Application of Adaptive Analysis to Rc Frames

APPL ICAT ION OF ADAPT IVE ANALYS IS TO

RE INFORCED CONCRETE FRAMES

By C. G. Karayannisfl B. A. lzzuddin, 2 and A. S. ElnashaP

ABSTRACT; This is the second of two papers concerned with the application of adaptive techniques to the nonlinear analysis of reinforced concrete frames. The first paper presented a new nonlinear elastic formulation capable of representing an entire reinforced concrete member with one element, and the present paper discusses the use of such a formulation within the framework of adaptive inelastic analysis. This is followed by a description of an inelastic cubic formulation based on the layered approach, which complements the elastic formulation within the proposed methodology. The concept of automatic mesh refinement is then outlined, and comments are made regarding the applicability of the elastic formulation under low amplitude dynamic or cyclic loading. Finally, the accuracy and efficiency of adaptive analysis is verified through static and dynamic analyses using the nonlinear analysis program ADAPTIC. Comparisons are made where appropriate with the results of existing analysis methods for reinforced concrete frames to illustrate the advantages of the proposed methodology.

INTRODUCTION

The need for accurate and simultaneously efficient tools for the nonlinear analysis of reinforced concrete frames forms the main motivation behind this work. Success in the formulation and application of adaptive analysis techniques for steel frames (Izzuddin and Elnashai 1993) has given a significant impetus to extending such concepts to the domain of reinforced concrete frames. Essentially, adaptive analysis, as propounded herein, uses the accuracy of the layered approach for inelastic frame analysis and ad- dresses its efficiency and modeling shortcomings through the sparing use of accurate " layered" elements. In that regard, analysis is always started using only one elastic element per member, where the elastic formulation uses explicit expressions for the generalized cross-sectional response, thus leading to considerable modeling advantages and computational savings. During analysis, the more elaborate, but accurate, elements based on the layered approach are inserted in appropriate regions of the structure where inelasticity occurs. Therefore, through the insertion of such elements where and when necessary, within the structure and during analysis, respectively, adaptive analysis achieves the objective of minimizing the computational demand while retaining a level of accuracy similar to that of an initially refined mesh of elements based on the layered approach.

It is therefore evident that adaptive analysis relies on the existence of an

~Asst. Prof., University of Thrace, Xanthi 67100, Greece; formerly, Academic Visitor, Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K.

2Lect. in Engrg. Computing, Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K.

3Reader in Earthquake Engrg., Civ. Engrg. Dept., Imperial College, London SW7 2BU, U.K.

Note. Discussion open until March 1, 1995. Separate discussions should be submitted for the individual papers in this symposium. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on August 19, 1993. This paper is part of the Journal of Structural Engineering, Vol. 120, No. 10, October, 1994. 9 ISSN 0733-9445/94/0010-2935/$2.00 + $.25 per page. Paper No. 6751.

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efficient formulation that is capable of modeling the elastic behavior of beam-column members accurately using only one element. Although the response of reinforced concrete members is inherently inelastic even at low strain levels, simplifying assumptions allow such inelasticity to be neglected when the extreme fiber strains are tensile or moderately compressive. The first component of adaptive analysis is therefore a powerful elastic formulation that can model the effects of tensile cracking and nonlinear compressive response of concrete at low strain levels, the various cross-sectional shapes of reinforced concrete members, and the variation of steel reinforcement along the member length.

The second component of adaptive analysis is an accurate formulation used for modeling the loading and unloading responses of the inelastic parts of beam-column members. For reinforced concrete frames, the effects of concrete crushing and cracking, concrete confinement, and the postyield behavior of steel reinforcement must be included.

Hereafter, two formulations satisfying the mentioned requirements of adaptive analysis are described. This is followed by an exposition of the processes involved in adaptive analysis and the underlying criteria for the execution of automatic mesh refinement. Finally, an extended application example, depicting a real reinforced concrete frame subjected to static and seismic actions, is presented to demonstrate the concepts discussed, as well as the accuracy and extreme efficiency of the proposed methodology.

ELASTIC QUARTIC ELEMENT

A new formulation, presented in Izzuddin et al. (1994) is used for modeling an entire reinforced concrete member in the elastic range using only one element. The formulation is based on a quartic shape function for the transverse displacements, and models the effects of concrete tensile cracking and the nonlinear elastic compressive response. It is also capable of representing the variation of reinforcement along the member length as well as various cross-sectional configurations.

The proposed elastic formulation is derived in an Eulerian local system, where the strain states within the element are Completely defined by generalized axial strain and curvature along the element reference axis. A quartic shape function is used for the transverse displacements (Fig. 1) given by

(gt 4 tg) 3 v(x) = [2L(02 - 0~) + 16t] + [L(02 + 0,)]

- [ L (02-0 , ) + 8t] (L )2 - [L (02 + 0,)] ( L )+ t (1)

The element curvatures are obtained from this shape function, whereas the generalized axial strain is obtained through an iterative process satisfying the constant-axial-force criterion, as detailed in Izzuddin et al. (1994).

In the formulation of the generalized cross-sectional response, a linear elastic model for steel is adopted, and a parabolic cracking model for concrete is assumed as given by the following equation:


(a)

~ _ ~ , , 1 _ v(x) 2 2

I - L/2 -I- L/2 V l A ~1

X

(b)

FIG. 1. Local Freedoms and Forces of Quartic Formulation

t % = kf~ 2 s--~ + s__x_~ ; -eco-< ec- 0 where k = confinement factor;fc = compressive strength; and ~o = crushing strain of concrete.

The range of applicability of the above concrete model is for tensile strains and compressive strains up to the crushing strain ecO. The quartic element is not intended to accurately model the behavior of reinforced concrete members with strains beyond the crushing strain of concrete or the yield strain of steel. If higher strains are imposed, the quartic element would be replaced by the more accurate, albeit more computationally expensive, element based on the layered approach through a process of automatic mesh refinement, as discussed in later sections.

Finally, the quartic element can model a variety of typical reinforced concrete cross sections through their decomposition into rectangular zones and reinforcement layers, as shown in Fig. 2. The response of each rectangular zone is formulated explicitly, and assembly methods are used to model the generalized response of the cross section. The variation of reinforcement along the element length is modeled by virtue of the ability to vary the number of Gauss points used in the numerical integration of the governing equations.

INELASTIC CUBIC ELEMENT

This formulation, described in detail in Izzuddin, (1991), is intended for representing the inelastic cyclic response of short lengths of reinforced concrete members. This is performed through the use of the layered approach, where the response of element cross sections is assembled from the responses


dl d2

ke

Z'

D ~G

Y

9 a

+-----h2---+ - -b 1 -

L-

jS +

Y

ki- ke As,i

+ 9 Lo Y

k i , k e : Confinement factors

FIG. 2, Representation of Column Section by Rectangular Areas

D 4

W

Unconfined Confined Concrete Layers Concrete Layers Steel Layers \ \

-I- I- m

FIG. 3. Decomposition of Column Cross-Section into Layers

of individual layers for which realistic material stress-strain relationships are applied. The decomposition of a typical reinforced concrete column cross section into layers is shown, as an example, in Fig. 3.

As for the elastic quartic formulation, the inelastic formulation is derived in an Eulerian local system, where the strain states within the element are completely defined by generalized axial strain and curvature along the element reference axis. A cubic shape function is used for the transverse displacements, hence the name cubic formulation. Since this formulation is intended to represent short lengths of reinforced concrete members, the generalized axial strain is assumed to be constant along the element length. Although this may not satisfy the constant-axial-force criterion, especially in the inelastic range, the variation of the axial force along short lengths is so small that an involved iterative procedure to enforce a constant axial force is not justifiable on efficiency grounds.

Only two Gauss points are used for the numerical integration of the governing equations of the cubic formulation (Fig. 4), an assumption con- sistent with the modeling of short lengths of reinforced concrete members. At each Gauss point, the inelastic generalized response of the element cross section is assembled from contributions of individual layers for which inelastic cyclic material stress-strain relationships are applied. Various models for steel and concrete were implemented for use with the cubic formulation.


x ' Gauss Po in t

" - - - - ";'= '," L""' A,r"' -- L ' ~ ~ , 2" ,4~ ! 2"v~ , , I I I I

9 l I i i

' L /2 ' L /2 ' k ', i ~ " v . . . . . i I I I I

I

FIG. 4.

X"

Degrees of Freedom and Gauss Points of Cubic Formulation

T ___

FIG. 5. Bilinear Kinematic Model for Steel

D,,

E

On the steel side, the bilinear model with kinematic hardening (Fig. 5) and the more accurate multisurface model presented by Popov and Petersson (1978) Was applied. For concrete, the model of Karsan and Jirsa (1969) (Fig. 6) and a more advanced model by Madas and Elnashai (1992) accounting for passive confinement effects were included.

Through the use of the layered approach, the inelastic cubic formulation is capable of representing the spread of inelasticity within the member cross section and along the member length. However, in the context of a conventional approach for the nonlinear analysis of reinforced concrete frames, this would necessitate a very fine mesh of cubic elements all over the structure, since the locations of inelasticity are not known a priori. Hence, this would require an excessive computational effort. Adaptive analysis techniques are used in this work to address the inefficiency of the layered approach without compromising its accuracy, as discussed in the following section.

AUTOMATIC MESH REFINEMENT

In the present work, the application of adaptive techniques to the nonlinear analysis of reinforced concrete frames is realized through a process


Stress FIG. 6. Inelastic Model for Concrete (Karsan and Jirsa 1969)

of automatic mesh refinement performed in the context of inelastic analysis. As outlined earlier, this involves the use of an elastic formulation capable of accurately modeling a whole member with one element and the automatic refinement of elastic elements into inelastic elements after detection of material inelasticity. This is demonstrated in Fig. 7 for a typical multistory frame on the structural and element levels. Analysis is always started using one elastic quartic element per member, each of which is checked for inelasticity during analysis in predefined regions along its length. If such regions become inelastic, the elastic element is removed and replaced ap- propriately by new quartic elastic and cubic inelastic elements.

Fig. 8 outlines the proposed automatic mesh refinement process in the context of an incremental iterative solution procedure for determining the nonlinear structural response. The departure from the traditional nonlinear analysis approach resides in the inclusion of an inelasticity check for each of the elastic elements after global equilibrium is achieved followed by automatic remeshing for elastic elements in which material inelasticity is detected, as discussed hereafter.

Inelasticity Check After global equilibrium is achieved for an incremental load step, each

elastic quartic element is checked to establish whether the assumption of elastic material, which underlies the element formulation, is still valid. The inelasticity check is performed along the quartic element length at locations corresponding to the Gauss points of potential cubic inelastic elements used in automatic mesh refinement, as shown in Fig. 9.

The first step in the inelasticity check is to determine the bending moment and axial force for the cross section under consideration, taking into account the beam-column effect. In this respect, the choice of the Eulerian system for the formulation of the quartic element proves to be convenient, since the expression for the cross section bending moment as a function of the element end moments M1 and M2 and axial force F can be readily obtained as

c x - Fv (x )

where v(x) = transverse displacement defined in (1).

(3)


Elastic zones Inelastic zones

t 1 & t~ = time / pseudo- time

(a)

/ /

IrfilSal Mod~, Model at (tt) Model at (t2)

)

I t--e Elastic element

t ~ Pre-defmedzones

Inelastic element

t 1 & t 2 = time/pseudo- time

(b)

/nitial element Elements at (tl) Elements at (t2)

FIG. 7. Automatic Mesh Refinement on: (a) Structural Level; and (b) Level of Element (*)

The next step involves the evaluation of whether the assumptions underlying the formulation of the generalized cross-section response are violated at the cross section under consideration, namely, whether concrete has exceeded the crushing strain (-ec0) or steel has exceeded the yield strain (+__ ey). While the strain distribution across the cross section can be determined for a given moment M and axial force F, this requires an iterative procedure that poses huge computational demands, especially considering that the inelasticity check has to be performed for all elastic elements and at a number of locations along the element length. A more direct approach based on the formulation of a yield surface in the M-F domain is adopted herein, where a dosed piecewise linear curve is constructed from 16 M-F yield point pairs, as shown in Fig. 10. These points are chosen such that a


INPUT -Initial mesh of elastic elements - Pr e- defined zones for inelasticity checks -Boundary conditions - Load ing reg ime

LOADING Apply next incremental load / ti~e step ]

ITERATION literate for current mesh iln--"---~ equilibrium I

fo rga;plirm~ahl~S ehlOm~Yt, ~

FIG. 8, Automatic Mesh Refinement Procedure

H

I

Quartic element Pre-defined zones of potential cubic elements Gauss points of potential cubic elements

] JL . . ; ,

FIG. 9. Predefined Zones for Inelasticity Check

reasonably accurate and conservative representation of the yield surface is obtained, with each of the points associated with a specific linear strain distribution across the cross section for which the strains in the steel and/or concrete are at the inelastic limit, as defined by Table 1. The de- termination of the M-F pair corresponding to a given strain distribution is obtained in accordance with the generalized cross-sectional response formulation, as presented in Izzuddin et al. (1994).

The yield surface is used to check whether the combination of moment


(o) /~\ --t- t---

/ ] ~ RSeyn~oi :teriCa~t

7

.~.~__ ~-~ 10 ~9 8

(+)

F (-)

16/O// x"-.~ 2 [Q DQI ~N Asymmetric

1 5 ; ~3 Reinforcement

1 4 ~ < 4

13 \ ! " 5

* " -~1 ,?'7 M 10 N ~ 9 9 8

; (+) w 9

FIG. 10. Piecewise Linear Yield Surface

M, determined from (3), and axial force F for the current cross section results in a violation of the elasticity assumption. Inelasticity is detected if any of the following expressions, defined in terms of M, F, and the 16 yield surface point pairs, is satisfied

F ----- F1 (4a)

F -> F9 (4b)

Ft-

TABLE 1. DefinRion of Yield Points in Terms of Cross-Sectional Strain States

Compressive Yield point Tensile steel Concrete steel Curvature

(1) (2) (3) (4) (5)

1

2 and 16

3 and 15

4 and 14

5 and 13

6 and 12

7 and 11

8 and 10

9

e=O

e = 0 .5E-y

E ~ ~y

E; = E;y

E = Ey

g : Ey

E = Ey

E : Ey

E ~ - -EcO

E ~ - -Eco

E ~ - -Ec t 3

E ~ - -eco

~ - 0.8e~0

~ - 0.6e~o

~ - 0.4~o

e~0

E = Ey

E 2> - -Ey

E ::> - -Ey

E ~ - -Ey

E :> - -Ey

-> --0.8Ey

--> --0.6%

--> -- 0.4%

e_>0

Causing maximum compressive strain at centroid

Positive for point (2); Negative for point (16) Positive for point (3); Negative for point (15) Positive for point (4); Negative for point (14) Positive for point (5); Negative for point (13) Positive for point (6); Negative for point (12) Positive for point (7); Negative for point (11) Positive for point (8); Negative for point (10) 0

F j

9 Node ~ Elastic quarfic element

, Detection of inelasticity ~ _'2 Inelastic cubic element

FIG. 11. Remeshing of Quartic Element: (a) Before; and (b) After

nodes, the replacement of inelastic zones by layered cubic elements, and the use of one quartic element to model a group of adjacent elastic zones.

Creation of Nodes Since the connectivity of elements, hence their contribution to global

structural resistance and stiffness, is defined by their end nodes, the remeshing process involves the creation of new nodes that are used by the connectivity definition of each of the newly created elements. For each of these new nodes, global displacement values must be established so that the stress states within the new mesh correspond closely to those in the original element at the end of the last equilibrium step. This is of utmost importance for the convergence of the nonlinear solution procedure during the subsequent incremental load steps. The global displacements of a new node can be readily obtained in terms of the nodal displacements of the original quartic element and the position of the new node along the element length. This should take into account the quartic shape function for the local transverse displacements and the distribution of the generalized axial strains along the original element length.

Insertion of Cubic Elements The creation of a new cubic element involves the specification of its end

nodes and its geometric and material properties, with the strain and stress states of the element defined implicitly by these properties and the global nodal displacements discussed earlier. The geometric properties of the element include its length and direction cosines as well as its cross-sectional characteristics defined by the areas and locations of the layers used in monitoring material stresses and strains. The material properties include the material model and the model parameters for each of the monitoring layers, with appropriate distinction made between layers corresponding to steel reinforcement and those representing confined and unconfined concrete.

Some of the difficulties in creating a new cubic element are related to


the fact that the cross-sectional properties of the new element depend on its position within the original quartic element domain, since the quartic formulation allows a variation in the reinforcement scheme along the length. This is further complicated by the difference in the geometric cross-sectional entities that the original quartic element and the new cubic element use. As discussed earlier, the quartic element uses rectangular areas for discre- tising the cross section, whereas the cubic element uses a significant number of layers for the same purpose. Such complications can be overcome with an adequate cross-reference between the original quartic element and arrays storing the properties of potential cubic elements.

qew Quartic Elements A new elastic quartic element is created when only some zones within

the original element become inelastic, with one or more zones remaining in the elastic range. In such a case, new inelastic cubic elements are used for the inelastic ones according to the previous section, and one or more new elastic quartic elements are created to model the elastic zones. The number of new quartic elements is determined by the layout of the elastic zones, where only one quartic element is used to model a set of adjacent elastic regions.

As for the cubic elements, the creation of a new quartic element involves the specification of its end nodes and its geometric and material properties. The geometric properties include the element length and direction cosines as well as the cross-sectional characteristics, which can be obtained from the nodal positions and the cross-sectional configuration of the original element, respectively. The material properties include the parameters of the material models used to represent the steel reinforcement as well as the confined and unconfined concrete, which can be determined from the corresponding properties of the original element.

An additional requirement for creating a new quartic element is the specification of the zones where inelasticity is to be checked during the later stages of the analysis. These zones can be obtained from the predefined zones of the original element and the location of the new element within the original element domain. Although the new element uses the same cross- sectional properties of the original element, a well-organized data structure is required to facilitate establishing the variation of the cross section along the new element length from that of the original element. This is important in view of the fact that the quartic formulation allows a varying reinforcement scheme along its length.

Advantages of Automatic Mesh Refinement The application of automatic mesh refinement in the context of adaptive

analysis, as discussed previously, leads to significant modeling and computational benefits. On the modeling side, the task of the structural analyst is significantly reduced, since the structural discretisation can be performed using only one element per member. With the conventional finite-element layered approach, the analyst has to represent each member with a number of inelastic elements, usually more than five, since the locations of inelasticity are not known before analysis.

Moreover, computing demand of the analysis is significantly reduced, since the expensive inelastic layered elements are introduced when and where necessary, during analysis and within the structure, respectively. An extreme scenario is for a structure that does not exhibit any inelasticity. In


such a case, apart from the inelasticity check that is computationaUy inex- pensive, adaptive analysis requires the same computational effort as that of elastic analysis. This represents huge computational savings in comparison with the conventional approach, which would use a large number of inelastic layered elements unnecessarily. The other extreme is for a structure that becomes completely inelastic from the first load step. In such a case, adaptive analysis requires the same computational effort as the conventional approach. Most analyses lie between these two extremes, often closer to the former scenario, which confirms that adaptive analysis is computationally superior to the conventional approach and cannot be less efficient.

APPLICATION

The previously discussed adaptive analysis methodology was implemented in the nonlinear analysis program ADAPTIC (Izzuddin and Elnashai 1989) running on Silicon Graphics workstations with 16 mflops and 60 specmarks. A two-story reinforced concrete frame with a ground-floor mezzanine, shown in Fig. 12, is chosen to demonstrate the accuracy and efficiency of the proposed methodology through a series of static and dynamic analyses. The frame column dimensions are the same from ground to upper story with

7- 3.00 m

2.50 m

3.00 m

,[ , / . j . . . ,

i i i i

3~18+3518

( -n03 3~18+3~18

(a) r i i i i

' 4401'8+4918'

4918+4~18

4918+4918

4918+4918

5018+5r

70018+70018 / / J / J / / / / J J / / / . / J

6.00 m

s0018+50018

4~18+4918

4918+4918

70018+70018 / J / / / / / / / / / / I / J . ,

6.00 m

4~16+40016

0-07 4916+4916

40016+4916

5,16+5,16 / l ' /

(~

(25/65)i /5,18 20018 5,18 \ / /\J ~1,18 5,18

FIG. 12. Geometric Configuration of Reinforced Concrete Frame: (a) Columns; and (b) Beams

2947

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kP3

kP2

kP~

7,

m4 m3 m4

m2 I

m 2

~/ / / / / / / / / / .

ml

m2

" / / / / / / / / / / ,

Ground excitation

m2

7

2 g = 9.81 m/see

P1 = 79.1 kN P2 = 226.5 kN P3 = 194.4 kN

m 1 = 13.4 tons m 2 = 11.9 tons m 3 = 9.3 tons m 4 = 5.7 tons

FIG. 13. Applied Loading on Reinforced Concrete Frame

the middle columns having a larger section, and all beams have the same dimensions and reinforcement layout. The frame constituent materials include concrete with crushing strength fc = 20 N/ram 2 and reinforcement steel with yield stress offy = 400 N/mmZ and strain hardening of ~ = 1%.

The reinforced concrete frame is studied under the action of static transverse loading as well as earthquake ground motion (see Fig. 13), with consideration given to the elastic and inelastic response, as discussed in the following sections.

Static Elastic Case The elastic response of the R/C flame to the transverse loading, assuming

no vertical loads, is obtained using four different approaches:

1. Conventional linear elastic approach with idealized cross sections, where concrete cracking is neglected, and the contribution of steel reinforcement to the cross-section stiffness is accounted for.

2. Conventional linear elastic approach differing from the previous approach in that the contribution of steel reinforcement to the stiffness is ignored, hence accounting only for the gross concrete configuration within all cross sections.

3. Conventional linear elastic approach assuming fully cracked cross sections all over the structure.

4. Nonlinear elastic approach using the elastic quartic formulation proposed in Izzuddin et al. (1994).

With all the approaches, the flame is modeled using one element per member, and the transverse load is increased proportionally to the load factor k. Comparisons in Fig. 14 demonstrate large differences between the results of the various conventional linear elastic approaches, with the fully cracked approach providing a better estimate of the response than the other two approaches. The difference of 28% between the conventional fully cracked approach and the proposed nonlinear approach using the quartic formulation is mainly attributed to the fact that the columns on the right side of the flame are under the action of compressive forces; hence, the assumption of fully cracked cross sections cannot be justified.

The frame is reanalyzed under the action of the transverse load but with


. . . . . . . . . Ide2disedsections C C O Fully-cracked

. . . . . Gross concrete Quartic formulation

1

0.9

0.8-

i 0.7- 0.6-

0.5-

~ 0.4-

"~ 0.3-

0.2-

0.I

FIG. 14.

I I I I I I I I I

0 10 20 3o 4o 5o 60 70 so 9o loo

Top Floor Displacement (mm)

Static Elastic Response of Reinforced Concrete Frame

t=

S

o= o

FIG. 15.

1-

0.9-

0.8-

0.7-

0.6-

0.5-

0.4-

0.3-

0.2-

0.1-

0 0

.. . . . . . . . With ver~cal

I I I I I I I I

10 20 30 40 50 60 70 80

Top Floor Displacement (ram)

Effect of Vertical Loading on Static Elastic Response


the vertical preload. The three conventional linear elastic approaches yield the same result whether or not the vertical load is included, whereas the

nonlinear approach accounts for the stiffening effect of the vertical ads due to crack closures in the columns, where a 9% difference is observed

in the results shown in Fig. 15.

Static inelastic Case Well before the attainment of the full transverse load (k = 1), the rein-

forced concrete frame undergoes inelastic deformation mainly due to the yielding of steel reinforcement. This is investigated using two approaches. The first is the conventional approach using an initially refined mesh of five inelastic cubic elements per member (i.e., 65 elements in total). The second approach is based on the automatic mesh refinement technique discussed earlier in the present paper, where analysis is started using one elastic quartic element per member (i.e., 13 elements in total), with each elastic element having five predefined zones for inelasticity checks.

The results for a proportional application of the transverse load without vertical loads are shown in Fig. 16, where the predictions using an initially refined and the automatically refined meshes are identical. Consideration of the central processing unit (CPU) time requirements of both approaches in Fig. 17 demonstrates the considerable computational savings achieved with the automatic mesh refinement process, where a CPU time reduction of 45% is attained at no loss in accuracy. The jump in the rate of CPU time demand in the displacement range (50-150 mm) for the automatic mesh refinement procedure is mainly due to the introduction of inelastic cubic elements at this stage. The lower rate of CPU time demand after a displacement of 150 mm is attributed to the need for only 13 inelastic cubic elements, instead of 65 elements with the initially refined mesh; as shown

t _

Y:

0

0.9

0.8

0.7

0,6

0.5

0.4

0.3

0.2

0.1

0

~f" w v v

ally-refmed

/ [ I

0 25 50 75 100" 125 150 175 200 225 250


FIG. 16. Static Inelastic Response of Reinforced Concrete Frame


25-

20

5-

15 ,r

.E_ [ . , .

~a

. . . . . . . . , InitiaUy.ref'med ..**~

,,.~

e'* ....." r - ......

~

.o**

I I I | " I I " | I " " ~ I

25 50 75 100 125 150 175 200 225 250


FIG. !7, CPU Time Demand for Static Inelastic Case

, Elastic qua~ elements

Inelastic cubic elements

i'i .......... i---il .... ...... ' i

FIG. 18. Deflected Shape with Automatic Mesh Refinement for Static Inelastic Case


t _

,d

0.9

0.8-

0.7-

0.6-

0.5

0.4

0.3

0.2

0.1

0

om - I 9 9 9 umi l IQ Ig l JG 9 9 I I 9

I I I I I I I I I

0 25 50 75 100 125 150 175 200 225 250

Top Floor Displacement (ram)

FIG. 19. Effect of Vertical Loading on Static Inelastic Response

in Fig. 18. For cases where inelasticity is less spread out in proportion to the overall structure and occurs at a much later stage during analysis, much greater computational savings can be achieved through the application of automatic mesh refinement.

The frame is reanalyzed using the automatic mesh refinement process with the vertical preload. The results depicted in Fig. 19 show a 10% increase in the frame resistance at a displacement of 100 ram, with such a difference reducing at larger displacements. This is attributed to the fact that, while the vertical loads enhance the moment capacity of column sections, such loads also delay the onset of yield in the column tensile reinforcement.

Seismic Elastic Case The reinforced concrete frame is analyzed under the combined actions

of vertical loading and earthquake excitation, where the acceleration signal of Fig. 20, scaled by a factor of 0.5, is applied in the horizontal direction at the ground level. The elastic response of the frame is obtained using the two conventional approaches (gross concrete and fully cracked cross sections) discussed previously as well as the proposed nonlinear approach based on the quartic formulation.

Comparisons in Fig. 21 show significant differences in the prediction of the proposed approach and the conventional approach based on gross concrete cross sections and demonstrate the shortening in the response period with the latter approach. This is justified by the overstiff response prediction of such an approach, as observed earlier in the static elastic case. A similar comparison is undertaken in Fig. 22, where it is shown that not only is the response period elongated with the conventional approach based on fully cracked cross sections, but also the frame drift increases by 65%. This is attributed to the inability of the fully cracked conventional approach to


0.3

0.2

~ 0.1 -

o-

"~ -0.1-

4).2

/

-0 .~ I I I I I I

0 2 4 6 8 10 12

T ime (sec)

FIG. 20. Earthquake Acceleration Signal

account for the existence of compressive forces in columns on alternating sides of the frame during excitation.

Seismic Inelastic Case The inelastic response of the frame to the acceleration signal of Fig. 20,

scaled by a factor of 2, is obtained using the initially refined mesh and the automatic mesh refinement approaches, as for the static inelastic case discussed earlier. The comparison in Fig. 23 demonstrates excellent agreement between the two approaches, thus justifying the accuracy of the automatic mesh refinement approach. The slight disagreement at low response am- plitudes is attributed to the inability of the elastic quartic formulation to model the hysteretic energy dissipation exhibited by concrete. This can be remedied, if deemed necessary, by adopting the suggestions made in the previous section with regard to the choice of the yield surface.

Consideration of the CPU time requirements of the two approaches in Fig. 24 shows the great computational savings achieved by the automatic mesh refinement process, where a 75% reduction in CPU time is attained. This is because only nine inelastic elements are introduced with the automatic mesh refinement approach, as depicted in Fig. 25, and after 4 s of excitation time. This contrasts with 65 inelastic elements used from the start of analysis with the initially refined mesh approach.


d ' 'i . . . . Quarfic formulation . . . . . . . . . Gross concrete 25

20

15-

10-

5-

o

~ -10

-15

.20

-25 I I I I I I 2 4 6 8 ,0 12

Time (see)

FIG, 21. Seismic Elastic Response of Reinforced Concrete Frame

25

20

15

5

-5

o ~ -10

-15

-20

-25

FIG. 22.

Quallic formulation . . . . . . . . . Fully-cracked

=

F

q ! I 0 :2 4 6

Time (sec)

' : 9 i:.

. i~ . :

i , 2 8 10 1

Seismic Elastic Response of Reinforced Concrete Frame


60

4O

-20-

~, -40-

FIG. 23.

Automatically-ref'med . . . . . . . . . Initially-refined

-80 I I I I | I 0 2 4 6 8 10 12

Time (see)

Seismic Inelastic Response ot Reinforced Concrete Frame

1000

9OO

8OO

700

50O

400

L) 300

200,

100-

0 0

FIG. 24.

Automatically-refined

I m l l a m u = " " .

I I I I / I

2 4 6 8 10 12

T ime (sec)

CPU Time Demand for Seismic Inelastic Case


9 - O- - 9 Initial shape

Elastic quartic elements

Inelastic cubic elements

i - i - w v

J= A ~1=

A . . . . . . . . . . . . . O

FIG. 25. Mesh Configuration with Automatic Mesh Refinement for Seismic In- elastic Case

CONCLUSIONS

Thepresent paper addressed the application of adaptive analysis to the inelastic analysis of reinforced concrete frames. The advantages of the proposed methodology were discussed, and the significant modeling and computational savings were highlighted.

An accurate elastic formulation for reinforced concrete members, representing the first component of adaptive analysis, was described, and its range of applicability in nonlinear analysis was discussed. This was followed by a description of an inelastic formulation based on the layered approach that is capable of modeling accurately the response of reinforced concrete members, including the effects of concrete cracking and crushing as well as reinforcement yielding.

The concept of automatic mesh refinement was also appraised, and a yield surface was used to specify the range of application of the elastic formulation. The computational and modeling advantages of adaptive analysis were highlighted, particularly in view of the ability of the proposed methodology to retain the accuracy of an initially refined mesh. The slight inaccuracies associated with the inability of the elastic formulation to model hysteretic energy dissipation were pointed out, and a remedy based on a modified yield surface was suggested, even though it was considered un- necessary.

Finally, the paper presented a number of examples of a reinforced concrete frame under static and seismic loading conditions. The results obtained using the nonlinear analysis program ADAPTIC demonstrated the accuracy of the proposed formulations, which was maintained when using automatic mesh refinement. It was also shown that adaptive analysis, while providing considerable modeling advantages, leads to significant computational say-


ings, with 75% reduction in CPU time achieved for the example under consideration.

APPENDIX I. REFERENCES

lzzuddin, B. A. (1991). "Nonlinear dynamic analysis of flamed structures," PhD thesis Imperial College, London, England.

Izzuddin, B. A., and Elnashai, A. S. (1989), "ADAPTIC: A program for the adaptive dynamic analysis of space frames." Rep. No. ESEE-89/7, Imperial College, Lon- don, England.

Izzuddin, B. A., and Elnashai, A. S. (1993). "Adaptive space frame analysis: part II, a distributed plasticity approach." Struct. and Build. J., 99(3), 317-326.

Izzuddin, B. A., Karayannis, C. G., and Elnashai, A. S. (1994). "Advanced non- finear formulation for reinforced concrete beam-columns." J. Struct. Engrg., ASCE, 120(10), 2913-2934.

Karsan, I. D., and Jirsa, J. O. (1969). "Behavior of concrete under compressive loadings." J. Struct. Div., ASCE, 95(12), 2543-2563.

Madas, P., and Elnashai, A. S. (1992). "A new passive confinement model for the analysis of concrete structures subjected to cyclic and transient dynamic loading." J. Earthquake Engrg. and Struct. Dynamics, 21,409-431.

Popov, E. P., and Petersson, H. (1978). "Cyclic metal plasticity: experiments and theory." J. Engrg. Mech. Div., ASCE, 104(6), 1371-1388.

APPENDIX II. NOTATION

The following symbols are used in this paper:

F; = axial force for yield surface point i; fc = local forces (M1, M2, F, T) T of quartic element; k = concrete confinement factor; L = element length;

Mi = Bending moment for yield surface point i; uc = local displacements (01, 02, A, t) r of quartic element;

v(x) = transverse displacements of quartic element; ec = direct strain in concrete;

e~0 = direct crushing strain of concrete; and gc = direct stress in concrete.


Documents

Application of Adaptive Analysis to Rc Frames