- Email: [email protected]
C frames
SALAH
E. EL-~~~~~Y~
and WAI-FM C&N$
~~~~~~fy of Engineering, Et-Mansoura
University, ENansoura, Egypt $%hoot d CivilEngineering, Fwdw University, West tafayette, IN 47907, U.S.A.
infemaf parameter axial ~~~~~tyof a member ffexural rigidity of a member
sti&ess matrix memberlength bendingmoment axial farce ~~id~t~--~~a~t maxims
~~~dj~-ta~~~t Taurus d~sp~a~e~t in X’-dir~%i~t~ joint dis~~~~rneilt in Y-direction joint
joint coordinatein X-direction joint coerdinate in Y-direction ordinate direction coordinatedirection joint rotation
~~~SC~j~F~
to ve~i~al loads (Fig. I@, The second example is a brittle Frame II, where flexural hinges are expected to Form in the mfumns. Frame II was Arst subjected to vertieal loads under service ~ondi~~o~s~ then, the vertical loads wore held constant while et lateral load was applied up to failure (Fig. 2a). The two frames were tested previously by Ernst et al. [4]. fn the foIjowing sections, the models used in the present analysis for ~~~~~~~ ~0~~~~~~~~ material ~on~inea~t~ and joint ~~xibility are first discussed. Then, fhe analytica results are compared with the ex~~mental data cited, FinalIy, the effects of parameters of n~~~ineariti~ and joint fle~b~~ityon the behavior of frames are examined,
Large dis~la~ments may have either a so~ening effect or a sti~~~i~g e&et on the behavior and strength of structures or members. Also, the action iand i=f iteration number member of the applied loads might be affected by the struc;: initial ture’s deformations. Hence, geometric nonlinearity fiexwat becomes a si~i~c~t factctr whets the level of loading taxgent is su~~ient to cause large disp~a~ments. This is u&mate especialfy true in the ease of slender members subjected to axial compression* or memhrs subjected to refatively large lateral loads. In this study, two different aspects of geometric oo~~n~~ty are conThe! b&avior of reinforced concrete frames is essen- sidered. These are the stability analysis considering tially controlled by three key parameters: geometric the P-4 effect of the member and thB overall strucnonlinearity, material nonhnearity and joint flexi* tural stability considering the P-A effect. The axial b&y, In this paper the effects of these ~~~e~rs on ~formation of a member due to geometric deformathe bebatior of frames is inv~~~t~. The models tions was found to be ~s~~~~~~~ in rein&e& reported reeent& in f&-3]are used to simulate these concrete rnern~~ because of their deep s&ions, and parameters of nonlin~~ties, and the Mness method therefore it is not discussed in this study. is applied to obtain numerical solutions. 2.1.1, ~~~~~~ ~~u~~~~~~ (P-3 @&et)+The member fn this paper we shall discuss two ~nrn~~l local stiffness matrix, K,, is mod&d by the stSness examples. One is a ductile Frame I, where flexural functions, &, fb,?,Qi3and 4, as given in Table I, to hinges are expected to form in the beams, subjected account for the P-6 effect. The devetopment of these f
eurrent
a
axial
1204
SALAH E.
EL-MEIWALLY and WA{-FAHCHEN
(a)
x
Experimental
0
Geometry + material
-
Geometry + joint
0
Materiot+
0.6
Gemetry
+ joint
Laterw
defkctwm,
A,
+ moterio\ $ join?
24
12
Vertitut
Geometry+rmteriat
-
pint
dr Qmnetry
0
0
AC
defhxtion,
(in
)
iin)
Fig. I. (a) Ductile frame under gravity loads. (b) Ductile frame under gravity and lateral loads.
functions and their modification for the member’s stiffness matrix are given in detail in [S, 61. EA i 0
0
12EI FQIl
0
6EI
F42 4EI
0
F43
--
EA
0
L
0
12EZ
0
--
0
--
L3
6EI 41
p2
6EI L2
42
q&4
Km =
--
EA L 0
0
0
12EZ --+I
0
EA
0
T0
12EI -+’
0
--
0 6EI -- L2 42
6EI L2
df2
L
where EA is the member axial rigidity, EZ is the flexural rigidity, and L is the length of the member. 2.1.2. Structural stability (P-A e&t). The joint global coordinates are modified after each iteration to account for the current joint displaced position. This reflects the load-displacement interaction of the structure. Since the member equilibrium matrix and hence the member stiffness matrix are functions of the current nodal displacements, the member stiffness matrix must be developed with respect to the current displaced coordinates of joints. If we denote the joint undeformed coordinates in the X and Y directions by x and y, and the joint displacements at iteration i by ui and u, in the X and Y dir~tions respectively, then the joint coordinates
for the i + 1 iteration &+I
are x,, , and yi+ 1
=x+u,
Yi+r ‘Y
+ui.
(2) (3)
The transformation matrix used to transfer the member stiffness matrix from the member local coordinates into the structural global coordinates is obtained from the deformed geometry of the structure. The joint deformed coordinates are therefore used in calculating this transformation matrix. 2.2. Material nonlinearity To make a direct computation of both flexural rigidity and axial rigidity due to any load combi-
span
nation, a n&hod ~r~se~t~d in f2], and described briefly here, is used. This method is found to be more economical in camputational time and much easier to use in nonlinear analysis than other methods available [7-Q], For the stress--strain ~Iationship of concrete in compression, the curve by Sohman and Yu fItI], shown in Fig. 3, is adopted in this analysis. For steel, an elastic-~rfectIy plast& stress-strain ~~at~o~sh~p is assumed. 2.2.f. Bctsic ~~~~~ti~~~. (if A linear strain distribution across the depth of a section is adopted. (2) The axial force is applied at the section centroid when calculating the applied moment. (3) Shear deformations are neglected. (4) Tensile strength of concrete can be accounted for in caiculating tbe section strength; however, in the present analysis concrete strength in tension was negkcted.
12.2. Ax&al ~~~e~~~~~ r~g~~~~~-~e~~~r ~~i~~ The secant axial rigidity, S,, and secant fIexura1 rigidity, S,, are defined as
where P is the axial force acting at the section centroid, M is the bending moment appiied to the section, L, is the axial strain of the section due to the axial load P and measured at the section centroid, and 4 is the section curvature. To calculate S, and S, due to the load vector (P,, Mb >‘* the corresponding section deformation vector, fq, &)‘, rmst be ~~u~at~ first. If the se&ion defo~ation vector due to the initial bd
Table 1. Stiffness functions for a beam subjected to axial foree Function 4%
Compression
Condition of axial load Tension
(k.L jJ sin KL
(k.&~sinh kt
WC
1%
Z&TO
t
SALAHE. EL-METWALLYand WAI-FAH CHEN
1206
.& = Cross-sectionat orea of tmnsverse reinforcement kIC = Areo of concrete under compression & = Area of bound concrete under compression # = b, or 0.7d,, whichever is greater 6, = breadth of bound concrete cross-section ~7, = Effective depth of bound concrete cross-section 5 = Longitudinal soocina of transverse reinforcement: _ S0 2 Lon itudinal spacmg at which trowverse .3 rem orcement is nut effecti confining the concrete (10 in. 7 in ucyr = Standard 6 x 12 in. cylinder strength
Strain
9
=(l.4&
5
f; = 0.9+,,,,
_o,45)
&(.%-5-J A,t S+ O.ccx?e~s’
(i+o.o59)
=ete= 2 f;tEc
0.0025 (ii-q)
Qcr’
dcf= 0.0045
(l+O.859)
Fig. 3. Adopted stress-strain curve for concrete confined by rectangular hoops [IO]. vector fPo, M,)’ is known as {co, Cp,}‘,the following relations can be written
From eqns (6) and (7) it is obvious that the incremental deformation vector {AC,AQt)’ corresponds to the incremental load vector (AP, AM}‘. The relationship between an infinitesimal load vector and an infinitesimal deformation vector can be formulated as follows
(8)
Solving eqn (11) for the deformation vector, we have
Equation (12) represents the solution algorithms for the incremental deformation vector. The elements of the section stiffness matrix can be evaluated numerically by using, for example, the finite difference method [2]. This can be achieved by assuming an incremental change in the strain and curvature at the current state of strain and calculating the corresponding incremental axial load and bending moment. Then, the elements of the stiffness matrix can be calculated. 2.2.3. Axial andflexural rigidity-tangent modulus. The tangent axial rigidity, T,, and the tangent flexural rigidity, T,, are detmed as (13)
(9) Or, in matrix form
[~~I=
~~
= KI
~~
[
dc d4
1
[~~1
(10)
T, and T, are determined at the current state of strain, and can be calculated by using the central difference method. The central difference reflects the material properties at the current state of strain better than either the forward difference or the backward difference; therefore, it is preferred.
2.3. Joint flexibility
where [K,] is the sectional tangent stiffness matrix. Replacing the infinitesimal changes in the load and deformation vectors by (finite) incremental changes, eqn (IO) has the form
(11)
In framed structures, inelastic deformations are essentially concentrated at several critical regions. These include the member ends (joints) and other regions where high bending moments may take place. An accurate prediction of the nonlinear behavior of these framed structures depends to a large extent on the reliable representation of these concentrated
1207
Nonlinear behavior of R/C frames
inelastic regions. Hence, an accurate modeling of the beam-column connections represents an important part in the study of the behavior of reinforced concrete frames. In this study, the joint model proposed recently in [3], and described briefly here, is used. This model is simple to use and easy to implement in the computer analysis of reinforced concrete frames. The implementation of the joint flexibility in the member stiffness matrix is discussed in (9, 111. 2.3.1. The joint model. In this study the joint is modeled mechanically as a concentrated rotational spring. The development of the spring stiffness is based on the thermodynamics of irreversible processes. The spring stiffness, K, is a function of the following three parameters. 1. Initial rotational stiffness of the connection (K,,). This can be computed for the connection with a linear elastic behavior of the materials. 2. Ultimate moment capacity of the connection (M,). This can be computed for the connection using the limit analysis of perfect plasticity. 3. Internal variable (a), which introduces the dissipated energy to the model formulation.
Thermodynamics is used to define these variables and to develop the general form of the constitutive equation of the spring. For the development of the proposed joint model, the following basic assumptions were made. 1. Loading and unloading curves are smooth functions; hence, the free energy and dissipated energy are both smooth functions. 2. The free energy and dissipated energy are both unbounded at failure; i.e., for the structure to collapse these two quantities are not finite. The moment-rotation relation of the joint was developed in [3], and is given by the following equation
(15) where 0 is the joint rotation, M is the moment, and 1 is the load factor, I = M/M,. The parameter ‘a’ takes different values with the maximum at zero moment (a,) and decreases gradually with increasing load factor. However, for simplicity ‘a’ can be assumed constant, which would still give satisfactory results. This will require the selection of one loading point to evaluate ‘a’. One particular point is most suitable for the evaluation of ‘u’ because it is well defined compared with other points. This is the loading point, when tension reinforcement in the member where flexural hinge is expected to form, starts to yield. By assuming ‘a’ as constant,
the energy dissipation takes the following simplified form [3] @ =$
E [na(nu +2)(ln1+ II n-l
1)+2] ~na+2 x ~
(na + 2)2’
(16)
The dissipated energy results from the inelastic behavior of the joint due to the following three factors: (1) the deterioration of the bond strength between the reinforcing steel bars and the concrete; (2) the cracks in the concrete; and (3) the inelastic behavior of the materials composing the joint, concrete and steel. In [3], the calculation of the dissipated energy due to each of these three factors is given in detail. 3. ANALYTICAL RESULTS AND EXPERIMENTAL DATA
Both Frames I and II are analysed for five different sets of analysis parameters as shown in Table 2. The results from the experiments and the present analysis are shown in Fig. la for Frame I, and Fig. 2a for Frame II, respectively. Also, Frame I is analysed for lateral and service vertical load conditions. The gravity loads are applied first; then, the lateral loads are applied gradually up to failure (Fig. 1b). Frame II is analysed under vertical loads that are applied gradually up to failure (Fig. 2b). For the ductile Frame I, which is designated as b40 in the paper by Ernst er al. [4], the first analysis, with rigid joint, and the fifth analysis, with flexible joint, match very well with the experimental data. The analysis was performed up to the maximum load only without attempting to predict the softening branch of the load-displacement curve. Therefore, the ultimate deflection is not shown in the results presented here. The second analysis, with the linear elastic material, gives a very good envelope for the measured load-deflection curve. The vertical load carrying capacity of the frame is underestimated by the third analysis, and even more so by the fourth analysis. For the brittle Frame II, which is designated as 2d9h in Ernst’s paper, both the first and third analysis
match very well with the experimental
results. The
Table 2. Set of analysis parameters Analysis
Geometry
Material
Joint
1
N
2 3 4
N
N E N
R F F F F
w
L
N N
N N
t Clear span length is used. R, rigid; F, flexible; L, linear; N, nonlinear; E, linear elastic cracked section.
1208
SALAH
E. EL-METWALLY and WAI-FAHCHEN
strength of the frame is, however, overestimated by the second analysis and underestimated by the fourth analysis. The fifth analysis slightly overestimates the strength of the frame. 4. 4.1.
DISCUSSION OF THE RESULTS FROM THE FRAMES ANALYSED
Material nonlinearity
Nonlinear constitutive relationships of the crosssection influence significantly the overall behavior of the frame. In Fig. la, the first analysis for the ductile frame, considering geometric and material nonlinearities, matches well with the experimental results. The fifth analysis, considering both material and geometric nonlinearity with flexible joints and using the clear span length of the members, also gives very good results. However, the second analysis with linear elastic constitutive relations and considering the geometric nonlinearity and joint flexibility, gives a good bilinear representation of the load-deflection curve. This can be explained only by the influence of the material properties on the values of the parameters of the joint model. From the analysis of Frame I under the combined vertical and lateral loads (Fig. lb), and by assuming that either the first or the fourth analysis will give the correct load-deflection relation, and finally by comparing either the first or the fourth analysis with the second analysis, we can observe the effect of material nonlinearity on the behavior of the frame. The strength of the brittle Frame II is overestimated by the second analysis because the material nonlinearity is ignored (Fig. 2a). Also, by comparing the second analysis of this frame under vertical loads (Fig. 2b) with either the first or the second analysis, it is obvious that the use of linear constitutive relations results in a much larger ultimate load of the frame than the correct one. This dictates the necessity of employing nonlinear constitutive relations in the analysis of stiff frames. 4.2. Geometric nonlinearit? Geometric nonlinearity coupled with either the material nonlinearity or the joint flexibility is adequate to predict the response of a ductile frame under vertical loads, Under the combined vertical and lateral loads, the load-deflection curve cannot be predicted accurately without considering the geometric nonlinearity. This is due to the significance of the P-A effect under this load condition. It is necessary to consider the geometric nonlinearity with the material nonlinearity in order to obtain a more accurate result for the response of a brittle frame under lateral loads. Under vertical loads only, by comparing the third analysis with either the first or the fourth analysis, the effect of geometric nonlinearity is noticeable.
4.3. Joint JIexibility The fifth analysis, which accounts for the two parameters of the nonlinearities and the joint flexibility, and performed with the clear span length of the members, gives a very good prediction of the load-displacement curve of the ductile frame. For the brittle frame this analysis slightly overestimates the strength of the frame. This shows the necessity of including the joint flexibility in the analysis if the clear span length of the members is used in the analysis instead of the center to center dimensions. The response of a ductile frame under vertical loads only is predicted reasonably well by the second analysis which accounts for both joint flexibility and geometric nonlinearity. In this case, the flexural hinge is expected to form in the beam which has a smaller flexural strength than the column. This also implies the presence of moments at the joints which are higher than those under the concentrated loads. The beam has a flexural strength near the ends that is greater than in the middle third where the loads are applied. The joints and the middle third of the beam reach their full strength simultaneously or nearly so. When the joints are near their full capacity, the tangent stiffness of the joint drops to almost zero. If the middle third of the beam cannot carry any additional moments, the joints will soon have to reach their full strength, causing a faihtre mechanism to form. In the second analysis, the ductile frame under the combined vertical and lateral loads behaves very similarly to the case of vertical loads only with some difference. The mechanism forms when the left joint (where higher moment is expected to occur first) reaches most of its strength and the P-A effect imposes additional moments on the frame that contribute to the failure. This indicates that a good upper value on the strength of the frame can be obtained by the second analysis. The response of the brittle frame under lateral loads when the second analysis is used can be explained in a similar manner to that for the ductile frame under lateral loads. In the case of a brittle frame, the beam is stronger than the column. With the assumption of flexible joints in the second analysis for vertical loads only, the beam is expected to behave in an intermediate stage between simply supported and fixed-ended beams. In other words, the moments at the middle third of the beam are always greater than the moments at the ends of the beam. Therefore, the joints do not reach their full strength until the middle of the beam reaches very high moments due to the linear elastic assumption for the material. The frame does not fail until the joints reach or near their full capacity. This in turn reduces the restraint of the column drastically causing the column to reach its buckling strength and finally causing the frame to collapse.
1209
Nonlinear behavior of R/C frames 5. SUMMARY AND CONCLUSIONS
The behavior of both ductile and brittle reinforced concrete frames has been examined under both a vertical, and a combined vertical and lateral load condition. The geometric and material nonlinearities, and the joint flexibility, are considered. Five different combinations of these parameters are analysed. The analytical results are compared with experimental data. Comprehensive state-of-the-art summaries on the recent and past progress in research on reinforced concrete plane frames are given elsewhere (12, 19. From the analysis of these sample problems that included both the ductile and the brittle frames under vertical loads and the combined vertical and lateral loads, the following conclusions can be drawn. 1. The most refined analysis is achieved by including both geometric and material nonlinearities as well as the joint flexibility. if this analysis is performed with the clear span of the members, it gives a good match with ex~~mental results for ductile frames, but it slightly overestimates the strength of stiff frames. On the other hand, such analysis would underestimate the strength of the frame if performed with center to center span length of members, 2. For an analysis considering the geometric and material nonlinearities with the assumption of rigid joint, an accurate prediction of the load~efo~ation curve can be obtained by using the member’s span length from center to center. 3. It was determined that material nonlinearity plays a major role in controlling the behavior of frames. This role is either direct through the section rigidity, or indirect through the values of the parameters of the joint model. Therefore, reasonable results may be obtained by considering material nonlinearity only in the analysis for some cases; e.g., short frames under vertical loads. 4. Joint flexibility coupled with geometric nonlinearity gives a good estimate of the strength and deformations of ductile frames under vertical loads. Such analysis gives an upper bound on the strength of ductile and brittle frames under a combined vertical and lateral load condition with a difference of 20%. S. The implementation of geometric nonIinea~ty in the analvsis can be significant for an accurate
prediction of the strength and the deformations of the structure under different load conditions, particularly in the case of tall frames under lateral loads or in frames where slender columns exist.
REFERENCES
E. El-Metwally, Nonlinear analysis of reinforced concrete frames. Ph.D. Thesis, School of Civil Engineer-
I. S.
ing, Purdue University, West Lafavette. IN (1986). 2 S.-E. El-Metwally and W. F. Chen, Genera&.x-l stress-strain relations for reinforced concrete sections. Techni~ul Report No. CE-STR-87-6, School of Civil Engineering, Purdue University, West Lafayette, IN (1987). 3. S. E. El-Metwally and W. F. Chen, Moment-rotation modeling of R/C beam-column connections. ACI Struct. J. July-August, 384-394 (1988). 4. G. C. Ernst, G. M. Smith, A. R. Riveland and D. N. Pierce, Basic reinforced concrete frame performance under vertical and laterai loads. ACZ J. 70, 261-269 (1973). 5. R. K. Livesly, Matrix method of StructuralAnalysis, 2nd Edn. Pergamon Press, New York (1975). 6. W. Weaver Jr. and J. M. Gem, Matrix Analysis of Framed Structures, 2nd Edn. Van Nostrand, New York (1980). I. R. J. Bedell and D. P. Abrams, Scale relationships of concrete columns. Structural Research Series No. 8302, Dept. of Civil, Environmental, and Architectural Engineering, College of Engineering and Applied Science, University of Colorado, Boulder, CO (Jan. 1983). 8. E. 0. Pfrang, C. P. Siess and M. A. Sozen, Load-moment-curvature characteristics of reinforced concrete cross sections. ACI J. 61,763-778 (1964). 9. M. S. L. Roufaiel and C. Meyer, Analysis of damaged concrete frame buildings. Technical Report No. NSF-CEE-81-21359-f, Dept. of Civil Engineering and Engineering Mechanics, Columbia University, New York (May 1983). to. M. T. M. Soliman and C. W. Yu, The flexural stress-strain relationships of concrete confined by rectangular transverse minforcement. Msg. Concrefe Res. 19, 223-238 (1967). 11. S. E. El-Metwally and W. F. Chen, Nonlinear behavior of R/C frames. Technical Report No. CE-STR-87-5, School of Civil Engineering, Purdue University, West Lafayette, IN (1987). 12. A. L. L. Baker and A. M. N. Amarakone, Inelastic hyperstatic frames analysis. Proc. Int. Symp. of the Flexural Mechanics of Reinforced Concrete, ASCE-AU,
pp. 85-142. Miami (Nov. 1964). 13. J. Blaauwendraad and A. K. De Groat, Progress in research on reinforced concrete plane frames. Heron Zs, No. 2 (1983).
E. EL-~~~~~Y~
and WAI-FM C&N$
~~~~~~fy of Engineering, Et-Mansoura
University, ENansoura, Egypt $%hoot d CivilEngineering, Fwdw University, West tafayette, IN 47907, U.S.A.
infemaf parameter axial ~~~~~tyof a member ffexural rigidity of a member
sti&ess matrix memberlength bendingmoment axial farce ~~id~t~--~~a~t maxims
~~~dj~-ta~~~t Taurus d~sp~a~e~t in X’-dir~%i~t~ joint dis~~~~rneilt in Y-direction joint
joint coordinatein X-direction joint coerdinate in Y-direction ordinate direction coordinatedirection joint rotation
~~~SC~j~F~
to ve~i~al loads (Fig. I@, The second example is a brittle Frame II, where flexural hinges are expected to Form in the mfumns. Frame II was Arst subjected to vertieal loads under service ~ondi~~o~s~ then, the vertical loads wore held constant while et lateral load was applied up to failure (Fig. 2a). The two frames were tested previously by Ernst et al. [4]. fn the foIjowing sections, the models used in the present analysis for ~~~~~~~ ~0~~~~~~~~ material ~on~inea~t~ and joint ~~xibility are first discussed. Then, fhe analytica results are compared with the ex~~mental data cited, FinalIy, the effects of parameters of n~~~ineariti~ and joint fle~b~~ityon the behavior of frames are examined,
Large dis~la~ments may have either a so~ening effect or a sti~~~i~g e&et on the behavior and strength of structures or members. Also, the action iand i=f iteration number member of the applied loads might be affected by the struc;: initial ture’s deformations. Hence, geometric nonlinearity fiexwat becomes a si~i~c~t factctr whets the level of loading taxgent is su~~ient to cause large disp~a~ments. This is u&mate especialfy true in the ease of slender members subjected to axial compression* or memhrs subjected to refatively large lateral loads. In this study, two different aspects of geometric oo~~n~~ty are conThe! b&avior of reinforced concrete frames is essen- sidered. These are the stability analysis considering tially controlled by three key parameters: geometric the P-4 effect of the member and thB overall strucnonlinearity, material nonhnearity and joint flexi* tural stability considering the P-A effect. The axial b&y, In this paper the effects of these ~~~e~rs on ~formation of a member due to geometric deformathe bebatior of frames is inv~~~t~. The models tions was found to be ~s~~~~~~~ in rein&e& reported reeent& in f&-3]are used to simulate these concrete rnern~~ because of their deep s&ions, and parameters of nonlin~~ties, and the Mness method therefore it is not discussed in this study. is applied to obtain numerical solutions. 2.1.1, ~~~~~~ ~~u~~~~~~ (P-3 @&et)+The member fn this paper we shall discuss two ~nrn~~l local stiffness matrix, K,, is mod&d by the stSness examples. One is a ductile Frame I, where flexural functions, &, fb,?,Qi3and 4, as given in Table I, to hinges are expected to form in the beams, subjected account for the P-6 effect. The devetopment of these f
eurrent
a
axial
1204
SALAH E.
EL-MEIWALLY and WA{-FAHCHEN
(a)
x
Experimental
0
Geometry + material
-
Geometry + joint
0
Materiot+
0.6
Gemetry
+ joint
Laterw
defkctwm,
A,
+ moterio\ $ join?
24
12
Vertitut
Geometry+rmteriat
-
pint
dr Qmnetry
0
0
AC
defhxtion,
(in
)
iin)
Fig. I. (a) Ductile frame under gravity loads. (b) Ductile frame under gravity and lateral loads.
functions and their modification for the member’s stiffness matrix are given in detail in [S, 61. EA i 0
0
12EI FQIl
0
6EI
F42 4EI
0
F43
--
EA
0
L
0
12EZ
0
--
0
--
L3
6EI 41
p2
6EI L2
42
q&4
Km =
--
EA L 0
0
0
12EZ --+I
0
EA
0
T0
12EI -+’
0
--
0 6EI -- L2 42
6EI L2
df2
L
where EA is the member axial rigidity, EZ is the flexural rigidity, and L is the length of the member. 2.1.2. Structural stability (P-A e&t). The joint global coordinates are modified after each iteration to account for the current joint displaced position. This reflects the load-displacement interaction of the structure. Since the member equilibrium matrix and hence the member stiffness matrix are functions of the current nodal displacements, the member stiffness matrix must be developed with respect to the current displaced coordinates of joints. If we denote the joint undeformed coordinates in the X and Y directions by x and y, and the joint displacements at iteration i by ui and u, in the X and Y dir~tions respectively, then the joint coordinates
for the i + 1 iteration &+I
are x,, , and yi+ 1
=x+u,
Yi+r ‘Y
+ui.
(2) (3)
The transformation matrix used to transfer the member stiffness matrix from the member local coordinates into the structural global coordinates is obtained from the deformed geometry of the structure. The joint deformed coordinates are therefore used in calculating this transformation matrix. 2.2. Material nonlinearity To make a direct computation of both flexural rigidity and axial rigidity due to any load combi-
span
nation, a n&hod ~r~se~t~d in f2], and described briefly here, is used. This method is found to be more economical in camputational time and much easier to use in nonlinear analysis than other methods available [7-Q], For the stress--strain ~Iationship of concrete in compression, the curve by Sohman and Yu fItI], shown in Fig. 3, is adopted in this analysis. For steel, an elastic-~rfectIy plast& stress-strain ~~at~o~sh~p is assumed. 2.2.f. Bctsic ~~~~~ti~~~. (if A linear strain distribution across the depth of a section is adopted. (2) The axial force is applied at the section centroid when calculating the applied moment. (3) Shear deformations are neglected. (4) Tensile strength of concrete can be accounted for in caiculating tbe section strength; however, in the present analysis concrete strength in tension was negkcted.
12.2. Ax&al ~~~e~~~~~ r~g~~~~~-~e~~~r ~~i~~ The secant axial rigidity, S,, and secant fIexura1 rigidity, S,, are defined as
where P is the axial force acting at the section centroid, M is the bending moment appiied to the section, L, is the axial strain of the section due to the axial load P and measured at the section centroid, and 4 is the section curvature. To calculate S, and S, due to the load vector (P,, Mb >‘* the corresponding section deformation vector, fq, &)‘, rmst be ~~u~at~ first. If the se&ion defo~ation vector due to the initial bd
Table 1. Stiffness functions for a beam subjected to axial foree Function 4%
Compression
Condition of axial load Tension
(k.L jJ sin KL
(k.&~sinh kt
WC
1%
Z&TO
t
SALAHE. EL-METWALLYand WAI-FAH CHEN
1206
.& = Cross-sectionat orea of tmnsverse reinforcement kIC = Areo of concrete under compression & = Area of bound concrete under compression # = b, or 0.7d,, whichever is greater 6, = breadth of bound concrete cross-section ~7, = Effective depth of bound concrete cross-section 5 = Longitudinal soocina of transverse reinforcement: _ S0 2 Lon itudinal spacmg at which trowverse .3 rem orcement is nut effecti confining the concrete (10 in. 7 in ucyr = Standard 6 x 12 in. cylinder strength
Strain
9
=(l.4&
5
f; = 0.9+,,,,
_o,45)
&(.%-5-J A,t S+ O.ccx?e~s’
(i+o.o59)
=ete= 2 f;tEc
0.0025 (ii-q)
Qcr’
dcf= 0.0045
(l+O.859)
Fig. 3. Adopted stress-strain curve for concrete confined by rectangular hoops [IO]. vector fPo, M,)’ is known as {co, Cp,}‘,the following relations can be written
From eqns (6) and (7) it is obvious that the incremental deformation vector {AC,AQt)’ corresponds to the incremental load vector (AP, AM}‘. The relationship between an infinitesimal load vector and an infinitesimal deformation vector can be formulated as follows
(8)
Solving eqn (11) for the deformation vector, we have
Equation (12) represents the solution algorithms for the incremental deformation vector. The elements of the section stiffness matrix can be evaluated numerically by using, for example, the finite difference method [2]. This can be achieved by assuming an incremental change in the strain and curvature at the current state of strain and calculating the corresponding incremental axial load and bending moment. Then, the elements of the stiffness matrix can be calculated. 2.2.3. Axial andflexural rigidity-tangent modulus. The tangent axial rigidity, T,, and the tangent flexural rigidity, T,, are detmed as (13)
(9) Or, in matrix form
[~~I=
~~
= KI
~~
[
dc d4
1
[~~1
(10)
T, and T, are determined at the current state of strain, and can be calculated by using the central difference method. The central difference reflects the material properties at the current state of strain better than either the forward difference or the backward difference; therefore, it is preferred.
2.3. Joint flexibility
where [K,] is the sectional tangent stiffness matrix. Replacing the infinitesimal changes in the load and deformation vectors by (finite) incremental changes, eqn (IO) has the form
(11)
In framed structures, inelastic deformations are essentially concentrated at several critical regions. These include the member ends (joints) and other regions where high bending moments may take place. An accurate prediction of the nonlinear behavior of these framed structures depends to a large extent on the reliable representation of these concentrated
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Nonlinear behavior of R/C frames
inelastic regions. Hence, an accurate modeling of the beam-column connections represents an important part in the study of the behavior of reinforced concrete frames. In this study, the joint model proposed recently in [3], and described briefly here, is used. This model is simple to use and easy to implement in the computer analysis of reinforced concrete frames. The implementation of the joint flexibility in the member stiffness matrix is discussed in (9, 111. 2.3.1. The joint model. In this study the joint is modeled mechanically as a concentrated rotational spring. The development of the spring stiffness is based on the thermodynamics of irreversible processes. The spring stiffness, K, is a function of the following three parameters. 1. Initial rotational stiffness of the connection (K,,). This can be computed for the connection with a linear elastic behavior of the materials. 2. Ultimate moment capacity of the connection (M,). This can be computed for the connection using the limit analysis of perfect plasticity. 3. Internal variable (a), which introduces the dissipated energy to the model formulation.
Thermodynamics is used to define these variables and to develop the general form of the constitutive equation of the spring. For the development of the proposed joint model, the following basic assumptions were made. 1. Loading and unloading curves are smooth functions; hence, the free energy and dissipated energy are both smooth functions. 2. The free energy and dissipated energy are both unbounded at failure; i.e., for the structure to collapse these two quantities are not finite. The moment-rotation relation of the joint was developed in [3], and is given by the following equation
(15) where 0 is the joint rotation, M is the moment, and 1 is the load factor, I = M/M,. The parameter ‘a’ takes different values with the maximum at zero moment (a,) and decreases gradually with increasing load factor. However, for simplicity ‘a’ can be assumed constant, which would still give satisfactory results. This will require the selection of one loading point to evaluate ‘a’. One particular point is most suitable for the evaluation of ‘u’ because it is well defined compared with other points. This is the loading point, when tension reinforcement in the member where flexural hinge is expected to form, starts to yield. By assuming ‘a’ as constant,
the energy dissipation takes the following simplified form [3] @ =$
E [na(nu +2)(ln1+ II n-l
1)+2] ~na+2 x ~
(na + 2)2’
(16)
The dissipated energy results from the inelastic behavior of the joint due to the following three factors: (1) the deterioration of the bond strength between the reinforcing steel bars and the concrete; (2) the cracks in the concrete; and (3) the inelastic behavior of the materials composing the joint, concrete and steel. In [3], the calculation of the dissipated energy due to each of these three factors is given in detail. 3. ANALYTICAL RESULTS AND EXPERIMENTAL DATA
Both Frames I and II are analysed for five different sets of analysis parameters as shown in Table 2. The results from the experiments and the present analysis are shown in Fig. la for Frame I, and Fig. 2a for Frame II, respectively. Also, Frame I is analysed for lateral and service vertical load conditions. The gravity loads are applied first; then, the lateral loads are applied gradually up to failure (Fig. 1b). Frame II is analysed under vertical loads that are applied gradually up to failure (Fig. 2b). For the ductile Frame I, which is designated as b40 in the paper by Ernst er al. [4], the first analysis, with rigid joint, and the fifth analysis, with flexible joint, match very well with the experimental data. The analysis was performed up to the maximum load only without attempting to predict the softening branch of the load-displacement curve. Therefore, the ultimate deflection is not shown in the results presented here. The second analysis, with the linear elastic material, gives a very good envelope for the measured load-deflection curve. The vertical load carrying capacity of the frame is underestimated by the third analysis, and even more so by the fourth analysis. For the brittle Frame II, which is designated as 2d9h in Ernst’s paper, both the first and third analysis
match very well with the experimental
results. The
Table 2. Set of analysis parameters Analysis
Geometry
Material
Joint
1
N
2 3 4
N
N E N
R F F F F
w
L
N N
N N
t Clear span length is used. R, rigid; F, flexible; L, linear; N, nonlinear; E, linear elastic cracked section.
1208
SALAH
E. EL-METWALLY and WAI-FAHCHEN
strength of the frame is, however, overestimated by the second analysis and underestimated by the fourth analysis. The fifth analysis slightly overestimates the strength of the frame. 4. 4.1.
DISCUSSION OF THE RESULTS FROM THE FRAMES ANALYSED
Material nonlinearity
Nonlinear constitutive relationships of the crosssection influence significantly the overall behavior of the frame. In Fig. la, the first analysis for the ductile frame, considering geometric and material nonlinearities, matches well with the experimental results. The fifth analysis, considering both material and geometric nonlinearity with flexible joints and using the clear span length of the members, also gives very good results. However, the second analysis with linear elastic constitutive relations and considering the geometric nonlinearity and joint flexibility, gives a good bilinear representation of the load-deflection curve. This can be explained only by the influence of the material properties on the values of the parameters of the joint model. From the analysis of Frame I under the combined vertical and lateral loads (Fig. lb), and by assuming that either the first or the fourth analysis will give the correct load-deflection relation, and finally by comparing either the first or the fourth analysis with the second analysis, we can observe the effect of material nonlinearity on the behavior of the frame. The strength of the brittle Frame II is overestimated by the second analysis because the material nonlinearity is ignored (Fig. 2a). Also, by comparing the second analysis of this frame under vertical loads (Fig. 2b) with either the first or the second analysis, it is obvious that the use of linear constitutive relations results in a much larger ultimate load of the frame than the correct one. This dictates the necessity of employing nonlinear constitutive relations in the analysis of stiff frames. 4.2. Geometric nonlinearit? Geometric nonlinearity coupled with either the material nonlinearity or the joint flexibility is adequate to predict the response of a ductile frame under vertical loads, Under the combined vertical and lateral loads, the load-deflection curve cannot be predicted accurately without considering the geometric nonlinearity. This is due to the significance of the P-A effect under this load condition. It is necessary to consider the geometric nonlinearity with the material nonlinearity in order to obtain a more accurate result for the response of a brittle frame under lateral loads. Under vertical loads only, by comparing the third analysis with either the first or the fourth analysis, the effect of geometric nonlinearity is noticeable.
4.3. Joint JIexibility The fifth analysis, which accounts for the two parameters of the nonlinearities and the joint flexibility, and performed with the clear span length of the members, gives a very good prediction of the load-displacement curve of the ductile frame. For the brittle frame this analysis slightly overestimates the strength of the frame. This shows the necessity of including the joint flexibility in the analysis if the clear span length of the members is used in the analysis instead of the center to center dimensions. The response of a ductile frame under vertical loads only is predicted reasonably well by the second analysis which accounts for both joint flexibility and geometric nonlinearity. In this case, the flexural hinge is expected to form in the beam which has a smaller flexural strength than the column. This also implies the presence of moments at the joints which are higher than those under the concentrated loads. The beam has a flexural strength near the ends that is greater than in the middle third where the loads are applied. The joints and the middle third of the beam reach their full strength simultaneously or nearly so. When the joints are near their full capacity, the tangent stiffness of the joint drops to almost zero. If the middle third of the beam cannot carry any additional moments, the joints will soon have to reach their full strength, causing a faihtre mechanism to form. In the second analysis, the ductile frame under the combined vertical and lateral loads behaves very similarly to the case of vertical loads only with some difference. The mechanism forms when the left joint (where higher moment is expected to occur first) reaches most of its strength and the P-A effect imposes additional moments on the frame that contribute to the failure. This indicates that a good upper value on the strength of the frame can be obtained by the second analysis. The response of the brittle frame under lateral loads when the second analysis is used can be explained in a similar manner to that for the ductile frame under lateral loads. In the case of a brittle frame, the beam is stronger than the column. With the assumption of flexible joints in the second analysis for vertical loads only, the beam is expected to behave in an intermediate stage between simply supported and fixed-ended beams. In other words, the moments at the middle third of the beam are always greater than the moments at the ends of the beam. Therefore, the joints do not reach their full strength until the middle of the beam reaches very high moments due to the linear elastic assumption for the material. The frame does not fail until the joints reach or near their full capacity. This in turn reduces the restraint of the column drastically causing the column to reach its buckling strength and finally causing the frame to collapse.
1209
Nonlinear behavior of R/C frames 5. SUMMARY AND CONCLUSIONS
The behavior of both ductile and brittle reinforced concrete frames has been examined under both a vertical, and a combined vertical and lateral load condition. The geometric and material nonlinearities, and the joint flexibility, are considered. Five different combinations of these parameters are analysed. The analytical results are compared with experimental data. Comprehensive state-of-the-art summaries on the recent and past progress in research on reinforced concrete plane frames are given elsewhere (12, 19. From the analysis of these sample problems that included both the ductile and the brittle frames under vertical loads and the combined vertical and lateral loads, the following conclusions can be drawn. 1. The most refined analysis is achieved by including both geometric and material nonlinearities as well as the joint flexibility. if this analysis is performed with the clear span of the members, it gives a good match with ex~~mental results for ductile frames, but it slightly overestimates the strength of stiff frames. On the other hand, such analysis would underestimate the strength of the frame if performed with center to center span length of members, 2. For an analysis considering the geometric and material nonlinearities with the assumption of rigid joint, an accurate prediction of the load~efo~ation curve can be obtained by using the member’s span length from center to center. 3. It was determined that material nonlinearity plays a major role in controlling the behavior of frames. This role is either direct through the section rigidity, or indirect through the values of the parameters of the joint model. Therefore, reasonable results may be obtained by considering material nonlinearity only in the analysis for some cases; e.g., short frames under vertical loads. 4. Joint flexibility coupled with geometric nonlinearity gives a good estimate of the strength and deformations of ductile frames under vertical loads. Such analysis gives an upper bound on the strength of ductile and brittle frames under a combined vertical and lateral load condition with a difference of 20%. S. The implementation of geometric nonIinea~ty in the analvsis can be significant for an accurate
prediction of the strength and the deformations of the structure under different load conditions, particularly in the case of tall frames under lateral loads or in frames where slender columns exist.
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E. El-Metwally, Nonlinear analysis of reinforced concrete frames. Ph.D. Thesis, School of Civil Engineer-
I. S.
ing, Purdue University, West Lafavette. IN (1986). 2 S.-E. El-Metwally and W. F. Chen, Genera&.x-l stress-strain relations for reinforced concrete sections. Techni~ul Report No. CE-STR-87-6, School of Civil Engineering, Purdue University, West Lafayette, IN (1987). 3. S. E. El-Metwally and W. F. Chen, Moment-rotation modeling of R/C beam-column connections. ACI Struct. J. July-August, 384-394 (1988). 4. G. C. Ernst, G. M. Smith, A. R. Riveland and D. N. Pierce, Basic reinforced concrete frame performance under vertical and laterai loads. ACZ J. 70, 261-269 (1973). 5. R. K. Livesly, Matrix method of StructuralAnalysis, 2nd Edn. Pergamon Press, New York (1975). 6. W. Weaver Jr. and J. M. Gem, Matrix Analysis of Framed Structures, 2nd Edn. Van Nostrand, New York (1980). I. R. J. Bedell and D. P. Abrams, Scale relationships of concrete columns. Structural Research Series No. 8302, Dept. of Civil, Environmental, and Architectural Engineering, College of Engineering and Applied Science, University of Colorado, Boulder, CO (Jan. 1983). 8. E. 0. Pfrang, C. P. Siess and M. A. Sozen, Load-moment-curvature characteristics of reinforced concrete cross sections. ACI J. 61,763-778 (1964). 9. M. S. L. Roufaiel and C. Meyer, Analysis of damaged concrete frame buildings. Technical Report No. NSF-CEE-81-21359-f, Dept. of Civil Engineering and Engineering Mechanics, Columbia University, New York (May 1983). to. M. T. M. Soliman and C. W. Yu, The flexural stress-strain relationships of concrete confined by rectangular transverse minforcement. Msg. Concrefe Res. 19, 223-238 (1967). 11. S. E. El-Metwally and W. F. Chen, Nonlinear behavior of R/C frames. Technical Report No. CE-STR-87-5, School of Civil Engineering, Purdue University, West Lafayette, IN (1987). 12. A. L. L. Baker and A. M. N. Amarakone, Inelastic hyperstatic frames analysis. Proc. Int. Symp. of the Flexural Mechanics of Reinforced Concrete, ASCE-AU,
pp. 85-142. Miami (Nov. 1964). 13. J. Blaauwendraad and A. K. De Groat, Progress in research on reinforced concrete plane frames. Heron Zs, No. 2 (1983).