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## Revista ingeniería de construcción

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*On-line version* ISSN 0718-5073

### Rev. ing. constr. vol.31 no.1 Santiago Apr. 2016

#### http://dx.doi.org/10.4067/S0718-50732016000100004

**CEINCI-LAB. A free software to find the seismic capacity curve of frames with ADAS or TADAS dissipators**

**R. Aguiar ^{1}*, D. Mora **, M. Rodríguez ***

* Universidad de Fuerzas Armadas ESPE, Quito. ECUADOR

** Escuela Politécnica Nacional, Quito. ECUADOR

**ABSTRACT**

CEINCI-LAB is a computer software developed using MATLAB for static or dynamic structural analysis, in a friendly way and simultaneously serves the user to reinforce structural knowledge. In this article the most important aspects are present to find the resistant seismic capacity curve of a reinforced concrete or steel plane frame, with ADAS or TADAS energy dissipators above Chevron Braces, using the Pushover technique. To whole dissipating-brace system two models are shown, the first is by two equivalent braces and the second is considering the dissipating element like a short element. For this last case, the dissipating element is analyzed in two ways, to the first the dissipating element stiffness matrix is found and to the second some rectangular segments of constant section are considered to model the dissipating element.

Keywords: ADAS or TADAS energy dissipator, chevron braces, pushover

**1. Introduction**

One way to systematically reinforce structures is with the use of ADAS (Added damping and stiffness) or TADAS (Triangular plate added damping and stiffness) dissipating devices on steel Chevron braces, as shown in Figure 1 (Whittaker et al. 1989; Tsai et al. 1993). The ADAS are made up of steel plates in the shape of an hour-glass, with dimensions *b*_{1} in the widest part and *b*_{2} in the narrowest section; whereas the shape of the TADAS is triangular with dimension *b*_{1}, in the widest part; in the case of the two dissipating devices,* h* is the height of the dissipating device and *t* is the thickness of one of the plates that may be steel or alloys of copper, zinc and aluminum (Heresi, 2012).

The shape of the ADAS dissipating devices allows the entire element to have plastic flexural double curvature and TADAS has a simple curvature. (Aguiar et al., 2015; Chistopupoulus C. y Filiatraul A., 2006). These dissipating devices increase the structural damping and rigidity.

This article presents the most important aspects of the use of the CEINCI-LAB computer system to obtain the resistant seismic capacity curve of concrete or steel frame, placing the dissipating devices mentioned on Chevron braces (in the shape of an inverted V); applying the monotonic pushover technique was use, applying lateral pressure to each floor until the structure reaches the point of collapse.

**Figure 1.** ADAS (right) and TADAS (left) energy dissipating devices on steel Chevron braces

**2. Models of dissipating devices-braces**

The upper part of Figure 2 presents the model of the equivalent brace; it actually involves two braces that are used with the brace-dissipating device set.

**(1)**

Where _{} is the equivalent axial rigidity of one of the braces; is the axial rigidity of the steel brace; is the secant rigidity (effective) of the bilineal diagram that defines the performance of the dissipating device; **Ø** is the angle formed by the equivalent brace on the horizontal axis. (Whitaker et al., 1989).

The lower part of Figure 2 shows that the brace-dissipating device set is composed of three elements: two steel braces and a dissipating device element. To the right of the figure, the global coordinates system is indicated for each of the elements; the steel brace is an element of flat support ( Kotulka (2007), y AISC-360, 2010.)

The two forms of the element’s rigidity matrix, known as A and B , were found for the dissipating element. In the first form the rigidity matrix was found to be an element in the variable section, whose geometry is defined by the ADAS or TADAS dissipating devices (see Figure 3.) (Tena 1997).

**Figure 2.** Models developed by the set of dissipating device-brace

**Figure 3.** Model 2 A; system of global coordinates of the dissipating element

However, in model B, the keystone methods were used, as shown in Figure 4; the rigidity matrix of each keystone is found, as if it were an element of the constant section; then the rigidity matrix of the direct assembly is obtained and finally the exterior coordinates are condensed as shown in Figure 4.

Table 1 describes the programs used for the two calculation models, indicated in Figure 2; Model 2 programs for A and B are indicated.

**Figure 4.** Model 2 B; keystones considered in the ADAS and TADAS dissipating devices

**Table 1.** Programs that determine the rigidity of the dissipating device over braces, according to the two calculation models

**3. Curvature-moment and rotation-moment rotation diagrams**

On the left of Figure 5, the moment-curvature diagram defines the elements’ non-lineal performance; the curve of the first quadrant corresponds to the case in which the traction structure is in the lower part, and the curve of the third quadrant is the opposite, since the traction structure is in the upper part. The diagram shows three zones: one elastic up to point Y, with rigidity , one plastic rigidity and a residual rigidity .

**Figure 5.** Moment- curvature and moment - rotation diagrams

**(2)**

Where , is the moment and curvature at the point of fluency, based on the work of *Y*. Park (1985) which includes theoretical and experimental support based on an essay of 400 elements. , are the moment and curvature in the last point located at the base of the recommendation of ASCE 41 from 2013; α is the relationship between the post-yield rigidity versus the elastic rigidity.

The right side of Figure 5 shows the important points of the moment-rotation diagram. Point B corresponds to the yield; C at the end, and segment , are the *R* point (residual moment). ASCE 41 provides the variables *a,b,c,* which are used to find points C and E; from the yield point, for some steel sections and for reinforced concrete.

Aguiar (2015) details the way to find the residual rigidity through structural analysis, which is valid up to a rotation less than or equal to *b* ; the *b* coefficient reports ASCE 41 and is the rotation at the yield point.

The step from rotation to curvature is done with longitudinal plastic , therefore when the section enters the non-lineal range, the plastic longitude is obtained based on the moment diagram (Ger and Cheng, 2012).

The calculation was described, only for yield; however for axial force, the steps are similar, but the yield moment is reduced due to axial weight (Li ,2007; the calculation is indicated in Aguiar et al. 2015). Table 2 describes the programs that define the non-lineal behavior of the elements, the different elements in the structure, and their contribution to the rigidity matrix of the structure.

**Figure 6.** Steel sections programmed in CEINCI-LAB

**Table 2.** Programs to find the elements contribution to the rigidity matrix structure: columns, beams, dissipating devices, street braces

**4. Monotonic Pushover**

The programs that are required for the static structure analysis with CEINCI-LAB are described in Aguiar (2014) so they will not be presented here. In the case of the non-linear analysis, what changes is the rigidity of the sections, according to the branch of the moment-curvature diagram where they are located (ATC 40 ,1996; Chopra ,2014 ) ; the programs for this were described in the prior two sections.

For the Pushover analysis, first the structure under vertical loads is resolved, and the moments and forces, in the elements in local coordinates, are obtained. Then the lateral load cycles start; for this, a very small basal shear is applied, and this is distributed in each of the floors according to the first vibration and weight mode, FEMA 440,2005.

Where is the basal shear used for each load cycle. *W _{i}* is the weight of the floor. Note that we are only working with the first vibration mode θ, so the sub-index

*i*corresponds to the floor.

If the basal shear used is very small, the calculation will be more exact but it will take more time to implement. If you want the basal shear , the user should enter the Pushover_2 program and set the value of the DeltaV variable, which is the shear . On the other hand, one of the collapse criteria in the program is that if the global shift is greater than 4% of the total height of the building, it will produce a collapse. The global shift is defined as the ratio between the maximum lateral displacement versus the total height. In the Pushover_2 program, the global shift variable is derg.

**(3)**

**Table 3.** Description of the programs to use Pushover

Table 3 presents the programs used to obtain the seismic resistance capacity curve.

The opening files required to use the programs are in the matrices described in Section 7 of this paper; then, you write the name of the main program such as Structure_Geom_2D_v2 y Pushover_2.

The program Structure_Geom_2D_v2 calls the various programs used to build the structure, which are detailed in Aguiar (2014). Other missing programs that haven’t been mentioned yet are Initialization, which sets some variables and equations at zero. Aguiar (2013) includes another complement to the programs used in the CEINCI-LAB computing system.

**5. Concrete structures analyzed**

On the left side of Figure 7, a 4- floor structure is presented, with columns of 50/50 cm. and beams of 40/40; this is the same for all floors. The vertical load on each floor is 0.75 T/m. On the right side of the Figure, typical structures of a column and beam are shown. The cross-sections of the Chevron braces are 100/100/10 mm. The elastic modulus of the concrete is 1500000 T/m2.

Steel dissipating devices have been installed on the first three floors, as indicated in Figure 8. The one on the first floor has 6 layers; the ones on the second and third floor have 4 layers, both for ADAS and TADAS. The thickness of the plates is 1 cm.; this size was used to fit within the non-lineal range.

The seismic resistance curve will be calculated, first with the ADAS , and then with the TADAS dissipating devices.

**Figure 7.** Analysis of reinforced concrete structure

**Figure 8.** Geometry of the dissipating devices considered for the first three floors. The left side shows ADAS and the right side shows TADAS

For Model 1 with equivalent braces, there are 26 degrees of freedom, as indicated in Figure 9. The left side of the Figure shows the numbering of the nodes and elements. Note that first numbering is assigned to all of the concrete elements and then to the equivalent braces; this way it is easier to sum the rigidity matrix of the concrete elements and the steel elements. The 4/4 mass program is used to find the mass matrix of this model.

Figure 10 shows Model 2, where the dissipating device is another element in the structure. In this case, the main coordinates (lateral) are 7; the mass and rigidity matrices are 7/7. The mass matrix elements (1,1): (3,3) and (5,5) are zero, since there is not weight at those points. The load gravitates to the floor, and therefore the program used is masas2.

The seismic resistance capacity curves with the ADAS dissipating devices are presented in Figure 11, and the TADA dissipaters in Figure 12. In both cases, the curves are shown that were calculated with with SAP 2000 and ETABS V15. Appendix A details how the dissipating devices were modeled for each of the cases. Note that the results found with SAP 2000 and ETABS V15, shown in these figures, are similar to those found with Model 1.

The objective of this article is to show the use of the CEINCI-LAB computer system in finding the seismic capacity curve in frames with ADAS or TADAS energy dissipating devices.

The names of the programs that calculate the seismic capacity curve with the equivalent braces model are: MON4p_ADASeq y MON4p_TADASeq. When the dissipating element is used, the programs are: MON4p_ADAS y MON4p_TADAS.

**Figure 9. **Geometry of the dissipating devices considered for the three first floors; ADAS shown on the left and TADAS shown on the right

**Figure 10.** Numbering of nodes, elements and degrees of freedom - Model 2

**Figure 11.** Seismic resistance capacity curves with ADAS dissipating devices

**Figure 12. **Seismic resistance capacity curves with TADAS dissipating devices

**6. Steel structure**

A heat-laminated steel structure of 6 floors is analyzed, with energy dissipating devices placed in the main span of each of the floors, as shown in Figure 13. They use profiles type "*I*" and "*H*." The first one corresponds to the flange width in inches and the second one is the weight by unit of length (lbr/ft). The sections of the profiles are symmetrical, which is different than the reinforced concrete structure analyzed in the prior section.

The energy dissipaters for the first three floors are the same, and they have 10 plates; the rest are different. Figure 14 indicates their geometry, both for ADAS and TADAS. The numbering of the nodes and elements used in the programs are MON6p_ADAS, MON6p_TADAS, MON6p_ADASeq, MON6p_TADASeq. The first two use a dissipating element and the last two use an equivalent brace. Figure 15 is for Model 1, with an equivalent brace. Figure 16 is for Model 2, which includes the energy dissipating device as another element within the structure. These figures also present the structures’ degree of freedom.

**Figure 13.** Steel structure with energy dissipating devices

**Figure 14. **Geometry of the dissipating devices used for the steel structure

**Figure 15.** Number of nodes, elements and degrees of freedom for Model 1

**Figure 16.** Numbering of nodes, elements, and degrees of freedom for Model 2

**Figure 17.** Seismic resistance capacity curve found in a steel structure with ADAS dissipating devices

**Figure 18.** Seismic resistance capacity curve found in a steel structure with TADAS dissipating devices

With the models 2A and 2B, practically the same curves are obtained; the difference is minimal, giving the impression of being just one curve. Model 1 shows lower values but they are similar to the values found with other models.

The ETABS V15C model provides results that are very similar to those found with other models, specifically models 1, 2A and 2B, especially for the case of ADAS. The ETABS 15D model also reports similar results. In Appendix A, there is a detailed description of the models used in the ETABS and SAP 2000 programs; in general, we can state that they generate similar values, comparable with those found with the CEINCI-LAB computer system.

The programs presented in this article show tables that indicate the sequence of the elements that get into the nonlinear range (with the moment in the moment-curvature diagram gets over the elastic zone) is shown, and has subroutines that display the sections where plastic hinges have been formed.

On the other hand, with these programs, we can do a seismic analysis of reinforced concrete structures or steel with braces in the shape of an inverted "V" without energy dissipaters or simply with braces.

Likewise we can obtain the seismic capacity resistance curve in structures that had ADAS or TADAS energy dissipaters installed on the beams, without a need for Chevron bracing, such as the Barlovento building that is being built in Lima, Peru.

**7. Data entry process**

The first step is to define the structural element sections: beams, columns and energy dissipating devices, if they are used. Each section definition should indicate the numerical code for the material and the cross-section type; these codes are shown in the table below:

**Table 4.** Definition of the structural elements

**7.1 Steel beams and/or columns**

To define a steel section, fields will be used that are mentioned below, with the name from matrix Table 1; each row represents the corresponding number of the element:

**Table 5.** Matrix Table1 to create steel elements

**Figure 19.** Section types for steel elements

**7.2 Concrete beams and/or columns**

To define a section of concrete, fields will be used that are indicated below in Table 1 Matrix; each row represents the number of the corresponding element:

**Table 6.** Table1 Matrix for reinforced concrete elements

**Figure 20.** Typical sections for elements of reinforced concrete

**Figure 21.** Information about the reinforcement of beams and columns

**7.3 ADAS and TADAS dissipating devices**

**Table 7. **Table1 Matrix to create ADAS dissipating devices

**Table 8.** Table1 Matrix to create TADAS dissipating devices

**8. Comments and conclusions**

The CEINCI-LAB computer system provides an easy-to-use way to find the resistant seismic capacity curve resistance of the flat frames of reinforced concrete or steel, with ADAS or TADAS energy dissipaters placed on top of Chevron bracing. The most important aspects of the theoretical framework were briefly presented, and the programs used to resolve these topics were covered, allowing the reader to have an understanding of these topics.

In the examples reviewed, the seismic capacity curve compares the basal shear with the maximum lateral displacement; the curves obtained by the equivalent brace model for the set of brace-dissipation devices report shear forces that are smaller than the model that considers the energy dissipation device as an additional element. Two models were presented: one corresponds to an analytical solution of the rigidity matrix of the ADAS dissipating device and the quasi-analytical solution for the TADAS dissipater, and the other is an approximate solution that considers segments of constant sections in the dissipating devices. For these two models, the examples show a close approximation of seismic capacity curves.

The programs related to the topics reviewed in this article are located in the dropbox, in the following website: www.espe.edu.ec. In this manner, we are contributing to the development of Seismic Engineering.

**9. Appendix A**

For the SAP2000 A and ETABS V15 C models: the energy dissipating device was modeled as an element type frame with that shape, an equivalent thichkness to the yield moment, and the behavior of the plastic hinges as flexo-compression from automatic tables of ASCE-41 code.

For the SAP2000 B and ETABS V15 B: the equivalent brace was remodeled as a link type multilinear element controlling their behavior by modeling strength – Axial strain curve of the form shown in Figure A1.

For the ETABS V15 C model: a multilineal link type element is modeled, controlling its performance with a Strength-Strain axial curve and the Shear-Strain curve as follows:

For the ETABS V15 D model: the equivalent brace was modeled as an element type pinned frames with an area of , controlling its performance with the allocation of plastic hinges; includes axial Strength-Strain curve models as indicated in models SAP2000 B and ETABS V15 B.

**Figure A1.** Information about the beam and concrete reinforcement

**Figure A2.** Strength-Axial Strain Curve, Shear-Strain in the shear direction curve, and Moment-Rotation curve

**10. References**

** AISC-360 (2010),** Specification for Structural Steel Buildings. American Institute of Steel Construction (AISC). [ Links ]

** Aguiar R., Rodríguez M., Mora D. (2015,1),** Análisis sísmico de estructuras con disipadores ADAS o TADAS, Presentado para su publicación en la Serie Monografías en Ingeniería Sísmica. Centro Internacional de Métodos Numéricos en Ingeniería CIMNE, Universidad Politécnica de Cataluña, Barcelona, España. [ Links ]

* Aguiar R., Mora D., Morales E., (2015,2),* "Peligrosidad sísmica de Quito y el método del espectro de capacidad con CEINCI-LAB", Revista Internacional de Ingeniería de Estructuras, 20 (1), 1-39. [ Links ]

* Aguiar R. (2014),* Análisis Matricial de Estructuras con CEINCI-LAB, Universidad de las Fuerzas Armadas, Instituto Panamericano de Geografía e Historia IPGH. Cuarta edición, 676 p. [ Links ]

* Aguiar R. (2012),* Dinámica de estructuras con CEINCI-LAB, Universidad de las Fuerzas Armadas, ESPE. Segunda Edición, 416 p. [ Links ]

* ASCE 41 (2013),* Seismic rehabilitation of existing buildings, American Society of Civil Engineers ASCE/SEI 41-13. [ Links ]

* ATC-40 (1996),* "Seismic evaluation and retrofit of concrete buildings", Applied Technology Council, Redwood City, California. [ Links ]

* Chistopupoulus C. y Filiatraul A. (2006),* Principles of passive Supplemental Damping and Seismic Isolation. IUSS Press, Instituto Universitario di Studi Superiori di Pavia, Pavia, Pavia,IT. [ Links ]

* Chopra A. K. (2014),* Dynamic of Structures Theory and application to earthquake engineering. 4th edition. [ Links ]

* FEMA 440 (2005),* Improvement of Nonlinear Static Seismic Analysis Procedures, Federal Emergency Management Agency, FEMA 440. [ Links ]

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* Holzer et al. (1975),* SINDER. A Computer Code for General Analysis of Two-Dimensional Reinforced Concrete Structures. Report. AFWL-TR-74-228 Vol. 1. Air Force Weapons Laboratory, Kirt-land, AFB, New Mexico.

* J.S. Chiou1, H.H. Yang2 y C.H. Chen (2008),* Plastic hinge setting for nonlinear pushover analysis of Pile foundations, The 14th World Conference on Earthquake Engineering. [ Links ]

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** Li Guo-Qiang (2007),** Advanced Analysis and Design of Steel Frames. [ Links ]

** Mora D. y Aguiar R. (2015),** "Modelación del diagrama momento-curvatura y momento rotación en secciones de acero estructural", Revista Ciencia, 17 (1), 101-128. [ Links ]

** Park Y. (1985),** Seismic damage analysis and damage-limiting design for reinforced concrete structures, Ph.D. Thesis, Department of Civil Civil Engineering, Univesity of Illinois at Urbana-Champaign. [ Links ]

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* Tena A. (1997)*, "Mathematical modelling of the ADAS energy dissipation device", Engineering Structures, Vol. 19, No. 10, pp. 811-821. [ Links ]

* Tena A. (2000),* "Modelado analítico de edificios con disipadores de energía", Revista de Ingeniería Sísmica, SMIS, No. 62, pp. 29-62, enero-junio. [ Links ]

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Profesor Asistente, School of Engineering, California State University, East Bay. Estados Unidos.

E-mail:rraguiar@espe.edu.ec

Fecha de Recepción: 21/08/2015 Fecha de Aceptación: 28/11/2015