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Testing and verification analysis by finite elements of reinforced concrete double cantilevers
Nuclear t:.ngineering and Design 79 (1984) 187-197 North-Holland, Amsterdam
TESTING AND VERIFICATION ANAINSIS CONCRETE DOUBLE CANTILEVERS
187
BY F I N I T E E L E M E N T S
OF REINFORCED
B.S. B R O W Z I N US. Nuclear Regulatory Commi.v,~'ion, Washington, D(" 20555. USA
S.S. T U L G A Jot&re. Apostal and Rltter Asvociates. Dal,i~'t~ille. RI 02854. USA
and O, B U Y U K O Z T U R K Mas'sachmetts Institute of Technology,, ('amhridge, MA 02139. USA
A three-dimensional nonlinear finite element analysis code developed at the Massachusetts Institute of Technology has been verified against tested specimens of double deep cantilevers. The stress field in deep cantilevers is essentially two dimensional. For this reason, the verification is of particular interest. The results obtained indicate a high degree of accuracy of the method based on a comparison of analyzed versus experimental steel stress in bars. The method may be used for the design of deep reinforced concrete sections.
I. Introduction The objective of this paper is to demonstrate the application and verification of the finite element computer code developed at the Massachusetts Institute of Technology (MIT) for the analysis of complex concrete structures, by comparing predicted and observed behaviors and response quantities of tested double cantilevers. The behavior of reinforced concrete members with large height-to-span ratios is substantially different from that of shallow beams. Structural members are customarily considered "'deep" when the ratio of the effective depth to the span is equal to or greater than one. Deep beams, corbels or brackets, and deep cantilevers belong to the category of deep members, Corbels are short members projecting from the columns, whereas deep cantilevers are members representing the part of the beams continued over a support. A cantilever can be a symmetrical structure projecting from the two sides of a support. Wall footings, supports for span structures at the top of bridge piers, and supports in buildings at the top of columns for precast beams are examples of deep cantilevers. The stress and strain fields in deep cantilevers are complex, and generally, assumptions adopted
in the conventional design of reinforced concrete members in flexure and shear may not be adequate. For this reason, analysis using the finite element method, which incorporates essential elements of concrete behavior such as nonlinear behavior of concrete in multiaxial stress state and cracking, and the verification of such an analysis method are of particular interest. Several researchers have reported tests on corbels. Extensive experimental investigations of corbels were conducted at Portland Cement Association (PCA) laboratories by Kriz and Raths [1]. Other work on corbels was performed by Mehmel and Beckner [2], Mehmel and Freitag [3], Sommerville [4], and Mattock [5]. Research on deep reinforced concrete double cantilevers has been reported by Browzin [6]. This paper presents the work performed on the finite element analysis of the cantilever specimens tested by Browzin. Formation and propagation of cracks, failure modes, and their relation to the amount of reinforcement are predicted by the finite element analysis and compared with the experimental results. Experiments have provided measurements of strains at a number of locations on reinforcing bars. Comparison of analytical results with the experimental measurements provided
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Deep cantilevers were tested By B.S. Browzm ~t~ the National Bureau of Standards (NBS) and at the Cast Institute of Technology (CIT) [6]. For convenience. cantilevers were tested in an upside down position, and the load was applied at the top of the specimens by, 600-kip (2670-kN) capacity testing machines with 15-kip (67-kN) increments in the NBS tests (specimens NI and N2) and with 30-kip (134-kN) increments in the C11 tests (specimens ('l to C5). Specimens CI and ('2 were duplicates of specimens N1 and N2. respectively. The overall dinaensions of the specimens were 36 by 25 ~ h\ 12 in (91 by 65 by 30.5 cm). and the span between the supports was 27 in (69 cm) (fig. 1). The centroid of reinforcement was at 1.5 in (4 cm) from the b o t t o m The structural arrangement of tested typical specimens is shown in figs. 2 and 3. Axes numbered I through 15 are used to identify cross sections of the specimen. The distance between axes 1. 3, 5, and 7 is 5 in (12.7) cm. Axis 8 corresponds to the symmetry plane. The face of specimens with support ,4 at the left-hand side is identified as face 1: the opposite one is identified as face 2 (fig. 3). The specimens rested on 1 by 4 bv 12 in, (2.5 by 10.2 by 30.5 cm) supporting plates, thenlselves bearing on rollers at each side of the span. For
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4Y2'~ 10 '/2'~3'4" 3'~:'-10V2"~4'/2' '-' 36"-Fig. 1. Sectional view of specimens. the means to assess the validity of the numerical method and to extend its application to the design of deep cantilevers of any configuration and reinforcing ratio. The verified finite element method, as described in this paper, has been used in M I T research performed for the U.S. Nuclear Regulatory Commission [7 9]. /7
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189
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the five specimens tested at CIT, two rollers in contact were used at each support as a precaution against possible escape of the rollers under loading action. The two specimens tested at NBS had one roller welded to the lower plate at support A and one fully movable roller at support B. The ready-mixed concrete was made with high-early-strength cement, Type Ill, with 3/16- to 3 / 8 - in (0.5- to 0.95-cm) silica gravel and regular sand. The proportion of cement, sand, and gravel by weight was 1 : 2 : 4 . A compressive strength of 3000 psi (21 N / r a m 2) was specified, and higher strength was obtained. The material properties of the reinforcing steel used in specimens N and C were, respectively: yield point, 45 300 and 45000 psi (312 and 310 N / m m 2 ) ; ultimate strength, 74000 and 77000 psi (510 and 531 N / m m 2 ) ; elastic modulus 25.8 × 106 psi (178 × 103 N / m m ~) and 29.2 × 10 6 psi (201 × 103 N / m m 2 ) . The cross-sectional areas of the reinforcement for specimens N1 and C1
were selected to provide a reinforcement ratio approximately corresponding to the design by working stress method at axis 8 (i.e., f, = 20 000 psi and f,, = 1350 psi, L = 138 N / m m 2 and f~ = 9.3 N/ram2). Specimens N1 and C1 were reinforced by two layers of reinforcing bars; the others had one layer of bars. The concrete material properties and reinforcement characteristics of the specimens are summarized in table 1. Electrical resistance strain gages SR-4 of Type AB-7, 3 / 8 in (0.95 cm) long, were located on the reinforcing bars (figs. 2 and 3). In specimens N1 and N2, strain gages were located at the top and bottom of each bar, 20 gages in all. In specimen NI, which had two layers of bars, the gages were attached on bars of the lower layer. In specimens C1 to C4, strain gages were located at the bottom of bars, three gages in each specimen. The strain readings were corrected to obtain the strain and stresses in all specimens at the same locations, i.e., at the intersection of axes 5, 8, and 11 with the centroid of the reinforcement. The test results are presented as load versus experimental stress in reinforcement bars at five locations: approximately at the quarter points of the span (axes 5 and 11), at the middle of the span (axis 8), and at the supports.
3. Analysis of the tested specimens 3.1. The analysis method
The finite element method (FEM) of analysis was used to predict the behavior of the tested specimens. The method approximates the solution to a structural problem by subdividing the continuum into smaller regions [10]. In this paper, the method as applied to
Table 1 Material properties and reinforcement characteristics for the specimens tested Specimen
N1 CI N2 C2 C3 C4 C5
Concrete compressive strength (ksi)
Size and number of bars
Area of reinforcement (sq. in)
Reinforcement ratios at cross sections Axes 5 and 11
Axis 8
5.74 4.91 5.95 5.27 3.93 3.93 3.93
~ 4,15 :~ 4,15 ~4,7 ~ 4,7 ~ 3,5 ~ 3,3
3.0 3.0 1.4 1.4 0.55 0.33 0.00
0.0172 0.0172 0.0081 0.0081 0.0032 0.0019 -
0.0119 0.0119 0.0056 0.0056 0.0022 0.0013
190
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concrete structures is briefly described, and the I i n i i e element modeling of the deep cantilever specimens is presented. The analysis of concrete structures requires simulation of various important nonlinearities, some ~,f which arise from the nonlinearity of the stress straip, relationship for concrete, the anisotropy in concrete resulting from varying biaxial or triaxial stress fields. the continuously changing topology of concrete resuhing from cracking, postcracking shear transfer m e c h a nisms, and yielding and subsequent plastic behavior of the steel reinforcement. For the present study, the computer program A R ( [ 11 ] developed for the nonlinear three-dimensional analysis of reinforced concrete structures is used. This program mcoroporates the above-mentioned nonlinear ef, fects. The program uses an iterative solution scheme to analyze the nonlinear structural behavior. With this scheme, the solution is obtained by ntaking successixe linear approximations until the constitutive laws and the conditions of equilibrium and compatibility are satisfied within specified limits. The program offers several options of constant and variable stiffness m e l h ods for forming material rigidity matrices. A mote detailed description of the computer program A R ( and the iterative nonlinear analysis of concrete is given b~ Buvukozturk and Shareef [11 ] and Tulga [t 2 L 3.2. t:'inite element idealization In the finite element idealization for concrete, threedimensional isoparametric solid elements are used. Each element has 15 integration points where stresses and strains are stored. In the analysis, if a principal stress at an integration point exceeds the concrete tensile strength, a crack is defined at that integration point in the direction perpendicular to that principal direction. If the total strain in that direction continues to be tensile, the rigidity matrix at that integration point is altered to reflect the new stress distribution. The released stresses are applied to the rest of the structure at the next iteration. Formation of one crack fixes that principal direction and constrains the other two principal directions to be perpendicular to the crack. If two cracks are present, all directions are fixed, and the third principal direction has to be perpendicular to the other two. Precracking at integration points can be included in the analysis by specifying the angles between the normals to the cracks and the global directions. For reinforcement, a two-dimensional reinforcement element is used. Reinforcement of concrete elements can be modeled in two directions by specifying the reinforcing bar cross-sectional area and spacing between
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4. Comparison of experimental and predicted response In this section, the results of the experiments and the analysis are given. The predicted reinforcement stresses
191
B.S. Browzin et a L / Testing and uerification analysis
at axes 5, 8, and 11 (figs. 2 and 3), cracking patterns, and final failure modes are compared with the experimental findings. Specimens N1 and C1 - - Both experimental and predicted responses for this specimen indicate linear elastic behavior for the structure up to a load of P = 90 kips (400 kN). With further increase of the load, cracking in the concrete starts at the lower integration points of elements 3 and 5. Finite element analysis results show that most cracking takes place between P = 135 kips (600 kN) and P = 165 kips (734 kN) (fig. 5) as observed in the testing (fig. 6). As shown in fig. 5, horizontal cracking at element 5 starts at P = 135 kips (600 kN). With increased loading ( P = 290 kips, 1290 kN), cracking propagates upward accompanied by increased horizontal cracking in the lower parts of the structure. At P = 350 kips (1557 kN), finite element analysis predicts crushing at the column head of the cantilever (element 1). This was not observed in the experiment because the specimens were heavily reinforced at that location to prevent premature local failure by crushing, In the test, the load of 405 kips (1801 kN) caused the formation of a major crack extending to the intersection of the column face, which indicated failure resulting from high tensile stresses in concrete. At this failure stage, reinforcement stresses were below the yield stress level. The formation and extension of the crack at the column face intersection and the failure load of 405 kips (1800 kN) were also predicted by the analysis. The
Fig. 6, Specimen N 1, face 2, photographed after failure.
locations of predicted major cracks with the observed cracks from fig. 6 superimposed are illustrated in fig. 5. Fig. 6 shows specimen N1 photographed after failure. Numbers at the tip of the cracks shown in the photo indicate the load levels when cracks first formed. The last strain gage readings were made at a load of 390 kips (1735 kN). Overall, the predicted crack propagation, failure mode, and ultimate load level are in good agreement with the observed behavior. In figs. 7, 8 and 9, the steel reinforcement stresses for
Observed~
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P=290 kips, Calculated crack pattern
Fig. 5. Crack history for specimen N1 at various load levels.
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the test specimens as predicted by the analysis are compared with those from the experiments. The reinforcement stresses predicted by the analysis (shown by dots on the graphs) are in good agreement with the experimental observations in all three locations: at axes 11 and 8 and Support B. More detailed comparison is provided below. Both the test and the analysis indicated a "diagonal tension" type of failure with no yielding of the steel. Specimen C1 exhibited behavior similar to that of specimen N1, with a failure load of 236 kips (1050 kN). This premature failure is attributed to the failure of the specimen head.
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Specimens N2 and C2 - - With these specimens, a linear behavior was observed up to a load of P = 90 kips (400 kN). The predicted limit for linear behavior is P = 75 kips (334 kN). Below this load no cracking occurred, and the stresses in reinforcing at axes 8 and II closely agree with the experimental results as seen in figs. 7, 8, and 9. The steel stresses at a load of 75 kips (334 kN) are in good agreement at axis 8, but at axes 5 and 11, the steel stress was approximately 2000 psi (13.8 N / m m 2) lower. In the experimental data, there was a release of strain at axes 5 and 11 probably caused by a sudden release of friction restraint at the supports. Stresses in specimen C2 are larger, possibly because of the different arrangement of the supports. At a load level of 100 kips (489 kN), the analysis predicted extensive cracking throughout element 3 and the lower portion of element 5 (fig. 10). The
• Observed ~
/I I I'P=110 kips, Calculated crack pattern
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Fig. 10. Crack history for specimen N2 at various load levels.
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194
B.S. Browzm et al. / T~'sting and i'erJt}c,,~f~(,i aplalv~
predicted steel stresses at axes 5 and 11 are in perfect agreement with the experimental results. At axis 8, the predicted reinforcement stresses are somewhat higher than those measured. Cracks propagated upward through the specimen, and diagonal cracks were predicted as the load was increased to 150 kips (667 kN) (fig. 10). This corresponds to a diagonal crack observed in the test at 135 kips (600 kN). The failure was predicted at P = 200 kips (890 kN) (fig. 10), which is in agreement with the observed behavior. The observed failure loads for specimens C2 and N2 were 224 kips and 254 kips (996 kN and 1130 kN), respectively. Specimen N2, photographed after failure, is shown in fig. 11. "lhe cause of failure is the formation of diagonal cracks with simultaneous yielding of the steel. Specimen C3 - - Testing and analysis of this specimen indicate the linear elastic limit to be P = 50 kips (222 kN). Initial cracking is predicted to take place at the middle of the structure. With increased loading, cracking extends upward, and horizontal cracking takes place in the vicinity of the supports. At a load of 120 kips (534 kN), yielding of steel reinforcement occurs and cracking extends to the intersection of the column face and the sloping face of the cantilever. Hence failure took place by diagonal tension in concrete following the yielding of the reinforcement. The observed failure mode and reinforcement stresses agree satisfactorily with the analytical results as illustrated in figs. 7, 8, and 9. Specimen C4 - - Three load increments are used in the finite element analysis of this low-reinforcement-
Fig. 11. Specimen N2, face 1, photographed after failure.
ratio specimen. The elastic limit ~ preducted t,~ he m t}~,u proximity of P - 4 0 kips (178 k N ) ('racking i,~ re stricted to the middle elements onl2, and at ~ load ,;f P = 74 kips (329 kN), the iniddle crack widen~ and causes yielding of the reinforcemcm, l h c same 2~idding mode of failure is observed experimentally. Somc~hat higher reinforcement stresses are predicted by the anal'~sis than were observed in the experiments, a~ illustrated in figs. 7. 8. and 9. Specimen C5 The elastic limit for the specimen with no reinforcement is predicted to be 30 kips (134 kN). The test resulted in failure of the specimen at a load of P:-: 33 kips (147 kN). Fhe collapse of the structure at this load was caused bv the formation of a major crack. The failure load as compared witt~ the failure load of 74 kips (329 kN) for specimen (i'4 shows that even an insignificant amount of reinforcement ira.proves the loading capacity considerably. In thi~ case~ a reinforcement of 0.13% at axis 8 (table 1) more than doubled the loading capacity of lhe structure.
5. Summary on the behavior of specimens with various reinforcing ratios 5.1. Numerical comparison o f F E M analyszv with test data
Data on stresses analyzed by FEM and experimental stresses are listed in table 2. By judging the quality of performed tests, the most successful is the testing of specimen N1. The development of strain and formation of cracks were gradual and typical as can be expected from general knowledge of the behavior of reinforced concrete structural members. Specimen N2 had a deficiency in supports, and consequently the load--stress curve shows abnormal behavior: it turns back at the load 110 kips (489 kN) (fig. 7). Specimens C3 and C4 with a low reinforcement ratio exhibited relatively small strains in the beginning of testing and consequently the accuracy of the measurements is low. It is satisfactory at failure ( - 1 % and - 2 5 % at axis 11). Consequently, the FEM analysis may best be compared with the test results on specimen N1. Of 20 observed strains, 14 are within 10% agreement with F E M analysis and 17 are within 20% agreement. ]'he three points with an error larger than 20% are those at the beginning of testing when stresses are low, i.e., below 5 ksi (35 N / m m 2) when the accuracy of measurement is low. These results prove the validity and accuracy of the proposed F E M analysis. With respect to specimen N2, comparison of the stresses for loads above 110 kips (489 kN), when the
195
B.S. Browzin et al. / Testing and ~)erification analysis Table 2 Comparison of stresses analyzed by finite element method (FEM) with experimental stresses (test) Quarter of span, axis 11 Load (kips)
Specimen 90 135 155 226 286 348 390
Middle of span, axis 8
Stresses FEM
Test
(ksi)
(ksi)
% error
Load (kips)
Support B
Stresses FEM
Test
(ksi)
(ksi)
% error
Load (kips)
Stresses FEM
Test
(ksi)
(ksi)
0.51 5.13 7.03 10.77 14.97 20.41 24.10
1.00 3.25 6.25 10.00 14.00 18.75 24.50
% error
NI 3.30 9.31 11.72 20.69 27.31 35.03 38.30
2.90 9.00 13.00 17.50 25.50 33.00 38.00
14 3 -10 18 7 6 1
90 170 226 287 348 410
3.45 12.41 18.55 25.17 33.10 41.45
5.00 13.50 20.50 27.50 34.00 42.25
- 31 8 10 -8 - 3 - 2
Specimen N2 50 1.45 75 2.69 110 14.76 150 23.25 200 34.35
2.75 5.50 10.00 21.50 35.20
47 51 48 8 - 2
50 75 110 150 200
3.59 8.55 18.48 28.62 40.61
6.00 9.75 15.00 23.50 35.50
- 40 - 12 23 22 14
Specimen ('3 58 1.93 90 24.83 120 41.66
4.75 15.75 42.00
59 58 -1
58 90 120
4.69 28.97 49.38
6.75 21.00 44.50
-31 40 11
Specimen C4 33 1.31 60 4.00
1.50 5.30
13 - 25
l o a d - s t r e s s curve becomes linear, shows that the error is within 8% (at axis 11) beginning with a load of 150 kips, and 14% (at axis 8) beginning with the load 200 kips (896 kN). At support B, the prediction is not good for specimen N2 because the experimental curve shows some abnormal behavior; i.e., the curve turns back to the left as do other curves of this specimen. For specimen C3 and C4 the prediction at support B is not good because of low stresses in reinforcement. For specimen C4 comparison is made for axis 11 only.
5. 2. Characteristics o f specimen behavior at failure The results obtained from the tests and the analysis at failure are included in table 2. F r o m the experiments, three types of failure were observed. The first one is the development of a diagonal crack forming along the compression trajectories approximately directed from the load to the supports. Along the crack line, the principal tension stresses in concrete are developed to a level that is higher than the tension resistance of concrete. Specimen NI failed by this mode. Failure of
90 135 155 226 287 348 395
49 58 12 8 7 9 - 2
specimen C1 was premature because of the failure of the column head. The second mode of failure resembles the first because similar diagonal cracks were formed. However, it differs from the first by the magnitude of stress developed in the steel reinforcement. In the first mode of failure (N1), the steel stresses at failure are below the yielding point (about 40.0 ksi, 276 N / r a m 2), (fig. 7). In the second mode, the stresses are near or at the yielding point (at 45.0 ksi, 310 N / m m 2 ) , (fig. 7). It appears that the yielding in steel and the final extension of the diagonal crack appear simultaneously. The primary cause may be attributed to the yielding of steel. Specimens N2, C2, and C3 failed by this mode. The third observed mode consisted of failure by formation of a crack at the middle of the specimen and the spreading of this crack to the column face. Failure occurred by yielding of the b o t t o m steel with simultaneous cracking of the concrete in tension. High ultimate nominal shear stresses were obtained at failure of the tested specimens (table 3); at cross sections corresponding to axes 5 and 11, these stresses are 1.73 ksi (11.9 N / m m 2) for specimen N1, 1.09 ksi
196
B.S. Browzin el aL / l>sttng a n d l,er{pcalum ana/~ ~,~
Table 3 Specimen characteristics at failure Specimen
Load at last recorded strain (ksi)
Failure load (ksi)
Steel stress from last recorded strain, axis 11 (ksi)
Steel stress by FEM analysis, axis 11 (ksi)
Ratio steel stres,s, FEM to recorded
Nominal shear stress at failure, cross >ections at axes 5 and 1 1 (ksi)
Mode of failure
173
By diagonal tensile stress of concrete (normal to maior cracks)
NI
390
405
38.0
38.3
1.01
C1
180
236
18.7
14.9
0.80
N2
225
225
40.9
39.9
0.98
1.09
By diagonal tensile stress of concrete normal to major cracks following the yielding of bottom reinforcement
C2
180
224
36.0
29.9
0.83
0.96
Same as N2
C3
120
149
42.0
41.7
0.99
0.64
Same as N2
C4
60
71
5.3
4.0
0.75
0.30
By near horizontal tensile stress of concrete in the middle following the fairure of bottom reinforcement
-
-
0.14
By near horizontal tensile stress of concrete in the middle
C5
33
(7.5 N / m m 2) for N2, a n d 0.96 ksi (6.6 N / m m 2) for C2. Specimen C3 with light reinforcement also exhibited a high n o m i n a l shear stress: 0.64 ksi (4.4 N / m m 2 ) . Even in specimen C4, the n o m i n a l shear stress value, 0.30 ksi (2.1 N / m m 2 ) , was high c o m p a r e d with the allowable stresses for reinforced concrete elements (near 0.10 ksi, 0.7 N / m m 2, with no reinforcement). In table 3, steel stresses from the last recorded strain are c o m p a r e d with stresses predicted by the finite elem e n t method. Recorded-to-predicted steel stress ratios are 1.03 a n d 0.98 for specimens N1 a n d N2, respectively, indicating excellent agreement between the tests a n d the analysis. Specimens C1 a n d C2 exhibited higher stresses in all ranges of loading p r o b a b l y because of the frictional effects at the supports causing restraints. C o m p a r i s o n for specimens C1 a n d C2 gives ratios of 0.80 a n d 0.83, respectively. Specimen C3 is characterized b y a ratio of 0.99, a n d specimen C4 by 0.75. It is seen that the finite element m e t h o d is well confirmed b y the experiments reported in this paper.
Premature failure of the column head
6. Conclusion The behavior of deep reinforced concrete cantilevers subjected to monotonically increasing vertical loads was studied experimentally. The tested specimens were used to c o m p a r e test results with the results obtained by the finite element analysis. C o m p a r i s o n s are made of steel stress levels, cracking behavior, and failure modes. Both analysis and experiments show that the failure mode of deep cantilevers in some tests is by diagonal tension, in others by the yielding of the b o t t o m reinforcement. The failure by diagonal tension in concrete occurred in specimens with a higher reinforcement ratio. In the specimens with a lower reinforcement ratio, the failure occurred by failure of b o t t o m reinforcement. Ultimate n o m i n a l shear stress recorded at failure is substantially higher than that assumed in concrete codes for members w i t h o u t diagonal reinforcement. It appears that diagonal reinforcement for the deep cantilevers tested is superfluous.
B.S. Browzin et aL / Testing and verification analysis
The finite element method verified against the tested specimens in this report may be used further to analyze a n d establish the true stress distribution in deep cracked reinforced concrete structural members, which may have a particularly useful application for bridge piers with cantilevers largely used at present in the United States. By using the finite element method, further improvement in the design of deep structural members may be achieved.
Acknowledgements
The testing at the Case Institute of Technology was performed by J. Schleich who then was a graduate student working u n d e r the supervision of B.S. Browzin. The testing at the National Bureau of Standards was performed by B.S. Browzin with the participation of R.G. Mathey and L. Catanio. E.L. Hill and M. Mejac of N R C edited this paper. The authors greatly appreciate the c o n t r i b u t i o n of the above persons. The analysis part of this research was performed by S.S. Tulga at the Massachusetts Institute of Technology u n d e r the supervision of O. Buyukozturk. The computer program for the finite element analysis reported in this p a p e r was developed at M I T as part of the work performed u n d e r C o n t r a c t No. 04-77-057 for the U.S. Nuclear Regulatory Commission.
Nomenclature
tensile stress in steel, = tensile strength of concrete, = compression strength of concrete, P = total load at the top of specimens, R/C = reinforced concrete, FEM = finite element method, cm centimeter, kN = kilonewton, N / m m 2 = newton per square millimeter.
f( ~'
197
References
[1] L.B. Kriz and C.M. Raths, Connections in precast concrete structures--strength of corbels, Journal, Prestressed Concrete Institute 10 (1) (February 1965) 16 61. [2] A. Mehmel and G. Beckner, Shear tests of short cantilevers, Der Bauingenieur 40, Heft 6 (1965) 224-231. [3] A. Mehmel and W. Freitag, Investigation of the strength of reinforced concrete corbels, Der Bauingenieur 42, Heft 10 (1967) 362 369. [4] W. Sommerville, The behavior and design of reinforced concrete corbels, Technical Report No. 42.472, Cement and Concrete Association, London (August 1972). [5] A.H. Mattock, Design proposals for reinforced concrete corbels, Journal, Prestressed Concrete Institute 21 (3) (May-June 1976) 2-26. [6] B.S. Browzin, Testing and design of longitudinal reinforcement in cantilevered bridge piers, Transportation Research Board, 903, Bridges and Culverts, National Research Council, National Academy of Sciences, Washington, DC (1983) pp. 27 35. [7] P. Leombruni, O. Buyukozturk and J.J. Connor, Analysis of shear transfer in reinforced concrete with application to containment wall specimens, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-1085 (October 1979). [8] Tsi-Ming Tseng, J. Calvo, O. Buyukozturk and J. Connor, Behavioral model for reinforced concrete panels under cyclic shear, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-2451 (December 1981). [9] J. Calvo, O. Buyukozturk and J.J. Connor, Design of reinforced concrete containment wall elements under combined action of shear and tension, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-3157 (February 1983). [10] O.C. Zienkiewics, The Finite Element Method (McGraw Hill, New" York, 1982). [11] O. Buyukozturk and S.S. Shareef, Constitutive modeling of concrete in finite element analysis, J. Comput. Struct. (1984). [12] S.S. Tulga, Nonlinear elastic finite element analysis of concrete structures, Master's degree thesis submitted to the Department of Mechanical Engineering, Massachusetts Institute of Technology (June 1979). [13] G. Winter and A.H. Nilson, Design of Concrete Structures, Eighth Edition (McGraw Book Co., New York, 1972).
TESTING AND VERIFICATION ANAINSIS CONCRETE DOUBLE CANTILEVERS
187
BY F I N I T E E L E M E N T S
OF REINFORCED
B.S. B R O W Z I N US. Nuclear Regulatory Commi.v,~'ion, Washington, D(" 20555. USA
S.S. T U L G A Jot&re. Apostal and Rltter Asvociates. Dal,i~'t~ille. RI 02854. USA
and O, B U Y U K O Z T U R K Mas'sachmetts Institute of Technology,, ('amhridge, MA 02139. USA
A three-dimensional nonlinear finite element analysis code developed at the Massachusetts Institute of Technology has been verified against tested specimens of double deep cantilevers. The stress field in deep cantilevers is essentially two dimensional. For this reason, the verification is of particular interest. The results obtained indicate a high degree of accuracy of the method based on a comparison of analyzed versus experimental steel stress in bars. The method may be used for the design of deep reinforced concrete sections.
I. Introduction The objective of this paper is to demonstrate the application and verification of the finite element computer code developed at the Massachusetts Institute of Technology (MIT) for the analysis of complex concrete structures, by comparing predicted and observed behaviors and response quantities of tested double cantilevers. The behavior of reinforced concrete members with large height-to-span ratios is substantially different from that of shallow beams. Structural members are customarily considered "'deep" when the ratio of the effective depth to the span is equal to or greater than one. Deep beams, corbels or brackets, and deep cantilevers belong to the category of deep members, Corbels are short members projecting from the columns, whereas deep cantilevers are members representing the part of the beams continued over a support. A cantilever can be a symmetrical structure projecting from the two sides of a support. Wall footings, supports for span structures at the top of bridge piers, and supports in buildings at the top of columns for precast beams are examples of deep cantilevers. The stress and strain fields in deep cantilevers are complex, and generally, assumptions adopted
in the conventional design of reinforced concrete members in flexure and shear may not be adequate. For this reason, analysis using the finite element method, which incorporates essential elements of concrete behavior such as nonlinear behavior of concrete in multiaxial stress state and cracking, and the verification of such an analysis method are of particular interest. Several researchers have reported tests on corbels. Extensive experimental investigations of corbels were conducted at Portland Cement Association (PCA) laboratories by Kriz and Raths [1]. Other work on corbels was performed by Mehmel and Beckner [2], Mehmel and Freitag [3], Sommerville [4], and Mattock [5]. Research on deep reinforced concrete double cantilevers has been reported by Browzin [6]. This paper presents the work performed on the finite element analysis of the cantilever specimens tested by Browzin. Formation and propagation of cracks, failure modes, and their relation to the amount of reinforcement are predicted by the finite element analysis and compared with the experimental results. Experiments have provided measurements of strains at a number of locations on reinforcing bars. Comparison of analytical results with the experimental measurements provided
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Deep cantilevers were tested By B.S. Browzm ~t~ the National Bureau of Standards (NBS) and at the Cast Institute of Technology (CIT) [6]. For convenience. cantilevers were tested in an upside down position, and the load was applied at the top of the specimens by, 600-kip (2670-kN) capacity testing machines with 15-kip (67-kN) increments in the NBS tests (specimens NI and N2) and with 30-kip (134-kN) increments in the C11 tests (specimens ('l to C5). Specimens CI and ('2 were duplicates of specimens N1 and N2. respectively. The overall dinaensions of the specimens were 36 by 25 ~ h\ 12 in (91 by 65 by 30.5 cm). and the span between the supports was 27 in (69 cm) (fig. 1). The centroid of reinforcement was at 1.5 in (4 cm) from the b o t t o m The structural arrangement of tested typical specimens is shown in figs. 2 and 3. Axes numbered I through 15 are used to identify cross sections of the specimen. The distance between axes 1. 3, 5, and 7 is 5 in (12.7) cm. Axis 8 corresponds to the symmetry plane. The face of specimens with support ,4 at the left-hand side is identified as face 1: the opposite one is identified as face 2 (fig. 3). The specimens rested on 1 by 4 bv 12 in, (2.5 by 10.2 by 30.5 cm) supporting plates, thenlselves bearing on rollers at each side of the span. For
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189
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the five specimens tested at CIT, two rollers in contact were used at each support as a precaution against possible escape of the rollers under loading action. The two specimens tested at NBS had one roller welded to the lower plate at support A and one fully movable roller at support B. The ready-mixed concrete was made with high-early-strength cement, Type Ill, with 3/16- to 3 / 8 - in (0.5- to 0.95-cm) silica gravel and regular sand. The proportion of cement, sand, and gravel by weight was 1 : 2 : 4 . A compressive strength of 3000 psi (21 N / r a m 2) was specified, and higher strength was obtained. The material properties of the reinforcing steel used in specimens N and C were, respectively: yield point, 45 300 and 45000 psi (312 and 310 N / m m 2 ) ; ultimate strength, 74000 and 77000 psi (510 and 531 N / m m 2 ) ; elastic modulus 25.8 × 106 psi (178 × 103 N / m m ~) and 29.2 × 10 6 psi (201 × 103 N / m m 2 ) . The cross-sectional areas of the reinforcement for specimens N1 and C1
were selected to provide a reinforcement ratio approximately corresponding to the design by working stress method at axis 8 (i.e., f, = 20 000 psi and f,, = 1350 psi, L = 138 N / m m 2 and f~ = 9.3 N/ram2). Specimens N1 and C1 were reinforced by two layers of reinforcing bars; the others had one layer of bars. The concrete material properties and reinforcement characteristics of the specimens are summarized in table 1. Electrical resistance strain gages SR-4 of Type AB-7, 3 / 8 in (0.95 cm) long, were located on the reinforcing bars (figs. 2 and 3). In specimens N1 and N2, strain gages were located at the top and bottom of each bar, 20 gages in all. In specimen NI, which had two layers of bars, the gages were attached on bars of the lower layer. In specimens C1 to C4, strain gages were located at the bottom of bars, three gages in each specimen. The strain readings were corrected to obtain the strain and stresses in all specimens at the same locations, i.e., at the intersection of axes 5, 8, and 11 with the centroid of the reinforcement. The test results are presented as load versus experimental stress in reinforcement bars at five locations: approximately at the quarter points of the span (axes 5 and 11), at the middle of the span (axis 8), and at the supports.
3. Analysis of the tested specimens 3.1. The analysis method
The finite element method (FEM) of analysis was used to predict the behavior of the tested specimens. The method approximates the solution to a structural problem by subdividing the continuum into smaller regions [10]. In this paper, the method as applied to
Table 1 Material properties and reinforcement characteristics for the specimens tested Specimen
N1 CI N2 C2 C3 C4 C5
Concrete compressive strength (ksi)
Size and number of bars
Area of reinforcement (sq. in)
Reinforcement ratios at cross sections Axes 5 and 11
Axis 8
5.74 4.91 5.95 5.27 3.93 3.93 3.93
~ 4,15 :~ 4,15 ~4,7 ~ 4,7 ~ 3,5 ~ 3,3
3.0 3.0 1.4 1.4 0.55 0.33 0.00
0.0172 0.0172 0.0081 0.0081 0.0032 0.0019 -
0.0119 0.0119 0.0056 0.0056 0.0022 0.0013
190
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concrete structures is briefly described, and the I i n i i e element modeling of the deep cantilever specimens is presented. The analysis of concrete structures requires simulation of various important nonlinearities, some ~,f which arise from the nonlinearity of the stress straip, relationship for concrete, the anisotropy in concrete resulting from varying biaxial or triaxial stress fields. the continuously changing topology of concrete resuhing from cracking, postcracking shear transfer m e c h a nisms, and yielding and subsequent plastic behavior of the steel reinforcement. For the present study, the computer program A R ( [ 11 ] developed for the nonlinear three-dimensional analysis of reinforced concrete structures is used. This program mcoroporates the above-mentioned nonlinear ef, fects. The program uses an iterative solution scheme to analyze the nonlinear structural behavior. With this scheme, the solution is obtained by ntaking successixe linear approximations until the constitutive laws and the conditions of equilibrium and compatibility are satisfied within specified limits. The program offers several options of constant and variable stiffness m e l h ods for forming material rigidity matrices. A mote detailed description of the computer program A R ( and the iterative nonlinear analysis of concrete is given b~ Buvukozturk and Shareef [11 ] and Tulga [t 2 L 3.2. t:'inite element idealization In the finite element idealization for concrete, threedimensional isoparametric solid elements are used. Each element has 15 integration points where stresses and strains are stored. In the analysis, if a principal stress at an integration point exceeds the concrete tensile strength, a crack is defined at that integration point in the direction perpendicular to that principal direction. If the total strain in that direction continues to be tensile, the rigidity matrix at that integration point is altered to reflect the new stress distribution. The released stresses are applied to the rest of the structure at the next iteration. Formation of one crack fixes that principal direction and constrains the other two principal directions to be perpendicular to the crack. If two cracks are present, all directions are fixed, and the third principal direction has to be perpendicular to the other two. Precracking at integration points can be included in the analysis by specifying the angles between the normals to the cracks and the global directions. For reinforcement, a two-dimensional reinforcement element is used. Reinforcement of concrete elements can be modeled in two directions by specifying the reinforcing bar cross-sectional area and spacing between
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4. Comparison of experimental and predicted response In this section, the results of the experiments and the analysis are given. The predicted reinforcement stresses
191
B.S. Browzin et a L / Testing and uerification analysis
at axes 5, 8, and 11 (figs. 2 and 3), cracking patterns, and final failure modes are compared with the experimental findings. Specimens N1 and C1 - - Both experimental and predicted responses for this specimen indicate linear elastic behavior for the structure up to a load of P = 90 kips (400 kN). With further increase of the load, cracking in the concrete starts at the lower integration points of elements 3 and 5. Finite element analysis results show that most cracking takes place between P = 135 kips (600 kN) and P = 165 kips (734 kN) (fig. 5) as observed in the testing (fig. 6). As shown in fig. 5, horizontal cracking at element 5 starts at P = 135 kips (600 kN). With increased loading ( P = 290 kips, 1290 kN), cracking propagates upward accompanied by increased horizontal cracking in the lower parts of the structure. At P = 350 kips (1557 kN), finite element analysis predicts crushing at the column head of the cantilever (element 1). This was not observed in the experiment because the specimens were heavily reinforced at that location to prevent premature local failure by crushing, In the test, the load of 405 kips (1801 kN) caused the formation of a major crack extending to the intersection of the column face, which indicated failure resulting from high tensile stresses in concrete. At this failure stage, reinforcement stresses were below the yield stress level. The formation and extension of the crack at the column face intersection and the failure load of 405 kips (1800 kN) were also predicted by the analysis. The
Fig. 6, Specimen N 1, face 2, photographed after failure.
locations of predicted major cracks with the observed cracks from fig. 6 superimposed are illustrated in fig. 5. Fig. 6 shows specimen N1 photographed after failure. Numbers at the tip of the cracks shown in the photo indicate the load levels when cracks first formed. The last strain gage readings were made at a load of 390 kips (1735 kN). Overall, the predicted crack propagation, failure mode, and ultimate load level are in good agreement with the observed behavior. In figs. 7, 8 and 9, the steel reinforcement stresses for
Observed~
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P=290 kips, Calculated crack pattern
Fig. 5. Crack history for specimen N1 at various load levels.
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B.S. Browzin et al. / Testing and verification analysis 1
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the test specimens as predicted by the analysis are compared with those from the experiments. The reinforcement stresses predicted by the analysis (shown by dots on the graphs) are in good agreement with the experimental observations in all three locations: at axes 11 and 8 and Support B. More detailed comparison is provided below. Both the test and the analysis indicated a "diagonal tension" type of failure with no yielding of the steel. Specimen C1 exhibited behavior similar to that of specimen N1, with a failure load of 236 kips (1050 kN). This premature failure is attributed to the failure of the specimen head.
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Specimens N2 and C2 - - With these specimens, a linear behavior was observed up to a load of P = 90 kips (400 kN). The predicted limit for linear behavior is P = 75 kips (334 kN). Below this load no cracking occurred, and the stresses in reinforcing at axes 8 and II closely agree with the experimental results as seen in figs. 7, 8, and 9. The steel stresses at a load of 75 kips (334 kN) are in good agreement at axis 8, but at axes 5 and 11, the steel stress was approximately 2000 psi (13.8 N / m m 2) lower. In the experimental data, there was a release of strain at axes 5 and 11 probably caused by a sudden release of friction restraint at the supports. Stresses in specimen C2 are larger, possibly because of the different arrangement of the supports. At a load level of 100 kips (489 kN), the analysis predicted extensive cracking throughout element 3 and the lower portion of element 5 (fig. 10). The
• Observed ~
/I I I'P=110 kips, Calculated crack pattern
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Fig. 10. Crack history for specimen N2 at various load levels.
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P=200 kips, Comparison, Calculated crack pattern and observed at failure (at 225 kips)
194
B.S. Browzm et al. / T~'sting and i'erJt}c,,~f~(,i aplalv~
predicted steel stresses at axes 5 and 11 are in perfect agreement with the experimental results. At axis 8, the predicted reinforcement stresses are somewhat higher than those measured. Cracks propagated upward through the specimen, and diagonal cracks were predicted as the load was increased to 150 kips (667 kN) (fig. 10). This corresponds to a diagonal crack observed in the test at 135 kips (600 kN). The failure was predicted at P = 200 kips (890 kN) (fig. 10), which is in agreement with the observed behavior. The observed failure loads for specimens C2 and N2 were 224 kips and 254 kips (996 kN and 1130 kN), respectively. Specimen N2, photographed after failure, is shown in fig. 11. "lhe cause of failure is the formation of diagonal cracks with simultaneous yielding of the steel. Specimen C3 - - Testing and analysis of this specimen indicate the linear elastic limit to be P = 50 kips (222 kN). Initial cracking is predicted to take place at the middle of the structure. With increased loading, cracking extends upward, and horizontal cracking takes place in the vicinity of the supports. At a load of 120 kips (534 kN), yielding of steel reinforcement occurs and cracking extends to the intersection of the column face and the sloping face of the cantilever. Hence failure took place by diagonal tension in concrete following the yielding of the reinforcement. The observed failure mode and reinforcement stresses agree satisfactorily with the analytical results as illustrated in figs. 7, 8, and 9. Specimen C4 - - Three load increments are used in the finite element analysis of this low-reinforcement-
Fig. 11. Specimen N2, face 1, photographed after failure.
ratio specimen. The elastic limit ~ preducted t,~ he m t}~,u proximity of P - 4 0 kips (178 k N ) ('racking i,~ re stricted to the middle elements onl2, and at ~ load ,;f P = 74 kips (329 kN), the iniddle crack widen~ and causes yielding of the reinforcemcm, l h c same 2~idding mode of failure is observed experimentally. Somc~hat higher reinforcement stresses are predicted by the anal'~sis than were observed in the experiments, a~ illustrated in figs. 7. 8. and 9. Specimen C5 The elastic limit for the specimen with no reinforcement is predicted to be 30 kips (134 kN). The test resulted in failure of the specimen at a load of P:-: 33 kips (147 kN). Fhe collapse of the structure at this load was caused bv the formation of a major crack. The failure load as compared witt~ the failure load of 74 kips (329 kN) for specimen (i'4 shows that even an insignificant amount of reinforcement ira.proves the loading capacity considerably. In thi~ case~ a reinforcement of 0.13% at axis 8 (table 1) more than doubled the loading capacity of lhe structure.
5. Summary on the behavior of specimens with various reinforcing ratios 5.1. Numerical comparison o f F E M analyszv with test data
Data on stresses analyzed by FEM and experimental stresses are listed in table 2. By judging the quality of performed tests, the most successful is the testing of specimen N1. The development of strain and formation of cracks were gradual and typical as can be expected from general knowledge of the behavior of reinforced concrete structural members. Specimen N2 had a deficiency in supports, and consequently the load--stress curve shows abnormal behavior: it turns back at the load 110 kips (489 kN) (fig. 7). Specimens C3 and C4 with a low reinforcement ratio exhibited relatively small strains in the beginning of testing and consequently the accuracy of the measurements is low. It is satisfactory at failure ( - 1 % and - 2 5 % at axis 11). Consequently, the FEM analysis may best be compared with the test results on specimen N1. Of 20 observed strains, 14 are within 10% agreement with F E M analysis and 17 are within 20% agreement. ]'he three points with an error larger than 20% are those at the beginning of testing when stresses are low, i.e., below 5 ksi (35 N / m m 2) when the accuracy of measurement is low. These results prove the validity and accuracy of the proposed F E M analysis. With respect to specimen N2, comparison of the stresses for loads above 110 kips (489 kN), when the
195
B.S. Browzin et al. / Testing and ~)erification analysis Table 2 Comparison of stresses analyzed by finite element method (FEM) with experimental stresses (test) Quarter of span, axis 11 Load (kips)
Specimen 90 135 155 226 286 348 390
Middle of span, axis 8
Stresses FEM
Test
(ksi)
(ksi)
% error
Load (kips)
Support B
Stresses FEM
Test
(ksi)
(ksi)
% error
Load (kips)
Stresses FEM
Test
(ksi)
(ksi)
0.51 5.13 7.03 10.77 14.97 20.41 24.10
1.00 3.25 6.25 10.00 14.00 18.75 24.50
% error
NI 3.30 9.31 11.72 20.69 27.31 35.03 38.30
2.90 9.00 13.00 17.50 25.50 33.00 38.00
14 3 -10 18 7 6 1
90 170 226 287 348 410
3.45 12.41 18.55 25.17 33.10 41.45
5.00 13.50 20.50 27.50 34.00 42.25
- 31 8 10 -8 - 3 - 2
Specimen N2 50 1.45 75 2.69 110 14.76 150 23.25 200 34.35
2.75 5.50 10.00 21.50 35.20
47 51 48 8 - 2
50 75 110 150 200
3.59 8.55 18.48 28.62 40.61
6.00 9.75 15.00 23.50 35.50
- 40 - 12 23 22 14
Specimen ('3 58 1.93 90 24.83 120 41.66
4.75 15.75 42.00
59 58 -1
58 90 120
4.69 28.97 49.38
6.75 21.00 44.50
-31 40 11
Specimen C4 33 1.31 60 4.00
1.50 5.30
13 - 25
l o a d - s t r e s s curve becomes linear, shows that the error is within 8% (at axis 11) beginning with a load of 150 kips, and 14% (at axis 8) beginning with the load 200 kips (896 kN). At support B, the prediction is not good for specimen N2 because the experimental curve shows some abnormal behavior; i.e., the curve turns back to the left as do other curves of this specimen. For specimen C3 and C4 the prediction at support B is not good because of low stresses in reinforcement. For specimen C4 comparison is made for axis 11 only.
5. 2. Characteristics o f specimen behavior at failure The results obtained from the tests and the analysis at failure are included in table 2. F r o m the experiments, three types of failure were observed. The first one is the development of a diagonal crack forming along the compression trajectories approximately directed from the load to the supports. Along the crack line, the principal tension stresses in concrete are developed to a level that is higher than the tension resistance of concrete. Specimen NI failed by this mode. Failure of
90 135 155 226 287 348 395
49 58 12 8 7 9 - 2
specimen C1 was premature because of the failure of the column head. The second mode of failure resembles the first because similar diagonal cracks were formed. However, it differs from the first by the magnitude of stress developed in the steel reinforcement. In the first mode of failure (N1), the steel stresses at failure are below the yielding point (about 40.0 ksi, 276 N / r a m 2), (fig. 7). In the second mode, the stresses are near or at the yielding point (at 45.0 ksi, 310 N / m m 2 ) , (fig. 7). It appears that the yielding in steel and the final extension of the diagonal crack appear simultaneously. The primary cause may be attributed to the yielding of steel. Specimens N2, C2, and C3 failed by this mode. The third observed mode consisted of failure by formation of a crack at the middle of the specimen and the spreading of this crack to the column face. Failure occurred by yielding of the b o t t o m steel with simultaneous cracking of the concrete in tension. High ultimate nominal shear stresses were obtained at failure of the tested specimens (table 3); at cross sections corresponding to axes 5 and 11, these stresses are 1.73 ksi (11.9 N / m m 2) for specimen N1, 1.09 ksi
196
B.S. Browzin el aL / l>sttng a n d l,er{pcalum ana/~ ~,~
Table 3 Specimen characteristics at failure Specimen
Load at last recorded strain (ksi)
Failure load (ksi)
Steel stress from last recorded strain, axis 11 (ksi)
Steel stress by FEM analysis, axis 11 (ksi)
Ratio steel stres,s, FEM to recorded
Nominal shear stress at failure, cross >ections at axes 5 and 1 1 (ksi)
Mode of failure
173
By diagonal tensile stress of concrete (normal to maior cracks)
NI
390
405
38.0
38.3
1.01
C1
180
236
18.7
14.9
0.80
N2
225
225
40.9
39.9
0.98
1.09
By diagonal tensile stress of concrete normal to major cracks following the yielding of bottom reinforcement
C2
180
224
36.0
29.9
0.83
0.96
Same as N2
C3
120
149
42.0
41.7
0.99
0.64
Same as N2
C4
60
71
5.3
4.0
0.75
0.30
By near horizontal tensile stress of concrete in the middle following the fairure of bottom reinforcement
-
-
0.14
By near horizontal tensile stress of concrete in the middle
C5
33
(7.5 N / m m 2) for N2, a n d 0.96 ksi (6.6 N / m m 2) for C2. Specimen C3 with light reinforcement also exhibited a high n o m i n a l shear stress: 0.64 ksi (4.4 N / m m 2 ) . Even in specimen C4, the n o m i n a l shear stress value, 0.30 ksi (2.1 N / m m 2 ) , was high c o m p a r e d with the allowable stresses for reinforced concrete elements (near 0.10 ksi, 0.7 N / m m 2, with no reinforcement). In table 3, steel stresses from the last recorded strain are c o m p a r e d with stresses predicted by the finite elem e n t method. Recorded-to-predicted steel stress ratios are 1.03 a n d 0.98 for specimens N1 a n d N2, respectively, indicating excellent agreement between the tests a n d the analysis. Specimens C1 a n d C2 exhibited higher stresses in all ranges of loading p r o b a b l y because of the frictional effects at the supports causing restraints. C o m p a r i s o n for specimens C1 a n d C2 gives ratios of 0.80 a n d 0.83, respectively. Specimen C3 is characterized b y a ratio of 0.99, a n d specimen C4 by 0.75. It is seen that the finite element m e t h o d is well confirmed b y the experiments reported in this paper.
Premature failure of the column head
6. Conclusion The behavior of deep reinforced concrete cantilevers subjected to monotonically increasing vertical loads was studied experimentally. The tested specimens were used to c o m p a r e test results with the results obtained by the finite element analysis. C o m p a r i s o n s are made of steel stress levels, cracking behavior, and failure modes. Both analysis and experiments show that the failure mode of deep cantilevers in some tests is by diagonal tension, in others by the yielding of the b o t t o m reinforcement. The failure by diagonal tension in concrete occurred in specimens with a higher reinforcement ratio. In the specimens with a lower reinforcement ratio, the failure occurred by failure of b o t t o m reinforcement. Ultimate n o m i n a l shear stress recorded at failure is substantially higher than that assumed in concrete codes for members w i t h o u t diagonal reinforcement. It appears that diagonal reinforcement for the deep cantilevers tested is superfluous.
B.S. Browzin et aL / Testing and verification analysis
The finite element method verified against the tested specimens in this report may be used further to analyze a n d establish the true stress distribution in deep cracked reinforced concrete structural members, which may have a particularly useful application for bridge piers with cantilevers largely used at present in the United States. By using the finite element method, further improvement in the design of deep structural members may be achieved.
Acknowledgements
The testing at the Case Institute of Technology was performed by J. Schleich who then was a graduate student working u n d e r the supervision of B.S. Browzin. The testing at the National Bureau of Standards was performed by B.S. Browzin with the participation of R.G. Mathey and L. Catanio. E.L. Hill and M. Mejac of N R C edited this paper. The authors greatly appreciate the c o n t r i b u t i o n of the above persons. The analysis part of this research was performed by S.S. Tulga at the Massachusetts Institute of Technology u n d e r the supervision of O. Buyukozturk. The computer program for the finite element analysis reported in this p a p e r was developed at M I T as part of the work performed u n d e r C o n t r a c t No. 04-77-057 for the U.S. Nuclear Regulatory Commission.
Nomenclature
tensile stress in steel, = tensile strength of concrete, = compression strength of concrete, P = total load at the top of specimens, R/C = reinforced concrete, FEM = finite element method, cm centimeter, kN = kilonewton, N / m m 2 = newton per square millimeter.
f( ~'
197
References
[1] L.B. Kriz and C.M. Raths, Connections in precast concrete structures--strength of corbels, Journal, Prestressed Concrete Institute 10 (1) (February 1965) 16 61. [2] A. Mehmel and G. Beckner, Shear tests of short cantilevers, Der Bauingenieur 40, Heft 6 (1965) 224-231. [3] A. Mehmel and W. Freitag, Investigation of the strength of reinforced concrete corbels, Der Bauingenieur 42, Heft 10 (1967) 362 369. [4] W. Sommerville, The behavior and design of reinforced concrete corbels, Technical Report No. 42.472, Cement and Concrete Association, London (August 1972). [5] A.H. Mattock, Design proposals for reinforced concrete corbels, Journal, Prestressed Concrete Institute 21 (3) (May-June 1976) 2-26. [6] B.S. Browzin, Testing and design of longitudinal reinforcement in cantilevered bridge piers, Transportation Research Board, 903, Bridges and Culverts, National Research Council, National Academy of Sciences, Washington, DC (1983) pp. 27 35. [7] P. Leombruni, O. Buyukozturk and J.J. Connor, Analysis of shear transfer in reinforced concrete with application to containment wall specimens, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-1085 (October 1979). [8] Tsi-Ming Tseng, J. Calvo, O. Buyukozturk and J. Connor, Behavioral model for reinforced concrete panels under cyclic shear, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-2451 (December 1981). [9] J. Calvo, O. Buyukozturk and J.J. Connor, Design of reinforced concrete containment wall elements under combined action of shear and tension, U.S. Nuclear Regulatory Commission, Massachusetts Institute of Technology, NUREG/CR-3157 (February 1983). [10] O.C. Zienkiewics, The Finite Element Method (McGraw Hill, New" York, 1982). [11] O. Buyukozturk and S.S. Shareef, Constitutive modeling of concrete in finite element analysis, J. Comput. Struct. (1984). [12] S.S. Tulga, Nonlinear elastic finite element analysis of concrete structures, Master's degree thesis submitted to the Department of Mechanical Engineering, Massachusetts Institute of Technology (June 1979). [13] G. Winter and A.H. Nilson, Design of Concrete Structures, Eighth Edition (McGraw Book Co., New York, 1972).