A structural optimization algorithm was researched including global displacements as decision variables. The algorithm was applied to planar reinforced concrete frames with nonlinear material behavior submitted to static loading. The flexural performance of the elements was evaluated as a function of the actual stress-strain diagrams of the materials. Formation of rotational hinges with strain hardening were allowed and the equilibrium constraints were updated accordingly. The adequacy of the frames was guaranteed by imposing as constraints required reliability indices for the members, maximum global displacements for the structure and a maximum system probability of failure

Hoit, Marc

Soeiro, Alfredo

English

NASA Technical Reports Server

N89-24631 
INTEGRATED OPTIMIZATION OF NONLINEAR R/C FRAMES 
WITH RELIABILITY CONSTRAINTS 
Alfredo Soeiro 
Dept. of Civil Engineering, Univ. Florida 
Gainesville, Florida 
Marc Hoit 
Dept. of Civil Engineering, Univ. Florida 
Gainesville, Florida 
PFZECEDENG PAGE BkkPdK NOT FILMED 
47 
https://ntrs.nasa.gov/search.jsp?R=19890015260 2020-03-20T03:00:58+00:00Z
Summary 
A structural optimization algorithm was researched including 
global displacements as decision variables. The algorithm was applied 
to planar reinforced concrete frames with nonlinear material behavior 
submitted to static loading. The flexural performance of the elements 
was evaluated as a function of the actual stress-strain diagrams of 
the materials. Formation of rotational hinges with strain hardening 
were allowed and the equilibrium constraints were updated accordingly. 
The adequacy of the frames was guaranteed by imposing as constraints 
required reliability indices for the members, maximum global 
displacements for the structure and a maximum system probability of 
failure. 
Previous Research 
Structural frame optimization problems have been usually 
formulated based on the cycling between two distinct phases: analysis 
and optimal design. The option described in this work combines both 
phases by the addition of the global displacements to the set of 
design variables, option researched by several authors (ref. 1 and 2). 
The main purpose of this strategy was to determine the benefits of 
extending the linear static formulation to nonlinear static structural 
problems using the secant stiffness method. The reason behind this 
research was the fact that the global stiffness changes created by the 
nonlinear behavior would be considered simultaneously with the changes 
of the element sizes, thus improving convergence. 
The first step of the research was to optimize elastic plane 
frames with elements with rectangular sections submitted to static 
loading. The objective function was the volume of the structure and 
constraints of the optimization problem were equalities representing 
global equilibrium of the structure and inequalities for the limits on 
global displacements and the maximum flexural stresses. The strategy 
adopted consisted of transforming the constrained problem in an 
unconstrained one using the method of the Augmented Lagrangian 
Multipliers (ref. 3 ) .  The unconstrained minimization was solved using 
the Hooke and Jeeves method. 
The results with this formulation were encouraging and the 
optimal solutions were found. The convergence rate was dependent on 
the initial design, scaling, penalty parameters and lagrangian 
multipliers values. The computational effort was considerable when 
compared with other explored techniques based on optimality criteria 
and mathematical programming methods. To improve the efficiency of 
the algorithm a gradient technique was implemented to solve the 
unconstrained minimization problem. This improvement was unsuccessful 
since the Augmented Lagrangian function was very steep, with large 
sensitivity to any small variation of the displacement variables. 
48 
Nonlinear Formulation 
The following logical objective was to extend this strategy to 
nonlinear reinforced concrete frames. The typical frame element has 
rectangular cross section and is doubly reinforced with equal amount 
of flexural steel on both sides. The model adopted for the inelastic 
reinforced concrete element was the one component model, where 
rotational springs are added to the ends of the elastic element to 
simulate the formation of plastic hinges at the extremities of the 
element (ref. 4). The stiffnesses of the linear elastic element 
stiffness and the springs was condensed using the flexibility 
formulation. 
The determination of the characteristics of each reinforced 
concrete section was based on the stress strain diagrams for the 
concrete and the reinforcing steel. The yielding and ultimate moments 
for each cross section were used to determine the characteristics of 
the springs for each element. The spring stiffness was considered 
infinite whenever the element moment was below the yielding moment. 
When the moment was above the yielding value the spring stiffness was 
updated and, since there was no incremental loading or unbalanced 
iteration of the structure, the secant stiffness was adopted for the 
spring stiffness (ref. 5 ) .  
ONE COMPONENT MODEL AND MATERIAL CHARACTERISTICS 
Linear E last ic  Elenent 
\ w i n g  with Secant Stiffness 
One Component Reinforced Concrete Element 
f Y 
f 9 - steel stress 
fy  - yielding stress 
strain 
> 
Steel Stress-Strain Diagram 
f - cnncrete stress 
strain 
0.002 > 
Concrete S tr ess-S t r a in Diagram 
49 
Procedure Implementation 
The objective function was the cost of the materials: concrete 
and steel. The constraints included the equilibrium equalities, 
maximum global displacements and maximum probability of failure for 
each element. The equilibrium constraints were evaluated every time a 
design variable changed with the corresponding updating of the spring 
stiffnesses. The values of the maximum displacements were dictated by 
serviceability constraints like maximum joint rotations or story 
drifts. The maximum element probabilities of failure were chosen with 
current practices of structural design codes (ref. 6). 
Flexural element actions are the most important in small and 
medium sized frames for the definition of section sizes and 
longitudinal steel. The evaluation of element reliability was based 
on the corresponding flexural failure function (ref. 7 ) .  The same 
approach was used to evaluate the system probability of failure. The 
basic variables considered were the compressive strength of concrete 
and the external loads. 
The system probability of failure was evaluated at the 
mechanism level at the end of each optimization cycle. If the value 
of the system probability of failure was not satisfactory the 
optimization was restarted using a different limit of element 
probability of failure for the elements involved in the failure 
mechanism. The system probability of failure was obtained using the 
beta-unzipping method (ref. 8). In summary, the elementary mechanisms 
of failure were determined using Watwood's method (ref. 9) and the 
correspondent failure functions formed. These mechanisms were then 
combined linearly and the related probabilities of failure calculated, 
while rejecting those combinations with values outside given 
intervals. 
SPRING SECANT STIFFNESS 
mu 
H 
4 
Spring Honent-Rotation Diagram 
B - Ultimate i m t  
Mj - Yielding Dorent 
K I  - Ik30 
ru - Ultimte ratetim 
y - Yielding rotetion 
Ksec - $ring stiffness for 
n > r q  K2 - CrwqVCrUly~ 
H - actual element moment 
r - actual elenent chord rotation 
Rotation r 
50 
Optimization Results 
The first approach to solve the optimization problem used the 
Augmented Lagrangian function and the Hooke and Jeeves method the 
unconstrained minimization technique. Considerable effort was put 
into this formulation with several options for the starting points, 
combinations of penalty parameters, scaling techniques and number of 
cycles. Three structures were tested with different levels of 
complexity. 
In some cases the values obtained were close with those 
corresponding to the expected optimal values. Reliability constraints 
were satisfied, displacements were within the limits and equality 
constraints were satisfied. However, convergence was difficult to 
obtain and largely dependent on several different choices made at the 
start of the optimization cycle. At the same time the element forces 
were not in accordance with the assumed secant spring stiffnesses 
showing lack of convergence of the nonlinear iteration process. 
To improve convergence of the nonlinear equilibrium of the 
structure an intermediate phase was created in the optimization 
process. The displacements were removed from the optimization cycle. 
This phase corresponded to the solution of the equilibrium equations 
every time a cycle of variable optimization in the Hooke and Jeeves 
was completed. The displacements were obtained from the equilibrium 
euuations assuming the values of the secant sprinq stiffnesses as 
those at the end of the cycling optimization. The results were 
nevertheless the same as before and for that reason another technique 
was implemented. 
as 
EXAMPLES TESTED 
I 
7 K  
51 
The Generalized Reduced Gradient method was chosen because of 
it's characteristic of solving iteratively a set of nonlinear 
equations. The algorithm used (ref. 10) performed very well for the 
elastic case with convergence in most of the cases. It proved to be 
almost insensitive to the initial design points. The extension to the 
nonlinear material behavior is however in process. A first phase of 
this extension consisted of assuming for the secant spring stiffness a 
yielding value, i.e., corresponding to the ratio of the yielding 
moment and the yielding rotation whenever the element moment was 
greater than the yielding moment. Equilibrium constraints were 
satisfied, element moments were in accordance with the assumed spring 
stiffnesses and the global displacements were in accordance with 
assumed spring stiffness values. 
The second phase of transforming the spring stiffness from the 
yielding stiffness to the secant stiffness is presently being 
researched. The initial design for the values presented using the 
secant spring stiffness formulation was obtained with the elastic 
stiffness version having as ultimate moment the yielding moment. The 
secant stiffness values were limited to a minimum value corresponding 
to the ratio of the ultimate moment and the ultimate rotation to 
prevent severe oscillations of these values. The results verified the 
equilibrium constraints within certain tolerance, the global 
displacements were those corresponding to the element spring 
stiffnesses, reliability constraints were satisfied and there was an 
improvement of the cost of the structure. All elements have yielded 
correspondinq to the expected results from a optimal configuration. 
However, there are discrepancies in the moments at the joints which 
shows a insufficient convergence of the equilibrium constraints. 
RESULTS WITH YIELDING STIFFNESS AND SECANT STIFFNESS 
0 - Spring 1 5  K 
b - base ( in 1; h - height (in 1; As - steel area ( in2 1: * - lower bounds. 
52 
I 
Directions Headed 
The Augmented Lagrangian function together with the Hooke and 
Jeeves for the unconstrained minimization although insensitive to 
discontinuities of the constraints is not the best choice when 
compared with the Generalized Reduced Gradient method. More research 
has to be done on the robustness of the convergence of the nonlinear 
iteration process (ref. 11). 
A possible improvement would consist of a similar enhancement 
to the one applied in the Augmented Lagrangian formulation with an 
intermediate solution of the displacements during the optimization 
cycle. Another possible improvement would be the use of a cycling 
procedure where the spring stiffness values were kept constant during 
the optimization and updated at the end with consequent optimization 
cycle until there was stabilization of the spring stiffness values. 
The integrated approach proved itself adequate for the elastic 
stiffness if an adequate mathematical programming technique is chosen. 
It is logical to expect that it will probably perform well in the case 
of inelastic stiffness if adequately integrated with some kind of 
nonlinear structural analysis technique. 
REFERE:NCES 
1. Schmit, Lucien A. Jr.; and Fox, Richard L.: An Integrated 
Approach to Structural Synthesis and Analysis. AIAA Journal, 
I Vol. 3, NO. 6, 1960, pp. 1104-1112. 
I 
2. Haftka, Raphael T.: Simultaneous Analysis and Design. AIAA 
Journal, Vol. 23, No. 7, 1985, pp. 1099-1103. 
3. Soeiro, Alfredo: Integrated Analysis and Optimal Design. Thesis 
I for the Degree of Master of Engineering, University of Florida, 
Gainesville, 1986. 
~ 
4. Charney, Finley A.: Correlation of the Analytical and Experimental 
Seismic Response of a 1/5th-Scale Seven-Story Reinforced Concrete 
Frame-Wall Structure. PhD Dissertation, University of 
California, Berkeley, 1986. 
5. Chajes, Alexander; and Churchill, James E.: Nonlinear Frame 
Analysis by Finite Element Methods. ASCE Journal of Structural 
Engineering, Vol. 113, No. 6, 1987, pp. 1221-1235. 
I 6. Augusti, G.; Baratta, A.; and Casciatti, F.: Probabilistic Methods 
I in Structural Engineering. Chapman and Hall, New York, 1984. 
7. Leporatti, Ezio: The Assessment of Structural Safety. Research 
Studies Press, Oreqon, 1979. 
8. Thoft-Christensen, Palle; and Morotsu, Yoshisada.: Application of 
Structural Systems Reliability Theory. Springer-Verlag, Berlin, 
1986. 
I 9. Watwood, Vernon: Mechanism Generation for Limit Analysis of 
Frames. ASCE Journal of Structural Division, Vol. 109, 1979, 
pp.1-15. 
10. Lasdon, L. S.; Warren, A. D. Warren; Jain, A.; and Ratner, M.: 
Design and Testing of a Generalized Reduced Gradient Code for 
Nonlinear Programming. ACM Transactions on Mathematical 
Software, Vol. 4, No.1, 1978, pp. 34-50. 
I 
11. Soeiro, Alfredo: Integrated Optimization of Reinforced Concrete 
I Frames with Reliability Constraints. PhD Dissertation, 
University of Florida, Gainesville. (To be printed in 1989.) 
54 
I 


Integrated optimization of nonlinear R/C frames with reliability constraints

Alfredo Soeiro

M. Hoit

Repositório Aberto da Universidade do Porto

N89-24631 INTEGRATED OPTIMIZATION OF NONLINEAR R/C FRAMES WITH RELIABILITY CONSTRAINTS Alfredo Soeiro Dept. of Civil Engineering, Univ. Florida Gainesville, Florida Marc Hoit Dept. of Civil Engineering, Univ. Florida Gainesville, Florida PFZECEDENG PAGE BkkPdK NOT FILMED 47 Summary A structural optimization algorithm was researched including global displacements as decision variables. The algorithm was applied to planar reinforced concrete frames with nonlinear material behavior submitted to static loading. The flexural performance of the elements was evaluated as a function of the actual stress-strain diagrams of the materials. Formation of rotational hinges with strain hardening were allowed and the equilibrium constraints were updated accordingly. The adequacy of the frames was guaranteed by imposing as constraints required reliability indices for the members, maximum global displacements for the structure and a maximum system probability of failure. Previous Research Structural frame optimization problems have been usually formulated based on the cycling between two distinct phases: analysis and optimal design. The option described in this work combines both phases by the addition of the global displacements to the set of design variables, option researched by several authors (ref. 1 and 2). The main purpose of this strategy was to determine the benefits of extending the linear static formulation to nonlinear static structural problems using the secant stiffness method. The reason behind this research was the fact that the global stiffness changes created by the nonlinear behavior would be considered simultaneously with the changes of the element sizes, thus improving convergence. The first step of the research was to optimize elastic plane frames with elements with rectangular sections submitted to static loading. The objective function was the volume of the structure and constraints of the optimization problem were equalities representing global equilibrium of the structure and inequalities for the limits on global displacements and the maximum flexural stresses. The strategy adopted consisted of transforming the constrained problem in an unconstrained one using the method of the Augmented Lagrangian Multipliers (ref. 3 ) .  The unconstrained minimization was solved using the Hooke and Jeeves method. The results with this formulation were encouraging and the optimal solutions were found. The convergence rate was dependent on the initial design, scaling, penalty parameters and lagrangian multipliers values. The computational effort was considerable when compared with other explored techniques based on optimality criteria and mathematical programming methods. To improve the efficiency of the algorithm a gradient technique was implemented to solve the unconstrained minimization problem. This improvement was unsuccessful since the Augmented Lagrangian function was very steep, with large sensitivity to any small variation of the displacement variables. 48 Nonlinear Formulation The following logical objective was to extend this strategy to nonlinear reinforced concrete frames. The typical frame element has rectangular cross section and is doubly reinforced with equal amount of flexural steel on both sides. The model adopted for the inelastic reinforced concrete element was the one component model, where rotational springs are added to the ends of the elastic element to simulate the formation of plastic hinges at the extremities of the element (ref. 4). The stiffnesses of the linear elastic element stiffness and the springs was condensed using the flexibility formulation. The determination of the characteristics of each reinforced concrete section was based on the stress strain diagrams for the concrete and the reinforcing steel. The yielding and ultimate moments for each cross section were used to determine the characteristics of the springs for each element. The spring stiffness was considered infinite whenever the element moment was below the yielding moment. When the moment was above the yielding value the spring stiffness was updated and, since there was no incremental loading or unbalanced iteration of the structure, the secant stiffness was adopted for the spring stiffness (ref. 5 ) .  ONE COMPONENT MODEL AND MATERIAL CHARACTERISTICS Linear E last ic  Elenent \ w i n g  with Secant Stiffness One Component Reinforced Concrete Element f Y f 9 - steel stress fy  - yielding stress strain > Steel Stress-Strain Diagram f - cnncrete stress strain 0.002 > Concrete S tr ess-S t r a in Diagram 49 Procedure Implementation The objective function was the cost of the materials: concrete and steel. The constraints included the equilibrium equalities, maximum global displacements and maximum probability of failure for each element. The equilibrium constraints were evaluated every time a design variable changed with the corresponding updating of the spring stiffnesses. The values of the maximum displacements were dictated by serviceability constraints like maximum joint rotations or story drifts. The maximum element probabilities of failure were chosen with current practices of structural design codes (ref. 6). Flexural element actions are the most important in small and medium sized frames for the definition of section sizes and longitudinal steel. The evaluation of element reliability was based on the corresponding flexural failure function (ref. 7 ) .  The same approach was used to evaluate the system probability of failure. The basic variables considered were the compressive strength of concrete and the external loads. The system probability of failure was evaluated at the mechanism level at the end of each optimization cycle. If the value of the system probability of failure was not satisfactory the optimization was restarted using a different limit of element probability of failure for the elements involved in the failure mechanism. The system probability of failure was obtained using the beta-unzipping method (ref. 8). In summary, the elementary mechanisms of failure were determined using Watwood's method (ref. 9) and the correspondent failure functions formed. These mechanisms were then combined linearly and the related probabilities of failure calculated, while rejecting those combinations with values outside given intervals. SPRING SECANT STIFFNESS mu H 4 Spring Honent-Rotation Diagram B - Ultimate i m t  Mj - Yielding Dorent K I  - Ik30 ru - Ultimte ratetim y - Yielding rotetion Ksec - $ring stiffness for n > r q  K2 - CrwqVCrUly~ H - actual element moment r - actual elenent chord rotation Rotation r 50 Optimization Results The first approach to solve the optimization problem used the Augmented Lagrangian function and the Hooke and Jeeves method the unconstrained minimization technique. Considerable effort was put into this formulation with several options for the starting points, combinations of penalty parameters, scaling techniques and number of cycles. Three structures were tested with different levels of complexity. In some cases the values obtained were close with those corresponding to the expected optimal values. Reliability constraints were satisfied, displacements were within the limits and equality constraints were satisfied. However, convergence was difficult to obtain and largely dependent on several different choices made at the start of the optimization cycle. At the same time the element forces were not in accordance with the assumed secant spring stiffnesses showing lack of convergence of the nonlinear iteration process. To improve convergence of the nonlinear equilibrium of the structure an intermediate phase was created in the optimization process. The displacements were removed from the optimization cycle. This phase corresponded to the solution of the equilibrium equations every time a cycle of variable optimization in the Hooke and Jeeves was completed. The displacements were obtained from the equilibrium euuations assuming the values of the secant sprinq stiffnesses as those at the end of the cycling optimization. The results were nevertheless the same as before and for that reason another technique was implemented. as EXAMPLES TESTED I 7 K  51 The Generalized Reduced Gradient method was chosen because of it's characteristic of solving iteratively a set of nonlinear equations. The algorithm used (ref. 10) performed very well for the elastic case with convergence in most of the cases. It proved to be almost insensitive to the initial design points. The extension to the nonlinear material behavior is however in process. A first phase of this extension consisted of assuming for the secant spring stiffness a yielding value, i.e., corresponding to the ratio of the yielding moment and the yielding rotation whenever the element moment was greater than the yielding moment. Equilibrium constraints were satisfied, element moments were in accordance with the assumed spring stiffnesses and the global displacements were in accordance with assumed spring stiffness values. The second phase of transforming the spring stiffness from the yielding stiffness to the secant stiffness is presently being researched. The initial design for the values presented using the secant spring stiffness formulation was obtained with the elastic stiffness version having as ultimate moment the yielding moment. The secant stiffness values were limited to a minimum value corresponding to the ratio of the ultimate moment and the ultimate rotation to prevent severe oscillations of these values. The results verified the equilibrium constraints within certain tolerance, the global displacements were those corresponding to the element spring stiffnesses, reliability constraints were satisfied and there was an improvement of the cost of the structure. All elements have yielded correspondinq to the expected results from a optimal configuration. However, there are discrepancies in the moments at the joints which shows a insufficient convergence of the equilibrium constraints. RESULTS WITH YIELDING STIFFNESS AND SECANT STIFFNESS 0 - Spring 1 5  K b - base ( in 1; h - height (in 1; As - steel area ( in2 1: * - lower bounds. 52 I Directions Headed The Augmented Lagrangian function together with the Hooke and Jeeves for the unconstrained minimization although insensitive to discontinuities of the constraints is not the best choice when compared with the Generalized Reduced Gradient method. More research has to be done on the robustness of the convergence of the nonlinear iteration process (ref. 11). A possible improvement would consist of a similar enhancement to the one applied in the Augmented Lagrangian formulation with an intermediate solution of the displacements during the optimization cycle. Another possible improvement would be the use of a cycling procedure where the spring stiffness values were kept constant during the optimization and updated at the end with consequent optimization cycle until there was stabilization of the spring stiffness values. The integrated approach proved itself adequate for the elastic stiffness if an adequate mathematical programming technique is chosen. It is logical to expect that it will probably perform well in the case of inelastic stiffness if adequately integrated with some kind of nonlinear structural analysis technique. REFERE:NCES 1. Schmit, Lucien A. Jr.; and Fox, Richard L.: An Integrated Approach to Structural Synthesis and Analysis. AIAA Journal, I Vol. 3, NO. 6, 1960, pp. 1104-1112. I 2. Haftka, Raphael T.: Simultaneous Analysis and Design. AIAA Journal, Vol. 23, No. 7, 1985, pp. 1099-1103. 3. Soeiro, Alfredo: Integrated Analysis and Optimal Design. Thesis I for the Degree of Master of Engineering, University of Florida, Gainesville, 1986. ~ 4. Charney, Finley A.: Correlation of the Analytical and Experimental Seismic Response of a 1/5th-Scale Seven-Story Reinforced Concrete Frame-Wall Structure. PhD Dissertation, University of California, Berkeley, 1986. 5. Chajes, Alexander; and Churchill, James E.: Nonlinear Frame Analysis by Finite Element Methods. ASCE Journal of Structural Engineering, Vol. 113, No. 6, 1987, pp. 1221-1235. I 6. Augusti, G.; Baratta, A.; and Casciatti, F.: Probabilistic Methods I in Structural Engineering. Chapman and Hall, New York, 1984. 7. Leporatti, Ezio: The Assessment of Structural Safety. Research Studies Press, Oreqon, 1979. 8. Thoft-Christensen, Palle; and Morotsu, Yoshisada.: Application of Structural Systems Reliability Theory. Springer-Verlag, Berlin, 1986. I 9. Watwood, Vernon: Mechanism Generation for Limit Analysis of Frames. ASCE Journal of Structural Division, Vol. 109, 1979, pp.1-15. 10. Lasdon, L. S.; Warren, A. D. Warren; Jain, A.; and Ratner, M.: Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming. ACM Transactions on Mathematical Software, Vol. 4, No.1, 1978, pp. 34-50. I 11. Soeiro, Alfredo: Integrated Optimization of Reinforced Concrete I Frames with Reliability Constraints. PhD Dissertation, University of Florida, Gainesville. (To be printed in 1989.) 54 I 

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Integrated optimization of nonlinear R/C frames with reliability constraints

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Repositório Aberto da Universidade do Porto