TITLE NO. 64-14
Finite Element Analysis of Reinforced Concrete Beams By D. NGO and A. C. SCORDELIS
The basic concept of using the finite element method of analysis in constructing an analytical model for the study of the behavior of reinforced concrete members is discussed. The finite elements chosen to represent the concrete, the steel reinforcement, and the bond links between the concrete and the steel reinforcement are described. Several examples of singly reinforced concrete beams on simple supports with different idealized cracking patterns are analyzed and results are presented for comparison and discussion. The effect of the assumed stiffness of the bond links is also ?xamined bri~fly. No general conclusions regardmg the behav1or of the reinforced concrete beams und;r load are attempted in the present investigat.on. The purpose of the paper is to demonstrate the feasibility and to explore the potentialities as well as the difficulties of using the finite element method, with an ultimate aim of developing a general analytical method for the study of reinforced concrete members in the full range of loading.
Keywords: beams {structural); bond {reinforced conc~ete); co':"puters; cracking (fracturing); de-
flect.on; matm methods (structural); reinforced concrete; stress concentration; structural analysis.
• STUDIES OF THE RESPONSE OF reinforced concrete members to load have occupied the attention of many investigators during the present century. For example, hundreds of papers have been written on the subject of shear strength of reinforced concrete members alone. Unfortunately. because of the complexities involved, no basic analytical approach has been developed which can be used to accurately determine the internal stress distribution in the concrete and in the steel reinforcement of a member throughout its loading history. Some of the complexities of the problem are enumerated below: 1. The structural system is three dimensional and is composed of two different materials, concrete and steel.
152
2. The structural system has a continuously changing character due to the cracking of the concrete under increasing load. 3. Effects of dowel action in the steel reinforcement, bond between the steel reinforcement and ~oncrete, and bond slip are difficult to incorporate mto a general analytical m-odel. 4. The stress-strain relationship for concrete is nonlinear and is a function of many variables. 5. Concrete deformations are influenced by creep and shrinkage and are time-dependent.
In view of such great complexities, an analysis to determine principal stresses throughout such a member by a direct application of the classical theories of continuum mechanics is virtually impossible. However, if these detailed stress distributions in the concrete, both in the compression zone and in the tensile zone between cracks, as well as in the steel reinforcement, could be determined for various load levels, studies of the basic behavior of these members would be immeasurably aided. The general approach to be followed in this paper has been made possible by two recent developments: first, the high-speed digital computer and second, the concept of the finite element approach for the analysis of structural systems. The finite element method consists of replacing the actual structural system by a system of finite elements interconnected at nodal points. In the case of a reinforced concrete member these finite elements have to be selected so as to accurately represent the behavior of the concrete and the steel reinforcement and their interaction through bond. This finite element system may then be analyzed by either a displacement or force method of analysis to yield the nodal point displacements and the internal stresses and deformations in the concrete and the steel reinforcement. In the initial studies described herein
' ACI JOURNAL I MARCH 1967
...0•
two-dimensional analyses of reinforced concrete beams with defined crack patterns are developed in which the concrete and the steel are assumed to have linearly elastic stress-strain relationships. The concrete and steel are represented by twodimensional triangular finite elements and account is taken of bond slip by use of finite spring elements designated as bond links between the steel and concrete, spaced along the bar length. For illustrative purposes, results are presented for several examples of reinforced concrete beams on simple supports under third-point loading. These beams were identical in all respects except for their assumed idealized crack patterns and thus the effect of cracking on the resulting stress distribution can be studied. In addition, for one of the crack patterns assumed, several different values were assumed for the stiffness of bond links to study its effect. The present paper is an initial report on a continuing investigation of the application of the finite element method to the analysis of reinforced concrete members. Ultimate purpose of this research is to make feasible the detailed analytical study of the behavior of reinforced concrete members through their entire elastic, inelastic, and ultimate ranges. The finite element method, used in conjunction with a digital computer, makes it possible to eventually incorporate into the analytical model: bond slip, nonlinear material properties, dowel action, failure criteria for concrete under combined stress, a trace of progressive cracking in the concrete up to failure, load history, and time-dependent effects. The accuracy of the analytical model used is, of course, dependent on obtaining extensive experimental data which can be used to develop realistic analytical relationships for each of these effects. FINITE ELEMENT METHOD
The finite element method can be defined as a general method of structural analysis in which a continuous solid is replaced by a finite number
of elements interconnected at a finite number of nodal points. With such an idealization, a problem in solid mechanics is transformed into a related problem of an articulated structure which can be analyzed by the standard methods of structural analysis. Excellent discussions of the method can be found in papers by Clough,! Wilson 2 and others, and thus only a general description of the method will be presented here, to facilitate the discussion of the beam problems under consideration. Fig. la shows an arbitrary plate type member, loaded in its own plane, which has been subdivided into small triangular finite elements for purposes of analysis. The thickness of the member is taken as unity in this two-dimensional problem in which the internal stress distribution is desired. The selection of the mesh size for the elements depends on the accuracy desired, the finer the mesh the better the results, but at the same time the larger the computation effort required. While, in general, any element shape can be used, the triangular elements are found to be very convenient, since they can be easily fitted to a curved or irregular boundary as well as to a gradual gradation in size throughout the member as desired. Thus a finer mesh can be used in regiorts where the stress gradient is expected to be large. Elements with an aspect ratio, height/base. in the neighborhood of unity are preferable, and any extreme variation from unity should be avoided wherever possible to improve the accuracy of the solution. The key step which influences the accuracy of the finite element analysis is the determination of the stiffness matrix of the triangular elements. A number of different stiffness matrices have been proposed. The one used in the studies in this paper is described and developed in detail by Wilson2 and has been used successfully on a wide variety of plane stress problems. The element is assumed to have two degrees of freedom
rz X Fig. I a-Plate type member subdivided into triangular elements
ACI JOURNAl I MARCH 1967
Fig. I b-Typical triangular finite element
153
at each nodal point, a horizontal and a vertical displacement component (Fig 1b). To develop the element stiffness matrix, displacement functions which are continuous over the entire element are first selected in terms of the nodal point displacements such that any straight line before deformation remains a straight line after deformation. In this way full compatibility is preserved between adjacent elements at their edges when the assembled structure is subjected to loads and displacements. The displacement functions specified above are satisfied by having constant strains E,, Ey, and Yxv throughout each element and consequently for a linearly elastic material, constant stresses a,, ay, and 1':rv thoughout each element. Once the displacement functions have been chosen, the entire analysis can be stated by the following matrix equations: 1. Strains E are expressed in terms of nodal point displacements r through a displacement transformation matrix A: {E} =
[A] {r}
=
[C] {E}
(2)
3. Stresses a at the edges of the element are replaced by equivalent stress resultants or nodal point forces S through a force transformation matrix B {S}
=
[B] {a}
tion matrix A, the element stiffness matrix k can be defined as follows: [k]
=
[B) [C] [A] {r}
(4) Since the force transformation matrix B is equal to the transpose of the displacement transforma-
[B] [C] [A]
=
[AF' [C] [A]
(5)
and Eq. (4) can be rewritten in the compact form: {S} =
[k] {r}
(6)
The stiffness matrix K of the entire system shown in Fig. 1 (a) can then be assembled by directly adding the contribution of each individual element stiffness k into its proper location. The resulting equation relates the external nodal point forces R to the nodal point displacements r of the total system. {R} =
(3)
4. By substitution: {S} = [B] {a} = [B] [C) {E}
ACI member A. C. Scordelis is professor of civil engineering, Department of Civil Engineering, University of California, Berkeley. Professor Scordelis has been a faculty member since 1949. He has been noted for his research work with reinforced and prestressed concrete and for numerous technical papers. Currently, he is a member of ACI-ASCE Committee 421, Design of Reinforced Concrete Slabs, ACI Committee 435, Deflection of Concrete Building Structures, and ACI Committee 334, Concrete Shell Design and Construction.
(1)
2. Stresses a are related to strains by a stressstrain law C: {a}
ACI member D. Ngo is a graduate student in civil engi· neering, University of California, Berkeley. He received a bachelor's degree in architecture in 1962 and an MS in 1966 from the University of California. He also obtained a BS in 1964 from San Jose State College where he served as graduate assistant. He has worked as an architectural draftsman and a computer programmer. Currently, he is participating in research on the application of digital computers to the solution of structural problems in reinforced concrete.
[K] {r}
(7)
The K matrix has a band form, and by careful numbering of the nodal points the band width of the matrix can be reduced to a minimum, which in turn reduces the computational effort and computer storage required when a direct stiffness solution is used.
TYPICAL TI2.1AN0ULAQ CONCRETE ELEMENT NODAL POINT
STEEL REIN FORC.EMENT
TRIAN C1ULAR
ELEMENT Fig. 2-Finite element idealization
154
ACI JOURNAL I MARCH 1967
To take advantage of the band form of the K matrix a recursive procedure, similar to that presented by Clough, Wilson and King 4 for tridiagonal matrices, is used for the solution of Eq. (7). This solution gives the unknown displacements r for a given set of forces R and boundary conditions from which the stresses a in each element can be determined using Eq. (1) and (2). {a}
=
[C] {E}
=
[C] [A] {r}
(8)
Eq. (7) represents a large set of simultaneous equations which requires a large amount of computation for solution. Digital computers with large storage capacities and fast computation speeds are required to treat most practical problems because of the large number of finite elements required for realistic idealizations. IDEALIZATION OF REINFORCED CON,CRETE BEAMS
The finite element idealization of a singly reinforced concrete beam is shown in Fig. 2. The beam is subjected to third point loading and is on simple supports. The concrete and the steel reinforcement are both decomposed into systems of triangular elements. Any cracking that takes place in the concrete can be simply represented by separating the concrete elements on either side of the crack. This is done by assigning a different nodal point number on each side of the crack. Physically, the two nodal points at the crack may still occupy the same point in space. In other words, the topological properties of the beam can be varied in any manner without altering its geometrical properties. This particular feature of the finite element method is believed to be one of its major advantages in the study of reinforced concrete members.
v
As pointed out before, one of the difficulties in constructing an analytical model of a reinforced concrete member is due to the composite action of the steel and concrete. Perfect bonding between steel and concrete can only exist at an early stage under low load intensity. As the load is increased, cracking as well as breaking of bond inevitably occurs, and a certain amount of bond slip will take place in the beam, all of which will in turn affect the stress distributions in the concrete and steel. To account for these effects, an additional finite element linking the concrete and steel must be used. The linkage element (Fig. 3) can be conceptually thought of as consisting of two linear springs parallel to a set of orthogonal axes H and V. For generality, the linkage element can be oriented at any arbitrary angle (} with the horizontal axis of the beam. Note that the linkage element has no physical dimension at all, and only its mechanical properties are of importance. This is completely in line with the basic concept of the finite element idealization of the reinforced concrete beam. Since the linkage element has no actual dimensions, it can be placed anywhere in the beam without disturbing the beam geometry. The linkage element can be used to connect the steel and concrete elements, or even for connections between steel and steel or concrete and concrete elements when such necessity arises, provided that the spring stiffness can be properly defined in each case. To incorporate the linkage element into the finite element computer program, it is necessary to develop the stiffness matrix of the linkage element. Let the springs in the H and V directions have stiffness Kn and K,, respectively. Positive direction of displacements () are shown in Fig. 3. The stress-strain relationship is given by: (9)
H where
E"
and Ev are relative displacements between
MODIFIED CONCRETE
Fig. 3-Linkage element
ACI JOURNAL I MARCH 1967
Fig. 4--Analytical model
155
Points I and J in the H and V directions and are positive when they are tension. The strains and the displacements are related by the displacement transformation matrix A: {E} =
[A] {1\}
or -S
c
-c
-s
(10)
where c = cos 8 and s = sin 8. By noting that the force transformation matrix B is equal to the transpose of the displacement transformation matrix A, the stiffness of the linkage element can be obtained from: [k] =
[AF [C] [A]
l
-c -s c s
sl [ K,
-c -s
0
c
-s -c
l
K~tc 2
OJ
Kv
+ Kvs
2
Khsc- Kvsc - Knc 2 - Kvs 2 - Knsc KvsC
+
- Knc 2 - Kvs 2 - K 11 sc Kvsc K 11 c 2 + Kvs 2 K,sc - K,sc
+
c
-s
~]
K"sc- Kvsc K"s~
- K"sc - Kns 2
+ K,c~ + K,sc -
Kvc~
(11)
With the aid of the linkage element, it is now possible to construct an analytical model for the study of reinforced concrete beams. The final model adopted, after several exploratory models had been tested in the investigation, is schematically shown in Fig. 4. Using transformed section concepts, the actual beam is converted to a beam of unit width with modified moduli of elasticity being used for the steel reinforcement and also for the concrete at the steel level to account for the reduction of concrete volume displaced by the steel reinforcement. Thus the actual three-dimensional reinforced concrete beam is replaced by an approximate two-dimensional, plane stress analytical model. The steel elements are connected to the concrete elements through linkage elements at nodal points, except at midspan and at the ends of the beam. At these points the linkage elements are omitted and the steel elements are connected directly to the con-
156
crete, since no relative movement between the steel and the concrete is expected to occur because of symmetry at midspan and anchorage of the steel at the ends. The linkage elements in the analytical model can be thought of as bond links which represent the bonding between the steel and concrete elements. At the same time, the bond links can permit a certain amount of slippage to take place during the transfer of the stresses from steel to concrete. The angle {) for the linkage elements becomes zero in the arrangement shown in Fig. 4, and the two springs in each linkage element represent the horizontal and vertical stiffness of the bond links. If the connection of the steel reinforcement to the concrete beam proper by means of such bond links can be accepted as a close approximation of the interaction between the two materials in a reinforced concrete beam, then the resulting horizontal forces in the bond links will give a measure of the bond stress distribution along the beam under each stage of loading and cracking conditions. The remaining question, of course, is the determination of the two spring stiffnesses in the linkage elements. The true relationship between bond slip (relative movement between steel and concrete) and bond stress is a complex one which is affected by many factors. For simplicity in the present study, a linear relationship between bond slip and bond stress was assumed, which means the bond slip at any section of the beam is directly proportional to the bond stress. In the examples described later the value taken for this proportionality, which may be called a "slip modulus," was calculated on the basis of experimental results from load-slip measurements on reinforced concrete cylinders, subjected to axial load through a single reinforcing bar embedded in the concrete cylinder. While this method of determination of the slip modulus for use in a beam problem is open to question and requires further investigation, it offers an initial trial value for the horizontal spring stiffness K" to be used in the present study. In one of the beam examples described later, three different cases were run in which Kn was varied to study its effect on the behavior of the beam. The vertical spring stiffness K, is even more difficult to determine, and this aspect of beam behavior has received practically no attention in the field of concrete research. In an actual beam, bonding in the vertical direction, transverse to the longitudinal steel reinforcement, may be important when the dowel action or the "pressdown" effect becomes pronounced after large crack openings have taken place on the beam. In this situation, the bonding depends not only on
ACI JOURNAL I MARCH 1967
the chemical adhesion and the mechanical interlocking between the steel and concrete, but also on how well the surrounding concrete is "holding" the steel from vertical separation. This is, of course, a three-dimensional problem. Despite all of these complexities, it appears reasonable to believe that in the range where cracking is not excessive and horizontal splitting due to the "press-down" effect has not yet occurred, the vertical movement of the steel is very small and its effect may be neglected. Thus a very large value of Kv was arbitrarily assigned to each linkage element in the present study as an initial trial.. Physically, this means that the steel and the concrete are almost rigidly connected in the vertical direction. The validity of this assumption is again open to question and requires further investigation. COMPUTER INPUT AND OUTPUT
Through the use of a digital computer and a properly written program, the only manual work involved in the finite element analysis is the preparation of the input data for each beam under investigation. In some cases, a computer program can also be written to generate the necessary input data, and the effort in preparing the required input for each problem is reduced to minimum. In general, the input data that has to be furnished for each reinforced concrete beam problem in the finite element analysis is as follows: 1. Nodal point coordinates: x and y of each point.
2. Element properties: Note that each element can have properties different from the others. The value of the modulus of elasticity E can be modified by the usual elastic transformation principle to compensate ..for any changes from the actual beam during the idealization from a three-dimensional to a two-dimensional problem. (a) Concrete: E and (b) Steel: E and
v
v
(c) Bond link: K,, Kv and angle of orientation(} 3. Boundary conditions: supporting points 4. Loading: any combination of x and y loads at each nodal point The output obtained from the computer is as follows: 1. Displacements: x and y components at each
nodal point. 2. Concrete or steel stresses in each element: (a) ax, cry, and T.,·y (b) Principal stresses a1, a2 , and angle a between x axis and principal stress a 1 . 3. H and V forces in each bond link: The horizontal bond force can be considered as the bond stress resultant in the beam problems. It should be noted that the stresses obtained in
the finite element analysis for any two dimensional element are constant within the element itself. Thus while full compatibility of displacements is preserved between adjacent elements
48"
l
!
l
SEAM SPfGIMfN WIDTH • 12"
* * ~---------I---------~
~--~--J-:-~---~--~
~-------- -Lt ------- ~
~--?-?'--l-J1 !--'4~--~
t;EAM
0-1
SEAM A-l
BEAM
SEAM
C-1
D-f
Fig. 5-Beam examples analyzed
ACI JOURNAL I MARCH 1967
157
at their edges, the stresses on either side of the common edge will be different. The finer the mesh used for the element subdivision, the smaller this difference should be. These stress discontinuities, especially for coarse meshes, are an undesirable feature in interpreting results, and in the present study nodal point stresses calculated from the surrounding element stresses by a special averaging process 2 were plotted instead of element stresses. Further studies are being conducted to determine the best and most accurate method for plotting final stresses. The computer program used in the present study is capable of treating structures with up to 1700 finite elements and 1000 nodal points. Any single nodal point may have up to eight adjacent nodal points. The maximum difference between any two nodal point numbers in a single element cannot exceed 50.
EXAMPLES For the purpose of illustration, a series of five beams were analyzed. The beams are identical in dimensions, reinforcement, loading condition, and mechanical properties (Fig. 5). The beam data are summarized below: Length: 162 in. Width: 12 in. Height: 22 in. Effective depth of steel: 19.5 in.
~
Steel: Two #9 bars Span: 144 in. Loading: 100.0 lb load at each third point SteelE: 30 x 106 psi Concrete E: 3 x 106 psi Stiffness Kh: 2.2 x 106 lb per in. (bond links at 6-in. intervals along the span)
Different idealized cracking patterns are assumed in each beam. The beam specimens are shown in Fig. 5 and are described as follows: 1. Beam 0-1: As a basis for comparison of the results obtained in other cracked beams, this beam is assumed to be uncracked under the given load. 2. Beam A-1: Vertical cracks are assumed to take place in the region of maximum moment. In reality, the locations of the cracks are random and can be anywhere within the constant maximum moment zone for the third point loaded beam. For the sake of simplicity and maintaining the symmetrical configuration, two vertical flexural cracks have been assigned to the beam. 3. Beam B-1: Two diagonal cracks are assumed to occur, one on each side of the midspan. Experiments have indicated that the diagonal cracks are generally curved toward the loading points.~ Thus the diagonal cracks are assumed to start at the middle of the shear span and point toward the loads. Each
I
g L.ONC11TUPINAL. STRESS IN U)NC.fl.ETE'
AVEf2.AC1E 5TI:.E~S IN STffL
ysi
50N D fORa Ch5 Fig. 6-Beam 0-1
158
ACI JOURNAL I MARCH 1967
1000
I
"
I
C6s
I
I
I
I
I
e f q h t j rt LONC11TUDINAL STiESS IN CONCRETE PC
AVERA~E
STRESS IN STEEL
J5i
Fig. 7-Beam A-I
1000
~.s
I
'3 LON~ITUOINAL
I
I
f1,
/(,
141.-
1000 (f:zs
STRESS IN CONCRETE
AVERA~f
STRESS IN STEEL
ysi
Fig. 8-Beam B-1
ACI JOURNAL I MARCH 1967
159
I
I
I
I
I
c i:< e f g ALONC11TUDINAL STRESS IN
AVE RA~f
I
J
t
k
CONC~ETE
ST2E~S
1000 cfo5
IN .STEE.L ,s~
Fig. 9-Beam C-1
I
c.
J...
,..,
I
I
e
f
L.ON~ITUDI NAL.
I
q
I
fi
I
i
I.
J
t
~
STRESS IN CONCRETE
AVERA~E :.TRE~S IN ~TEEL
1000 c:hs
psi
Fig. 10-Beam D-1
160
ACI JOURNAL I MARCH 1967
diagonal crack is idealized by two straight lines. Diagonal cracks in curvilinear form can be closely reproduced in the finite element model with the triangular elements. However, it was felt that such a refinement served no particular purpose in this preliminary investigation. 4. Beam C-1: Both vertical and diagonal cracks appear in this beam. Note that the dimension and location of the cracks are kept identical to those in Beams A-1 and B-1. 5. Beam D-1: Assuming the cracking becomes more intensive, additional cracks are imposed in this beam.
RESULTS AND DISCUSSION Selected analytical results are presented for each beam in Fig. 6 to 10. Note that due to symmetry, only half of the beam from the midspan to the right support is shown in each case. The distribution of longitudinal normal stresses in the concrete are shown on the upper portion of the figure. The distribution of the steel stresses and the bond forces are shown directly below. Several points are of interest to observe in the results presented. 1. In Beam 0-1 (Fig. 6) where there is no cracking, the longitudinal stress distribution in the concrete at any cross section is almost linear, and the neutral axis stays at a relatively constant position along the span. The linear increase in the steel stress within the constant shear zone, and the leveling of this stress in the constant moment zone are also evident from the Fig. 6. The bond forces, which represent the changes in the steel stress along the beam, are relatively small, since the stress in the steel increases rather gradually. In the area where the steel stress is constant, the bond forces become practically zero. The results obtained for the uncracked beam are very close to what would be expected from an ordinary linear elastic analysis based on transformed section concepts. 2. When there is a crack, such as in Beam A-1 and B-1 (Fig. 7 and 8), the steel stress may increase as much as 400 percent at the cracks. The bond forces also increase considerably near the cracks. Note also the change in sign of the bond forces on either side of a crack. The compressive stress in concrete at the cracked sections may be doubled in some cases. The distribution of the concrete stress is no longer linear in the vicinity of the cracks, and the tendency is to reduce the tensile stress in the concrete. The fluctuation of the neutral axis position along the span can also be seen in the cracked beams. 3. The effect of cracking is rather localized. The stress distributions, in steel and in concrete, tend
ACI JOURNAL I MARCH 1967
to return to those in the uncracked state once they are taken at sections a small distance away from the cracks. This can be seen by comparing Beams A-1 and B-1 (Fig. 7 and 8) with Beam 0-1 (Fig. 6) . This fact is not surprising if one recognizes Saint-Venant's principle. Photoelastic model studies on the similar cracked beams indicated the same evidence. 4. The localized effect of cracking can also be seen in the Beam C-1 (Fig. 9), in which the stress distributions around the vertical and diagonal cracks are similar to those in Beams A-1 and B-1 (Fig. 7 and 8). In other words, when the cracks are sufficiently far apart, their mutual influence is almost negligible. 5. In Beam D-1 (Fig. 10) where cracks are more closely spaced, the tensile stress in the concrete is greatly reduced. The tensile stress in the beam is now carried almost entirely by the steel reinforcement and assumes almost a constant value in the constant moment region in the middle third of the span. However, results for a finer mesh size with more bond links between the cracks at b and d would indicate some tensile stresses in the concrete in this region. The vertical deflections at mid-span of the various beams are shown in Fig. 11. The beams with more cracks will naturally deflect a greater amount, as shown in the graph. Comparing Beam B-1 with Beam A-1, the deflection due to a diagonal crack near the support is seen to be less than that due to a flexural crack near midspan. To study the effect of the horizontal spring stiffness K, of the bond links on the beam behavior, two additional cases were run using different values of K,. for Beam C-1. Both vertical flexural cracks and diagonal tension cracks are
.10
Kh = 2.2 "10 6 LB/IN.
:n
.08
:J:
v
z z .06 I
...
0
1-
v
Ill _J
LL
lj,
.04
u.J
0
z<( 0... ofl
I
.02
f-
~
-I~: I
<(
~
~ <(
<( IU
0
%
0
10
uJ
cO
1"""1
I
co ~
<( UJ 1:
10
I
I
v ::2
•"
<(
lU
cO
0. ~ <( UJ
cO
0 Fig. !!-Deflection comparison
161
present in this beam (Fig. 5). The resulting midspan deflections and the horizontal crack widths at the longitudinal steel level for both the vertical flexural crack and the diagonal tension crack are shown in Fig. 12. An almost linear relationship with Kh is observed for all three quantities. It can also be seen that substantial variations in the horizontal spring stiffness Kh have a relatively small effect on changes in beam deflections and crack widths. Consequently, the changes in the stress distributions (which are not shown) for the three cases were practically nil. However, no general conclusions should be drawn from the limited number of cases studied.
CONCLUSIONS The method of constructing an analytical model for a reinforced concrete beam which uses the finite element method of analysis has been described. Examples of beam analyses have also been presented for illustration. From the results it can be seen that the finite element analysis offers a complete picture of the stress distribution in the entire beam, which generally cannot easily be obtained by other analytical or experimental methods. The flexibility in altering the properties of the structure, as illustrated in the examples, makes the proposed analytical model favorable for studies of reinforced concrete
MIDSPAN DeFLECTION
i= \.) ILl -1
lL
w
0
4......
HO~IZONTAL C~CI'.
...._ ...._ ...._
::t
1-CI
STEEL LEVEL FOR DIA~ONAL - ....._ TENSION CRACK '"2t-...._
~ .0068
ACKNOWLEDGMENTS
..................
~
~
WIDTH AT
--....a.
.0067
v
.0049
HORIZONTAL CRACK. WIDTH AT STEEL LEVEL FOR VERTICAL
A.....
z
-...-...-...
~ ~04-7
~v
members in which changing material properties and continuous cracking occur under increasing load. The study presented herein is an initial report in which the number of variables has been purposely reduced to a minimum for a simpler and clearer understanding of the proposed analytical model and the general concept of the approach. Additional analyses, similar to those described in this paper, have been made for beams with web reinforcement and for axially loaded reinforced concrete prisms. More complete and complex computer programs, based on an incremental approach, are presently being developed at the University of California with the aim of incorporating nonlinear material properties, nonlinear bond slip versus stress relationships, failure criteria for concrete under combined stress and progressive cracking in the concrete. No tool is perfect in itself, and this is also true of the finite element method of analysis. Several factors should be kept in mind. First, the method is an approximate analytical procedure, whose accuracy depends on the fineness of the mesh size used. Second, the accuracy of the analytical results, when referred to the actual reinforced concrete member, are dependent on including all of the major influences in the analytical idealization of the actual member. Third, as more and more of these influences are incorporated into the analytical model the computational effort becomes so great that it begins to tax even the largest modern digital computers in terms of computation time and storage. However, new developments in digital computers will undoubtedly keep pace with this requirement. Fourth, and perhaps most important, considerable skill and knowledge is still required to interpret the analytical results properly and to make significant and meaningful conclusions from these results regarding the basic behavior of reinforced concrete members at various loading levels.
................
FLEXURAL CRACK "lt---...
The study reported in this paper was made possible by a University of California Faculty Research Grant. The generous support of the University's Computer Center, which provided the computer facilities, is also gratefully acknowledged. The plane stress programs, which were modified for use in the analysis described in this paper, were originally developed by E. L. Wilson and I. P. King at the University of California .
................ .............. ....0.
.0046
~ D04S~----~~--------~~----------L-~ 1.8XI0°
BOND
Z.Zl
Z.6XIO"
LINK STIFFNESS, Kn L~./IN.
Fig. 12-Effect of bond Iink stiffness, K, (Beam C-1 )
162
REFERENCES 1. Clough, R. W., "The Finite Element Method in Structural Mechanics," Stress Analysis, edited by 0. C. Zienkiewicz and G. S. Holister, John Wiley and Sons, Inc .. New York, 1965, pp. 85-119.
ACI JOURNAL I MARCH 1967
2. Wilson. E. L., "Finite Element Analysis of TwoDimensional Structures,'' Report 63-2, Department of Civil Engineering, University of California, June 1963, 72 pp. 3. Bresler. B .. and Scordelis, A. C., "Shear Strength of Reinforced Concrete Beams," ACI JoURNAL, Proceedings V. 60, No. 1, Jan. 1963, pp. 51-74. 4. Clough, R. W.; Wilson, E. L.; and King, I. P., "Large Capacity Multistory Frame Analysis Programs," Proceedings, ASCE, V. 89. ST4, Aug. 1963, pp. 179-204.
y el acero. Se analizan varios ejemplos de vigas de concreto simplemente reforzadas sobre apoyos simples con diferentes configuraciones idealizadas de agrietamiento, y se presentan los resultados para comparaci6n y discusi6n. Tambien se examina brevemente el efecto de la supuesta rigidez de la adherencia. No se intenta en la presente investigaci6n alcanzar conclusiones generales en relaci6n con el comportamiento de las vigas de concreto reforzado bajo carga. El prop6sito de este articulo es demostrar la factibilidad y explorar las posibilidades, asi como las dificultades, de usar el metodo de elementos finitos con un prop6sito ultimo dirigido hacia el desarrollo de un metodo analitico general para el estudio de miembros de concreto refrozados en toda la amplitud de cargas.
APPENDIX NOTATION Unless redefined where they appear, the letter symbols adopted for use in this paper are defined and listed below: [A] [B) [C] E [k]
[K] {r} {R} {S} a
{II} {E} fl
v {a} '!xy
=
a1, 02
Ox, Gy Fx,
Ey
'txy
K1,, K,
=
displacement transformation matrix force transformation matrix stress-strain relationship matrix modulus of elasticity element stiffness matrix structure stiffness matrix displacement matrix external load matrix element corner force matrix angle defining principal stress direction bond link displacement matrix strain matrix angle of orientation of linkage element Poisson's ratio stress matrix shearing strain principal stresses stress in x, y direction strain in x, y direction shear stress spring stiffness of bond link in Hand V directions
This paper was received by the Institute Aug. 31, 1966.
Sinopsis-Resume-Zusammenfassung
Analisis por Elementos Finitos de Vigas de Concreto Refor:z:ado Se discute el concepto basico de usar el metodo de analisis por elementos finitos para construir un modelo analitico para el estudio del comportamiento de miembros de concreto reforzado. Se describen los elementos finitos elegidos para representar al concreto, al acero de refuerzo y a la adherencia entre el concreto
ACI JOURNAL/ MARCH 1967
Calcul des poutres en beton arme par Ia methode des elements finis Les auteurs discutent les bases fondamentales de l' application de la methode des elements finis la conception d'un modele analytique, en vue de l'et1:1de du comportement des pieces en beton arme. Ils decrivent les elements finis choisis pour representer le beton, !'armature et la liaison d'adherence. Plusieurs exemples de poutres simple armature, sur appuis simples, avec divers schemas de fissuration idealises sont analyses et les resultats sont presentes en vue de leur comparaison et de leur discussion. L'influence de !'hypothese faite sur la raideur des liaisons d'adherence est egalement examinee brievement. Dans la presente recherche, les auteurs n'ont pas cherche a presenter de conclusions generales quant au comportement des poutres en beton arme sous charge. Le but de !'article est de montrer qu'il est possible d'utiliser la methode des elements finis et d'en explorer les possibilites ainsi que les difficultes. L'objectif final est de mettre au point une methode analystique generale, pour l'etude des pieces en beton arme dans toute l'etendue de leur mise en charge.
a
a
Die Berechnung von Stahlbetonbalken nach der "Finite Element" Methode Die Grundkonzeption der Anwendung der "finite element" ·Methode zum Aufbau eines Berechnungsmodelles fiir die Untersuchung des Verhaltens von Stahlbetonbauteilen wird diskutiert. Die Elemente werden beschrieben, die zur Darstellung des Betons, der StaMbewehrung und der Verbundglieder zwischen Beton und Stahlbewehrung Verwendung finden. Verschiedene Beispiele von frei aufliegenden Stahlbetonbalken mit verschiedenen idealisierten Rissbildern werden analy'siert und die Ergebnisse zum Vergleich und zur Diskussion dargestellt. Auch der Einfluss der Steifigkeit der Verbundglieder wird kurz untersucht. In der vorliegenden Untersuchung wird kein Versuch unternommen, allgemein gtiltige Schlussfolgerungen in Bezug auf das Verhalten der Stahlbetonbalken unter Last zu ziehen. Der Zweck dieser Arbeit ist es vielmehr, die Brauchbarkeit dieser Methode aufzuzeigen und sowohl den Anwendungsbereich als auch die Schwierigkeiten bei der Anwendung der "finite element" Methode zu untersuchen. Das Endziel ist, eine allgemein gi.iltige Berechnungsmethode zu entwickeln, mit der das Verhalten von Stahlbetongliedern im gesamten Belastungs•bereich studiert werden kann.
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