAU
.9737767E+05 .8521242E+05 .2801543E+04
ELEMENT 9
NODES 16 4 15
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
9 -.50619E-11 .16643E-33 .12082E-16 -.16688E-03 -.50063E-04 .13941E-09
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
9 -.50619E-11 .00000E+00 .12082E-16 -.16687E-03 -.50062E-04 .13941E-09
UPDATED PRESTRESS
SX SY TAU
.1203582E+06 .9210745E+05 -.1955792E-10
ELEMENT 10
NODES 1 15 4
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
10 .88504E-11 -.18497E-10 -.45423E-11 .10883E-03 -.52226E-03 -.52411E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
10 .88505E-11 -.18497E-10 -.45423E-11 .10884E-03 -.52226E-03 -.52411E-04
UPDATED PRESTRESS
SX SY TAU
.9686449E+05 .8350182E+05 -.8644788E+04
ELEMENT 11
NODES 4 5 1
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
11 .46574E-11 -.18474E-10 .14410E-10 -.29171E-04 -.56298E-03 .16627E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
11 .46575E-11 -.18474E-10 .14410E-10 -.29168E-04 -.56298E-03 .16627E-03
UPDATED PRESTRESS
SX SY TAU
.1446910E+06 .9783068E+05 -.1561558E+05
ELEMENT 12
NODES 2 1 5
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
12 -.87633E-11 -.36519E-10 -.17152E-10 -.65007E-03 -.12906E-02 -.19791E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
12 -.87633E-11 -.36519E-10 -.17152E-10 -.65008E-03 -.12906E-02 -.19791E-03
UPDATED PRESTRESS
SX SY TAU
.9784664E+05 .1446473E+06 -.1565376E+05
ELEMENT 13
NODES 3 2 5
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
13 -.87633E-11 -.36519E-10 .17152E-10 -.65007E-03 -.12906E-02 .19791E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
13 -.87633E-11 -.36519E-10 .17152E-10 -.65008E-03 -.12906E-02 .19791E-03
UPDATED PRESTRESS
SX SY TAU
.9784664E+05 .1446473E+06 .1565376E+05
ELEMENT 14
NODES 5 6 3
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
14 .46574E-11 -.18474E-10 -.14410E-10 -.29171E-04 -.56298E-03 -.16627E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
14 .46575E-11 -.18474E-10 -.14410E-10 -.29168E-04 -.56298E-03 -.16627E-03
UPDATED PRESTRESS
SX SY TAU
.1446910E+06 .9783068E+05 .1561558E+05
ELEMENT 15
NODES 23 3 6
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
15 .88504E-11 -.18497E-10 .45423E-11 .10883E-03 -.52226E-03 .52411E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
15 .88505E-11 -.18497E-10 .45423E-11 .10884E-03 -.52226E-03 .52411E-04
UPDATED PRESTRESS
SX SY TAU
.9686449E+05 .8350182E+05 .8644788E+04
ELEMENT 16
NODES 6 24 23
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
16 -.50619E-11 .14390E-32 .12152E-16 -.16688E-03 -.50063E-04 .14021E-09
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
16 -.50619E-11 .00000E+00 -.59973E-27 -.16687E-03 -.50062E-04 -.69199E-20
UPDATED PRESTRESS
SX SY TAU
.1203582E+06 .9210745E+05 -.1366110E-08
ELEMENT 17
NODES 4 16 17
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
17 -.50619E-11 .14309E-32 .12082E-16 -.16688E-03 -.50063E-04 .13941E-09
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
17 -.50619E-11 .00000E+00 -.59973E-27 -.16687E-03 -.50062E-04 -.69199E-20
UPDATED PRESTRESS
SX SY TAU
.1203582E+06 .9210745E+05 -.1366110E-08
ELEMENT 18
NODES 17 7 4
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
18 .88504E-11 -.18497E-10 .45423E-11 .10883E-03 -.52226E-03 .52411E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
18 .88505E-11 -.18497E-10 .45423E-11 .10884E-03 -.52226E-03 .52411E-04
UPDATED PRESTRESS
SX SY TAU
.9686449E+05 .8350182E+05 .8644788E+04
ELEMENT 19
NODES 5 4 7
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
19 .46574E-11 -.18474E-10 -.14410E-10 -.29171E-04 -.56298E-03 -.16627E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
19 .46575E-11 -.18474E-10 -.14410E-10 -.29168E-04 -.56298E-03 -.16627E-03
UPDATED PRESTRESS
SX SY TAU
.1446910E+06 .9783068E+05 .1561558E+05
ELEMENT 20
NODES 7 8 5
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
20 -.87633E-11 -.36519E-10 .17152E-10 -.65007E-03 -.12906E-02 .19791E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
20 -.87633E-11 -.36519E-10 .17152E-10 -.65008E-03 -.12906E-02 .19791E-03
UPDATED PRESTRESS
SX SY TAU
.9784664E+05 .1446473E+06 .1565376E+05
ELEMENT 21
NODES 8 9 5
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
21 -.87633E-11 -.36519E-10 -.17152E-10 -.65007E-03 -.12906E-02 -.19791E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
21 -.87633E-11 -.36519E-10 -.17152E-10 -.65008E-03 -.12906E-02 -.19791E-03
UPDATED PRESTRESS
SX SY TAU
.9784664E+05 .1446473E+06 -.1565376E+05
ELEMENT 22
NODES 6 5 9
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
22 .46574E-11 -.18474E-10 .14410E-10 -.29171E-04 -.56298E-03 .16627E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
22 .46575E-11 -.18474E-10 .14410E-10 -.29168E-04 -.56298E-03 .16627E-03
UPDATED PRESTRESS
SX SY TAU
.1446910E+06 .9783068E+05 -.1561558E+05
ELEMENT 23
NODES 9 25 6
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
23 .88504E-11 -.18497E-10 -.45423E-11 .10883E-03 -.52226E-03 -.52411E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
23 .88505E-11 -.18497E-10 -.45423E-11 .10884E-03 -.52226E-03 -.52411E-04
UPDATED PRESTRESS
SX SY TAU
.9686449E+05 .8350182E+05 -.8644788E+04
ELEMENT 24
NODES 24 6 25
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
24 -.50619E-11 .16722E-33 .12152E-16 -.16688E-03 -.50063E-04 .14021E-09
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
24 -.50619E-11 .00000E+00 .12152E-16 -.16687E-03 -.50062E-04 .14021E-09
UPDATED PRESTRESS
SX SY TAU
.1203582E+06 .9210745E+05 -.1860152E-10
ELEMENT 25
NODES 7 17 18
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
25 .88504E-11 -.44298E-14 .18238E-10 .29173E-03 .87385E-04 .21044E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
25 .88505E-11 -.43817E-14 .18238E-10 .29173E-03 .87388E-04 .21044E-03
UPDATED PRESTRESS
SX SY TAU
.9737767E+05 .8521242E+05 .2801543E+04
ELEMENT 26
NODES 18 19 7
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
26 .00000E+00 .18421E-10 .86649E-11 .18219E-03 .60729E-03 .99979E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
26 .00000E+00 .18421E-10 .86649E-11 .18219E-03 .60729E-03 .99979E-04
UPDATED PRESTRESS
SX SY TAU
.8521326E+05 .9737754E+05 .2802885E+04
ELEMENT 27
NODES 19 20 8
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
27 .00000E+00 .35996E-10 -.20169E-17 .35601E-03 .11867E-02 -.23272E-10
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
27 .00000E+00 .35996E-10 .00000E+00 .35601E-03 .11867E-02 .00000E+00
UPDATED PRESTRESS
SX SY TAU
.9211395E+05 .1203798E+06 .0000000E+00
ELEMENT 28
NODES 8 7 19
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
28 -.87633E-11 .18403E-10 .26405E-10 -.10689E-03 .52004E-03 .30467E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
28 -.87633E-11 .18403E-10 .26405E-10 -.10689E-03 .52004E-03 .30467E-03
UPDATED PRESTRESS
SX SY TAU
.8351018E+05 .9685909E+05 .8657109E+04
ELEMENT 29
NODES 20 21 8
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
29 .00000E+00 .35996E-10 -.20169E-17 .35601E-03 .11867E-02 -.23272E-10
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
29 .00000E+00 .35996E-10 -.20169E-17 .35601E-03 .11867E-02 -.23272E-10
UPDATED PRESTRESS
SX SY TAU
.9211395E+05 .1203798E+06 .2927084E-10
ELEMENT 30
NODES 9 8 21
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
30 -.87633E-11 .18403E-10 -.26405E-10 -.10689E-03 .52004E-03 -.30467E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
30 -.87633E-11 .18403E-10 -.26405E-10 -.10689E-03 .52004E-03 -.30467E-03
UPDATED PRESTRESS
SX SY TAU
.8351018E+05 .9685909E+05 -.8657109E+04
ELEMENT 31
NODES 21 22 9
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
31 .00000E+00 .18421E-10 -.86649E-11 .18219E-03 .60729E-03 -.99979E-04
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
31 .00000E+00 .18421E-10 -.86649E-11 .18219E-03 .60729E-03 -.99979E-04
UPDATED PRESTRESS
SX SY TAU
.8521326E+05 .9737754E+05 -.2802885E+04
ELEMENT 32
NODES 25 9 22
LINEAR INCREMENTAL ANALYSIS
EX EY GA SX SY TAU
32 .88504E-11 -.44298E-14 -.18238E-10 .29173E-03 .87385E-04 -.21044E-03
INCREMENTAL NONLINEAR ANALYSIS
EX EY GA SX SY TAU
32 .88505E-11 -.43817E-14 -.18239E-10 .29173E-03 .87388E-04 -.21044E-03
UPDATED PRESTRESS
SX SY TAU
.9737767E+05 .8521242E+05 -.2801543E+04
Stop - Program terminated.
6.6 Problems.
1. For the plane stress cantilever Example 6.1 study the effect of changing the grid size (element density) in an attempt to obtain reasonable deep beam stresses. (Timoshenko and Goodier, 1951, pp. 361 ff)
2. Discuss the removal of rigid body rotations for the cases of a truss bar and a plane stress triangular element. (Hint: for the truss bar see figures 1.13 and 1.16)
3. An aluminum ( psi, =0.3) prestressed sheet is shown in Figure 6.6. The sheet is L=4 inches square, 0.1 inches thick and there is a uniform initial prestress of 1000 psi. Using the method of load incrementing find the buckling load.(Ans: between 430 lbs and 440 lbs.)
4. For fabric structures applications, rather than buckling, fabric "wrinkling" would be the appropriate phenomenon. Fabric wrinkling can be modeled by monitoring the principal stresses as the structure is loaded. When the initial prestress which is of course tension is taken to zero in terms of either principal stress during the load application, the isotropic finite element stiffness can be changed to an anisotropic one with only stiffness in the direction of the principal tensile stress. (This is the "tension field" concept used in plate girders.) Modify MEMNL.FOR to include principal stresses and monitoring of the loss of prestress and find for the prestressed sheet of Figure 6.6 at what load do wrinkles occur and their angle of inclination. (Ans: P=240 lbs at an angle of 27 to the horizontal. Up to a load of 600 lbs wrinkles do not form in the other triangular facets.)
Figure 6.6. Finite Element Grid for the Prestressed Sheet
5. Model the 24-bar truss of Example 3.6 using nonlinear membrane plane stress elements and find the buckling load. Comment on the expected and attained closeness of the membrane results vs. truss results.
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