FLEXURAL BEHAVIOR OF REINFORCED AND PRESTRESSED CONCRETE BEAMS USING FINITE ELEMENT ANALYSIS by Anthony J. Wolanski, B.S. A Thesis submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Master of Science Milwaukee, Wisconsin May, 2004
87
Embed
Flexural Behavior of Reinforced and Pre Stressed Concrete Beams Using Finite Element Analysis
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
FLEXURAL BEHAVIOR OF
REINFORCED AND PRESTRESSED CONCRETE BEAMS
USING FINITE ELEMENT ANALYSIS
by
Anthony J. Wolanski, B.S.
A Thesis submitted to the Faculty of the Graduate School,
Marquette University, in Partial Fulfillment of
the Requirements for the Degree of
Master of Science
Milwaukee, Wisconsin May, 2004
PREFACE
Several methods have been utilized to study the response of concrete structural
components. Experimental based testing has been widely used as a means to analyze
individual elements and the effects of concrete strength under loading. The use of finite
element analysis to study these components has also been used.
This thesis is a study of reinforced and prestressed concrete beams using finite
element analysis to understand their load-deflection response. A reinforced concrete
beam model is studied and compared to experimental data.
The parameters for the reinforced concrete model were then used to model a
prestressed concrete beam. Characteristic points on the load-deformation response curve
predicted using finite element analysis were compared to theoretical (hand-calculated)
results.
Conclusions were then made as to the accuracy of using finite element modeling
for analysis of concrete. The results compared well to experimental and hand calculated.
ACKNOWLEDGMENTS
This research was performed under the supervision of Dr. Christopher M. Foley. I am
extremely grateful for the guidance, knowledge, understanding, and numerous hours
spent helping me complete this thesis. Appreciation is also extended to my thesis
committee, Dr. Stephen M. Heinrich and Dr. Baolin Wan, for their time and efforts.
I would like to thank my parents, John and Sue Wolanski, my brother, John
Wolanski, and my sister, Christine Wolanski for their understanding, encouragement and
support. Without my family these accomplishments would not have been possible.
The 93 in. dimension for the X-coordinates is the mid-span of the beam. Due to
symmetry, only one loading plate and one support plate are needed. The support is a 3 in.
x 5 in. x 1 in. steel plate, while the plate at the load point is 6 in. x 5 in. x 1 in. The
dimensions for the plate and support are shown in Table 3.6. The combined volumes of
the plate, support, and beam are shown in Figure 3.9. The FE mesh for the beam model
is shown in Figure 3.10.
Figure 3.9 – Volumes Created in ANSYS
Steel Support
Steel Loading Plate
Concrete Beam
Figure 3.10 – Mesh of the Concrete, Steel Plate, and Steel Support
Link8 elements were used to create the flexural and shear reinforcement.
Reinforcement exists at a plane of symmetry and in the beam. The area of steel at the
plane of symmetry is one half the normal area for a #5 bar because one half of the bar is
cut off. Shear stirrups are modeled throughout the beam. Only half of the stirrup is
modeled because of the symmetry of the beam. Figure 3.11 illustrates that the rebar
shares the same nodes at the points that it intersects the shear stirrups. The element type
number, material number, and real constant set number for the calibration model were set
for each mesh as shown in Table 3.7.
Steel Plate Element Width 1.25 in.
Concrete Element Length 1.5 in.
Concrete Element Width 1.25 in.
Concrete Element Height 1.2 in.
Steel Plate Element Length 1.5 in.
Steel Support Element Width 1.25 in.
Steel Support Element Length 1.5 in.
Figure 3.11 – Reinforcement Configuration
Table 3.7 – Mesh Attributes for the Model
Model Parts Element Type
Material Number
Real Constant Set
Concrete Beam 1 1 1 Steel Plate 2 3 N/A Steel Support 2 3 N/A Rebar at Center of Cross-Section 3 2 3 Rebar 2.5 in. of Cross-Section 3 2 2 Stirrup at Center of Beam 3 2 5 Other Stirrups 3 2 4
3.2.5 Meshing
To obtain good results from the Solid65 element, the use of a rectangular mesh is
recommended. Therefore, the mesh was set up such that square or rectangular elements
#3 Shear Stirrups
#5 Bar Reinforcement located 2.5 in. from the end of the Cross-Section
Shared nodes of Stirrups and Rebar
#5 Bar Reinforcement at Plane of Symmetry
Stirrup at Plane of Symmetry
were created (Figure 3.10). The volume sweep command was used to mesh the steel
plate and support. This properly sets the width and length of elements in the plates to be
consistent with the elements and nodes in the concrete portions of the model.
The overall mesh of the concrete, plate, and support volumes is shown in Figure
3.10. The necessary element divisions are noted. The meshing of the reinforcement is a
special case compared to the volumes. No mesh of the reinforcement is needed because
individual elements were created in the modeling through the nodes created by the mesh
of the concrete volume. However, the necessary mesh attributes as described above need
to be set before each section of the reinforcement is created.
3.2.6 Numbering Controls
The command merge items merges separate entities that have the same location. These
items will then be merged into single entities. Caution must be taken when merging
entities in a model that has already been meshed because the order in which merging
occurs is significant. Merging keypoints before nodes can result in some of the nodes
becoming “orphaned”; that is, the nodes lose their association with the solid model. The
orphaned nodes can cause certain operations (such as boundary condition transfers,
surface load transfers, and so on) to fail. Care must be taken to always merge in the order
that the entities appear. All precautions were taken to ensure that everything was merged
in the proper order. Also, the lowest number was retained during merging.
3.2.7 Loads and Boundary Conditions
Displacement boundary conditions are needed to constrain the model to get a unique
solution. To ensure that the model acts the same way as the experimental beam,
boundary conditions need to be applied at points of symmetry, and where the supports
and loadings exist.
The symmetry boundary conditions were set first. The model being used is
symmetric about two planes. The boundary conditions for both planes of symmetry are
shown in Figure 3.12.
Figure 3.12 – Boundary Conditions for Planes of Symmetry
Constraint in the z-direction
Constraint in the x-direction
Nodes defining a vertical plane through the beam cross-section centroid defines a plane
of symmetry. To model the symmetry, nodes on this plane must be constrained in the
perpendicular direction. These nodes, therefore, have a degree of freedom constraint UX
= 0. Second, all nodes selected at Z = 0 define another plane of symmetry. These nodes
were given the constraint UZ = 0.
The support was modeled in such a way that a roller was created. A single line of
nodes on the plate were given constraint in the UY, and UZ directions, applied as
constant values of 0. By doing this, the beam will be allowed to rotate at the support. The
support condition is shown in Figure 3.13.
Figure 3.13 – Boundary Condition for Support
Support roller condition to allow rotation
The force, P, applied at the steel plate is applied across the entire centerline of the
plate. The force applied at each node on the plate is one tenth of the actual force applied.
Figure 3.14 illustrates the plate and applied loading.
Figure 3.14 – Boundary Conditions at the Loading Plate
3.2.8 Analysis Type
The finite element model for this analysis is a simple beam under transverse loading. For
the purposes of this model, the Static analysis type is utilized.
The Restart command is utilized to restart an analysis after the initial run or load
step has been completed. The use of the restart option will be detailed in the analysis
portion of the discussion.
Loading Applied on the Plate
Boundary Conditions at Plate
The Sol’n Controls command dictates the use of a linear or non-linear solution for
the finite element model. Typical commands utilized in a nonlinear static analysis are
shown in Table 3.8.
Table 3.8 – Commands Used to Control Nonlinear Analysis
Analysis Options Small DisplacementCalculate Prestress Effects No Time at End of Loadstep 5120 Automatic Time Stepping On Number of Substeps 1 Max no. of Substeps 2 Min no. of Substeps 1 Write Items to Results File All Solution Items Frequency Write Every Substep
In the particular case considered in this thesis the analysis is small displacement and
static. The time at the end of the load step refers to the ending load per load step. Table
3.8 shows the first load step taken (e.g. up to first cracking). The sub steps are set to
indicate load increments used for this analysis. The commands used to control the solver
and output are shown in Table 3.9.
Table 3.9 – Commands Used to Control Output
Equation Solvers Sparse Direct Number of Restart Files 1 Frequency Write Every Substep
All these values are set to ANSYS (SAS 2003) defaults. The commands used for the
nonlinear algorithm and convergence criteria are shown in Table 3.10. All values for the
nonlinear algorithm are set to defaults.
Table 3.10 – Nonlinear Algorithm and Convergence Criteria Parameters
Line Search Off DOF solution predictor Prog Chosen Maximum number of iteration 100 Cutback Control Cutback according to predicted number of iter. Equiv. Plastic Strain 0.15 Explicit Creep ratio 0.1 Implicit Creep ratio 0 Incremental displacement 10000000 Points per cycle 13
Set Convergence Criteria Label F U Ref. Value Calculated calculated Tolerance 0.005 0.05 Norm L2 L2 Min. Ref. not applicable not applicable
The values for the convergence criteria are set to defaults except for the tolerances. The
tolerances for force and displacement are set as 5 times the default values. Table 3.11
shows the commands used for the advanced nonlinear settings. The program behavior
upon nonconvergence for this analysis was set such that the program will terminate but
not exit. The rest of the commands were set to defaults.
Table 3.11 – Advanced Nonlinear Control Settings Used
Program behavior upon nonconvergence Terminate but do not exit Nodal DOF sol'n 0 Cumulative iter 0 Elapsed time 0 CPU time 0
3.2.9 Analysis Process for the Finite Element Model
The FE analysis of the model was set up to examine three different behaviors: initial
cracking of the beam, yielding of the steel reinforcement, and the strength limit state of
the beam. The Newton-Raphson method of analysis was used to compute the nonlinear
response.
The application of the loads up to failure was done incrementally as required by
the Newton-Raphson procedure. After each load increment was applied, the restart
option was used to go to the next step after convergence. A listing of the load steps, sub
steps, and loads applied per restart file are shown in Table 3.12.
Table 3.12 – Load Increment for Analysis of Finite Element Model
Figure 4.2 – Load vs. Deflection Curve for Prestressed Concrete Model
Initial Cracking
Steel Yielding
Failure
Zero Deflection
DecompressionSelf-Weight
Initial Cracking
Linear Region
Secondary Linear Region
SEE FIGURE BELOW
Prestressing
The deflection in ANSYS (SAS 2003) corresponds well to the calculated value. The
stresses in the extreme fibers were looked at in the beam also. Since camber is occurring
in the beam (Figure 4.3), the controlling stress exists in the extreme top fiber.
Figure 4.3 – Deflection due to prestress
The hand-calculated top fiber stress and the top fiber stress found using FEA are also
shown in Table 4.4. The values correlate very well.
A phenomenon that occurred when the prestress was added was bursting in the
concrete where the prestress force is being applied. This phenomenon can be seen in
Figure 4.4. The contour plot shows that at the time of prestressing, there is a maximum
Camber due to prestressing force
Deflection (in.)
stress in the concrete where bursting occurs. The bursting phenomenon is explained in
detail in Nawy (2000).
Figure 4.4 – Bursting Phenomenon
Also, localized cracking occurs in the concrete, as shown in Figure 4.5. However, this
does not impact the solution because no other cracking occurs in that area after the initial
application of prestress. When a traditional level of prestress was applied
( 159,840pef psi= ), bursting zone cracking was extensive – a converged solution was not
possible to obtain. This is the reason for the reduction in pef by the factor of 3.
Bursting Effect at Center of Beam
Tendon Level
Figure 4.5 – Localized Cracking From Effective Prestress Application
4.2.2 Self-Weight
Adding the self-weight gives a deflection value which corresponds well to the calculated
deflection in Appendix B (Table 4.4). The FEA calculated top fiber stress at this stage
also correlates well with hand-calculations.
4.2.3 Zero Deflection
Hand-calculations (Appendix B) determined the load at which the deflection of the
centerline was zero. Table 4.4 indicates very good correlation between the hand
calculated values and the FEA.
Cracking that occurs from bursting
4.2.4 Decompression
One definition of decompression denotes the point of loading where the stress at the
bottom fiber of the concrete beam moves from compression due to the prestress to
tension from superimposed loading. Table 4.4 indicates very good correlation at this
level of loading as well.
4.2.5 Initial Cracking
Initial cracking is defined to be the loading at which the extreme tension fiber reaches the
modulus of rupture. Initial cracking of the beam in the FE model occurs at load 7850 lbs.
The hand-calculated load where cracking occurs is 7538 lbs. (Table 4.4 and Appendix B).
This is just past the modulus of rupture of the beam of 520 psi. The stress increases up to
539 psi at the centerline when the first crack occurs. This first crack occurs in the
constant moment region, and is a flexural crack.
4.2.6 Secondary Linear Region
In the secondary linear region of the response, significantly more cracking occurs as more
load is applied the beam as seen in Figure 4.6. Cracking increases in the constant
moment region, and the beam begins cracking out towards the supports with at a load of
12,000 lbs. Additional flexural cracking occurs in the beam at 20,000 lbs. Also, diagonal
tension cracks are beginning to form in the model.
Figure 4.6 – Cracking at 12,000 and 20,000 lbs.
4.2.7 Behavior of Steel Yielding and Beyond
Yielding of the prestress steel is defined as 0.85 puf for this model. Figure 4.2 illustrates
the load level at the point of yielding of the prestress steel. At this point in the response,
the displacements of the beam begin to increase at a higher rate as more load is applied.
The ability of the beam to distribute load throughout the cross-section has diminished
Flexural CracksBursting Cracks
Diagonal Tension Cracks
greatly. Therefore, greater deflections occur at the beam centerline. Increasing flexural
cracks and diagonal tension cracks form as the beam approaches failure (Figure 4.6).
4.2.8 Flexural Limit State
At a load of 28,823 lbs. unresolvable non-convergence of the nonlinear algorithm occurs,
indicating that cracking throughout the entire constant moment region has occurred.
Figure 4.7 illustrates the excessive cracking in the beam.
Figure 4.7 – Cracking at Flexural Capacity
Cracking at Failure
Hand calculations (Appendix B) predicted that the flexural capacity of the beam would
correspond to 27,587 lbs. (Table 4.4). The FEA prediction (28,823 lbs. in Table 4.4)
corresponds very well with the hand calculations. The stress in the prestressing steel at
failure predicted using FEA was 264,822 psi (Table 4.4). Using strain compatibility the
stress in the prestressing steel at failure was 254,520 psi (Appendix B), which
corresponds well to the FEA prediction.
The application of the moderate prestressing, 3pef , assumed in the present case
increased the limit load for the beam from 16,310 lbs. (Chapter 3) to 28,823 lbs. (near
doubling). The benefits of prestressing are apparent.
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5.0 Introduction
The use of the finite element method to analyze reinforced and prestressed concrete
beams was evaluated. A reinforced concrete beam model was calibrated to experimental
data, and predictions of initial cracking, yielding of the steel and flexural failure of the
beam were compared to the experimental results. A prestressed concrete beam model
was also created. Initial prestress, addition of self-weight, zero deflection point,
decompression, initial cracking, yielding of steel, and flexural failure were then studied
and compared to theoretical values obtained via accepted methods of hand calculation.
5.1 Conclusions
The following conclusions can be stated based on the evaluation of the analyses of the
calibration model and the prestressed concrete beam.
(1) Deflections and stresses at the centerline along with initial and progressive
cracking of the finite element model compare well to experimental data obtained
from a reinforced concrete beam.
(2) The failure mechanism of a reinforced concrete beam is modeled quite well using
FEA, and the failure load predicted is very close to the failure load measured
during experimental testing.
(3) For the prestressed concrete beam, camber due to the initial prestress force and
after application of the self-weight of the beam compares well to hand-computed
values. Also, a bursting effect was seen in the FE model.
(4) Deflections and stresses at the zero deflection point and decompression are
modeled well using a finite element package.
(5) The load applied to cause initial cracking of the prestressed concrete beam
compares well with hand calculations.
(6) Flexural failure of the prestressed concrete beam is modeled well using a finite
element package, and the load applied at failure is very close to hand calculated
results.
5.2 Recommendations for Future Work
The literature review and analysis procedure utilized in this thesis has provided useful
insight for future application of a finite element package as a method of analysis. To
ensure that the finite element model is producing results that can be used for study, any
model should be calibrated with good experimental data. This will then provide the
proper modeling parameters needed for later use.
While modeling the prestressed beam, relaxation losses due to prestress, creep,
shrinkage, and elastic shortening were lumped together in a single load step. Individual
modeling of these losses could be included in future research. Many prestressed
structural components have non-symmetric geometries and loadings. Therefore, non-
symmetric geometries and loadings should be analyzed using finite element analysis with
prestress for further study. Higher strength concrete, the bursting effect, and the transfer
length of the prestressing steel are all candidates for future research.
REFERENCES
American Concrete Institute (1978), Douglas McHenry International Symposium on Concrete and Concrete Structures, American Concrete Institute, Detroit, Michigan. Branson, D.E.; Meyers, B.L.; and Kripanarayanan, K.M. (1970), “Loss of Prestress, Camber and Deflection of Noncomposite and Composite Structures Using Different Weight Concrete,” Iowa State Highway Comission, Report No. 70-6, Aug. Buckhouse, E.R. (1997), “External Flexural Reinforcement of Existing Reinforced Concrete Beams Using Bolted Steel Channels,” Master’s Thesis, Marquette University, Milwaukee, Wisconsin. Faherty, K.F. (1972), “An Analysis of a Reinforced and a Prestressed Concrete Beam by Finite Element Method,” Doctorate’s Thesis, University of Iowa, Iowa City. Fanning, P. (2001), “Nonlinear Models of Reinforced and Post-tensioned Concrete Beams,” Electronic Journal of Structural Engineering, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland, Sept.12. Kachlakev, D.I.; Miller, T.; Yim, S.; Chansawat, K.; Potisuk, T. (2001), “Finite Element Modeling of Reinforced Concrete Structures Strengthened With FRP Laminates,” California Polytechnic State University, San Luis Obispo, CA and Oregon State University, Corvallis, OR for Oregon Department of Transportation, May. Janney, J.R. (1954), “Nature of Bond in Pre-tensioned Prestressed Concrete,” Journal of the ACI, Proceedings, Vol.50, No.5, May. MacGregor, J.G. (1992), Reinforced Concrete Mechanics and Design, Prentice-Hall, Inc., Englewood Cliffs, NJ. McCurry, D., Jr. and Kachlakev, D.I (2000), “Strengthening of Full Sized Reinforced Concrete Beam Using FRP Laminates and Monitoring with Fiber Optic Strain Guages” in Innovative Systems for Seismic Repair and Rehabilitation of Structures, Design and Applications, Technomic Publishing Co., Inc., Philadelphia, PA, March. Nawy, E.G., (2000), Prestressed Concrete: A Fundamental Approach, Prentice-Hall, Inc., Upper Saddle River, NJ SAS (2003) ANSYS 7.1 Finite Element Analysis System, SAS IP, Inc. Shing, P.B. and Tanabe, T.A., Ed. (2001), Modeling of Inelastic Behavior of RC Structures Under Seismic Loads, American Society of Civil Engineers.
Tavarez, F.A., (2001), “Simulation of Behavior of Composite Grid Reinforced Concrete Beams Using Explicit Finite Element Methods,” Master’s Thesis, University of Wisconsin-Madison, Madison, Wisconsin. Willam, K., and Tanabe, T.A., Ed. (2001), Finite Element Analysis of Reinforced Concrete Structures, American Concrete Institute, Farmington Hills, MI. Willam, K.J. and Warnke, E.P. (1974), “Constitutive Model for Triaxial Behaviour of Concrete,” Seminar on Concrete Structures Subjected to Triaxial Stresses, International Association of Bridge and Structural Engineering Conference, Bergamo, Italy, p.174.
APPENDICES
The following appendices have been included for reference: