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Matrix formulation of static analysis of cable structures
MATRIX FORMULATION OF STATIC ANALYSIS OF CABLE STRUCTURES JACEKPIl?lWAK Technical University
of Poznah, Poznafi. Poland
(Received 20 December 1976; received for publication
14 July 1977)
-The
paper concerns the static analysis of the cable stmcturcs. Energy approach is used in the analysis. Matrixfomudadoa of thebasic expressions for the total potential energy and its gradiint is given. It is proposed in the paper to utiliz the Huang’s reverse algorithm Enclosed is a nttntcricai example.
for finding stationary
1. ~hc paper eonecms static analysis of cable structures
based on the principle of minimum of tbe total potential energy and with the use of numerical minimization m&hod. General theory of the direct energy minimization approach to static analysis of cable structures was developed by Buchholdt d al.[l-4), who used steepest descent, conjugate gradients (CG) and DavidonFletcher-Powell (DFP) aJgotithms for finding minimum point of the total potential energy. The CCL DFP algorithms and FiiMeCormick’s revised variable metric methods wcrc used by Murray and Willcms[6,71 in their investigations on static bchaviour of tension structures. In the present work Huang’s reverse algorithm[S] is applii for finding stationary point of the total potential cncrgy of the cable struoturc. The basic advantage of the algorithm is that it cnablcs to bypass unidimcnsional minim&&n whioh is rather troublesome when using CG or DFP algorithms. In Section 2 of the paper problem of static analysis of cable st~ctur6s is stated, basic expressions for the total potential energy and its gradient arc recalled and matrix formulation of the expressions arc presented. Section 3 contains a brief description of the Huang’s reverse algorithm (HRA). In Section 4 a simple cable net is studied numerically with the USCof the HRA.
point of the total potential energy.
2.1 Total potential energy The total potential energy Q of any structure is the sum of strain energy U and potential V of the external forces n=U+V.
(1)
The strain energy of cable structure is the sum of energy stored in all members of the structure and is given by
(2) where U,.,,, --initial energy stored in clement m (as a result of prestrcss), PO.m = uutial ’ ’ force in the element, E,,, = moduli of elasticity. A,,, = cross sectional area of the clement, L, = lengthof the element, e, = elongation of the element causedby external forces. The elongatione,,, of the member m, incident with the nodesj and n, is given by e, = &
WL
- &I
+ tx, - X,&L
+ 12(X”, - 4,) + k,
-
- x,J
x,,m, - 4)
+ w2, - x,,)+k, - X,*)1(&* - x,Jl
(3)
in which X,, XjY, XJz and X,, X., X,,, are coordinates of the nodes j and n, respectively, and x,~, xlrr xl,, xnI, x,,, x., arc displacementsof the nodes. We can rewrite cqn (3) in the following matrix form:
4 PnonLSMsrATE+rnNT The problem which WCdiscuss is to find displacements of the nodes of cable structure subjected to the action of external loads and to calculate internal forces in the members of the structure when the following assumptions are fulhllcdz ti) structure is built of the straight mcmbcrs whiih arc abk to carry exclusively tensile forces, (ii) the member material is linearly elastic, (iii) the ext6mal forces act only in the nodes, (iv) confiyrrtion of the structure and initial prcstrcss conditions guarantee that in all members only tensile forces exist, the values of whi6h do not exceed elastic limit. We can formulate our problem in the form of the following minimization problem: find point x = X* such.
e, = &-
[ZX’H,(ii,‘x)
?I!
t (ii,lx)‘(H,‘x)]
(4)
where
is the node coordinate matrix of order (in the three dimensionalcase) 3 Wx1 ( W = total number of nodes). x = {x,lx, v&r. * . q&.q*
.
. . %v&wqw*l
is the node displacement matrix of order 3J x I (J = mnbcr of nodes undergoingdisplacements),H, is the 3 W X 3 incidencematrix of the element m. and H, is the
that 4x*)= w~min. where x is the column matrix of node displacementsand w(x) is the value of the total potential energy of the structure. 39
J.
40
PIETRZAK
35 X3 submatrix of IL. The way of building of incidence matrices and their submatrices is illustrated in Fig. 1. The superelements hi, or order 3 X 3, of the matrix H, are if the element is incident with the node i and the orientation of the element is FROM the node i,
I I=
‘,
’
where
[1
The potential V of external forces is V = -F’x
(6)
where if the element is incident with the node i and the orientation of the element is TOWARDS the node i, otherwise.
hi =
To1 1
L
PO.,
.
.
F,F&,
,
F,*F,&,}
is
the column matrix of external forces. Thus, the total potential energy of the structure can be written as ?r= Uo+Po’e+Ae’Ke-Px. 2
(71
I .2 Gradient of total potential energy The gradient Vr of the total potential energy is given by
J
Introducing now matrices PO= PO.,
F = (F,,F,,F,,
. .
PO.ML Vn=VU-F
(8)
where the gradient VU of the strain energy is [ I]:
ari ;;;:
and matrix em.. e,)
e={e,...
CNI ;;;;
the element e,,, of which is given by (4), we can rewrite expression (2) for strain energy in the form U = Li,+P,‘e+~e’Ke
au KY ... LW
(5)
vu=
;;;;; dU g
dU g
-I 0 0 0
Y
I-!
I
0 l
0 0 0
0 0 0 element I
i
cn
0
li
-I
2
0
aU
0
2
0 0
“, 0 0
0
,:i;l’ll ; 0 o=[ I OO 0
element
element
2 Fig. I.
q
-~~(x,-x,~)+(x”,-x,~)I]
q-2
[(X”, - x,)+(x”,
- 41
I
fg(-~~(x.,-x,)+(x.,--*lr)1).
H=[H ,... H,...HM],
0
ii = [ii,. . .I?, . . ii&41
lo
.OJ
I”
In eqn (9): ? represents summation with respect to nodes n which are directly connected with the node j; Pin is internal force in the ekment m, incident with nodes j and II, and its value is Pi, = P,,, = PO.,,,+ (EA,IL,) e, ; L,. = L, is the length of the element m connecting nodes j and n. Introducing matrices
&.O *
{ -~I(x.,-x,*)+(x.,-xi,)l)
I 0
0 4’
Ix,,
0
5E
0
al ax,
orientation
r
i
au ix,
Arbitrary selected
‘x-
9)
.
-
I
I-
12
and
T = It,.
t,.
t,,,],
41
Staric analysis of cable structures ‘IIte
where
increase Ax is defined by Axi =: -pi sign (VP; q,)q,,
and M is the number of elements in the structure, we can rewrite eqn (9) in compact matrix form VU = fiT(H’X + ii’x).
(II)
3. SOLUTION OF PROBLEM The problem of finding stationary point x* of s, formulated in Section 1, can be solved with the use of various numerical minimization algorithms. It is suggested here to use for the purpose Huang’s reverse algorithm which offers important advantage of bypassing unidimensional minimization. (In the case of static analysis of cable structure, unidimensional minimization process involved in e.g. CG or DFP algorithms, is rather troublesome. It requires, in each iteration step, finding the smallest value of the third order algebraic equation [ I]). The major steps in the Huang’s reverse algorithm are: (i) determination of search direction q, from poir.t xi; (ii) determination of trial increase Axi in the direction qi: (iii) evaluation of function ri+t = ?T(x, + Ax,); (iv) modification of Axi until condition ?ri+, < q i.e. ?r(xi + Axi)< *(xi) is fulfilled: (v) final determination of the point Xi+, = Xit Ax,; (vi) evaluation of gradient Va&, = Vr(x, +Axi); (vii) determination of the new search direction qi+l from the point xi+,. etc.; (viii) termination if Vri:tVWi+,
where p, is a quantity in range 0 < p, C (&iv and (c(~)~ may be chosen as (16)
(10)
Substitution of eqn (IO) into eqn (8) gives the matrix expression for the gradient of the total potential energy PO = fiT(H’X + fi’x) - F.
(IS)
where RI(, = lower bound of n. Necessary modifications of Ax, in the present work were carried out accordingly to the rule Axi = (AxJS). 4. NUMElllCAL EXAMPLE The matrix expressions for the total potential energy and its gradient as well as the Huang’s reverse algorithm were implemented to the ODRA 1204computer program. Results of computation of a simple cable net, shown in Fig. 2, are given in Table 1. Point x, = (0 0 .OOS0 0 .OlO0 0 .OOS0 0 O}was assumed as the starting point for iteration and the condition VT;+, .Vr,+, < IO-” was taken as the termination cri-
(12)
where Bt is a square matrix. (In our three dimensional cable problem matrix Bi is of order 3J x 31.) For i = 0 onecantakeB,=Ill...Il.fori+O 5kN
Bi =Bi_l+
Ci_lCI_I Cl_,(V7ri -V7r_,)
)X
(13)
5.6
where
1
Ci_, =Axi_,-B,_,(Vn,
-VT+,).
12.4.1.7
I.Om
1
11.3.2.9 I.Om
1
Fig. 2.
(14) Table I.
Elements
Forces (kN)
Nodes
Directions
I 2 3 4 5 6 7 8 9 IO II I2
29.266 29.266 25.370 25.370 29.32 I 25.362 29.229 25.387 25.387 29.228 25.363 29.321
I I I 2 2 2 3 3 3 4 4 4
x Y
.? x Y I x Y 2 x Y i
Displacements 0.054 0.006 I.514 0.058 0.058 4.999 0.006 o.os4 I.514 0.003 0.003 0.753
[cm]
I.Om
10.9
J. PIETRZAK
42
terion. The lower bound of the function was approximately estimated as -0.05 and substituted into eqn (16). Computation involved 827 and 237, function and gradient evaluations, respectively. Acknowlcdgemmfs-The author is grateful to Mgr. R. Trafas for coding the program.
4.
5. 6.
REFERENCES
1. H. A. Buchholdt, ~format~n of prestresJed cable-nets. Actu ~oIy$ec~nicu~~inaeica Ci 38 (1966). 2. H. A. Buchholdt. M. Davies and M. J. L. Hussey, The anatysis of cable nets. 1. Inst. Mafhs Apptics 4.339-358 (l!W. 3. H. A. Buchholdt, The Newton-Raphson approach to skeletal
7.
assemblies having significant displacements. Acra Polyfechnica Scandinauica Ci 72, (1971). H. A. Buehholdt and B. R. McMiilan, iterative methods for the solution of pretensioned cable structures and pinjointed assemblies having scent geometrical displacements. fASS Pa&k Symp. PM iI, Tokyo and Kyoto (17-23 Oct. 1971). H. Y. Huang, Methods of dual matrices for function minimization, I. Optimization 7Xeory and Applicafions, 13(S). 519537 (1974). T. M. Murray and N. Willems. Application of direct energy minimization to the static analysis of cable supported structures. Studiesin ~inee~ ~~chanics, Report-No. 33. The U~ve~itv of Kansas. (1~0). T. M. Murray and N. -Willems, Analysis of inelastic suspension structures. J. Stmt. Mechnnics, ASCE. (Proc. Paper 8574).2791-2806(1971).
of Poznah, Poznafi. Poland
(Received 20 December 1976; received for publication
14 July 1977)
-The
paper concerns the static analysis of the cable stmcturcs. Energy approach is used in the analysis. Matrixfomudadoa of thebasic expressions for the total potential energy and its gradiint is given. It is proposed in the paper to utiliz the Huang’s reverse algorithm Enclosed is a nttntcricai example.
for finding stationary
1. ~hc paper eonecms static analysis of cable structures
based on the principle of minimum of tbe total potential energy and with the use of numerical minimization m&hod. General theory of the direct energy minimization approach to static analysis of cable structures was developed by Buchholdt d al.[l-4), who used steepest descent, conjugate gradients (CG) and DavidonFletcher-Powell (DFP) aJgotithms for finding minimum point of the total potential energy. The CCL DFP algorithms and FiiMeCormick’s revised variable metric methods wcrc used by Murray and Willcms[6,71 in their investigations on static bchaviour of tension structures. In the present work Huang’s reverse algorithm[S] is applii for finding stationary point of the total potential cncrgy of the cable struoturc. The basic advantage of the algorithm is that it cnablcs to bypass unidimcnsional minim&&n whioh is rather troublesome when using CG or DFP algorithms. In Section 2 of the paper problem of static analysis of cable st~ctur6s is stated, basic expressions for the total potential energy and its gradient arc recalled and matrix formulation of the expressions arc presented. Section 3 contains a brief description of the Huang’s reverse algorithm (HRA). In Section 4 a simple cable net is studied numerically with the USCof the HRA.
point of the total potential energy.
2.1 Total potential energy The total potential energy Q of any structure is the sum of strain energy U and potential V of the external forces n=U+V.
(1)
The strain energy of cable structure is the sum of energy stored in all members of the structure and is given by
(2) where U,.,,, --initial energy stored in clement m (as a result of prestrcss), PO.m = uutial ’ ’ force in the element, E,,, = moduli of elasticity. A,,, = cross sectional area of the clement, L, = lengthof the element, e, = elongation of the element causedby external forces. The elongatione,,, of the member m, incident with the nodesj and n, is given by e, = &
WL
- &I
+ tx, - X,&L
+ 12(X”, - 4,) + k,
-
- x,J
x,,m, - 4)
+ w2, - x,,)+k, - X,*)1(&* - x,Jl
(3)
in which X,, XjY, XJz and X,, X., X,,, are coordinates of the nodes j and n, respectively, and x,~, xlrr xl,, xnI, x,,, x., arc displacementsof the nodes. We can rewrite cqn (3) in the following matrix form:
4 PnonLSMsrATE+rnNT The problem which WCdiscuss is to find displacements of the nodes of cable structure subjected to the action of external loads and to calculate internal forces in the members of the structure when the following assumptions are fulhllcdz ti) structure is built of the straight mcmbcrs whiih arc abk to carry exclusively tensile forces, (ii) the member material is linearly elastic, (iii) the ext6mal forces act only in the nodes, (iv) confiyrrtion of the structure and initial prcstrcss conditions guarantee that in all members only tensile forces exist, the values of whi6h do not exceed elastic limit. We can formulate our problem in the form of the following minimization problem: find point x = X* such.
e, = &-
[ZX’H,(ii,‘x)
?I!
t (ii,lx)‘(H,‘x)]
(4)
where
is the node coordinate matrix of order (in the three dimensionalcase) 3 Wx1 ( W = total number of nodes). x = {x,lx, v&r. * . q&.q*
.
. . %v&wqw*l
is the node displacement matrix of order 3J x I (J = mnbcr of nodes undergoingdisplacements),H, is the 3 W X 3 incidencematrix of the element m. and H, is the
that 4x*)= w~min. where x is the column matrix of node displacementsand w(x) is the value of the total potential energy of the structure. 39
J.
40
PIETRZAK
35 X3 submatrix of IL. The way of building of incidence matrices and their submatrices is illustrated in Fig. 1. The superelements hi, or order 3 X 3, of the matrix H, are if the element is incident with the node i and the orientation of the element is FROM the node i,
I I=
‘,
’
where
[1
The potential V of external forces is V = -F’x
(6)
where if the element is incident with the node i and the orientation of the element is TOWARDS the node i, otherwise.
hi =
To1 1
L
PO.,
.
.
F,F&,
,
F,*F,&,}
is
the column matrix of external forces. Thus, the total potential energy of the structure can be written as ?r= Uo+Po’e+Ae’Ke-Px. 2
(71
I .2 Gradient of total potential energy The gradient Vr of the total potential energy is given by
J
Introducing now matrices PO= PO.,
F = (F,,F,,F,,
. .
PO.ML Vn=VU-F
(8)
where the gradient VU of the strain energy is [ I]:
ari ;;;:
and matrix em.. e,)
e={e,...
CNI ;;;;
the element e,,, of which is given by (4), we can rewrite expression (2) for strain energy in the form U = Li,+P,‘e+~e’Ke
au KY ... LW
(5)
vu=
;;;;; dU g
dU g
-I 0 0 0
Y
I-!
I
0 l
0 0 0
0 0 0 element I
i
cn
0
li
-I
2
0
aU
0
2
0 0
“, 0 0
0
,:i;l’ll ; 0 o=[ I OO 0
element
element
2 Fig. I.
q
-~~(x,-x,~)+(x”,-x,~)I]
q-2
[(X”, - x,)+(x”,
- 41
I
fg(-~~(x.,-x,)+(x.,--*lr)1).
H=[H ,... H,...HM],
0
ii = [ii,. . .I?, . . ii&41
lo
.OJ
I”
In eqn (9): ? represents summation with respect to nodes n which are directly connected with the node j; Pin is internal force in the ekment m, incident with nodes j and II, and its value is Pi, = P,,, = PO.,,,+ (EA,IL,) e, ; L,. = L, is the length of the element m connecting nodes j and n. Introducing matrices
&.O *
{ -~I(x.,-x,*)+(x.,-xi,)l)
I 0
0 4’
Ix,,
0
5E
0
al ax,
orientation
r
i
au ix,
Arbitrary selected
‘x-
9)
.
-
I
I-
12
and
T = It,.
t,.
t,,,],
41
Staric analysis of cable structures ‘IIte
where
increase Ax is defined by Axi =: -pi sign (VP; q,)q,,
and M is the number of elements in the structure, we can rewrite eqn (9) in compact matrix form VU = fiT(H’X + ii’x).
(II)
3. SOLUTION OF PROBLEM The problem of finding stationary point x* of s, formulated in Section 1, can be solved with the use of various numerical minimization algorithms. It is suggested here to use for the purpose Huang’s reverse algorithm which offers important advantage of bypassing unidimensional minimization. (In the case of static analysis of cable structure, unidimensional minimization process involved in e.g. CG or DFP algorithms, is rather troublesome. It requires, in each iteration step, finding the smallest value of the third order algebraic equation [ I]). The major steps in the Huang’s reverse algorithm are: (i) determination of search direction q, from poir.t xi; (ii) determination of trial increase Axi in the direction qi: (iii) evaluation of function ri+t = ?T(x, + Ax,); (iv) modification of Axi until condition ?ri+, < q i.e. ?r(xi + Axi)< *(xi) is fulfilled: (v) final determination of the point Xi+, = Xit Ax,; (vi) evaluation of gradient Va&, = Vr(x, +Axi); (vii) determination of the new search direction qi+l from the point xi+,. etc.; (viii) termination if Vri:tVWi+,
where p, is a quantity in range 0 < p, C (&iv and (c(~)~ may be chosen as (16)
(10)
Substitution of eqn (IO) into eqn (8) gives the matrix expression for the gradient of the total potential energy PO = fiT(H’X + fi’x) - F.
(IS)
where RI(, = lower bound of n. Necessary modifications of Ax, in the present work were carried out accordingly to the rule Axi = (AxJS). 4. NUMElllCAL EXAMPLE The matrix expressions for the total potential energy and its gradient as well as the Huang’s reverse algorithm were implemented to the ODRA 1204computer program. Results of computation of a simple cable net, shown in Fig. 2, are given in Table 1. Point x, = (0 0 .OOS0 0 .OlO0 0 .OOS0 0 O}was assumed as the starting point for iteration and the condition VT;+, .Vr,+, < IO-” was taken as the termination cri-
(12)
where Bt is a square matrix. (In our three dimensional cable problem matrix Bi is of order 3J x 31.) For i = 0 onecantakeB,=Ill...Il.fori+O 5kN
Bi =Bi_l+
Ci_lCI_I Cl_,(V7ri -V7r_,)
)X
(13)
5.6
where
1
Ci_, =Axi_,-B,_,(Vn,
-VT+,).
12.4.1.7
I.Om
1
11.3.2.9 I.Om
1
Fig. 2.
(14) Table I.
Elements
Forces (kN)
Nodes
Directions
I 2 3 4 5 6 7 8 9 IO II I2
29.266 29.266 25.370 25.370 29.32 I 25.362 29.229 25.387 25.387 29.228 25.363 29.321
I I I 2 2 2 3 3 3 4 4 4
x Y
.? x Y I x Y 2 x Y i
Displacements 0.054 0.006 I.514 0.058 0.058 4.999 0.006 o.os4 I.514 0.003 0.003 0.753
[cm]
I.Om
10.9
J. PIETRZAK
42
terion. The lower bound of the function was approximately estimated as -0.05 and substituted into eqn (16). Computation involved 827 and 237, function and gradient evaluations, respectively. Acknowlcdgemmfs-The author is grateful to Mgr. R. Trafas for coding the program.
4.
5. 6.
REFERENCES
1. H. A. Buchholdt, ~format~n of prestresJed cable-nets. Actu ~oIy$ec~nicu~~inaeica Ci 38 (1966). 2. H. A. Buchholdt. M. Davies and M. J. L. Hussey, The anatysis of cable nets. 1. Inst. Mafhs Apptics 4.339-358 (l!W. 3. H. A. Buchholdt, The Newton-Raphson approach to skeletal
7.
assemblies having significant displacements. Acra Polyfechnica Scandinauica Ci 72, (1971). H. A. Buehholdt and B. R. McMiilan, iterative methods for the solution of pretensioned cable structures and pinjointed assemblies having scent geometrical displacements. fASS Pa&k Symp. PM iI, Tokyo and Kyoto (17-23 Oct. 1971). H. Y. Huang, Methods of dual matrices for function minimization, I. Optimization 7Xeory and Applicafions, 13(S). 519537 (1974). T. M. Murray and N. Willems. Application of direct energy minimization to the static analysis of cable supported structures. Studiesin ~inee~ ~~chanics, Report-No. 33. The U~ve~itv of Kansas. (1~0). T. M. Murray and N. -Willems, Analysis of inelastic suspension structures. J. Stmt. Mechnnics, ASCE. (Proc. Paper 8574).2791-2806(1971).