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Optimal design of prestressed trusses
0045.7949192S5.00+0.00
compurers &strucrures Vol.43. No. 4, pp. 741-744. 1992
Q 1992PergamonPressLtd
Printedin Great Britain.
OPTIMAL
DESIGN R.
Faculty
OF PRESTRESSED LEVY
and A.
TRUSSES
HANAOR
of Civil Engineering, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel (Received 29 March 1991)
Abstract-This paper is concerned with the effect of prestress on the minimum weight design of singly loaded trusses which are required to satisfy stress constraints. Use is made of realizability theorems to derive two distinct formulations--the equilibrium formulation and the hybrid formulation, in a linear programming form. The former has the member forces as unknown variables whereas the latter searches directly for the prestressing system and calculates the member forces a posteriori. A 72-bar double layered grid is used as an example to show weight savings of up to 37%.
INTRODUCTION
A=N6
The minimum weight design of singly loaded trusses of fixed geometry which are required to satisfy stress constraints was discussed as early as 1900 by Cilley [l] who showed that the optimal design is fully stressed and statically determinate. Prestressing as a means of weight reduction was used in two cases [2, 31 and its benefits demonstrated. Recently Spillers and Levy [4] studied two loadings in prestressed truss design proving fully stressedness and presenting an iterative scheme for attaining the optimum. Because of the fully stressedness property it might appear that prestressing has no meaning as far as weight is concerned in singly loaded trusses. However, in civil engineering design, a large class of structures requires a predetermined ratio of areas of cross-section (same for simplicity). Double layered grids, being of this class, are considered in this paper and here prestressing can reduce weight due to the fact that one member governs the design while all others are highly understressed. The paper presents two distinct techniques for the optimal design of prestressed space structures and specifically attains and investigates optimal designs of a 72-bar double layered grid.
(member/nodal
displacement
(3) equations),
where Fis the member force matrix; P, the nodal load matrix; K the primitive stiffness matrix; A, the member displacement matrix; D, ‘the lack of fit’ variable matrix; 6, the nodal displacement matrix; and N, the generalized incidence matrix. Note that the primitive stiffness matrix is diagonal and positive definite with the ith diagonal element Ki= A,E/L,, where Ai is the cross-sectional area of member i, E is Young’s modulus and L, is the length of member i. The structural optimization problem can now be stated as find K, F, 6, and A to minimize
ZKi(Ap)*
subject to RF =p
(4) (5)
F=K(A-D)
(6)
A=N6
(7)
IF,lGK,Ai
(8)
-K,
(9)
and FORMULATION
The problem of minimizing the weight of singly loaded elastic prestressed trusses is indeed a nonlinear structural optimization problem. For its proper synthesis the equations of statics are first introduced below in the form of the node method [S]. These are h+F=P
(1)
Here 4 is the absolute value of the allowable displacement for member i and eqn (4) represents volume since CK,(A;)* = C(A,Ei/Li)(auL,/E)* = (a”)*/E . .ZA,L, N volume. For convenience it is assumed at this stage that Q”, the allowable stress, is the same for both tensile and compressive members.
(nodal equilibrium) F=K(A-D) (constitutive
equations)
Equilibrium formulation (2)
Realizability through prestressing enables one to reduce the general nonlinear programming problem [eqns (4)-(9)] to a linear programming problem that 741
R. LEVYand A. HANAOR
142
accounts for equilibrium only. It states that for any given force system F, satisfying equilibrium, fiF = P, and any given stiffness vector, K, satisfying 1F,I 2 K,Ap there exists a prestressing system, D, such that the design is realizable. In loose terms ‘realizing’ a design means the reproduction of a given force system using the equations of structures while satisfying safety requirements. The validity of this statement can be argued by taking 6 = A = 0 and choosing D, = - F, /K). For the problem at hand, where all members possess the same cross-sectional areas, the structure is designed for a given optimal force system, F, as K,=kmax\F,l. l
(IO)
By designing in accordance to eqn (lo), eqns (4)-(lo), reduce to minimize
max 1F, (
subject to
flF = P.
G to replace F by imposing a lower bound, B, on F sothatF=G-Band-B
S
subject to
[ -%I{;}={
:Z;}.
(12)
Here I is the identity matrix and e a vector with entries e, = 1. The tableau in eqn (12) is very easily generated automatically and its formulation is advantageous when different allowable stresses for compression and tension are encountered. Reformulation is unnecessary. One can simply adjust the lower bound and solve a sequence of linear programs until the ratio of the maximum compressive force to the maximum tensile force is the same as the ratio of the corresponding allowable stresses. Hybrid formulation
(11)
An optimal solution to eqn (I 1) implies an optimum force system, F, for which a prestressing system, D, can be found to realize the design and, therefore, satisfies the original nonlinear programming problem. Prestressing can thus, be dismissed for posterior considerations and interest lies in that force system, of all possible force systems, that satisfies equilibrium where the maximum absolute force is minimal. It should, nevertheless, be pointed out here that the minimum number of bars that have to be prestressed to realize an optimal force system is the same as the number of redundant bars in the structure. This is because eqns (6) and (7) have to be satisfied. If more than T members are assigned zero prestress, with T being the number of bars in a statically determinate configuration, then eqn (7) which is rewritten as
The equations of structures in partitioned form are pivotal to the hybrid formulation [7]. By assuming a priori that prestressing is imposed on redundant members only it is possible to obtain explicit expressions of force and nodal displacements in terms of ‘lack of fit’ variables. The optimization problem is then cast as a linear program whose solution yields a prestressing system for given EA /Li. Uniform scaling will finally determine the design. When partitioned, eqns (l)-(3) take the form of (13) FT= KrA,
(14)
FL = KL(AL- DL)
(15)
(16) Equations
(13)-(16) can be rearranged to read
might not have a solution. Here L stands for the number of redundant members.
F=AP+BD,
(17)
h=UP+VD,,
(18)
where A=
fi;’
- fl;‘fiL,K~’
+ NLN;‘K~‘R,‘fiL]-‘]NLN~’
[K;’ + N,N;‘K,‘m;‘~~]-‘[N,N~‘~~‘]
The minimax nature of eqn (11) poses a slight difficulty which can be overcome by introducing a positive scalar, S, and demanding that S be greater than or equal to )Fi I [6]. Now readily available programs for the solution of linear programming problems will usually not handle absolute value signs and require nonnegativity of design variables. By setting S > F, and S > -F, to replace the constraint, S 3 1F, I, and introducing a new nonnegative variable
[-Ki’
U = [N;‘K;‘fi;’ X [Ki’
K;‘&‘)
1
- NtN,‘KF’S;‘RJ
(19)
]
(20)
- N;‘K;‘&‘RL
+ N,N;‘K;‘~:r’~L]-‘NLN~‘K~‘~~‘]
(21)
and V = NF’KF’$J’~~
x [KL’ + NLN:‘K;‘fi;‘&]-‘.
(22)
Optimal design of prestressed trusses
743
Fig. 1. Plan, elevation, and oblique views of the double layered grid.
If member, r, is identified as the critical member governing the design and eqn (18) is neglected due to arguments of realizability then the optimization problem for the design of prestressed trusses having equal cross-sectional areas can be written as minimize subject to
analysis is performed to obtain the optimal force system and the design is then scaled appropriately. With the introduction of a new nonnegative variable, Q, to replace D, and a lower bound, R, eqn (23) transforms to minimize
-A,P-B,Q AP+BD,>F,
-4Q
subject to
AP+BD,
(23) EXAMPLGA 72-BAR DOUBLE LAYERED GRID
Here F, and F, are the allowable forces in compression and tension, respectively. Equation (23) is formulated for a given AE/L, and the assumption that a compression member governs. When an optimal prestressing system D, is found a linear elastic
1 .oo
I
1
I
1.25
1.50
1.75
Allowable
.
2.00
stress ratio
Fig. 2. Weight saving curve for the double layered grid.
This section presents the design of a 72-bar double layered grid with prestress and demonstrates the advantages of its use in terms of weight savings. Figure 1 shows three views of the truss under investigation. It is composed of 72 members having equal lengths of 54 in. A load of 13,534 lb acts on the center node in the upward direction. The center node is free to move in the vertical direction only while all edge nodes are restrained in that direction. One corner node is additionally restrained in a perpendicular direction to a main diagonal in the plan view in that plane. Moreover it is assumed that Young’s modulus, E, is 16 x IO6lb/in* and that the density of material is constant. Table 1 presents optimal designs of the double layered grid with and without prestress for a ratio, tl, of the allowable stress in tension (40,000 psi) to that of compression (20,000 psi) of 2.0. All values in that table are in inches and pounds. Only one eighth of the structure is tabulated due to symmetry.
R. LEVYand A. HANAOR
744
Table 1. Optimal design with and without prestress
Bar I
2 3 4 5 6 7 8 9 10 11 I2
Optimal force no prestress (lb)
Optimal force with prestress (lb)
Prestress (in)
2871 4785 - 4426 -1914 479 -239 1196 1077 1436 -598 718 - 1436
5583 4785 -2791 798 1595 -798 399 -798 2791 -2791 -1196 -2791
0 0 0 0 -0.1205 0 0 0 -0.3085 0 0 0
A = 0.2213 in2 Volume = 860.4 in’
A = 0.13955 in2 Volume = 542.6 in3
tensile force from the design without prestress does not change but rather the absolute maximum compressive forces reduce in value. Beyond c( = 1.6 maximum tensile forces increase and absolute maximum compressive forces decrease. CONCLUSION
Linear programming is an exact representation of the singly loaded truss design with prestress and can be looked at as a ‘relaxed’ state of the nonlinear programming formulation. Double layered grids have been shown to enjoy lighter designs due to prestress with weight savings of up to 37% for a ratio, TV,of allowable tensile to compressive stresses of 2.0. Acknowledgement-The authors wish to thank Appel for his technical assistance.
The optimal design without prestress is obtained by performing an elastic analysis using eqns (l)-(3) with D f 0 and A, = 1. The area is then corrected according to max{A, (‘), A co} where Ai’) = l/a; max Fi and A y) = l/a;. max & and c and t represent compression and tension, respectively. For the design with prestress the IMSL routine ZX3LP [8] is used to solve for the optimal force system. The area is obtained in the same way as for the design without prestress and prestressing system is then found to realize the optimal force using D, = NLNf’K:‘FL and D, = 0. Weight savings for this case of CI= 2.0 amount to 36.9%. The study is taken further to investigate the effect of tl on weight reduction resulting in the curve shown in Fig. 2. Note that the design with prestress of Table 1 is one point on that curve. Two distinct portions in the curve are evident. Up to r = 1.6 the maximum
Mr L.
REFERENCES
1. F. H. Cilley, The exact design of statically indeterminate frameworks, an exposition of its possibility, but futility. Tram ASCE 43, 353443 (1900). 2. L. D. Hofmeister and L. P. Felton, Prestressing in structural synthesis. AIAA Jnl 8, 363-364 (1970). 3. L. P. Felton and M. W. Dobbs, On optimized prestressed trusses. AIAA Jnl 15, 1037-1039 (1977). 4. W. R. Spillers and R. Levy, Truss design: two loading conditions with prestress. J. Struct. Engng, ASCE 110, 677687 (1984). 5. W. R. Spillers, Automated Structural Analysis: An Infroduction. Pergamon Press, New York (1972). 6. M. B. Fuchs, Minimax in structural design. J. Sfrucf. Engng, ASCE 109, 1107-1117 (1983). 7. A. Hanaor, and R. Levy, Imposed lack of fit as a means of enhancing space truss design. Space Structures 1, 1477154 (1985). 8. IMSL Library User’s Manual, Vol. 4 (1984).
compurers &strucrures Vol.43. No. 4, pp. 741-744. 1992
Q 1992PergamonPressLtd
Printedin Great Britain.
OPTIMAL
DESIGN R.
Faculty
OF PRESTRESSED LEVY
and A.
TRUSSES
HANAOR
of Civil Engineering, Technion-Israel Institute of Technology, Technion City, Haifa 32000, Israel (Received 29 March 1991)
Abstract-This paper is concerned with the effect of prestress on the minimum weight design of singly loaded trusses which are required to satisfy stress constraints. Use is made of realizability theorems to derive two distinct formulations--the equilibrium formulation and the hybrid formulation, in a linear programming form. The former has the member forces as unknown variables whereas the latter searches directly for the prestressing system and calculates the member forces a posteriori. A 72-bar double layered grid is used as an example to show weight savings of up to 37%.
INTRODUCTION
A=N6
The minimum weight design of singly loaded trusses of fixed geometry which are required to satisfy stress constraints was discussed as early as 1900 by Cilley [l] who showed that the optimal design is fully stressed and statically determinate. Prestressing as a means of weight reduction was used in two cases [2, 31 and its benefits demonstrated. Recently Spillers and Levy [4] studied two loadings in prestressed truss design proving fully stressedness and presenting an iterative scheme for attaining the optimum. Because of the fully stressedness property it might appear that prestressing has no meaning as far as weight is concerned in singly loaded trusses. However, in civil engineering design, a large class of structures requires a predetermined ratio of areas of cross-section (same for simplicity). Double layered grids, being of this class, are considered in this paper and here prestressing can reduce weight due to the fact that one member governs the design while all others are highly understressed. The paper presents two distinct techniques for the optimal design of prestressed space structures and specifically attains and investigates optimal designs of a 72-bar double layered grid.
(member/nodal
displacement
(3) equations),
where Fis the member force matrix; P, the nodal load matrix; K the primitive stiffness matrix; A, the member displacement matrix; D, ‘the lack of fit’ variable matrix; 6, the nodal displacement matrix; and N, the generalized incidence matrix. Note that the primitive stiffness matrix is diagonal and positive definite with the ith diagonal element Ki= A,E/L,, where Ai is the cross-sectional area of member i, E is Young’s modulus and L, is the length of member i. The structural optimization problem can now be stated as find K, F, 6, and A to minimize
ZKi(Ap)*
subject to RF =p
(4) (5)
F=K(A-D)
(6)
A=N6
(7)
IF,lGK,Ai
(8)
-K,
(9)
and FORMULATION
The problem of minimizing the weight of singly loaded elastic prestressed trusses is indeed a nonlinear structural optimization problem. For its proper synthesis the equations of statics are first introduced below in the form of the node method [S]. These are h+F=P
(1)
Here 4 is the absolute value of the allowable displacement for member i and eqn (4) represents volume since CK,(A;)* = C(A,Ei/Li)(auL,/E)* = (a”)*/E . .ZA,L, N volume. For convenience it is assumed at this stage that Q”, the allowable stress, is the same for both tensile and compressive members.
(nodal equilibrium) F=K(A-D) (constitutive
equations)
Equilibrium formulation (2)
Realizability through prestressing enables one to reduce the general nonlinear programming problem [eqns (4)-(9)] to a linear programming problem that 741
R. LEVYand A. HANAOR
142
accounts for equilibrium only. It states that for any given force system F, satisfying equilibrium, fiF = P, and any given stiffness vector, K, satisfying 1F,I 2 K,Ap there exists a prestressing system, D, such that the design is realizable. In loose terms ‘realizing’ a design means the reproduction of a given force system using the equations of structures while satisfying safety requirements. The validity of this statement can be argued by taking 6 = A = 0 and choosing D, = - F, /K). For the problem at hand, where all members possess the same cross-sectional areas, the structure is designed for a given optimal force system, F, as K,=kmax\F,l. l
(IO)
By designing in accordance to eqn (lo), eqns (4)-(lo), reduce to minimize
max 1F, (
subject to
flF = P.
G to replace F by imposing a lower bound, B, on F sothatF=G-Band-B
S
subject to
[ -%I{;}={
:Z;}.
(12)
Here I is the identity matrix and e a vector with entries e, = 1. The tableau in eqn (12) is very easily generated automatically and its formulation is advantageous when different allowable stresses for compression and tension are encountered. Reformulation is unnecessary. One can simply adjust the lower bound and solve a sequence of linear programs until the ratio of the maximum compressive force to the maximum tensile force is the same as the ratio of the corresponding allowable stresses. Hybrid formulation
(11)
An optimal solution to eqn (I 1) implies an optimum force system, F, for which a prestressing system, D, can be found to realize the design and, therefore, satisfies the original nonlinear programming problem. Prestressing can thus, be dismissed for posterior considerations and interest lies in that force system, of all possible force systems, that satisfies equilibrium where the maximum absolute force is minimal. It should, nevertheless, be pointed out here that the minimum number of bars that have to be prestressed to realize an optimal force system is the same as the number of redundant bars in the structure. This is because eqns (6) and (7) have to be satisfied. If more than T members are assigned zero prestress, with T being the number of bars in a statically determinate configuration, then eqn (7) which is rewritten as
The equations of structures in partitioned form are pivotal to the hybrid formulation [7]. By assuming a priori that prestressing is imposed on redundant members only it is possible to obtain explicit expressions of force and nodal displacements in terms of ‘lack of fit’ variables. The optimization problem is then cast as a linear program whose solution yields a prestressing system for given EA /Li. Uniform scaling will finally determine the design. When partitioned, eqns (l)-(3) take the form of (13) FT= KrA,
(14)
FL = KL(AL- DL)
(15)
(16) Equations
(13)-(16) can be rearranged to read
might not have a solution. Here L stands for the number of redundant members.
F=AP+BD,
(17)
h=UP+VD,,
(18)
where A=
fi;’
- fl;‘fiL,K~’
+ NLN;‘K~‘R,‘fiL]-‘]NLN~’
[K;’ + N,N;‘K,‘m;‘~~]-‘[N,N~‘~~‘]
The minimax nature of eqn (11) poses a slight difficulty which can be overcome by introducing a positive scalar, S, and demanding that S be greater than or equal to )Fi I [6]. Now readily available programs for the solution of linear programming problems will usually not handle absolute value signs and require nonnegativity of design variables. By setting S > F, and S > -F, to replace the constraint, S 3 1F, I, and introducing a new nonnegative variable
[-Ki’
U = [N;‘K;‘fi;’ X [Ki’
K;‘&‘)
1
- NtN,‘KF’S;‘RJ
(19)
]
(20)
- N;‘K;‘&‘RL
+ N,N;‘K;‘~:r’~L]-‘NLN~‘K~‘~~‘]
(21)
and V = NF’KF’$J’~~
x [KL’ + NLN:‘K;‘fi;‘&]-‘.
(22)
Optimal design of prestressed trusses
743
Fig. 1. Plan, elevation, and oblique views of the double layered grid.
If member, r, is identified as the critical member governing the design and eqn (18) is neglected due to arguments of realizability then the optimization problem for the design of prestressed trusses having equal cross-sectional areas can be written as minimize subject to
analysis is performed to obtain the optimal force system and the design is then scaled appropriately. With the introduction of a new nonnegative variable, Q, to replace D, and a lower bound, R, eqn (23) transforms to minimize
-A,P-B,Q AP+BD,>F,
-4Q
subject to
AP+BD,
(23) EXAMPLGA 72-BAR DOUBLE LAYERED GRID
Here F, and F, are the allowable forces in compression and tension, respectively. Equation (23) is formulated for a given AE/L, and the assumption that a compression member governs. When an optimal prestressing system D, is found a linear elastic
1 .oo
I
1
I
1.25
1.50
1.75
Allowable
.
2.00
stress ratio
Fig. 2. Weight saving curve for the double layered grid.
This section presents the design of a 72-bar double layered grid with prestress and demonstrates the advantages of its use in terms of weight savings. Figure 1 shows three views of the truss under investigation. It is composed of 72 members having equal lengths of 54 in. A load of 13,534 lb acts on the center node in the upward direction. The center node is free to move in the vertical direction only while all edge nodes are restrained in that direction. One corner node is additionally restrained in a perpendicular direction to a main diagonal in the plan view in that plane. Moreover it is assumed that Young’s modulus, E, is 16 x IO6lb/in* and that the density of material is constant. Table 1 presents optimal designs of the double layered grid with and without prestress for a ratio, tl, of the allowable stress in tension (40,000 psi) to that of compression (20,000 psi) of 2.0. All values in that table are in inches and pounds. Only one eighth of the structure is tabulated due to symmetry.
R. LEVYand A. HANAOR
744
Table 1. Optimal design with and without prestress
Bar I
2 3 4 5 6 7 8 9 10 11 I2
Optimal force no prestress (lb)
Optimal force with prestress (lb)
Prestress (in)
2871 4785 - 4426 -1914 479 -239 1196 1077 1436 -598 718 - 1436
5583 4785 -2791 798 1595 -798 399 -798 2791 -2791 -1196 -2791
0 0 0 0 -0.1205 0 0 0 -0.3085 0 0 0
A = 0.2213 in2 Volume = 860.4 in’
A = 0.13955 in2 Volume = 542.6 in3
tensile force from the design without prestress does not change but rather the absolute maximum compressive forces reduce in value. Beyond c( = 1.6 maximum tensile forces increase and absolute maximum compressive forces decrease. CONCLUSION
Linear programming is an exact representation of the singly loaded truss design with prestress and can be looked at as a ‘relaxed’ state of the nonlinear programming formulation. Double layered grids have been shown to enjoy lighter designs due to prestress with weight savings of up to 37% for a ratio, TV,of allowable tensile to compressive stresses of 2.0. Acknowledgement-The authors wish to thank Appel for his technical assistance.
The optimal design without prestress is obtained by performing an elastic analysis using eqns (l)-(3) with D f 0 and A, = 1. The area is then corrected according to max{A, (‘), A co} where Ai’) = l/a; max Fi and A y) = l/a;. max & and c and t represent compression and tension, respectively. For the design with prestress the IMSL routine ZX3LP [8] is used to solve for the optimal force system. The area is obtained in the same way as for the design without prestress and prestressing system is then found to realize the optimal force using D, = NLNf’K:‘FL and D, = 0. Weight savings for this case of CI= 2.0 amount to 36.9%. The study is taken further to investigate the effect of tl on weight reduction resulting in the curve shown in Fig. 2. Note that the design with prestress of Table 1 is one point on that curve. Two distinct portions in the curve are evident. Up to r = 1.6 the maximum
Mr L.
REFERENCES
1. F. H. Cilley, The exact design of statically indeterminate frameworks, an exposition of its possibility, but futility. Tram ASCE 43, 353443 (1900). 2. L. D. Hofmeister and L. P. Felton, Prestressing in structural synthesis. AIAA Jnl 8, 363-364 (1970). 3. L. P. Felton and M. W. Dobbs, On optimized prestressed trusses. AIAA Jnl 15, 1037-1039 (1977). 4. W. R. Spillers and R. Levy, Truss design: two loading conditions with prestress. J. Struct. Engng, ASCE 110, 677687 (1984). 5. W. R. Spillers, Automated Structural Analysis: An Infroduction. Pergamon Press, New York (1972). 6. M. B. Fuchs, Minimax in structural design. J. Sfrucf. Engng, ASCE 109, 1107-1117 (1983). 7. A. Hanaor, and R. Levy, Imposed lack of fit as a means of enhancing space truss design. Space Structures 1, 1477154 (1985). 8. IMSL Library User’s Manual, Vol. 4 (1984).