Andres Lahe
The EST Method
Examples in structural analysis
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Tallinn 2014
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Copyright: Andres Lahe, 2014
l∖vspace*1mm
This book is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported
License. To view a copy of this license, visit http://creativecommons.org/licenses/
by-sa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco,
California, 94105, USA.
Program excerpts in the book are subject to the terms of the GNU GPL programs License
2.0 or later. To view a copy of the GNU General Public License license, visit http:
//www.gnu.org/licenses/old-licenses/gpl-2.0.html or write to the Free Software
Foundation, Inc. 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
Mente et manu
The EST method1 is a method for solving boundary value problems for the structural
analysis of frames, beams and trusses. Differential equations are considered here
together with a set of boundary conditions:
∙ compatibility equations of the displacements at nodes,
∙ joint equilibrium equations,
∙ side conditions (hinges),
∙ restrictions on support displacements.
The EST method programs written in GNU Octave language assemble and solve
sparse systems of equations with unknown member-end displacements, member-end
forces and support reactions. The analysis technique is illustrated with numerous
examples accompanied with GNU Octave programs.
Andres Lahe
1
./ESTmethod.pdf.
4
Table of Contents
List of Figures
7
List of Tables
9
1 First-order structural analysis
1.1 Computation of frames with the EST method . . . . . . . . . . . . . . .
1.2 Computation of beams with the EST method . . . . . . . . . . . . . . .
1.3 Computation of trusses with the EST method . . . . . . . . . . . . . . .
11
11
25
32
A Matrices
A.1 Sparse matrices and GNU Octave . . . . . . . . . .
A.1.1 Introduction to sparse matrices . . . . . . .
A.1.2 Creating sparse matrices . . . . . . . . . .
A.1.3 Sparse matrix functions in the EST method
A.2 Transformation matrices . . . . . . . . . . . . . . .
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B Work and work-energy theorem
51
B.1 Work done by internal and external forces . . . . . . . . . . . . . . . . . 51
C Computer programs for the EST method
55
C.1 Programs for first-order analysis . . . . . . . . . . . . . . . . . . . . . . . 55
Bibliography
71
Index
75
5
6
TABLE OF CONTENTS
List of Figures
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1.17
1.18
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
Structural systems of a two-span frame . . . . . . . . . . . . . . . .
Numeration of nodes and members of a two-span frame . . . . . . .
Elements of a two-span frame . . . . . . . . . . . . . . . . . . . . .
Sparsity pattern of matrix spA of a two-span frame . . . . . . . . .
Frames with shear force hinge . . . . . . . . . . . . . . . . . . . . .
Numeration of nodes and members of frames with shear force hinge
Elements of a frame with shear force hinge . . . . . . . . . . . . . .
Sparsity pattern of matrix spA of a frame with shear force hinge . .
Three-hinged frames . . . . . . . . . . . . . . . . . . . . . . . . . .
Numeration of nodes and members of a three-hinged frame . . . . .
Elements of a three-hinged frame . . . . . . . . . . . . . . . . . . .
Sparsity pattern of matrix spA of a three-hinged frame . . . . . . .
Continuous beam II . . . . . . . . . . . . . . . . . . . . . . . . . . .
Elements of continuous beam II . . . . . . . . . . . . . . . . . . . .
Sparsity pattern of matrix spA of continuous beam II . . . . . . . .
Numeration of nodes and members of continuous beam II . . . . . .
Multispan hinged beams . . . . . . . . . . . . . . . . . . . . . . . .
Elements of a Gerber beam . . . . . . . . . . . . . . . . . . . . . .
Numeration of nodes and members of Gerber beams . . . . . . . . .
Sparsity pattern of matrix spA of the Gerber beam . . . . . . . . .
The trusses EST . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Free-body diagrams of the trusses with joint numbers . . . . . . . .
Sparsity pattern of matrix spA of truss 1N15WFI . . . . . . . . . .
Planar trusses II . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Numeration of nodes and members of trusses II . . . . . . . . . . .
Sparsity pattern of matrix spA of truss 1N27 . . . . . . . . . . . . .
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12
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33
34
35
37
38
39
A.1
A.2
A.3
A.4
Sparsity pattern of matrix spA . . . .
Assembly sequence of a Gerber beam
Coordinate transformation . . . . . .
Direction cosines of an element . . .
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B.1 Bar member a–b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7
8
LIST OF FIGURES
List of Tables
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
Dimensions and loads of a two-span frame . . . . . . .
Dimensions and loads of a frame with shear force hinge
Dimensions and loads of a three-hinged frame . . . . .
Dimensions of continuous beam II . . . . . . . . . . .
Loads of continuous beam II . . . . . . . . . . . . . . .
Beam dimensions . . . . . . . . . . . . . . . . . . . . .
Beam loads. Sections 𝑘 and 𝑖 . . . . . . . . . . . . . .
Loads and dimensions of trusses . . . . . . . . . . . . .
Distributed dead and live loads of trusses II . . . . . .
Dimensions of trusses. Nodes 𝑖 and 𝑘 . . . . . . . . . .
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11
15
20
25
25
27
28
32
36
36
10
LIST OF TABLES
1. First-order structural analysis
1.1
Computation of frames with the EST method
Example 1.1. A two-span frame (Fig. 1.1). Computation of the displacements,
internal forces 𝑀 , 𝑄, 𝑁 and support reactions.
Initial data are given in Table 1.1 (B denotes load case numbers), and free-body
diagrams are shown in Figs. 1.1 and 1.2.
The free-body diagram number N (circled numbers 1 , ..., 0 shown in Figs. 1.1
and 1.2) conforms with the numbers of GNU Octave programs for the EST method. The
programs can be downloaded from
spESTframeNLaheDefWFI.m.zip1
1. spESTframe1LaheDefWFI.m
2. spESTframe2LaheDefWFI.m
3. spESTframe3LaheDefWFI.m
4. spESTframe4LaheDefWFI.m
5. spESTframe5LaheDefWFI.m
6. spESTframe6LaheDefWFI.m
7. spESTframe7LaheDefWFI.m
8. spESTframe8LaheDefWFI.m
9. spESTframe9LaheDefWFI.m
0. spESTframe10LaheDefWFI.m
In Fig. 1.3, the elements, and in Fig. 1.4, the sparsity pattern of matrix spA of a
two-span frame are shown.
Table 1.1. Dimensions and loads of a two-span frame2
1 2 3 4 5 6 7 8 9
B
𝑙 [𝑚]
6 8 10 6 8 10 6 8 10
ℎ [𝑚]
3 4 4 4 5 5 3 4 4
𝐼1 /𝐼2
2 2 2 2 2 3 3 3 3
𝑝1 [𝑘𝑁/𝑚] 12 0 0 14 0 0 8 0 0
𝑝2 [𝑘𝑁/𝑚] 0 14 0 0 8 0 0 16 0
𝑝3 [𝑘𝑁/𝑚] 0 0 8 0 0 12 0 0 14
𝐹1 [𝑘𝑁 ] 50 0 0 40 0 0 30 0 0
𝐹2 [𝑘𝑁 ]
0 50 0 0 40 0 0 30 0
𝐹3 [𝑘𝑁 ]
0 0 50 0 0 40 0 0 30
1
2
10
12
5
3
10
0
0
20
0
0
./octavePrograms/spESTframeNLaheDefWFI.m.zip.
http://digi.lib.ttu.ee/opik_eme/ylesanded.pdf#page=39. Web. 06 Jan. 2014.
11
1. First-order structural analysis
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1.1 Computation of frames with the EST method
X
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Figure 1.2. Numeration of nodes and members of a two-span frame
7
C 8 [92]
14
1. First-order structural analysis
spESTframe1DefWFI
−10
Numeration of displacements and forces
u w fi N Q M at the end
1 2 3 4 5 6
13 14 15 16 17 18
25 26 27 28 29 30
37 38 39 40 41 42
49 50 51 52 53 54
61 62 63 64 65 66
73 74 75 76 77 78
u w fi N Q M at the beginning
7 8 9 10 11 12
19 20 21 22 23 24
31 32 33 34 35 36
43 44 45 46 47 48
55 56 57 58 59 60
67 68 69 70 71 72
79 80 81 82 83 84
−8
−6
Support reactions: 85 86 87 88 89 90 91 92
−4
3
−2
4
3
22
6 78
4
1
0
5
6
5
1
7
2
−4
−2
0
2
4
6
8
10
12
14
16
Figure 1.3. Elements of a two-span frame
spy(spA) − sparse matrix spA(92,92) non−zero elements [3%]
0
Basic equations of frame 1−42
10
20
30
40
Compatibility equations of displacements 43−59
50
60
Joint equilibrium equations 60−82
70
80
Side conditions 83−84
Restrictions on supports displacements 85−92
90
0
10
20
30
40
50
60
70
80
90
Figure 1.4. Sparsity pattern of matrix spA of a two-span frame
1.1 Computation of frames with the EST method
15
Example 1.2. A frame with shear force hinge (Fig. 1.5). Computation of the
displacements, internal forces 𝑀 , 𝑄, 𝑁 and support reactions.
Initial data are given in Table 1.2 (B denotes load case numbers), and free-body
diagrams are shown in Figs. 1.5 and 1.6.
The free-body diagram number NM (circled numbers 16 , 27 , 38 , 49 , and 50
shown in Figs. 1.5 and 1.6) conforms with the numbers of GNU Octave programs for
the EST method. The programs can be downloaded from
spESTframeNMWFI.m.zip 3
16. spESTframe16WFI.m
27. spESTframe27WFI.m
38. spESTframe38WFI.m
49. spESTframe49WFI.m
50. spESTframe50WFI.m
Table 1.2. Dimensions and loads of a frame with shear force hinge4
3
B
1
2
3
4
5
6
7
8
9
0
l [m]
5
6
8
9
10
5
6
8
9
10
h [m]
4
5
6
7
8
4
5
6
7
8
𝐼1 /𝐼2
2
3
2
3
2
3
2
3
2
3
𝑝1 [kN/m]
8
0
10
0
12
0
8
0
10
0
𝑝2 [kN/m]
0
10
0
12
0
14
0
16
0
10
𝐹1 [kN]
10
0
15
0
20
0
12
0
14
0
𝐹2 [kN]
0
14
0
16
0
18
0
20
0
16
./octavePrograms/spESTframeNMWFI.m.zip.
Compiled by Andrus Räämet, PhD: http://staff.ttu.ee/~raamet/Failid/Joumkodutoo.pdf.
Web. 06 Jan. 2014.
4
16
1. First-order structural analysis
16
F2
11
00
00
11
l
11
00
00
11
l
3 8
11 I1
00
00
11
00
11
00
11
00
11
00 0.6l
11
00
11
00
p11
F1
2
00
11
00
11
I2
00
11
00
11
00
11
00
11
00
11
11
00
00
11
F2
p
111111111111111
000000000000000
000000000000000
111111111111111
00
11
I
00
11
2 7
I2
111
000
l
I2
0.5h
F1
111
000
I1
I
11
00
00
11
2
l
l
111
000
4 9
1
I2
1111111111111111
0000000000000000
0000000000000000
1111111111111111
I1
0.6l
h
F1
I2
I2
111
000
000
111
l
11
00
1
0
0
1
0
1
0
1
p2
0
I2 1
0
1
0
1
0
1
0
1
0
1
0
1
11
00
00
11
F1
I2
l
111
000
0.8h
I1
11
00
p1
I1
0.6h
h
0.6l
F2
111
000
000
111
l
p
111111111111111
000000000000000
000000000000000
111111111111111
F2
I1
5 0
I2
0.6l
I1
1
1
h
11
00
00
11
00
p11
00
11
2
00
11
00 I2
11
00
11
00
11
00
11
0.5h
0.8h
1
p
111111111111111
000000000000000
000000000000000
111111111111111
F2
h
2
l
Figure 1.5. Frames with shear force hinge4
l
11
00
00
11
00
11
00
11
p2
00
11
00
I11
2
00
11
00
11
00
11
00
11
11
00
0.8h
I
111
000
000
111
1
0
0
1
0
1
0
1
0
1
0
1
0
I2 1
0
1
p2
0
1
0
1
0
1
0
1
0.8h
F1
I2
0.6l
0.6h
0.8h
I1
I1
0.5h
1
h
p
111111111111111
000000000000000
000000000000000
111111111111111
1.1 Computation of frames with the EST method
17
X
16
4
2
4
x
z
5
x
x
11C
00
3
3
C 2 [62]
5 C 4 [64]
C 6 [66]
111
000
1
Z
z
z
z
z
1
6
x
x
2
1 [61]
11
00
C 3 [63]
C 5 [65]
2 7
2
3 8
3 [63]
C 5 [65]
x
z
x
z
4 [64]
5
11
00
3
5
11
00
00
11
C 5 [65]
C 2 [62]
5
x
z
z
C 2 [62]
z
3
1
x
x
3 C
111
000
000
111
4
1 C [61]
3 C
11
00
00 C [63] 111
11
000
5
z
z
3
1 [61]
6
x
x
4
z
C
1
11
00
00
11
x
1
z
2
6
x
z
x
4
2
1
4
2
C 6 [66]
C 6 [66]
C 4 [64]
4 9
2
2
x
x
z
x
z
C 2 [62]
3
111
000
000C
111
z
5
x
111
000
x
1 C 1 [61]
C 3 [63]
C
1
11
00
00
11
6
z
z
z
1
z
x
1 [61]
C 2 [62]
3
3
111
000
000
111
C 3 [63]
6
5
C 4 [64]
5
11
00
00
11
x
4
x
2
x
4
x
z
1
2
z
z
5 0
4
4
C 6 [66]
C 5 [65]
5 C 5 [65]
11
00
3
4 [64]
C 6 [66]
Figure 1.6. Numeration of nodes and members of frames with shear force hinge
18
1. First-order structural analysis
In Fig. 1.7, the elements, and in Fig. 1.8, the sparsity pattern of matrix spA of a
frame with shear force hinge are represented.
spESTframe16WFI
−10
Numeration of displacements and forces
−8
−6
u w fi N Q M at the beginning
7 8 9 10 11 12
19 20 21 22 23 24
31 32 33 34 35 36
43 44 45 46 47 48
55 56 57 58 59 60
u w fi N Q M at the end
1 2 3 4 5 6
13 14 15 16 17 18
25 26 27 28 29 30
37 38 39 40 41 42
49 50 51 52 53 54
Support reactions: 61 62 63 64 65 66
−4
4
4
2
6
2
−2
3
5
3
5
1
0
2
−4
1
−2
0
2
4
6
8
10
Figure 1.7. Elements of a frame with shear force hinge
12
14
1.1 Computation of frames with the EST method
0
19
spy(spA,14) − sparse matrix spA(66,66) non−zero elements [3.9%]
Basic equations of frame
1−30
10
20
30
Compatibility equations of displacements 31− 42
40
Joint equilibrium equations
43−57
50
Side conditions 58−60
60
Restrictions on support displacements
0
10
20
30
40
61−66
50
60
Figure 1.8. Sparsity pattern of matrix spA of a frame with shear force hinge
20
1. First-order structural analysis
Example 1.3. A three-hinged frame (Fig. 1.9). Computation of the displacements,
internal forces 𝑀 , 𝑄, 𝑁 and support reactions.
Initial data are given in Table 1.3 (B denotes load case numbers), and free-body
diagrams are shown in Figs 1.9 and 1.10.
The free-body diagram number K (circled numbers 1 , ..., 0 shown in Figs. 1.9
and 1.10) conforms with the numbers of GNU Octave programs for the EST method.
The programs can be downloaded from
spESTframe3hingeKNQM.m.zip 5
1. spESTframe3hinge1NQM.m
2. spESTframe3hinge2NQM.m
3. spESTframe3hinge3NQM.m
4. spESTframe3hinge4NQM.m
5. spESTframe3hinge5NQM.m
6. spESTframe3hinge6NQM.m
7. spESTframe3hinge7NQM.m
8. spESTframe3hinge8NQM.m
9. spESTframe3hinge9NQM.m
0. spESTframe3hinge10NQM.m
In Fig. 1.11, the elements, and in 1.12, the sparsity pattern of matrix spA of a
three-hinged frame are shown.
Table 1.3. Dimensions and loads of a three-hinged frame6
5
B
1
2
3
4
5
6
7
8
9
0
l [m]
6
8
10
6
8
10
6
8
10
6
h [m]
4
5
6
4
5
6
4
5
6
4
𝜉
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.75
F [kN]
35
30
25
30
25
20
25
20
15
20
p [kN/m]
24
22
20
22
20
18
20
18
16
26
./octavePrograms/spESTframe3hingeKNQM.m.zip.
Compiled by Andrus Räämet, PhD: http://staff.ttu.ee/~raamet/Failid/Raamikodutoo.
pdf. Web. 06 Jan. 2014.
6
1.1 Computation of frames with the EST method
2
q
111111111111111111111
000000000000000000000
l/2
h
F
11
00
002m
11
q
11111111111111111111111
00000000000000000000000
00000000000000000000000
11111111111111111111111
111
000
2m
11
00
2m
F
l/2
l/2
111
000
000
111
2m
q
1111111111111
0000000000000
0000000000000
1111111111111
0
ξh
h
l
l/2
111111111111111111111
000000000000000000000
11
00
002m
11
F
111
000
2m
l/2
q
111
000
000
111
2m
l
h
ξh
112m
00
F
ξh
ξh
F
8
111111111111111111111
000000000000000000000
111
000
2m
q
1111111111111
0000000000000
0000000000000
1111111111111
111
000
2m
q
l/2
h
l/2
l/2
6
ξh
h
7
l/2
F
112m
00
F
111
000
2m
1111111111111
0000000000000
111
000
2m
q
1111111111111111111111
0000000000000000000000
0000000000000000000000
1111111111111111111111
5
q
h
l/2
l/2
ξh
h
112m
00
l/2
11
00
002m
11
h
11111111111111111111111
00000000000000000000000
F
ξh
4
q
3
9
11
00
00
11
ξh
l/2
ξh
ξh
h
l/2
111
000
000
111
2m
11111111111111
00000000000000
F
F
111
000
000
111
2m
q
h
1
21
112m
00
Figure 1.9. Three-hinged frames6
l/2
l/2
112m
00
1. First-order structural analysis
x
x
x
z
z
C 4 [40]
x
z
2
111
000
000C [38]
111
6
5
6
111 C [39]
000
000C [40]
111
z
x
z
z
z
x
x
111
000
000
111
4
z
1
7 C 3 [39]
x
6
x
1
2
5
x
111
000
000C [38]
111
C 1 [37]
4
3
2
3
z
5
z
1
4
6
x
2
2
5
z
Z
4
3
2
3
z
X
1
x
22
1
3
7
4
2
4
x
x
x
z
1
1 C 1 [37]
z
x
z
z
x
z
z
z
z
x
z
z
x
x
7 C 3 [39]
111C
000
2 [38]
z
x
1
4
4 [40]
6
7
111C [38]
000
111C
000
2
4 [40]
6
5
x
x
111C
000
C 1 [37]
5
z
6
z
1
2
x
5
4
3
2
3
6
7 C 3 [39]
11C
00
2 [38]
z
z
4
111C
000
6
5
6
C 1 [37]
x
x
z
2
1
1
z
x
z
4
6
5
x
x
z
x
x
4
3
2
3
111C [40]
000
z
7 C 3 [39]
x
C 1 [37]
4
z
6
2 [38]
2
5
x
5
5
4
3
2
3
x
11C
00
4
6
z
2
5
z
1
1
4
3
2
3
x
3
C 3 [39]
4 [40]
8
2 [38]
4
4
3
x
z
x
x
z
x
2
4
5
5
z
z
z
x
z
x
z
x
x
z
7 C [39]
11
00
00C [40]
11
3
x
x
z
z
6
6
z
x
z
5
C 1 [37]
2
6
5
x
x
z
x
z
x
z
x
x
4
111
000
000C
111
3
6
z
2
5
1
1
0
x
4
3
4 [40]
4
z
2 [38]
2
3
111
000
000C
111
2
5
x
C 1 [37]
4
3
2
z
9
6
7 C 3 [39]
x
11
00
00C
11
5
z
1
1
4
3
6
x
2
5
z
3
4
3
2
x
7
7 C 3 [39]
111C
000
4 [40]
111
000
C 2 [38]
x
2 [38]
1
1 C 1 [37]
z
z
z
111C
000
6
x
1 C 1 [37]
z
1
6
7 C 3 [39]
111C
000
4 [40]
Figure 1.10. Numeration of nodes and members of a three-hinged frame
1.1 Computation of frames with the EST method
23
spESTframe3hinge1NQM
Numeration of forces
−8
N Q M at the end
1 2 3
7 8 9
13 14 15
19 20 21
25 26 27
31 32 33
N Q M at the beginning
4 5 6
10 11 12
16 17 18
22 23 24
28 29 30
34 35 36
−6
Support reactions: 37 38 39 40
−4
3
2
2
4
4
5
1
−2
0
2
−4
3
0
6
6
7
1
−2
5
2
4
6
8
Figure 1.11. Elements of a three-hinged frame
10
12
24
1. First-order structural analysis
spy(spA) − sparse matrix spA(40,40) non−zero elements [5.4%]
0
Basic equations of frame
1−18
5
10
15
20
Joint equilibrium equations 19−36
25
30
35
Side conditions 37−40
40
0
5
10
15
20
25
30
35
Figure 1.12. Sparsity pattern of matrix spA of a three-hinged frame
40
1.2 Computation of beams with the EST method
1.2
25
Computation of beams with the EST method
Example 1.4. A continuous beam (Fig. 1.13). Computation of the displacements,
internal forces 𝑀 , 𝑄 and support reactions.
Initial data. The dead load g is given in Table 1.5, where B denotes load case
numbers. The forces 𝐹𝑎 = 80 𝑘𝑁 and simultaneously acting 𝐹𝑏 = 60 𝑘𝑁 and 𝐹𝑐 = 40 𝑘𝑁
are live loads. Location of the points (see Fig. 1.13) a, b, c at which concentrated
loads 𝐹𝑎 , 𝐹𝑏 , and 𝐹𝑐 act is indicated by numbers in Table 1.5. Versions (A) of beam
dimensions are given in Table 1.4. The flexural rigidity EI is assumed to be constant
along the beam.
In Figs. 1.13 and 1.14, free-body diagrams are shown. The GNU Octave program for continuous beam II spESTbeam32LaheWFI.m can be downloaded from
spESTbeam32LaheWFI.m.zip 7 .
1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 17 18 19 2021 2223 24 25 26 27 2829 30
31
0
1
2
l
l1
3
l3
2
2m
Figure 1.13. Continuous beam II8
Table 1.4. Dimensions of continuous beam II8
A
𝑙1 [𝑚]
𝑙2 [𝑚]
𝑙3 [𝑚]
𝑖
1 2 3 4 5 6
8 10 8 10 8 10
10 10 8 8 10 8
8 8 10 8 10 10
0 0 0 1 1 1
7
6
8
8
1
8
8
6
8
2
9
8
8
6
2
0
6
6
8
2
Table 1.5. Loads of continuous beam II8
B
𝑔 [𝑘𝑁/𝑚]
𝑎
𝑏
𝑐
𝑘
1
12
3
12
17
24
2
14
4
12
16
22
3
16
2
13
18
26
4
18
26
14
18
8
5 6 7 8 9 0
12 14 16 12 14 16
27 18 17 26 24 23
14 2 2 3 4 4
17 7 6 8 8 7
4 6 28 16 14 12
In Fig. 1.14, nodes and elements of the continuous beam II are shown.
7
8
./octavePrograms/spESTbeam32LaheWFI.m.zip.
http://digi.lib.ttu.ee/opik_eme/ylesanded.pdf#page=39.
26
1. First-order structural analysis
X
2
111
000
x
z
x
z
C 1 [33]
11
00
4
111
000
z
2 [34]
3
4
x
2
3
C 4 [36]
C 3 [35]
5
z
1
x
Z
0110
1010 1
C
C 5 [37]
Figure 1.14. Elements of continuous beam II
In Fig. 1.15, the sparsity pattern of matrix spA, and in Fig. 1.16, the node and
member numbers of the continuous beam II are shown.
spy(spA) − sparse matrix spA(37,37) non−zero elements [6.9%]
0
Basic equations of frame 1−16
5
10
15
Compatibility equations of displacements 17−22
20
Joint equilibrium equations 23−32
25
30
Restrictions on support displacements 33−37
35
0
5
10
15
20
25
30
35
Figure 1.15. Sparsity pattern of matrix spA of continuous beam II
1.2 Computation of beams with the EST method
27
spESTbeam32LaheWFI
−10
Numeration of displacements and forces
−8
w fi Q M at the beginning
5 6 7 8
13 14 15 16
21 22 23 24
29 30 31 32
−6
−4
w fi Q M at the end
1 2 3 4
9 10 11 12
17 18 19 20
25 26 27 28
Support reactions: 33 34 35 36 37
−2
0
1
1
2
2
3
3
4 45
2
4
−4 −2
0
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32 34
Figure 1.16. Numeration of nodes and members of continuous beam II
Example 1.5. A multispan hinged beam (Fig. 1.17). Computation of the internal
forces 𝑀 , 𝑄 and support reactions.
Initial data. The free-body diagram numbers A (circled numbers 1 , ..., 0
shown in Figs. 1.17 and 1.19) conform with the numbers of GNU Octave programs for
the EST method. Beam dimensions for loading variant B are given in Table 1.6. The
dead load: uniform load 𝑔 = 16 𝑘𝑁/𝑚, forces 𝐹𝑘 = 60 𝑘𝑁 (section k in Table 1.7),
𝐹𝑖 = 60 𝑘𝑁 (section i in Table 1.7), and 𝐹𝑗𝑟 = 40 𝑘𝑁 (in moment hinge near the
support r, Table 1.6).
Table 1.6. Beam dimensions9
B
𝑙1 [𝑚]
𝑙2 [𝑚]
𝑙3 [𝑚]
𝑙4 [𝑚]
r
9
1
10
15
12
12
b
2
16
15
12
16
c
3
18
12
16
12
d
4
10
12
15
12
b
5
15
12
16
12
c
6
18
16
12
12
d
7
12
16
12
10
b
http://digi.lib.ttu.ee/opik_eme/ylesanded.pdf#page=19.
8
16
12
18
16
c
9
15
16
12
12
d
0
10
10
12
16
c
28
1. First-order structural analysis
The uniform load 𝑔 and forces 𝐹𝑘 , 𝐹𝑖 are element loads. The force 𝐹𝑗𝑟 is a nodal
load.
The free-body diagram numbers A conform with the numbers of GNU Octave
programs for a multispan hinged beam. The programs can be downloaded from
spESTGerberBeamNQM.m.zip 10
1. spESTGerberBeam1QM.m
2. spESTGerberBeam2QM.m
3. spESTGerberBeam3QM.m
4. spESTGerberBeam4QM.m
5. spESTGerberBeam5QM.m
6. spESTGerberBeam6QM.m
7. spESTGerberBeam7QM.m
8. spESTGerberBeam8QM.m
9. spESTGerberBeam9QM.m
0. spESTGerberBeam10QM.m
Table 1.7. Beam loads. Sections 𝑘 and 𝑖9
𝐵⇓ ∖ 𝐴⇒
k
i
10
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
6
16
7
8
17
9
6
18
8
19
12
1
3
13
2
1
12
3
1
2
2
12
14
6
12
14
8
12
14
7
12
7
6
12
9
8
13
7
8
14
7
3
16
11
6
17
13
8
18
11
6
13
11
16
12
13
17
11
12
18
17
19
4
11
6
8
12
6
8
13
6
12
8
17
1
11
18
2
13
19
3
7
2
./octavePrograms/spESTGerberBeamNQM.m.zip.
5
7
12
2
9
14
3
7
12
4
14
12
2
13
14
3
12
13
2
14
3
6
11
3
12
13
2
14
12
3
13
2
17
7
8
18
7
8
19
7
8
7
7
7
1
2
9
3
4
7
1
2
3
16
17
7
18
16
8
17
18
7
17
8
16
17
11
18
19
11
16
17
11
18
11
13
17
12
11
18
13
12
19
13
9
11
2
12
13
3
14
12
2
13
3
17
18
7
17
18
8
17
18
7
17
0
18
16
13
18
16
11
18
16
13
18
11
12
17
13
12
17
11
12
18
13
1.2 Computation of beams with the EST method
1a
2
2
2
3
3
3
1
2
2
2
3
3
3
1
<
<
<
=
=
=
1
F
F
F
G
G
G
1
P
P
P
Q
Q
Q
a
Z
Z
[
1
Z
Z
Z
[
[
[
4
2
3
2
4
3
2
1
2
3
2
3
1
2
:
:
:
98
:
;:
;:
7
7
9c
8
6
7
10 11
9c
8
6
7
5
6
7
10 11
10 11
b
7
10 11
UT
UT
^
^
]
b
6
7
_
_
4
5
6
7
`
`
a`
a`
17
19 e
ON
18
YX
YX
b
b
cb
cb
b
17
18
19 e
a
c
c
14d
15 16
17
18
19 e
10 11
12
13
14d
15 16
10 11
12
13
18
19 e
14d
17
15 16
17
18
19 e
20
21
20
21
20
21
20
21
20
21
20
21
20
21
ON
19 e
b
15 16
21
ED
9c
8
ED
X
9c
8
19 e
18
20
;
b
7
13
17
15 16
WV
12
18
N
WV
10 11
6
15 16
14d
17
ML
14d
13
19 e
D
ML
13
12
15 16
a
5
12
18
CB
14d
`
b
13
`
10 11
9c
8
12
17
;
CB
_^
5
15 16
9
14d
V
9c
8
13
L
_^
6
12
KJ
SR
5
14d
B
9c
8
13
A@
9c
8
12
9
KJ
]\
4
3
5
\
8a
8
98
IH
]\
4
7
8
8
A@
\
1
5
b
4
6
8
76
?>
b
4
3
5
6
76
5
b
]
7a
3
SR
[
6a
2
6
b
IH
4a
Z
4
54
?>
3a
[
3
4
54
5
2a
5
2
4
29
0
1
e
9a
1
2
3
4
,
$
-
%
5
b
,
-
6
7
9c
8
10 11
12
13
14d
$
%
'&
'&
(
)(
"
"
"
"
#"
#"
15
)(
16
17
18
19
20
0
1
21
0
.
1
0a
*
*
+
+
1
2
3
4
*
*
+
+
b
5
6
7
9c
8
!
!
!
!
10 11
12
13
l1
14d
#
l2
e
15
16
17
18
19
20 21
#
l3
l4
2m
Figure 1.17. Multispan hinged beams
spESTGerberBeam1QM
−20
Numeration of forces
Q M at the beginning
3 4
7 8
11 12
−15
15 16
19
23
27
31
−10
Support reactions: 33 34 35 36 37
−5
0
5
−10
20
24
28
32
Q M at the end
1 2
5 6
9 10
13 14
17 18
21 22
25 26
29 30
1
1
−5
0
5
2 23
10
15
3
20
4 45
25
30
5
35
6 67
40
45
7
50
Figure 1.18. Elements of a Gerber beam
55
8 89
60
65
70
/
.
/
1. First-order structural analysis
z
C 5 [37]
5 [37]
x
z
8
7
8
x
7
z
x
z
x
z
z
x
z
x
z
C
11
00
9
00
11
00
11
C 5 [37]
11
00
7
8
z
x
z
6
8
7
6 [38]
11
00
009C
11
00
11
6 [38]
x
6
9
z
4 [36]
6
11
00
00
11
8
11
00
00C
11
8
x
7
C 4 [36]
5
x
z
C 3 [35]
x
7
11
00
00C
11
6
5
11
00
11
00
9
z
x
x
z
x
z
x
z
x
z
4
8
8
x
z
3
z
11
00
7
C 3 [35]
4
x
11
00
00
11
5
5
9
7
6
6
8
118
00
00
11
z
5
9
C 5 [37]
C 4 [36]
4
4
3
x
z
C 2 [34]
6
8
11
00
00
11
x
x
11
00
2
z
x
z
x
z
x
z
x
z
x
z
x
z
1
x
3
2
5
C 3 [35]
3
z
5
11
00
00
11
x
4
C 2 [34]
1
6
8
z
z
4
7
7
C 4 [36]
5
9
C 5 [37]
6
116
00
00
11
11
00
00
11
x
x
11
00
00
11
2
z
C 1 [33]
x
z
C 1 [33]
3
2
x
1
111
000
3
C 2 [34]
1
0
2
11
00
00
11
5
C 3 [35]
2
C 1 [33]
111
000
000
111
11
00
z
z
1
9
3
C 2 [34]
C 1 [33]
1
111
000
000
111
11
00
2
4
4
7
C 4 [36]
115
00
00
11
3
7
6
11
00
00
11
8
8
C 5 [37]
6
C 3 [35]
2
x
x
1
5
4
4
7
z
z
1
5
9
x
x
7
z
3
3
C 2 [34]
C 1 [33]
11
00
00
11
C 3 [35]
2
112
00
00
11
x
z
1
1
111
000
000
111
z
x
C 2 [34]
7
6
C 4 [36]
4
4
11
00
00
11
x
3
11
00
00
11
C 5 [37]
6
5
C 3 [35]
3
11
00
00
11
8
8
z
2
11
00
00
11
5
11
00
00
11
7
7
C 4 [36]
4
4
11
00
x
z
3
9
8
C 5 [37]
6
6
C 3 [35]
2
C 1 [33]
8
x
1
111
000
000
111
5
11
00
00
11
8
7
7
C 4 [36]
5
C 2 [34]
1
111
000
4
4
3
11
00
00
11
z
2
2
C 1 [33]
6
3
C 2 [34]
1
5
C 3 [35]
3
11
00
00
11
11
00
x
2
2
1
111
000
000
111
11
00
6
6
9
C 5 [37]
z
1
C 1 [33]
4
11
00
5
5
8
8
11
00
00
11
C 4 [36]
4
4
7
11
00
00
11
x
111
000
000
111
3
3
C 2 [34]
1
6
7
z
z
C 1 [33]
5
C 3 [35]
2
2
6
x
x
1
3
z
C 2 [34]
1
111
000
x
2
4
11
00
00
11
5
z
C 1 [33]
3
11
00
00
11
4
x
2
3
z
2
x
1
111
000
000
111
1
z
1
x
30
C 4 [36]
C 5 [37]
Figure 1.19. Numeration of nodes and members of Gerber beams
In Fig. 1.20, the sparsity pattern of matrix spA of the Gerber beam is shown.
1.2 Computation of beams with the EST method
31
spy(spA) − sparse matrix spA(37,37) non−zero elements [5.6%]
0
Basic equations of frame 1−16
5
10
15
Joint equilibrium equations 17−31
20
25
30
Side conditions 32−37
35
0
5
10
15
20
25
30
35
Figure 1.20. Sparsity pattern of matrix spA of the Gerber beam
32
1. First-order structural analysis
1.3
Computation of trusses with the EST method
Example 1.6. Statically indeterminate planar trusses (Fig. 1.21). Computation of the displacements and internal forces 𝑁 .
Initial data. The trusses depicted in Fig. 1.21 are subjected to loads 𝐹1 , 𝐹2 , and
𝐹3 . Load values and dimensions of the truss are given in Table 1.8, where B denotes
load case numbers. The span is of length 𝑙 = 4𝑑 (4 equal panels, each of length 𝑑) and
of height ℎ = 𝑑 [REL83].
The free-body diagram numbers A are shown in Fig. 1.22. The programs can be
downloaded from
spESTtrussN15WFI.m.zip 11
spESTtruss1N15WFI.m
spESTtruss2N15WFI.m
spESTtruss3N15WFI.m
spESTtruss4N15WFI.m
spESTtruss5N15WFI.m
spESTtruss6N15WFI.m
spESTtruss7N15WFI.m
spESTtruss8N15WFI.m
spESTtruss9N15WFI.m
spESTtruss10N15WFI.m
Table 1.8. Loads and dimensions of trusses
B
𝑑 [𝑚]
𝐹1 [𝑘𝑁 ]
𝐹2 [𝑘𝑁 ]
𝐹3 [𝑘𝑁 ]
𝐴1 /𝐴2
𝐴1 /𝐴3
11
1
3.0
60
50
0
1.2
1.5
2
3.2
60
0
40
1.3
1.8
3
3.4
60
50
0
1.4
2.0
4
3.6
60
0
40
1.5
2.2
./octavePrograms/spESTtrussN15WFI.m.zip.
5
4.0
60
50
0
1.2
2.4
6
3.0
80
0
40
1.3
2.2
7
3.2
80
50
0
1.4
1.8
8
3.4
80
0
40
1.2
2.0
9
3.6
80
50
0
1.3
1.5
10
4.0
80
0
40
1.4
2.2
1.3 Computation of trusses with the EST method
F2
F1
1
4
4
2
3
2
1111
0000
0000
1111
1
8
6
7
9
5
5
11
00
00
11
6
1
3
d
d
2
2
1
1111
0000
0000
1111
4
4
3
8
6
7
9
5
1
5
11
00
00
11
6
3
d
d
2
4
3
1111
0000
0000
1111
1
5
5
3
4
6
5
2
15
17
2
14
9
11
00
00
11
d
6
20
8
10
F3
10
14
13
11
8
15
5
d
19
18
21
17
7
4
4
3
7
5
6
1
2
1
1111
0000
0000 d
1111
8
F3
6
12
9 11
10
5
8
11
00
00
11
h
3
d
14
d
F2
6
8
7
1111
0000
0000
1111
8
19
16
h
20
11
11
00
00
11
17
9
d
d
h
h/2
F3
12
8
16
11 13
9
10
7
5
h
10
15
14
1
17
11
00
00
11
h
9
d
d
d
F1
2
h
10
18
2
4
4
h/2
9
d
7
6
0
17
7
111
000
000
111
12
F1
9
11
00
00
11
d
13
4
5
1
14
d
17
14
F3
10 5
3
2
10
16
15
13
3
10
15
7
d
11
1
d
13
15
7
4
h
8
2
16
10
6
9
3
20
d
F2
111
000
000
111
8
11
F2
4
6
F3
5
11
00
00
11
d
h
11
00
00
11
d
12
9
3
d
4
d
6
7
6
5
3
1
9
16
d
F1
9
5
8
1
12
11
d
3
10
F2
4
F1
h
h/2
8
11
00
00
11
d
8
1
2
11
00
00
11
d
12
3
3
F3
11
12
14
12
16
18
22
24
19
25
9 11 13
15 17
21 23
8
14
20
7
9
13
d
2
11
17
9 16
8
7
3 5
1
4
1
111
000
000
111
12
19
18
F2
10
6
10
d
9
7
h
F3
14
3
1111
0000
0000
1111
2
7
d
13 15
d
4
1
13
10
7
F1
7
2
11
6
6
F1
10
16
7
d
1
111
000
000
111
5
4
8
12
4
1
9
11
00
00
11
d
2
h
F3
11
8
2
17
7
12
4
4
14
d
6
10
9
6
13
2
10
15
10
F2
F1
5
11
16
F2
F1
2
8
F2
F1
3
F3
12
33
3
F2
6
8
6
F3
8
12
11
7 9
5
d
10 13
5
1
2
7
3
1
111
000
000 d
111
Figure 1.21. The trusses EST
10
16
h/2
15
14
17
h
111
000
000
111
9
d
d
d
34
1. First-order structural analysis
X
1
2
Z
3
2
C 1 [137]
4
4
2
8
6
7
9
5
1
1
1111
0000
0000
1111
C [138]
8
12
11
13
10
11
00
00C5 [139]
11
6
3
2
16
15
14
7
3
10
2
17
2
11
00
00C
11
C 1 [201]
9
4 [140]
3
4
5
6
4
3 5
7
10
1
3
1
1111
0000
0000C [202]
1111
6
8
10
11
12
14
12
16
18
22
24
13
19
25
9 11
15 17
21 23
8
14
20
7
9
13
11
00
00C
11
2
111
000
000C
111
3 [203]
4 [204]
4
2
4
4
2
3
8
6
7
9
5
C 1 [137]
1
1111
0000
1
0000
1111
C [138]
6
3
2
8
12
11
11
00
00C5
11
3 [139]
6
14
10
2
15
17
2
14
9
11
00
00C
11
16
13
10
7
4
4
3
5
C 1 [137]
4 [140]
5
1
1111
0000
1
0000
1111
C [138]
8
6
7
6
3
2
8
12
9
11
13
10
5
11
00
00C [139]
11
7
16
10
15
17
14
9
111
000
000C
111
3
4 [140]
6
2
5
5
4
C 1 [177] 3
1
1111
0000
0000
1111
10
9
6
12
4
1
7
C 2 [178]
3
2
20
17
9 16
7
12
19
18
13 15
11
8
2
10
8
11
00
00C
11
3
1
C 1 [161]
11
3 [179]
7
5
4
4
8
6
2
1111
0000
1
0000
1111
C [162]
6
9
11
7
3
8
15
14
19
13
10 5
12
17
9
20
11
111
000
000C
111
3 [163]
7
2
10
18
16
8
4
6
5
2
6
9
7
4
1
C 1 [169]
2
8
10
13
11
8
15
5
12
19
18
12
3
3
10
14
21
17
7
4
4
9
16
20
1111
0000
0000
1111
C [170]
11
111
000
000
111
C [172]
2
4
6
1
12
8
16
11 13
9
3
10
7
5
10
15
14
2
C 3 [171] C 1 [137]
1
7
5
3
2
6
8
1111
0000
0000
1111
C [138]
17
C 3 [139]
111
000
000
111
C [140]
1
9
2
4
0
9
2
4
4
3
1
C 1 [137]
2
1
1111
0000
0000
1111
C [138]
2
8
7
5
6
3
6
12
9 11
10
5
8
2
10
16
13
3
1
17
7
4
4
15
14
2
C 3 [139] C [137]
1
9
1111
0000
0000
1111
C [140]
4
5
3
6
8
6
7 9
5
8
12
11
10 13
7
10
16
15
14
1
1111
0000
0000
1111
C [138]
2
Figure 1.22. Free-body diagrams of the trusses with joint numbers
17
C 3 [139]
111
000
000
111
C [140]
9
4
1.3 Computation of trusses with the EST method
35
spy(spA,14) − sparse matrix spA(140,140) non−zero elements [1.7%]
0
Basic equations of frame 1−68
20
40
60
Compatibility equations of displacements 69−116
80
100
Joint equilibrium equations 117−136
120
Restriction on support displacements
140
0
20
40
60
137−140
80
100
120
Figure 1.23. Sparsity pattern of matrix spA of truss 1N15WFI
140
36
1. First-order structural analysis
Example 1.7. Planar trusses (Fig. 1.24). Computation of the internal forces 𝑁 ,
influence line ordinates and support reactions. Computation of the maximum and minimum internal forces due to dead and live loads. Draw the influence line for the truss
member of panel k shown in Table 1.10.
Initial data. The simply supported trusses shown in Fig. 1.24 are subjected to
uniform distributed dead load 𝑔, uniform distributed live load 𝑝 (Table 1.9) and live load
𝐹𝑖 = 100 kN at node i shown in Table 1.10. The span is of length 𝑙 = 8𝑑 (8 equal
panels, each of length 𝑑) and of height ℎ, the distance between rafters is marked by the
letter a (Table 1.10).
In Figs. 1.24 and 1.25, free-body diagram numbers A are shown. The programs can
be downloaded from
spESTtrussN27.m.zip 12
spESTtruss1N27.m
spESTtruss2N27.m
spESTtruss3N27.m
spESTtruss4N27.m
spESTtruss5N27.m
spESTtruss6N27.m
spESTtruss7N27.m
spESTtruss8N27.m
spESTtruss9N27.m
spESTtruss10N27.m
Table 1.9. Distributed dead and live loads of trusses II13
A
𝑔 [𝑘𝑃 𝑎]
𝑝 [𝑘𝑃 𝑎]
1 2 3 4 5 6 7 8 9 0
3.0 3.0 3.0 3.0 3.0 4.0 4.0 4.0 4.0 4.0
0.75 0.75 0.75 0.75 0.75 1.0 1.0 1.0 1.0 1.0
Table 1.10. Dimensions of trusses. Nodes 𝑖 and 𝑘 13
B
𝑑 [𝑚]
ℎ [𝑚]
𝑎 [𝑚]
i
k
12
13
1 2 3 4 5 6 7 8 9 0
1.5 2.0 2.5 3.0 3.5 1.5 2.0 2.5 3.0 3.5
2.0 2.6 3.5 4.0 5.0 2.4 3.0 4.0 5.0 5.5
6 6 6 6 6 5 5 5 5 5
7 5 11 3 7 5 11 13 7 11
3 4 5 6 7 2 3 4 5 6
./octavePrograms/spESTtrussN27.m.zip.
http://digi.lib.ttu.ee/opik_eme/ylesanded.pdf#page=29.
1.3 Computation of trusses with the EST method
9
11
13
12
14
2
16
18
3
5
7
9
11
6
4
5
1
5
7
9
11
3
5
6
$
$
%
%
3
$
$
%
%
5
1
3
7
9
11
13
8
1
3
8
h
12
17
14
5
7
9
11
13
16
18
15
17
8
"
"
"
#"
#"
10
12
14
5
d
16
&
&
'&
'&
'&
!
!
!
'
'
!
!
!
15
7
9
11
!
1
3
13
16
18
7
9
11
13
4
17
16
#
6
8
10
12
14
#
16
#
18
0
#"
15
15
5
d
!
2
14
h
1/2 h
15
12
2
10
8
2
13
d
&
'
6
4
9
11
10
4
d
$
%
9
2
10
12
18
14
1
$
%
16
h
$
%
14
h
$
%
12
4
7
6
2
10
1/2 h
7
8
6
d
16
15
4
6
13
8
17
15
12
d
1
14
3
10
2
13
8
11
2
15
d
13
6
4
4
1
7/16 h
3/4 h
15/16 h
h
d
10
8
6
4
24/25 h
h
9
1/2 h
1
7
5
3
15
d
3
7
9/25 h
h
5
18
h
16
3
14
16
14
1
12
7/16 h
3/4 h
15/16 h
h
2
h
12
2
10
8
16/25 h
4
6
4
1/2 h
2
10
21/25 h
6
1/2 h
8
1
37
1
3
5
7
d
Figure 1.24. Planar trusses II13
9
11
13
15
17
38
1. First-order structural analysis
8
1
6
4
2
2
10
1
3
1
1
9
5
1
7
8
2
12
7
7
11
21
20
11
8
15
12
4
3
5
7
7
8
4
11
13
10
16
3
20
20
21
28
23
24
10
5
9
7
17
19
13
9
18
1
7
23
22
11
2
2
1
27
13
25
15
26
27 30 31 33
21
13
10 11
14
19
15
23
5
9
4
10
5
21
11
13
5
4
6
8
7
7
17
11
16
9
8 16 10
12
20
18
3
2
11
7
15
6
10
17
1
1
11111
00000
000000
00000 111111
11111
000000
111111
3
5
5
9
14
7
13
12
19
12
13
26
21
Figure 1.25. Numeration of nodes and members of trusses II
29
1111
0000
0000
1111
0000
1111
15 27 16
16
28
23
11
23
14
24
20
17
14
24 28
18
9
17
21 25
11
19 22
33
29
1111
0000
0000
1111
26
13
18
15
10
22
12
8
25
8
18
32
30
26 31
15
3
3
18
9
14
6
33
29
16
27
22
13
7
17
1111
0000
0000
1111
14
24
29
15
25
12
20
0
32
26 30 31
21
16
23
11
10
8
12
18
32
22
17
9
17
1111
0000
0000
1111
0000
1111
29
16
28
28
5
4
16
1
15
11111
00000
111111
000000
00000
11111
000000
111111
00000 111111
11111
000000
16
27
13
6
4
14
7
19
18
15
14
8
3
2
12
28
7
5
28 29
23
9
9
24
12
20
33
30
25
13
3
8
14
13
14
10
5
6
11111
00000
000000
00000 111111
11111
000000
111111
26
21
24
15
4
1
16
10
16
31
17
2
2
27
5
3
6
29
15
6
4
6
14
25
19
11
12
14 15
1
1
12
8
11
6
16
9
2
10
18
17
7
25
22
4
11111
00000
00000
11111
16
9
14
13
6
1
17 19
3
17
29
15
26
11
18
32
27
17
9
16
28
22
4
12
26
13
33
23
8
2
14
24
18
8
11111
00000
000000
00000 111111
11111
000000
111111
10
18
10
9
1
1
8
13
5
3
2
21
11
30
25
7
9
13
12
20
19
12
11
31
26
22
6
1
11111
00000
00000
11111
00000
11111
3
27
24
5
4
2
23
18
18
4
14 15
6
4
32
8
7
9
9
14
9
16
11
5
5
3
1
1
11111
00000
111111
00000 000000
11111
000000
111111
00000 111111
11111
000000
13
10
16
12
8
7
6
3
2
16
14
22
10
6
2
19
4
14
27 29
24 15 28
28
15
4
3
11111
00000
00000
11111
12
20
13
6
5
3
2
11
24
6
5
25
20
10
15
11 14
4
2
23
21
17
5
3
11111
00000
00000
11111
26
12
10
2
12
19
16
9
16
6
7
6
17
12
8
3
1
7
8
4
2
8
5
4
4
2
11 13
9
7
5
3
1
11111
00000
00000
11111
00000
11111
3
15
6
6
4
22
10
8
2
10
18
14
18
32
33
27 30 31
13
25
15
29
17
1111
0000
0000
1111
1.3 Computation of trusses with the EST method
39
spy(spA) − sparse matrix spA(32,32) non−zero elements [8.7%]
0
Joint equilibrium equations 1−29
5
10
15
20
25
30
Restriction on support displacements 30−32
0
5
10
15
20
25
30
Figure 1.26. Sparsity pattern of matrix spA of truss 1N27
40
1. First-order structural analysis
A. Matrices
A matrix type that stores only the values of non-zero elements and their row and column
indexes is generally called sparse 1 2 . For the storage and creation of sparse matrices we
use GNU Octave 3 4 .
A.1
A.1.1
Sparse matrices and GNU Octave
Introduction to sparse matrices
For calculating support reactions and interaction forces on statically determinate hinged
beams, also known as Gerber 5 6 beams (see Fig. A.2), we have a system of equilibrium
equations where the coefficient matrix is sparse. The sparsity pattern of this matrix
spA is shown in Fig. A.1.
spy(spA) − sparse matrix spA(8,8) non−zero elements [22%]
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
Figure A.1. Sparsity pattern of matrix spA
1
http://www.gnu.org/software/octave/doc/interpreter/Sparse-Matrices.html. Web. 08
August 2013.
2
http://en.wikipedia.org/wiki/Sparse_matrix. Web. 08 August 2013.
3
http://www.obihiro.ac.jp/~suzukim/masuda/octave/html3/octave_112.html#SEC216.
Web. 08 August 2013.
4
http://www.network-theory.co.uk/docs/octave3/octave_205.html. Web. 08 August 2013.
5
Heinrich Gerber (1832–1912), a German civil engineer and inventor of multispan hinged beams.
6
http://de.wikipedia.org/wiki/Heinrich_Gottfried_Gerber. Web. 08 August 2013.
41
42
A. Matrices
a)
0
1
000
111
000
111
0
1
000
111
0
1
0
1
000
111
0
1
0
1
000
111
0
1
0
1
000
111
0
1
0
1
000
111
0
1
0
1
1
000
111
0
1
0
1
000
111
0
1
0
1
111
000
0
0 1
1
F1= 20 kN
2
F3 = 30 kN
F2 = 40 kN
8 kN/m
1111111111111111111111111111111111111111111
0000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000
1111111111111111111111111111111111111111111
3
4
5
6
7
8
2m 2m
1
0
0
1
11111
00000
0
1
0
1
00000
11111
0
1
0
000001
11111
111111
000000
0
1
4m
4m
1
0
0
1
0
1
2m 2m
4m
9
1
0
0
1
11111
00000
0
1
0
1
00000
11111
0
1
0
1
00000
11111
111111
000000
0
1
2m
10
4m
X 7 = 44.00 kN
7 = 44.0 kN
X 6 = 39.00 kN
X 3= 125.00 kN
X 6 = 39.00 kN
1
0
0
1
0
1
0
1
X 1= − 236.00 kN.m
F1
b)
12
11
4m
1m
11
00
3
F2
F5 = 20 kN
1
0
0
1
11111
00000
0
1
0
1
00000
11111
0
1
0
1
00000
11111
111111
000000
0
1
F
q
1111111111111111
11 0000000000000000
00
11
00
0000000000000000
1111111111111111
0000000000000000
1111111111111111
1111
0000
11
00
X
q
1111111111111111111111111111
0000000000000000000000000000
0000000000000000000000000000
1111111111111111111111111111
0000000000000000000000000000
1111111111111111111111111111
0000
1111
00
11
0000
1111
00
11
H 1= 0
F4 = 100 kN
X 8 = − 34.00 kN
X 8 = − 43.00 kN
F5
F4
111
000
X 4= 90.00 kN
11
00
X 5= 64.00 kN
XA
Internal reactions
X 2 = 59.00 kN
XL
X 1 − X 5 − external reactions
X 6 − X 8 − internal reactions
Figure A.2. Assembly sequence of a Gerber beam
To prevent a large number of calculations, this system is decomposed in such a way
that an unknown force could be calculated directly with each equilibrium equation.
Consider the equilibrium equations for the beams in Fig. A.2:
beam 6–8
𝑋8 · 6 + 𝐹 3 · 2 + 𝑞 · 6 · 3 = 0
−𝑋7 · 6 + 𝐹3 · 4 + 𝑞 · 6 · 3 = 0
(A.1)
(A.2)
−𝑋8 · 10 − 𝑋4 · 8 + 𝐹4 · 4 − 𝐹5 · 1 = 0
−𝑋8 · 2 + 𝑋5 · 8 − 𝐹4 · 4 − 𝐹5 · 9 = 0
(A.3)
(A.4)
−𝑋7 · 10 + 𝑋3 · 8 − 𝐹2 · 4 − 𝑞 · 10 · 5 = 0
−𝑋7 · 2 − 𝑋6 · 8 + 𝐹2 · 4 + 𝑞 · 10 · (5 − 2) = 0
(A.5)
(A.6)
𝑋6 − 𝑋 2 + 𝐹 1 = 0
𝑋1 + 𝑋6 · 4 + 𝐹 1 · 4 = 0
(A.7)
(A.8)
Σ𝑀6 = 0;
Σ𝑀8 = 0;
beam 8–12
Σ𝑀11 = 0;
Σ𝑀9 = 0;
beam 3–6
Σ𝑀3 = 0;
Σ𝑀5 = 0;
beam 1–3
Σ𝑍 = 0;
Σ𝑀1 = 0;
A.1 Sparse matrices and GNU Octave
43
Now rewrite the systems of equations (A.1)–(A.8) in matrix form:
1.
2.
3.
4.
5.
6.
7.
8.
Σ𝑀6 = 0;
Σ𝑀8 = 0;
Σ𝑀11 = 0;
Σ𝑀9 = 0;
Σ𝑀3 = 0;
Σ𝑀5 = 0;
Σ𝑍 = 0;
Σ𝑀1 = 0
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎤
⎤⎡
𝑋1
· · · · · ·
·
6
⎥
⎢
· · · · · ·
−6
· ⎥
⎥ ⎢ 𝑋2 ⎥
⎥
⎥⎢
⎥
⎢
· · · −8 · ·
·
−10 ⎥
⎥ ⎢ 𝑋3 ⎥
⎢
⎥
· · · · 8 ·
·
−2 ⎥ ⎢ 𝑋4 ⎥
⎥
⎥=
⎥⎢
⎢ 𝑋5 ⎥
· · 8 · · · −10
· ⎥
⎥
⎥⎢
⎥
⎢
· · · · · −8 −2
· ⎥
⎥ ⎢ 𝑋6 ⎥
⎥
⎥⎢
· −1 · · · 1
·
· ⎦ ⎣ 𝑋7 ⎦
𝑋8
1 · · · · 4
·
·
⎡
=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
−⎢
⎢
⎢
⎢
⎢
⎢
⎣
𝐹3 · 2 + 𝑞 · 6 · 3
𝐹3 · 4 + 𝑞 · 6 · 3
𝐹4 · 4 − 𝐹5 · 1
−𝐹4 · 4 − 𝐹5 · 9
−𝐹2 · 4 − 𝑞 · 10 · 5
𝐹2 · 4 + 𝑞 · 10 · (5 − 2)
𝐹1
𝐹1 · 4
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(A.9)
We have obtained the sparse system (8 × 8) of equilibrium equations
A · X = −B
(A.10)
where
⎡
A=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎤
0
0 0
0 0
0
0
6
0
0 0
0 0
0 −6
0 ⎥
⎥
⎥
0
0 0 −8 0
0
0 −10 ⎥
⎥
0
0 0
0 8
0
0 −2 ⎥
⎥
⎥,
0
0 8
0 0
0 −10
0 ⎥
⎥
0
0 0
0 0 −8 −2
0 ⎥
⎥
⎥
0 −1 0
0 0
1
0
0 ⎦
1
0 0
0 0
4
0
0
B=
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
−204
−264
−380
580
560
−400
−20
−80
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(A.11)
The Gerber beam support reactions are calculated by hand in the reverse order to
that of the assembly sequence (see calculating order in Eqs. (A.1)–(A.9), and Fig. A.2).
The sparsity pattern of the matrix A of Eq. (A.11) is shown in Fig. A.1.
The non-zero elements of the matrix A can be represented as the row (iv), column
(jv) and data (sv) vectors in Eq. (A.12).
iv = [1 2
3
3 4
4 5
5
6
6
7 7 8 8]
jv = [8 7
4
8 5
8 3
7
6
7
2 6 1 6]
sv = [6 − 6 − 8 − 10 8 − 2 8 − 10 − 8 − 2 − 1 1 1 4]
(A.12)
44
A. Matrices
A.1.2
Creating sparse matrices
GNU Octave uses the compressed column format storage technique.7 There are several
modes to create a sparse matrix. Sparse matrices can be constructed from matrices or
vectors.
The function sparse(iv, jv, sv) has constructed a sparse matrix spA (shown in
computing diary A.1) from three vectors representing the row (iv), column (jv) and
data (sv) (see Eq. (A.12)). In this diary, the conversion of the sparse matrix spA to a
full matrix A (full(spA)) is shown.
Computing diary A.1
octave:1> iv = [1 2 3 3 4 4 5 5 6 6 7 7 8 8]
iv =
1
2
3
3
4
4
5
5
6
6
7
7
8
8
6
1
6
octave:2> jv = [8 7 4 8 5 8 3 7 6 7 2 6 1 6]
jv =
8
7
4
8
5
8
3
7
6
7
2
octave:3> sv = [6 -6 -8 -10 8 -2 8 -10 -8 -2 -1 1 1 4]
sv =
6
-6
-8
-10
8
-2
8
-10
-8
-2
-1
1
1
4
octave:4> spA=sparse(iv, jv, sv)
spA =
Compressed Column Sparse (rows = 8, cols = 8, nnz = 14 [22%])
(8,
(7,
(5,
(3,
(4,
(6,
(7,
1)
2)
3)
4)
5)
6)
6)
octave:5> A=full(spA)
A =
0
0
0
0
0
0
0
0
0
0
0
-8
0
0
0
0
0
0
8
0
0
0
0
0
0
-1
0
0
1
0
0
0
octave:6>
7
(8,
(2,
(5,
(6,
(1,
(3,
(4,
-> 1
-> -1
-> 8
-> -8
-> 8
-> -8
-> 1
0
0
0
8
0
0
0
0
0
0
0
0
0
-8
1
4
0
-6
0
0
-10
-2
0
0
6)
7)
7)
7)
8)
8)
8)
->
->
->
->
->
->
->
4
-6
-10
-2
6
-10
-2
6
0
-10
-2
0
0
0
0
http://www.obihiro.ac.jp/~suzukim/masuda/octave/html3/octave_113.html#SEC219.
Web. 08 August 2013.
A.1 Sparse matrices and GNU Octave
45
The non-zero elements of the matrix A of Eq. (A.11) can be represented as the
matrix A1 with columns iv, jv and sv, see Eq. (A.12):
⎡
A1 =
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
2
3
3
4
4
5
5
6
6
7
7
8
8
8 6
7 −6 ⎥
⎥
⎥
4 −8 ⎥
⎥
8 −10 ⎥
⎥
5 8 ⎥
⎥
⎥
8 −2 ⎥
⎥
3 8 ⎥
⎥
⎥
7 −10 ⎥
⎥
6 −8 ⎥
⎥
⎥
7 −2 ⎥
⎥
2 −1 ⎥
⎥
6 1 ⎥
⎥
⎥
1 1 ⎦
6 4
⎤
(A.13)
The function spconvert(A1) (given in computing diary A.2) constructs a sparse
matrix spA from the matrix A1.
Computing diary A.2
octave:1>
2
7
3
4
3
8
4
5
4
8
5
3
5
7
6
6
6
7
7
2
7
6
8
1
8
6
A1=[1
-6;
-8;
-10;
8;
-2;
8;
-10;
-8;
-2;
-1;
1;
1;
4]
8
6;
A1 =
1
2
3
3
4
4
5
5
6
6
7
7
8
8
8
7
4
8
5
8
3
7
6
7
2
6
1
6
6
-6
-8
-10
8
-2
8
-10
-8
-2
-1
1
1
4
octave:2> spA=spconvert(A1)
spA =
Compressed Column Sparse (rows = 8, cols = 8, nnz = 14 [22%])
(8,
(7,
(5,
(3,
(4,
(6,
(7,
1)
2)
3)
4)
5)
6)
6)
-> 1
-> -1
-> 8
-> -8
-> 8
-> -8
-> 1
(8,
(2,
(5,
(6,
(1,
(3,
(4,
6)
7)
7)
7)
8)
8)
8)
->
->
->
->
->
->
->
4
-6
-10
-2
6
-10
-2
46
A. Matrices
The function sparse(A) converts the full matrix A to a sparse matrix (see computing diary A.3).
Computing diary A.3
octave:6> A
A =
0
0
0
0
0
0
0
0
0
0
0
0
0
-1
1
0
0
0
0
0
8
0
0
0
0
0
-8
0
0
0
0
0
0
0
0
8
0
0
0
0
0
0
0
0
0
-8
1
4
0
-6
0
0
-10
-2
0
0
6
0
-10
-2
0
0
0
0
octave:7> spA=sparse(A)
spA =
Compressed Column Sparse (rows = 8, cols = 8, nnz = 14 [22%])
(8,
(7,
(5,
(3,
(4,
(6,
(7,
1)
2)
3)
4)
5)
6)
6)
(8,
(2,
(5,
(6,
(1,
(3,
(4,
-> 1
-> -1
-> 8
-> -8
-> 8
-> -8
-> 1
6)
7)
7)
7)
8)
8)
8)
->
->
->
->
->
->
->
4
-6
-10
-2
6
-10
-2
octave:8>
A.1.3
Sparse matrix functions in the EST method
There are several functions that manipulate sparse matrices: full, sparse, spconvert,
spfind, sprank, spy, speye, etc. 8
We shall introduce the GNU Octave function spA=spInsertBtoA(spA,M,N,spB)
(p. 69) written for the EST method. This function inserts a sparse matrix spB into
the sparse matrix spA, starting at row index M and column index N. The overlapping
elements of the matrices spA and spB are added together.
The insertion of the matrix B (Eq. (A.14)) into the sparse matrix spC is described
in computing diary A.4. There, the elements of the matrix C of value 2 (C(5,5) and
C(6,6)) have been obtained as the sum of overlapped elements of matrices spB and
spB1.
⎡
B=
8
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(A.14)
http://www.obihiro.ac.jp/~suzukim/masuda/octave/html3/octave_113.html#SEC222.
Web. 08 August 2013.
A.1 Sparse matrices and GNU Octave
Computing diary A.4
octave:1> iv = [1 2 3 4 5 6]
iv =
1
2
3
4
5
6
octave:2> jv = [1 2 4 3 5 6]
jv =
1
2
4
3
5
6
octave:3> sv = [1 1 1 1 1 1]
sv =sparse matrix spB
1
1
1
1
1
1
octave:4> spB=sparse(iv, jv, sv)
spB =
Compressed Column Sparse (rows = 6, cols = 6, nnz = 6 [17%])
(1,
(2,
(4,
(3,
(5,
(6,
1)
2)
3)
4)
5)
6)
->
->
->
->
->
->
1
1
1
1
1
1
octave:5> spB1=spB;
octave:6> spC=sparse(10,10)
spC =
Compressed Column Sparse (rows = 10, cols = 10, nnz = 0 [0%])
octave:7> spC=spInsertBtoA(spC,1,1,spB);
octave:8> spC=spInsertBtoA(spC,5,5,spB1)
spC =
Compressed Column Sparse (rows = 10, cols = 10, nnz = 10 [10%])
(1, 1) -> 1
(2, 2) -> 1
(4, 3) -> 1
(3, 4) -> 1
(5, 5) -> 2
(6, 6) -> 2
(8, 7) -> 1
(7, 8) -> 1
(9, 9) -> 1
(10, 10) -> 1
47
48
A. Matrices
octave:9> C=full(spC)
C =
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
octave:10>
Next we introduce the GNU Octave function spA=spSisestaArv(spA,M,N,V) (p. 69)
written for the EST method. This function inserts a numeric value V into the sparse
matrix spA at row index M and column index N.
A.2
Transformation matrices
Consider the two right-handed coordinate systems of Fig. A.3, defined by orthogonal
unit vectors i, j, k, and i* , j* , k* . Let xyz be global coordinates and 𝑥* 𝑦 * 𝑧 * a local
coordinate system.
→
The vector F in Fig. A.3 can be written as the sum of two vectors along the
coordinate axes i, k with magnitude 𝐹𝑥 , 𝐹𝑧 and along the coordinate axes i* , k* with
magnitude 𝐹𝑥* , 𝐹𝑧* .
→
→
→
F = 𝐹𝑥 · i + 𝐹𝑧 · k =
→
𝐹𝑥* i*
i*
β
+
→
𝐹𝑧* k* ,
k
α zz*
⎩
→
*
,
·k
cos α xx = cos α
α α xx
−β
x
α xz*
cos
α xz* = −cos β
αzx*
α
z
·i*
x*
i
F
k*
→
⎧
⎨
cos
z*
αzx*
= cos β
cos α zz* = cos α
Figure A.3. Coordinate transformation
⎧ →
⎨ ·
i
k
⎩ ·→
(A.15)
A.2 Transformation matrices
49
→
To find the components of the vector F of Eq. (A.15), we multiply this equation by
→
→
*
i and k* . The scalar products are:
→
→
→
→
→
→
F · i* = 𝐹𝑥* = 𝐹𝑥 · i · i* + 𝐹𝑧 · k · i*
→ →
→ →
→ →
F · k* = 𝐹𝑧* = 𝐹𝑥 · i · k* + 𝐹𝑧 · k · k*
(A.16)
where the scalar product of the two orthogonal vectors is zero.
→
→
To find the inverse transformation, we multiply Eq. (A.15) by i and k . The scalar
products are:
→
→
→
→
→
→
F · i = 𝐹𝑥 = 𝐹𝑥* · i* · i + 𝐹𝑧* · k* · i
→ →
→ →
→ →
F · k = 𝐹𝑧* = 𝐹𝑥* · i* · k + 𝐹𝑧* · k* · k
(A.17)
The scalar product of the two unit vectors is related to the cosine of the angle between
these vectors (Fig. A.3).
→
→
→
→
→
i · i* = i* · i = cos 𝛼𝑥𝑥* ,
→ →
→ →
k · k* = k* · k = cos 𝛼𝑧𝑧* ,
→
i · k* = cos 𝛼𝑥𝑧*
→ →
i* · k = cos 𝛼𝑧𝑥*
(A.18)
→
In Fig. A.3, we show the direction cosines of the vector F:
cos 𝛼𝑥𝑥* = cos 𝛼,
cos 𝛼𝑧𝑧* = cos 𝛼,
cos 𝛼𝑧𝑥* = cos 𝛽
cos 𝛼𝑥𝑧* = − cos 𝛽
(A.19)
Be careful using cosine and sine angles associated with coordinate system transformation: cos 𝛼𝑥𝑥* = cos 𝛼 and cos 𝛼𝑧𝑥* = cos 𝛽 (cos 𝛽 = cos (90 ∘ + 𝛼) = − sin 𝛼).
The length l and the direction cosines of an element can be calculated using coordinates 𝑥𝐴 , 𝑧𝐴 of the node at the beginning and coordinates 𝑥𝐿 , 𝑧𝐿 of the node at the
end of the element (Fig. A.4):
𝑙=
√︁
(𝑧𝐿 − 𝑧𝐴 )2 + (𝑥𝐿 − 𝑥𝐴 )2
(A.20)
xL
xA
∆x
The starting point
111111111
000000000
000000000
111111111
000000000
111111111
000000000
111111111
000000000 The end point
111111111
000000000
111111111
β
α
∆z
zL
zA
x
z
Figure A.4. Direction cosines of an element
50
A. Matrices
cos 𝛼 =
𝑥𝐿 − 𝑥𝐴
𝑙
(A.21)
cos 𝛽 =
𝑧𝐿 − 𝑧𝐴
𝑙
(A.22)
→
We now consider the transformation of the vector F components 𝐹𝑥 , 𝐹𝑧 of Eq. (A.15)
from the global xy coordinate system to the components 𝐹𝑥* , 𝐹𝑧* in the local 𝑥* 𝑦 *
coordinate system.
[︃
𝐹𝑥*
𝐹𝑧*
]︃
[︃
=
cos 𝛼 cos 𝛽
− cos 𝛽 cos 𝛼
]︃ [︃
𝐹𝑥
𝐹𝑧
]︃
(A.23)
Taking into account that cos 𝛽 = cos (90 ∘ + 𝛼) = − sin 𝛼, we can write the above
equation as
[︃
𝐹𝑥*
𝐹𝑧*
]︃
[︃
=
cos 𝛼 − sin 𝛼
sin 𝛼 cos 𝛼
]︃ [︃
𝐹𝑥
𝐹𝑧
]︃
(A.24)
→
The inverse transformation of the vector F components 𝐹𝑥* , 𝐹𝑧* of Eq. (A.15) from
the local 𝑥* 𝑦 * coordinate system to the components 𝐹𝑥 , 𝐹𝑧 of the global xy coordinate
system:
[︃
𝐹𝑥
𝐹𝑧
]︃
[︃
=
cos 𝛼 − cos 𝛽
cos 𝛽 cos 𝛼
]︃ [︃
𝐹𝑥*
𝐹𝑧*
]︃
(A.25)
Taking into account that cos 𝛽 = cos (90 ∘ + 𝛼) = − sin 𝛼, we can write Eq. (A.25) in
the form
[︃
𝐹𝑥
𝐹𝑧
]︃
[︃
=
cos 𝛼 sin 𝛼
− sin 𝛼 cos 𝛼
]︃ [︃
𝐹𝑥*
𝐹𝑧*
]︃
(A.26)
Comparing the transformation matrix (A.23) with that of (A.25), we can see that
they are transposed (rows and columns reversed). The multiplication of the matrix
(A.23) by a transpose of itself of Eq. (A.25) gives the identity matrix
[︃
cos 𝛼 cos 𝛽
− cos 𝛽 cos 𝛼
]︃ [︃
cos 𝛼 − cos 𝛽
cos 𝛽 cos 𝛼
]︃
[︃
=
1 0
0 1
]︃
(A.27)
We have thus proved that the matrices are orthogonal – an orthogonal matrix has the
property that its transpose equals the inverse.
B. Work and work-energy theorem
B.1
Work done by internal and external forces
The work-energy theorem in structural analysis: the sum of the work done by internal
and external forces is zero:
𝑊𝑖 + 𝑊𝑒 = 0
(B.1)
where 𝑊𝑖 is the work done by internal forces and 𝑊𝑒 is the work done by external forces.
The Green’s functional for a frame element is [KHMW10]
−
∫︁ 𝑏
𝑎
⏟
ˆ −
𝑁𝑥 𝜆𝑑𝑥
∫︁ 𝑏
𝑎
𝑀𝑦 𝜓ˆ𝑦 𝑑𝑥 +
⏞
𝑊𝑖 − 𝑤𝑜𝑟𝑘 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓 𝑜𝑟𝑐𝑒𝑠
+ [𝑁𝑥 𝑢ˆ]𝑏𝑎
⏟
+ [𝑄𝑦 𝑤ˆ +
𝑀𝑦 𝜙ˆ𝑦 ]𝑏𝑎
+
∫︁ 𝑏
𝑎
𝑞𝑥 (𝑥)ˆ
𝑢𝑑𝑥 + 𝐹𝑥𝑖 𝑢ˆ𝑖 +
∫︁ 𝑏
𝑎
(B.2)
𝑞𝑧 (𝑥)𝑤𝑑𝑥
ˆ + 𝐹𝑧𝑖 𝑤ˆ𝑖 = 0
⏞
𝑊𝑒 − 𝑤𝑜𝑟𝑘 𝑜𝑓 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓 𝑜𝑟𝑐𝑒𝑠
where we consider two systems (states) of forces associated with respective deformations
and displacements [BP13]:
𝑁𝑥 , 𝑀𝑦 are the internal axial force and bending moment of the first load state;
ˆ 𝜓ˆ𝑦 – axial and bending deformations of the second load state;
𝜆,
𝑁𝑥 |𝑏𝑎 , 𝑄𝑦 |𝑏𝑎 , 𝑀𝑦 |𝑏𝑎 – axial force, shear force and bending moment of the first load state
at boundaries a and b;
𝑏
𝑢ˆ|𝑎 , 𝑤|
ˆ 𝑏𝑎 , 𝜙ˆ𝑦 |𝑏𝑎 – longitudinal and transverse displacements, and the rotation of the cross
section of the second load state at boundaries a and b;
𝑞𝑥 (𝑥), 𝑞𝑧 (𝑥) – distributed loads of the first load state;
𝑢ˆ(𝑥), 𝑤(𝑥)
ˆ
– longitudinal and transverse displacements of the second load state;
𝐹𝑥𝑖 , 𝐹𝑧𝑖 – force components of the first load state, applied at point i in the x and z
directions, respectively;
𝑢ˆ𝑖 , 𝑤ˆ𝑖 – longitudinal and transverse displacements of point i of the second load state.
The first and second elements of Eq. (B.2) describe the work 𝑊𝑖 of internal forces:
𝑊𝑖 = −
∫︁ 𝑏
𝑎
ˆ −
𝑁𝑥 𝜆𝑑𝑥
51
∫︁ 𝑏
𝑎
𝑀𝑦 𝜓ˆ𝑦 𝑑𝑥
(B.3)
52
B. Work and work-energy theorem
The final four elements of Eq. (B.2) describe the work 𝑊𝑓 done by active forces:
𝑊𝑓 =
∫︁ 𝑏
𝑎
𝑞𝑥 (𝑥)ˆ
𝑢𝑑𝑥 + 𝐹𝑥𝑖 𝑢ˆ𝑖 +
∫︁ 𝑏
𝑎
𝑞𝑧 (𝑥)𝑤𝑑𝑥
ˆ + 𝐹𝑧𝑖 𝑤ˆ𝑖
(B.4)
The third and fourth elements of Eq. (B.2) describe the work 𝑊𝑏 done by boundary
forces (fixed-end forces and moments at joints1 , support reactions):
𝑊𝑏 = [𝑁𝑥 𝑢ˆ]𝑏𝑎 + [𝑄𝑦 𝑤ˆ + 𝑀𝑦 𝜙ˆ𝑦 ]𝑏𝑎
(B.5)
The external work 𝑊𝑒 can be divided into two parts:
∙ 𝑊𝑓 – work done by active forces, e.g. concentrated loads, uniformly distributed
loads;
∙ 𝑊𝑏 – work done by reaction forces, e.g. support reactions (Fig. ??), internal
reactions [WP960] (contact forces 2 ) (Fig. ??).
𝑊𝑒 = 𝑊𝑏 + 𝑊𝑓
(B.6)
Applying now Eq. (B.6) to (B.1), we obtain
𝑊𝑖 + 𝑊𝑏 + 𝑊𝑓 = 0
(B.7)
Equation (B.7) is a shortened form of Eq. (B.2).
With the elastic energy 𝑈𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 and dissipation energy 𝐷 existing, the work 𝑊𝑖
done by internal forces is in relation to the internal energy 𝑈𝑒𝑙𝑎𝑠𝑡𝑖𝑐 + 𝐷:
𝑊
⏟ ⏞𝑖
=−
𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑤𝑜𝑟𝑘
⏟𝑈⏞
𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦
−
⏟𝐷⏞
(B.8)
𝑑𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑒𝑛𝑒𝑟𝑔𝑦
Equations (B.2) and (B.7) are the basic methodical tools of structural analysis.
Figure B.1. Bar member a–b
1
The fixed-end forces and moments at joints are called the internal reactions [WP960] or the joint
contact forces [GN12].
2
A contact force is a force that acts at the points of contact between two objects [Rand07].
B.1 Work done by internal and external forces
53
Example B.1 (conservation of mechanical energy). We consider the bar from
Fig. B.1, subjected to the load F. The bar has been split into sections where the normal
forces 𝑁𝑎 and 𝑁𝑏 exerted on the member at cross-sections a and b are treated as external
loads [VrCty] or, to be more precise, as boundary forces.
No loads act on the bar member a–b, and so 𝑊𝑓 = 0 and 𝑊𝑖 ̸= 0 in Eq. (B.7). The
expression for energy conservation of the bar member a–b is
𝑊𝑖 + 𝑊𝑏 = 0
or
−
∫︁ 𝑏
𝑎
⏟
ˆ
𝑁𝑥 𝜆𝑑𝑥
⏞
𝑊𝑖 − 𝑤𝑜𝑟𝑘 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑓 𝑜𝑟𝑐𝑒𝑠
+
(B.9)
[𝑁𝑥 𝑢ˆ]𝑏𝑎
⏟
⏞
𝑊𝑏 − 𝑤𝑜𝑟𝑘 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝑠 [WP960]
=0
(B.10)
54
B. Work and work-energy theorem
C. Computer programs for the EST
method
The basic URLs (folder paths for programs):
C.1
D:/, E:/, F:/, S:/, Z:/
Programs for first-order analysis
Program C.1 (spESTframe1LaheDefWFI.m)1 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.2 (spESTframe2LaheDefWFI.m)2 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.3 (spESTframe3LaheDefWFI.m)3 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.4 (spESTframe4LaheDefWFI.m)4 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
1
./octavePrograms/spESTframe1LaheDefWFI.m
./octavePrograms/spESTframe2LaheDefWFI.m
3
./octavePrograms/spESTframe3LaheDefWFI.m
4
./octavePrograms/spESTframe4LaheDefWFI.m
2
55
56
C. Computer programs for the EST method
Program C.5 (spESTframe5LaheDefWFI.m)5 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.6 (spESTframe6LaheDefWFI.m)6 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.7 (spESTframe7LaheDefWFI.m)7 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.8 (spESTframe8LaheDefWFI.m)8 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.9 (spESTframe9LaheDefWFI.m)9 11 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.10 (spESTframe10LaheDefWFI.m)10 11 – is used to compute the
displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Function C.1 (LaheFrameDFIm(baasi0,Ntoerkts,esQkoormus,esFjoud,
sSolmF,tsolm,tSiire,krdn,selem)) 11 55, 56, 58 – is used to assemble and solve
5
./octavePrograms/spESTframe5LaheDefWFI.m
./octavePrograms/spESTframe6LaheDefWFI.m
7
./octavePrograms/spESTframe7LaheDefWFI.m
8
./octavePrograms/spESTframe8LaheDefWFI.m
9
./octavePrograms/spESTframe9LaheDefWFI.m
10
./octavePrograms/spESTframe10LaheDefWFI.m
11
./octavePrograms/LaheFrameDFIm.m
6
C.1 Programs for first-order analysis
the boundary problem equations of a plane frame.
Called functions:
yzhqzm(baasi0,x,a,qx,qz,EA,EJ) 12 ,
InsertBtoA(A,I,J,IM,JN,B,M,N) 13 ,
spInsertBtoA(spA,IIv,IJv,spvF) 14 ,
spSisestaArv(spA,iv,jv,sv) 15 ,
SpTeisendusMaatriks2x2(NSARV,NEARV,VarrasN,krdn,selem) 16 ,
SpTeisendusMaatriks(NSARV,NEARV,VarrasN,krdn,selem) 17 ,
SpTeisendusUhikMaatriks2x2(VarrasN) 18 ,
SpTeisendusUhikMaatriks0x1v(VarrasN) 19 ,
SpTeisendusUhikMaatriks(VarrasN) 20 ,
SpToeReaktsioonZvektor(NSARV,NEARV,VarrasN,krdn,selem) 21 ,
SpToeReaktsioonXvektor(NSARV,NEARV,VarrasN,krdn,selem) 22 ,
SpToeSiirdeFiVektor(VarrasN) 23 ,
SpToeSiirdeUvektor(NSARV,NEARV,VarrasN,krdn,selem) 24 ,
SpToeSiirdeWvektor(NSARV,NEARV,VarrasN,krdn,selem) 25 ,
VardadSolmes(NSARV,NEARV,Solm,AB,ABB) 26 ,
VardaPikkus(NSARV,NEARV,krdn,selem) 27 ,
ylfhlin(baasi0,x,EA,GAr,EJ) 28 ,
ysplfhlin(baasi0,x,EA,GAr,EJ)) 29 ,
ysplvfmhvI(baasi0,x,l,EA,GAr,EJ) 30 ,
yzfzv(baasi0,x,a,Fx,Fz,EA,EJ) 31 ,
yzhqzm(baasi0,x,a,qx,qz,EA,EJ) 32 .
12
./octavePrograms/ESTFrKrmus.m
./octavePrograms/InsertBtoA.m
14
./octavePrograms/spInsertBtoA.m
15
./octavePrograms/spSisestaArv.m
16
./octavePrograms/SpTeisendusMaatriks2x2.m
17
./octavePrograms/SpTeisendusMaatriks.m
18
./octavePrograms/SpTeisendusUhikMaatriks2x2.m
19
./octavePrograms/SpTeisendusUhikMaatriks0x1v.m
20
./octavePrograms/SpTeisendusUhikMaatriks.m
21
./octavePrograms/SpToeReaktsioonZvektor.m
22
./octavePrograms/SpToeReaktsioonXvektor.m
23
./octavePrograms/SpToeSiirdeFiVektor.m
24
./octavePrograms/SpToeSiirdeUvektor.m
25
./octavePrograms/SpToeSiirdeWvektor.m
26
./octavePrograms/VardadSolmes.m
27
./octavePrograms/VardaPikkus.m
28
./octavePrograms/ylfhlin.m
29
./octavePrograms/ysplfhlin.m
30
./octavePrograms/ysplvfmhvI.m
31
./octavePrograms/yzfzv.m
32
./octavePrograms/yzhqzm.m
13
57
58
C. Computer programs for the EST method
Function C.2 (SisejoudPunktism(VardaNr,X,AlgPar,lvarras,selem,
esQkoormus,esFjoud,suurused)) 33 55, 56, 58 – is used to compute the displacements and forces of the element ’VardaNr’ at x = X.
Called function:
ESTFrKrmus(baasi0,xx,Li,Fjoud,qkoormus,EA,EI) 34 .
Program C.11 (spESTframe16WFI.m)35 15 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.12 (spESTframe27WFI.m)36 15 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.13 (spESTframe38WFI.m)37 15 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.14 (spESTframe49WFI.m)38 15 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.15 (spESTframe50WFI.m)39 15 – is used to compute the displacements and internal forces of a plane frame.
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.16 (spESTframe3hinge1WFI.m)40 – is used to compute the displacements and internal forces of a three-hinged frame.
33
./octavePrograms/SisejoudPunktism.m
./octavePrograms/ESTFrKrmus.m
35
./octavePrograms/spESTframe16WFI.m
36
./octavePrograms/spESTframe27WFI.m
37
./octavePrograms/spESTframe38WFI.m
38
./octavePrograms/spESTframe49WFI.m
39
./octavePrograms/spESTframe50WFI.m
40
./octavePrograms/spESTframe3hinge1WFI.m
34
C.1 Programs for first-order analysis
59
Called functions:
LaheFrameDFIm.m,
SisejoudPunktism.m.
Program C.17 (spESTframe3hinge1NQM.m)41 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.18 (spESTframe3hinge2NQM.m)42 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.19 (spESTframe3hinge3NQM.m)43 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.20 (spESTframe3hinge4NQM.m)44 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.21 (spESTframe3hinge5NQM.m)45 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.22 (spESTframe3hinge6NQM.m)46 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
41
./octavePrograms/spESTframe3hinge1NQM.m
./octavePrograms/spESTframe3hinge2NQM.m
43
./octavePrograms/spESTframe3hinge3NQM.m
44
./octavePrograms/spESTframe3hinge4NQM.m
45
./octavePrograms/spESTframe3hinge5NQM.m
46
./octavePrograms/spESTframe3hinge6NQM.m
42
60
C. Computer programs for the EST method
Program C.23 (spESTframe3hinge7NQM.m)47 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.24 (spESTframe3hinge8NQM.m)48 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.25 (spESTframe3hinge9NQM.m)49 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Program C.26 (spESTframe3hinge10NQM.m)50 20
internal forces of a three-hinged frame.
Called functions:
LaheFrame3hingeNQM.m
Sisejoud3LraamiPnktism.m.
– is used to compute the
Function C.3 (LaheFrame3hingeNQM(Ntoerkts,esQkoormus,esFjoud,
sSolmF,tsolm,krdn,selem)) 51 59, 60 – is used to assemble and solve the boundary problem equations of a statically determinate plane frame.
Called functions:
ESTSKrmus(xx,Li,Fjoud,qkoormus) 52 ,
InsertBtoA(A,I,J,IM,JN,B,M,N) 53 ,
spInsertBtoA(spA,IIv,IJv,spvF) 54 ,
spSisestaArv(spA,iv,jv,sv) 55 ,
SpTeisendusMaatriks2x2D(NSARV,NEARV,VarrasN,krdn,selem) 56 ,
SpTeisendusMaatriksD(NSARV,NEARV,VarrasN,krdn,selem) 57 ,
47
./octavePrograms/spESTframe3hinge7NQM.m
./octavePrograms/spESTframe3hinge8NQM.m
49
./octavePrograms/spESTframe3hinge9NQM.m
50
./octavePrograms/spESTframe3hinge10NQM.m
51
./octavePrograms/LaheFrame3hingeNQM.m
52
./octavePrograms/ESTSKrmus.m
53
./octavePrograms/InsertBtoA.m
54
./octavePrograms/spInsertBtoA.m
55
./octavePrograms/spSisestaArv.m
56
./octavePrograms/SpTeisendusMaatriks2x2D.m
57
./octavePrograms/SpTeisendusMaatriksD.m
48
C.1 Programs for first-order analysis
61
SpTeisendusUhikMaatriks0x1v(VarrasN) 58 ,
SpTeisendusUhikMaatriks2x2(VarrasN) 59 ,
SpTeisendusUhikMaatriks(VarrasN) 60 ,
VardadSolmesD(NSARV,NEARV,Solm,AB,ABB) 61 ,
VardaPikkusD(NSARV,NEARV,krdn,selem) 62 ,
ylSfhlin(x) 63 ,
yspSlfhlin(x) 64 ,
yspSlvfmhvI(x) 65 ,
yzSfzv(x,a,Fx,Fz) 66 ,
yzShqz(x,qx,qz) 67 .
Function C.4 (Sisejoud3LraamiPnktism(VardaNr,X,AlgPar,lvarras,
esFjoud,esQkoormus,suurused)) 68 59, 60 – is used to compute the displacements
and internal forces of the element ’VardaNr’ at x = X.
Called function:
ESTSKrmu(xx,Li,Fjoud,qkoormus) 69 .
Program C.27 (spESTbeam32LaheWFI.m)70 25 – is used to compute the displacements and internal forces of a beam.
Called functions:
LaheBeamDFI.m
SisejoudTalaPunktis.m.
Function C.5 (LaheBeamDFI(baasi0,Ntoerkts,esQkoormus,esFjoud,
sSolmF,tsolm,tSiire,krdn,selem)) 71 61 – is used to assemble and solve the boundary problem equations of a beam.
Called functions:
VardaPikkusT(NSARV,NEARV,krdn,selem) 72 ,
yspTlvfmhvI(baasi0,x,l,GAr,EJ) 73 ,
yspTlfhlin(baasi0,x,GAr,EJ) 74 ,
58
./octavePrograms/SpTeisendusUhikMaatriks0x1v.m
./octavePrograms/SpTeisendusUhikMaatriks2x2.m
60
./octavePrograms/SpTeisendusUhikMaatriks.m
61
./octavePrograms/VardadSolmesD.m
62
./octavePrograms/VardaPikkusD.m
63
./octavePrograms/ylSfhlin.m
64
./octavePrograms/yspSlfhlin.m
65
./octavePrograms/yspSlvfmhvI.m
66
./octavePrograms/yzSfzv.m
67
./octavePrograms/yzShqz.m
68
./octavePrograms/Sisejoud3LraamiPnktism.m
69
./octavePrograms/ESTFrKrmus.m
70
./octavePrograms/spESTbeam32LaheWFI.m
71
./octavePrograms/LaheBeamDFI.m
72
./octavePrograms/VardaPikkusT.m
73
./octavePrograms/yspTlvfmhvI.m
74
./octavePrograms/yspTlfhlin.m
59
62
C. Computer programs for the EST method
ESTtalaKrmus(baasi0,xx,Li,Fjoud,qkoormus,EI) 75 ,
yzTfzv(baasi0,x,a,Fz,EJ) 76 ,
yzThqz(baasi0,x,qz,EJ) 77 ,
VardadSolmesT(NSARV,NEARV,Solm,AB,ABB) 78 ,
SpTeisendusUhikMaatriks1x0(VarrasN) 79 ,
SpTeisendusUhikMaatriks2x2(VarrasN) 80 ,
SpToeSiirdeWvektorT(VarrasN) 81 ,
SpToeSiirdeFiVektorT(VarrasN) 82 ,
ylTfhlin(baasi0,x,GAr,EJ) 83 ,
spInsertBtoA(spA,IIv,IJv,spvF) 84 ,
spSisestaArv(spA,iv,jv,sv) 85 ,
InsertBtoA(A,I,J,IM,JN,B,M,N) 86 .
Function C.6 (SisejoudTalaPunktis(VardaNr,X,AlgPar,lvarras,selem,
esFjoud,esQkoormus,suurused)) 87 61 – is used to compute the displacements and
internal forces of the element ’VardaNr’ at x = X.
Called function:
ESTtalaKrmus.m
.
Function C.7 (ESTtalaKrmus(baasi0,xx,Li,Fjoud,qkoormus,EI))88 62
used to compute the loading vector (q + F) for a continuous beam.
Called functions:
yzThqz(baasi0,x,qz,EJ) 89 ,
yzTfzv(baasi0,x,a,Fz,EJ) 90 .
– is
Program C.28 (spESTGerberBeam1QM.m)91 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
75
./octavePrograms/ESTtalaKrmus.m
./octavePrograms/yzTfzv.m
77
./octavePrograms/yzThqz.m
78
./octavePrograms/VardadSolmesT.m
79
./octavePrograms/SpTeisendusUhikMaatriks1x0.m
80
./octavePrograms/SpTeisendusUhikMaatriks2x2.m
81
./octavePrograms/SpToeSiirdeWvektorT.m
82
./octavePrograms/SpToeSiirdeFiVektorT.m
83
./octavePrograms/ylTfhlin.m
84
./octavePrograms/spInsertBtoA.m
85
./octavePrograms/spSisestaArv.m
86
./octavePrograms/InsertBtoA.m
87
./octavePrograms/SisejoudTalaPunktis.m
88
./octavePrograms/ESTtalaKrmus.m
89
./octavePrograms/yzThqz.m
90
./octavePrograms/yzTfzv.m
91
./octavePrograms/spESTGerberBeam1QM.m
76
C.1 Programs for first-order analysis
63
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.29 (spESTGerberBeam2QM.m)92 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.30 (spESTGerberBeam3QM.m)93 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.31 (spESTGerberBeam4QM.m)94 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.32 (spESTGerberBeam5QM.m)95 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.33 (spESTGerberBeam6QM.m)96 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.34 (spESTGerberBeam7QM.m)97 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
92
./octavePrograms/spESTGerberBeam2QM.m
./octavePrograms/spESTGerberBeam3QM.m
94
./octavePrograms/spESTGerberBeam4QM.m
95
./octavePrograms/spESTGerberBeam5QM.m
96
./octavePrograms/spESTGerberBeam6QM.m
97
./octavePrograms/spESTGerberBeam7QM.m
93
64
C. Computer programs for the EST method
Program C.35 (spESTGerberBeam8QM.m)98 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.36 (spESTGerberBeam9QM.m)99 28 – is used to compute the internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
Program C.37 (spESTGerberBeam10QM.m)100 28
internal forces of a Gerber beam.
Called functions:
LaheGerberBeamQM.m
SsjoudGrbrTalaPnktis.m.
– is used to compute the
Function C.8 (LaheGerberBeamQM(Ntoerkts,esQkoormus,esFjoud,
sSolmF,tsolm,krdn,selem)) 101 63, 63 – is used to assemble and solve the boundary
problem equations of a statically determinate beam.
Called functions:
InsertBtoA(A,I,J,IM,JN,B,M,N) 102 ,
spInsertBtoA(spA,IIv,IJv,spvF) 103 ,
spSisestaArv(spA,iv,jv,sv) 104 ,
SpTeisendusUhikMaatriks1x0(VarrasN) 105 ,
SpTeisendusUhikMaatriks2x2(VarrasN) 106 ,
VardadSolmesDT(NSARV,NEARV,Solm,AB,ABB 107 ,
VardaPikkusDT(NSARV,NEARV,krdn,selem) 108 ,
ylSTfhlin(x) 109 ,
yspSTlfhlin(x) 110 ,
yspSTlvfmhvI(x) 111 ,
98
./octavePrograms/spESTGerberBeam8QM.m
./octavePrograms/spESTGerberBeam9QM.m
100
./octavePrograms/spESTGerberBeam10QM.m
101
./octavePrograms/LaheGerberBeamQM.m
102
./octavePrograms/InsertBtoA.m
103
./octavePrograms/spInsertBtoA.m
104
./octavePrograms/spSisestaArv.m
105
./octavePrograms/SpTeisendusUhikMaatriks1x0.m
106
./octavePrograms/SpTeisendusUhikMaatriks2x2.m
107
./octavePrograms/VardadSolmesDT.m
108
./octavePrograms/VardaPikkusDT.m
109
./octavePrograms/ylSTfhlin.m
110
./octavePrograms/yspSTlfhlin.m
111
./octavePrograms/yspSTlvfmhvI.m
99
C.1 Programs for first-order analysis
65
yzSTfzv(x,a,Fz) 112 ,
yzSThqz(x,qz) 113 .
Function C.9 (SsjoudGrbrTalaPnktis(VardaNr,X,AlgPar,lvarras,
esFjoud,esQkoormus,suurused)) 114 63, 63 – is used to compute the displacements
and internal forces of the element ’VardaNr’ at x = X.
Called function:
ESTSTKrmus.m
.
Function C.10 (ESTSTKrmus(xx,Li,Fjoud,qkoormus))115 65 – is used to compute the loading vector (q + F) for a Gerber beam.
Called functions:
yzSThqz(x,qz) 116 ,
yzSTfzv(x,a,Fz) 117 .
Program C.38 (spESTtruss1N15WFI.m)118 32 – is used to compute the displacements and internal forces of a plane truss.
Called functions:
LaheTrussDFI.m
Program C.39 (spESTtruss2N15WFI.m)119 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.40 (spESTtruss3N15WFI.m)120 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.41 (spESTtruss4N15WFI.m)121 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
112
./octavePrograms/yzSTfzv.m
./octavePrograms/yzSThqz.m
114
./octavePrograms/SsjoudGrbrTalaPnktis.m
115
./octavePrograms/ESTSTKrmus.m
116
./octavePrograms/yzSThqz.m
117
./octavePrograms/yzSTfzv.m
118
./octavePrograms/spESTtruss1N15WFI.m
119
./octavePrograms/spESTtruss2N15WFI.m
120
./octavePrograms/spESTtruss3N15WFI.m
121
./octavePrograms/spESTtruss4N15WFI.m
113
66
C. Computer programs for the EST method
Program C.42 (spESTtruss5N15WFI.m)122 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.43 (spESTtruss6N15WFI.m)123 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.44 (spESTtruss7N15WFI.m)124 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.45 (spESTtruss8N15WFI.m)125 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.46 (spESTtruss9N15WFI.m)126 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Program C.47 (spESTtruss10N15WFI.m)127 32 – is used to compute the displacements and internal forces of a plane truss.
Called function:
LaheTrussDFI.m
Function C.11
(LaheTrussDFI(baasi0,Ntoerkts,sSolmF,tsolm, tSiire,krdn,selem)) 128 65, 65
– is used to assemble and solve the boundary problem equations for a truss.
Called functions:
VardaPikkusTr(NSARV,NEARV,krdn,selem) 129 ,
yspSRmhvI(baasi0,x,EA) 130 ,
122
./octavePrograms/spESTtruss5N15WFI.m
./octavePrograms/spESTtruss6N15WFI.m
124
./octavePrograms/spESTtruss7N15WFI.m
125
./octavePrograms/spESTtruss8N15WFI.m
126
./octavePrograms/spESTtruss9N15WFI.m
127
./octavePrograms/spESTtruss10N15WFI.m
128
./octavePrograms/LaheTrussDFI.m
129
./octavePrograms/VardaPikkusTr.m
130
./octavePrograms/yspSRmhvI.m
123
C.1 Programs for first-order analysis
67
yspSRhlin(baasi0,x,EA) 131 ,
VardadSolmesTr(NSARV,NEARV,Solm,AB,ABB) 132 ,
SpTeisendusMaatriksTr2x2(NSARV,NEARV,VarrasN,krdn,selem) 133 ,
SpTeisendusMaatriksTr2x1(NSARV,NEARV,VarrasN,krdn,selem) 134 ,
SpTeisendusUhikMaatriks0x1v(VarrasN) 135 ,
SpTeisendusUhikMaatriks2x2(VarrasN) 136 ,
SpToeSiirdeUvektorTr(NSARV,NEARV,VarrasN,krdn,selem) 137 ,
SpToeSiirdeWvektorTr(NSARV,NEARV,VarrasN,krdn,selem) 138 ,
spInsertBtoAvect(spA,IM,JN,spB) 139 ,
spInsertBtoA(spA,IIv,IJv,spvF) 140 ,
spSisestaArv(spA,iv,jv,sv) 141 ,
InsertBtoA(A,I,J,IM,JN,B,M,N) 142 .
Program C.48 (spESTtruss1N27.m)143 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.49 (spESTtruss2N27.m)144 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.50 (spESTtruss3N27.m)145 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
131
./octavePrograms/yspSRhlin.m
./octavePrograms/VardadSolmesTr.m
133
./octavePrograms/SpTeisendusMaatriksTr2x2.m
134
./octavePrograms/SpTeisendusMaatriksTr2x1.m
135
./octavePrograms/SpTeisendusUhikMaatriks0x1v.m
136
./octavePrograms/SpTeisendusUhikMaatriks2x2.m
137
./octavePrograms/SpToeSiirdeUvektorTr.m
138
./octavePrograms/SpToeSiirdeWvektorTr.m
139
./octavePrograms/spInsertBtoAvect.m
140
./octavePrograms/spInsertBtoA.m
141
./octavePrograms/spSisestaArv.m
142
./octavePrograms/InsertBtoA.m
143
./octavePrograms/spESTtruss1N27.m
144
./octavePrograms/spESTtruss2N27.m
145
./octavePrograms/spESTtruss3N27.m
132
68
C. Computer programs for the EST method
Program C.51 (spESTtruss4N27.m)146 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.52 (spESTtruss5N27.m)147 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.53 (spESTtruss6N27.m)148 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.54 (spESTtruss7N27.m)149 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.55 (spESTtruss8N27.m)150 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.56 (spESTtruss9N27.m)151 36 – is used to compute the internal forces
of a plane truss.
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.57 (spESTtruss10N27.m)152 36
forces of a plane truss.
146
./octavePrograms/spESTtruss4N27.m
./octavePrograms/spESTtruss5N27.m
148
./octavePrograms/spESTtruss6N27.m
149
./octavePrograms/spESTtruss7N27.m
150
./octavePrograms/spESTtruss8N27.m
151
./octavePrograms/spESTtruss9N27.m
152
./octavePrograms/spESTtruss10N27.m
147
– is used to compute the internal
C.1 Programs for first-order analysis
69
Called functions:
spSisestaArv.m
spInsertBtoA.m.
Program C.58 (spSisestaArv(spA,iv,jv,sv))153 68 – inserts the number sv into
sparse matrix spA, starting at row index iv and column index jv.
Program C.59 (spInsertBtoA(spA,IM,JN,spB))154 68 – inserts sparse matrix
spB into sparse matrix spA, starting at row index IM and column index JN. Overlapping
elements of matrices spA and spB were added together.
Program C.60 (InsertBtoA(A,I,J,IM,JN,B,M,N))155 57 – inserts matrix B (dimensions M, N) into matrix A (dimensions I, J) starting at row index IM and column
index JN.
153
./octavePrograms/spSisestaArv.m
./octavePrograms/spInsertBtoA.m
155
./octavePrograms/InsertBtoA.m
154
70
C. Computer programs for the EST method
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Zustandsgr%C3%B6ssen+Kr%C3%A4tzig&hl=en&sa=X&ei=LSsDUsjfAZH2sgatiIHgCQ&ved=
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74
BIBLIOGRAPHY
Index
active forces, 52
spESTframe4LaheDefWFI.m, 55
spESTframe50WFI.m, 58
spESTframe5LaheDefWFI.m, 56
spESTframe6LaheDefWFI.m, 56
spESTframe7LaheDefWFI.m, 56
spESTframe8LaheDefWFI.m, 56
spESTframe9LaheDefWFI.m, 56
spESTGerberBeam10QM.m, 64
spESTGerberBeam1QM.m, 62
spESTGerberBeam2QM.m, 63
spESTGerberBeam3QM.m, 63
spESTGerberBeam4QM.m, 63
spESTGerberBeam5QM.m, 63
spESTGerberBeam6QM.m, 63
spESTGerberBeam7QM.m, 63
spESTGerberBeam8QM.m, 64
spESTGerberBeam9QM.m, 64
spESTtruss10N15WFI.m, 66
spESTtruss10N27.m, 68
spESTtruss1N15WFI.m, 65
spESTtruss1N27.m, 67
spESTtruss2N15WFI.m, 65
spESTtruss2N27.m, 67
spESTtruss3N15WFI.m, 65
spESTtruss3N27.m, 67
spESTtruss4N15WFI.m, 65
spESTtruss4N27.m, 68
spESTtruss5N15WFI.m, 66
spESTtruss5N27.m, 68
spESTtruss6N15WFI.m, 66
spESTtruss6N27.m, 68
spESTtruss7N15WFI.m, 66
spESTtruss7N27.m, 68
spESTtruss8N15WFI.m, 66
spESTtruss8N27.m, 68
spESTtruss9N15WFI.m, 66
compatibility equations, 3
computer function
ESTSTKrmus.m, 65
ESTtalaKrmus.m, 62
LaheBeamDFI.m, 61
LaheFrame3hingeNQM.m, 60
LaheFrameDFIm.m, 56
LaheGerberBeamQM.m, 64
LaheTrussDFI.m, 66
Sisejoud3LraamiPnktism.m, 61
SisejoudPunktism.m, 58
SisejoudTalaPunktis.m, 62
SsjoudGrbrTalaPnktis.m, 65
computer program
InsertBtoA.m, 69
spESTbeam32LaheWFI.m, 61
spESTframe10LaheDefWFI.m, 56
spESTframe16WFI.m, 58
spESTframe1LaheDefWFI.m, 55
spESTframe27WFI.m, 58
spESTframe2LaheDefWFI.m, 55
spESTframe38WFI.m, 58
spESTframe3hinge10NQM.m, 60
spESTframe3hinge1NQM.m, 59
spESTframe3hinge1WFIm, 58
spESTframe3hinge2NQM.m, 59
spESTframe3hinge3NQM.m, 59
spESTframe3hinge4NQM.m, 59
spESTframe3hinge5NQM.m, 59
spESTframe3hinge6NQM.m, 59
spESTframe3hinge7NQM.m, 60
spESTframe3hinge8NQM.m, 60
spESTframe3hinge9NQM.m, 60
spESTframe3LaheDefWFI.m, 55
spESTframe49WFI.m, 58
75
76
spESTtruss9N27.m, 68
spInsertBtoA.m, 69
spSisestaArv.m, 69
contact forces, 52
direction cosines, 49
energy conservation, 53
equilibrium equation of beam, 42
forces,
active, 52
contact, 52
reaction, 52
full matrix, 46
Gerber beam, 41
global coordinates, 48
Green’s functional, 51
identity matrix, 50
internal reactions, 52
inverse matrix, 50
inverse transformation, 49, 50
joint equilibrium equations, 3
local coordinates, 48
orthogonal matrix, 50
reaction forces, 52
sparse matrix, 41, 44–46
sparse(iv,jv,sv), 44
sparsity pattern, 11, 18, 20, 26, 30
transformation matrix, 48
transpose matrix, 50
work,
boundary forces, 52
external forces, 51
internal forces, 51
INDEX