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.
TABLE OF CONTENTS
PAGE
LIST OF FIGURES . . . vi
LIST OF TABLES . . . viii
CHAPTER 1 – INTRODUCTION . . . 1
1.1 General . . . 1
1.2 Objectives and Outline of Thesis . . . 2
CHAPTER 2 – LITERATURE REVIEW AND SYNTHESIS . . . 4
2.0 Introduction . . . 4
2.1 Experiment-Based Testing of Concrete . . . 5
2.2 Finite Element Analysis . . . 6
2.3 Failure Surface Models for Concrete . . . 10
2.4 FE Modeling of Steel Reinforcement . . . 13
2.5 Direction for Present Research . . . 15
CHAPTER 3 – CALIBRATION MODEL . . . 16
3.0 Introduction . . . 16
3.1 Experimental Beam . . . 16
3.2 ANSYS Finite Element Model . . . 20
3.2.1 Element Types . . . 20
3.2.2 Real Constants . . . 22
3.2.4 Modeling . . . 30
3.2.5 Meshing . . . 33
3.2.6 Numbering Controls . . . 34
3.2.7 Loads and Boundary Conditions . . . 35
3.2.8 Analysis Type . . . 37
3.2.9 Analysis Process for the Finite Element Model . . . 40
3.3 Results . . . 42
3.3.1 Behavior at First Cracking . . . 42
3.3.2 Behavior at Initial Cracking . . . 43
3.3.3 Behavior Beyond First Cracking . . . 43
3.3.4 Behavior of Reinforcement Yielding and Beyond . . . 44
3.3.5 Strength Limit State . . . 46
3.3.6 Load-Deformation Response . . . 48
CHAPTER 4 – PRESTRESSED CONCRETE BEAM MODEL . . . 49
4.0 Introdution . . . 49
4.1 Finite Element Model . . . 49
4.1.1 Real Constants . . . 50
4.1.2 Material Properties . . . 51
4.1.3 Solution . . . 51
4.1.4 Analysis Process for the Finite Element Model . . . 53
4.2 Results . . . 54
4.2.1 Application of Effective Prestress . . . 54
4.2.3 Zero Deflection . . . 58
4.2.4 Decompression . . . 59
4.2.5 Initial Cracking . . . 59
4.2.6 Secondary Linear Region . . . 59
4.2.7 Behavior of Steel Yielding and Beyond . . . 60
4.2.8 Flexural Limit State . . . 61
CHAPTER 5 – CONCLUSIONS AND RECOMMENDATIONS . . . 63
5.0 Introduction . . . 63
5.1 Conclusions . . . 63
5.2 Recommendations for Future Work . . . 64
REFERENCES . . . 66
APPENDICES . . . 68
Theoretical Calculations for Calibration Model . . . 69
LIST OF FIGURES
FIGURE PAGE
2.1 Typical Cracking of Control Beam at Failure (Buckhouse 1997) . . . 5
2.2 Reinforced Concrete Beam With Loading (Faherty 1972) . . . 6
2.3 FEM Discretization for a Quarter of the Beam (Kachlakev, et al. 2001) . 8 2.4 Load vs. Deflection Plot (Kachlakev, et al. 2001) . . . 9
2.5 Typical Cracking Signs in Finite Element Models (Kachlakev, et al. 2001) . . . 10
2.6 Failure Surface of Plain Concrete Under Triaxial Conditions (Willam and Warnke 1974) . . . 11
2.7 Three Parameter Model (Willam and Warnke 1974) . . . 11
2.8 Models for Reinforcement in Reinforced Concrete (Tavarez 2001) . . . . 14
3.1 Loading and Supports for the Beam (Buckhouse 1997) . . . 16
3.2 Typical Detail for Control Beam Reinforcement (Buckhouse 1997) . . . . 17
3.3 Failure in Flexure (Buckhouse 1997) . . . 18
3.4 Load vs. Deflection Curve for Beam C1 (Buckhouse 1997) . . . 19
3.5 Solid 65 Element (ANSYS, SAS 2003) . . . 21
3.6 Solid 45 Element (ANSYS, SAS 2003) . . . 22
3.7 Link 8 Element (ANSYS, SAS 2003) . . . 23
3.8 Uniaxial Stress-Strain Curve . . . 27
3.9 Volumes Created in ANSYS . . . 31
3.11 Reinforcement Configuration . . . 33
3.12 Boundary Conditions for Planes of Symmetry . . . 35
3.13 Boundary Condition for Support . . . 36
3.14 Boundary Conditions at the Loading Plate . . . 37
3.15 1st Crack of the Concrete Model . . . 44
3.16 Cracking at 8,000 and 12,000 lbs. . . . 45
3.17 Increased Cracking After Yielding of Reinforcement . . . 46
3.18 Failure of the Concrete Beam . . . 47
3.19 Load vs. Deflection Curve Comparison of ANSYS and Buckhouse (1997) 48 4.1 Stress-Strain Curve for 270 ksi strand . . . 52
4.2 Load vs. Deflection Curve for Prestressed Concrete Model . . . 55
4.3 Deflection due to prestress . . . 56
4.4 Bursting Phenomenon . . . 57
4.5 Localized Cracking From Effective Prestress Application . . . 58
4.6 Cracking at 12,000 and 20,000 lbs. . . . 60
4.7 Cracking at Flexural Capacity . . . 61
A.1 Loading of Beam with Supports . . . 70
A.2 Transformed Cross-Section . . . 71
LIST OF TABLES
TABLE PAGE
3.1 Properties for Steel and Concrete (Buckhouse 1997) . . . 17
3.2 Test Data for Control Beam C1 (Buckhouse 1997) . . . 19
3.3 Element Types for Working Model . . . 21
3.4 Real Constants for Calibration Model . . . 23
3.5 Material Models for the Calibration Model . . . 25
3.6 Dimensions for Concrete, Steel Plate, and Steel Support Volumes . . . 31
3.7 Mesh Attributes for the Model . . . 33
3.8 Commands Used to Control Nonlinear Analysis . . . 38
3.9 Commands Used to Control Output . . . 38
3.10 Nonlinear Algorithm and Convergence Criteria Parameters . . . 39
3.11 Advanced Nonlinear Control Settings Used . . . 39
3.12 Load Increments for Analysis of Finite Element Model . . . 40
3.13 Deflection and Stress Comparisons At First Cracking . . . 43
3.14 Deflections of Control Beam (Buckhouse 1997) vs. Finite Element Model At Ultimate Load . . . 47
4.1 Real Constants for Prestressed Beam . . . 50
4.2 Values for Multilinear Isotropic Stress-Strain Curve . . . 52
4.3 Load Increments for Analysis of Prestressed Beam Model . . . 53
CHAPTER 1 INTRODUCTION
1.1 General
Concrete structural components exist in buildings and bridges in different forms. Understanding the response of these components during loading is crucial to the development of an overall efficient and safe structure.
Different methods have been utilized to study the response of structural components. Experimental based testing has been widely used as a means to analyze individual elements and the effects of concrete strength under loading. While this is a method that produces real life response, it is extremely time consuming, and the use of materials can be quite costly. The use of finite element analysis to study these
components has also been used. Unfortunately, early attempts to accomplish this were also very time consuming and infeasible using existing software and hardware.
In recent years, however, the use of finite element analysis has increased due to progressing knowledge and capabilities of computer software and hardware. It has now become the choice method to analyze concrete structural components. The use of
computer software to model these elements is much faster, and extremely cost-effective. To fully understand the capabilities of finite element computer software, one must look back to experimental data and simple analysis. Data obtained from a finite element analysis package is not useful unless the necessary steps are taken to understand what is happening within the model that is created using the software. Also, executing the
necessary checks along the way is key to make sure that what is being output by the computer software is valid.
By understanding the use of finite element packages, more efficient and better analyses can be made to fully understand the response of individual structural
components and their contribution to a structure as a whole. This thesis is a study of reinforced and prestressed concrete beams using finite element analysis to understand the response of reinforced and prestressed concrete beams due to transverse loading.
1.2 Objectives and Outline of Thesis
The objective of this thesis was to investigate and evaluate the use of the finite element method for the analysis of reinforced and prestressed concrete beams The following procedure was used to meet this goal.
First, a literature review was conducted to evaluate previous experimental and analytical procedures related to reinforced concrete components. Second, a calibration model using a commercial finite element analysis package (ANSYS, SAS 2003) was set up and evaluated using experimental data. A mild-steel reinforced concrete beam with flexural and shear reinforcement was analyzed to failure and compared to experimental results to calibrate the parameters in ANSYS (SAS 2003) for later analyses.
Based on the results obtained from the calibration model and the
analysis/modeling parameters set by this model, a prestressed concrete beam was analyzed from initial prestress to flexural failure. Deflections, stresses, and cracking of the concrete beam were analyzed at different key points along the way. These key points
include initial prestress, addition of self-weight, zero deflection point, decompression, initial cracking, yielding of steel, and failure.
Discussion of the results obtained for the calibration model and the prestressed concrete beam model is also provided. Conclusions regarding the analysis are then drawn and recommendations for further research are made.
CHAPTER 2
LITERATURE REVIEW AND SYNTHESIS
2.0 Introduction
To provide a detailed review of the body of literature related to reinforced and prestressed concrete in its entirety would be too immense to address in this thesis. However, there are many good references that can be used as a starting point for research (ACI 1978, MacGregor 1992, Nawy 2000). This literature review and introduction will focus on recent contributions related to FEA and past efforts most closely related to the needs of the present work.
The use of FEA has been the preferred method to study the behavior of concrete (for economic reasons). Willam and Tanabe (2001) contains a collection of papers concerning finite element analysis of reinforced concrete structures. This collection contains areas of study such as: seismic behavior of structures, cyclic loading of reinforced concrete columns, shear failure of reinforced concrete beams, and concrete-steel bond models.
Shing and Tanabe (2001) also put together a collection of papers dealing with inelastic behavior of reinforced concrete structures under seismic loads. The monograph contains contributions that outline applications of the finite element method for studying post-peak cyclic behavior and ductility of reinforced concrete columns, the analysis of reinforced concrete components in bridge seismic design, the analysis of reinforced concrete beam-column bridge connections, and the modeling of the shear behavior of reinforced concrete bridge structures.
The focus of these most recent efforts is with bridges, columns, and seismic design. The focus of this thesis is the study of non-prestressed and prestressed flexural members. The following is a review and synthesis of efforts most relevant to this thesis discussing FEA applications, experimental testing, and concrete material models.
2.1 Experiment-Based Testing Of Concrete
Buckhouse (1997) studied external flexural reinforcement of existing concrete beams. Three concrete control beams were cast with flexural and shear reinforcing steel. Shear reinforcement was placed in each beam to force a flexural failure mechanism.
All three beams were loaded with transverse point loads at third points along the beams. Loading was applied to the beams until failure occurred as shown in Figure 2.1.
The mode of failure characterized by the beams was compression failure of the concrete in the constant moment region (flexural failure). All failures were ductile, with
significant flexural cracking of the concrete in the constant moment region.
Load-deflection curves were plotted for each beam and compared to predicted ultimate loads. This thesis will utilize the experimental results of these control beam tests for calibration of the FE models.
2.2 Finite Element Analysis
Faherty (1972) studied a reinforced and prestressed concrete beam using the finite
element method of analysis. The two beams that were selected for modeling were simply supported and loaded with two symmetrically placed concentrated transverse loads (Figure 2.2).
The analysis for the reinforced concrete beam included: non-linear concrete properties, a linear bond-slip relation, bilinear steel properties, and the influence of progressive cracking of the concrete. The transverse loading was incrementally applied and ranged in magnitude from zero to a load well above that which initiated cracking. Because the loading and geometry of the beam were symmetrical, only one half of the beam was modeled using FEA. The finite element model produced very good results that compared well with experimental results in Janney (1954).
Faherty (1972) also analyzed a prestressed concrete beam that included: non-linear concrete properties, a non-linear bond slip relation with a destruction of the bond between the steel and concrete, and bilinear steel properties. The dead load, release of the prestressing force, the elastic prestress loss, the time dependent prestress loss, and the loss of tensile stress in the concrete as a result of concrete rupture were applied as single loading increments, whereas the transverse loading was applied incrementally. Only three finite element models of the prestressed beam were implemented (or used): two uncracked sections, and a partially cracked section. Symmetry was once again utilized.
These results for the prestressed beam showed that deflections computed using the finite element model were very similar to those observed by Branson, et al. (1970). However, the load-deflection curve past the cracking point was not generated because only three distinct cracking patterns were used for this analysis. It was recommended that additional analysis of the prestressed concrete beam should be undertaken after a procedure is developed for modeling the tensile rupture of the concrete. The model utilized in this research required the beam to be unloaded and the finite element model redefined as each crack is initiated or extended.
Kachlakev, et al. (2001) used ANSYS (SAS 2003) to study concrete beam members with externally bonded Carbon Fiber Reinforced Polymer (CFRP) fabric. Symmetry allowed one quarter of the beam to be modeled as shown in Figure 2.3.
Figure 2.3 – FEM Discretization for a Quarter of the Beam (Kachlakev, et al. 2001)
At planes of symmetry, the displacement in the direction perpendicular to the plane was set to zero. A single line support was utilized to allow rotation at the supports. Loads were placed at third points along the full beam on top of steel plates. The mesh was refined immediately beneath the load (Figure 2.3). No stirrup-type reinforcement was used.
The nonlinear Newton-Raphson approach was utilized to trace the equilibrium path during the load-deformation response. It was found that convergence of solutions for the model was difficult to achieve due to the nonlinear behavior of reinforced
(at points of linearity in the response) to small (when instances of cracking and steel yielding occurred). The load-deflection curve for the non-CFRP reinforced beam that was plotted shows reasonable correlation with experimental data (McCurry and Kachlakev 2000) as shown in Figure 2.4.
Figure 2.4 – Load vs. Deflection Plot (Kachlakev, et al. 2001)
Also, concrete crack/crush plots were created at different load levels to examine the different types of cracking that occurred within the concrete as shown in Figure 2.5.
The different types of concrete failure that can occur are flexural cracks, compression failure (crushing), and diagonal tension cracks. Flexural cracks (Figure 2.5a) form vertically up the beam. Compression failures (Figure 2.5b) are shown as circles. Diagonal tension cracks (Figure 2.5c) form diagonally up the beam towards the loading that is applied.
Figure 2.5 – Typical Cracking Signs in Finite Element Models: a)Flexural Cracks, b)Compressive Cracks, c)Diagonal Tensile Cracks (Kachlakev, et al. 2001)
This study indicates that the use of a finite element program to model
experimental data is viable and the results that are obtained can indeed model reinforced concrete beam behavior reasonably well.
2.3 Failure Surface Models For Concrete
Willam and Warnke (1974) developed a widely used model for the triaxial failure surface of unconfined plain concrete. The failure surface in principal stress-space is shown in Figure 2.6. The mathematical model considers a sextant of the principal stress space because the stress components are ordered according to σ1≥σ2 ≥σ3. These stress components are the major principal stresses.
The failure surface is separated into hydrostatic (change in volume) and deviatoric (change in shape) sections as shown in Figure 2.7. The hydrostatic section forms a
meridianal plane which contains the equisectrix σ1=σ2 =σ3 as an axis of revolution (see Figure 2.6). The deviatoric section in Figure 2.7 lies in a plane normal to the equisectrix (dashed line in Figure 2.7).
Figure 2.6 – Failure Surface of Plain Concrete Under Triaxial Conditions (Willam and Warnke 1974)
The deviatoric trace is described by the polar coordinates r , and θ where r is the position vector locating the failure surface with angle, θ . The failure surface is defined as: 1 1 1 ( ) a a cu cu z f r f σ τ θ + = (2.1) where: a
σ and τa = average stress components
z= apex of the surface cu
f = uniaxial compressive strength
The opening angles of the hydrostatic cone are defined by ϕ1 and ϕ2. The free parameters of the failure surface z and r , are identified from the uniaxial compressive strength ( fcu), biaxial compressive strength ( fcb), and uniaxial tension strength ( ft)
The Willam and Warnke (1974) mathematical model of the failure surface for the concrete has the following advantages:
1. close fit of experimental data in the operating range;
2. simple identification of model parameters from standard test data; 3. smoothness (e.g. continuous surface with continuously varying tangent
planes);
4. convexity (e.g. monotonically curved surface without inflection points). Based on the above criteria, a constitutive model for the concrete suitable for FEA implementation was formulated.
This constitutive model for concrete based upon the Willam and Warnke (1974) model assumes an appropriate description of the material failure. The yield condition can
be approximated by three or five parameter models distinguishing linear from non-linear and elastic from inelastic deformations using the failure envelope defined by a scalar function of stress ( ) 0f σ = through a flow rule, while using incremental stress-strain relations. The parameters for the failure surface can be seen in Figure 2.7.
During transition from elastic to plastic or elastic to brittle behavior, two numerical strategies were recommended: proportional penetration, which subdivides proportional loading into an elastic and inelastic portion which governs the failure surface using integration, and normal penetration, which allows the elastic path to reach the yield surface at the intersection with the normal therefore solving a linear system of equations. Both of these methods are feasible and give stress values that satisfy the constitutive constraint condition. From the standpoint of computer application the normal penetration approach is more efficient than the proportional penetration method, since integration is avoided.
2.4 FE Modeling of Steel Reinforcement
Tavarez (2001) discusses three techniques that exist to model steel reinforcement in finite element models for reinforced concrete (Figure 2.8): the discrete model, the embedded model, and the smeared model.
The reinforcement in the discrete model (Figure 2.8a) uses bar or beam elements that are connected to concrete mesh nodes. Therefore, the concrete and the reinforcement mesh share the same nodes and concrete occupies the same regions occupied by the reinforcement. A drawback to this model is that the concrete mesh is restricted by the
location of the reinforcement and the volume of the mild-steel reinforcement is not deducted from the concrete volume.
(a) (b)
(c)
Figure 2.8 – Models for Reinforcement in Reinforced Concrete (Tavarez 2001): (a) discrete; (b) embedded; and (c) smeared
The embedded model (Figure 2.8b) overcomes the concrete mesh restriction(s) because the stiffness of the reinforcing steel is evaluated separately from the concrete elements. The model is built in a way that keeps reinforcing steel displacements compatible with the surrounding concrete elements. When reinforcement is complex, this model is very advantageous. However, this model increases the number of nodes and degrees of freedom in the model, therefore, increasing the run time and computational cost.
The smeared model (Figure 2.8c) assumes that reinforcement is uniformly spread throughout the concrete elements in a defined region of the FE mesh. This approach is used for large-scale models where the reinforcement does not significantly contribute to the overall response of the structure.
Fanning (2001) modeled the response of the reinforcement using the discrete model and the smeared model for reinforced concrete beams. It was found that the best modeling strategy was to use the discrete model when modeling reinforcement.
2.5 Direction for Present Research
The literature review suggested that use of a finite element package to model reinforced and prestressed concrete beams was indeed feasible. It was decided to use ANSYS (SAS 2003) as the FE modeling package. A reinforced concrete beam with reinforcing steel modeled discretely will be developed with results compared to the experimental work of Buckhouse (1997). The load-deflection response of the experimental beam will be compared to analytical predictions to calibrate the FE model for further use. A second analysis of a prestressed concrete beam will also be studied. The different stages of the response of a prestressed concrete beam are computed using FEA and compared to results generated using hand computations.
CHAPTER 3 CALIBRATION MODEL
3.0 Introduction
This chapter discusses the calibration of the finite element model using experimental load-deformation behavior of a concrete beam provided in Buckhouse (1997). The use of ANSYS (SAS 2003) to create the finite element model is also discussed. All the
necessary steps to create the calibrated model are explained in detail and the steps taken to generate the analytical load-deformation response of the member are discussed.
3.1 Experimental Beam
Buckhouse (1997) studied a method to reinforce a concrete beam for flexure using external structural steel channels. The study included experimental testing of control beams that can be used for calibration of finite element models. The width and height of the beams tested were 10 in. and 18 in., respectively. As shown in Figure 3.1, the length
of the beam was 15 ft.-6 in. with supports located 3 in. from each end of the beam allowing a simply supported span of 15 ft. The mild steel flexural reinforcements used were 3-#5 bars and shear reinforcements included #3 U-stirrups. Cover for the rebar was set to 2 in. in all directions. The layout of the reinforcement is detailed in Figure 3.2.
Figure 3.2 – Typical Detail for Control Beam Reinforcement (Buckhouse 1997)
The steel yield stress, 28-day compressive stress of concrete, and area of steel reinforcement are included in Table 3.1.
Table 3.1 – Properties for Steel and Concrete (Buckhouse 1997)
Area of Steel (in.2) 0.93
Yield Stress of Steel, fy
(psi) 60,000
28-Day Compressive Strength of Concrete, fc'
(psi)
Two 50-kip capacity load cells were placed at third points, or 5 ft. from each support on steel bearing plates (Figure 3.1). Data acquisition equipment was used to record applied loading, beam deflection at the midspan, and strain in the internal flexural reinforcement. The beam was loaded to flexural failure (Figure 3.3).
Figure 3.3 – Failure in Flexure (Buckhouse 1997)
Vertical cracks first formed in the constant moment region, extended upward, and then out towards the constant shear region with eventual crushing of the concrete in the constant moment region as shown in Figure 3.3. Test data for the beam is summarized in Table 3.2.
The theoretical ultimate load for the beam was calculated to be 14,600 lbs (Buckhouse 1997). Table 3.2 shows the experimental ultimate load determined was 16,310 lbs. The ultimate loading corresponded to the nominal flexural capacity of the cross-section being reached. A plot of load versus deflection for control beam C1 (Buckhouse 1997) is shown in Figure 3.4.
Table 3.2 – Test data for control beam C1 (Buckhouse 1997)
Avg. Load at 1st
Crack (lbs.) 4,500
Avg. Failure Load, P
(lbs.) 16,310
Avg. Centerline Deflection at Failure
(in.) 3.65
Mode of Failure compression failure of concrete
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Avg. Centerline Deflection (in.)
Av
g. Loa
d, P
(lbs
.)
Figure 3.4 – Load vs. Deflection Curve for Beam C1 (Buckhouse 1997)
C1 theoretical ultimate load (14,600 lbs.) A B C
Point A - First Cracking Point B - Steel Yielding Point C - Failure Nonlinear Region
The plot shows the linear behavior before first cracking (point A). A second slope corresponding to the cracked section is followed until point B where the flexural reinforcement yields. The cracked moment of inertia with yielding internal reinforcement then defines the stiffness until flexural failure at point C.
3.2 ANSYS Finite Element Model
The FEA calibration study included modeling a concrete beam with the dimensions and properties corresponding to beam C1 tested by Buckhouse (1997). Due to the symmetry in cross-section of the concrete beam and loading, symmetry was utilized in the FEA, only one quarter of the beam was modeled.
To create the finite element model in ANSYS (SAS 2003) there are multiple tasks that have to be completed for the model to run properly. Models can be created using command prompt line input or the Graphical User Interface (GUI). For this model, the GUI was utilized to create the model. This section describes the different tasks and entries into used to create the FE calibration model.
3.2.1 Element Types
The element types for this model are shown in Table 3.3. The Solid65 element was used to model the concrete. This element has eight nodes with three degrees of freedom at each node – translations in the nodal x, y, and z directions. This element is capable of plastic deformation, cracking in three orthogonal directions, and crushing. A schematic of the element is shown in Figure 3.5.
Table 3.3 – Element Types For Working Model
Material Type ANSYS Element Concrete Solid65 Steel Plates and
Supports Solid45
Steel
Reinforcement Link8
Figure 3.5 – Solid 65 Element (SAS 2003)
A Solid45 element was used for steel plates at the supports for the beam. This element has eight nodes with three degrees of freedom at each node – translations in the nodal x, y, and z directions. The geometry and node locations for this element is shown
in Figure 3.6. The descriptions for each element type are laid out in the ANSYS element library (SAS 2003).
Figure 3.6 – Solid 45 Element (SAS 2003)
A Link8 element was used to model steel reinforcement. This element is a 3D spar element and it has two nodes with three degrees of freedom – translations in the nodal x, y, and z directions. This element is also capable of plastic deformation. This element is shown in Figure 3.7.
3.2.2 Real Constants
The real constants for this model are shown in Table 3.4. Note that individual elements contain different real constants. No real constant set exists for the Solid45 element.
Figure 3.7 – Link 8 Element (SAS 2003)
Table 3.4 – Real Constants For Calibration Model
Real Constant Set Element Type Constants
Real Constants for Rebar 1 Real Constants for Rebar 2 Real Constants for Rebar 3 Material Number 0 0 0 Volume Ratio 0 0 0 Orientation Angle 0 0 0 1 Solid 65 Orientation Angle 0 0 0 Cross-sectional Area (in.2) 0.31 2 Link8 Initial Strain (in./in.) 0 Cross-sectional Area (in.2) 0.155 3 Link8 Initial Strain (in./in.) 0 Cross-sectional Area (in.2) 0.11 4 Link8 Initial Strain (in./in.) 0 Cross-sectional Area (in.2) 0.055 5 Link8 Initial Strain (in./in.) 0
Real Constant Set 1 is used for the Solid65 element. It requires real constants for rebar assuming a smeared model. Values can be entered for Material Number, Volume Ratio, and Orientation Angles. The material number refers to the type of material for the reinforcement. The volume ratio refers to the ratio of steel to concrete in the element. The orientation angles refer to the orientation of the reinforcement in the smeared model (Figure 2.8c). ANSYS (SAS 2003) allows the user to enter three rebar materials in the concrete. Each material corresponds to x, y, and z directions in the element (Figure 3.5). The reinforcement has uniaxial stiffness and the directional orientation is defined by the user. In the present study the beam is modeled using discrete reinforcement. Therefore, a value of zero was entered for all real constants which turned the smeared reinforcement capability of the Solid65 element off.
Real Constant Sets 2, 3, 4, and 5 are defined for the Link8 element. Values for cross-sectional area and initial strain were entered. Cross-sectional areas in sets 2 and 3 refer to the reinforcement of 3-#5 bars. Due to symmetry, set 3 is half of set 2 because one-half the center bar in the beam is cut off. Cross-sectional areas in sets 4 and 5 refer to the #3 stirrups. Once again set 5 is half of set 4 because half of the stirrup at the mid-span of the beam is cut off resulting from symmetry. A value of zero was entered for the initial strain because there is no initial stress in the reinforcement.
3.2.3 Material Properties
Parameters needed to define the material models can be found in Table 3.5. As seen in Table 3.5, there are multiple parts of the material model for each element.
Table 3.5 – Material Models For the Calibration Model
Material Model
Number Element Type Material Properties
Linear Isotropic EX 3,949,076 psi PRXY 0.3 Multilinear Isotropic Strain Stress Point 1 0.00036 1421.7 Point 2 0.0006 2233 Point 3 0.0013 3991 Point 4 0.0019 4656 Point 5 0.00243 4800 Concrete ShrCf-Op 0.3 ShrCf-Cl 1 UnTensSt 520 UnCompSt -1 BiCompSt 0 HydroPrs 0 BiCompSt 0 UnTensSt 0 TenCrFac 0 1 Solid65 Linear Isotropic EX 29,000,000 psi PRXY 0.3 2 Solid45 Linear Isotropic EX 29,000,000 psi PRXY 0.3 Bilinear Isotropic Yield Stss 60,000 psi
Tang Mod 2,900 psi
3 Link8
Material Model Number 1 refers to the Solid65 element. The Solid65 element requires linear isotropic and multilinear isotropic material properties to properly model concrete. The multilinear isotropic material uses the von Mises failure criterion along with the Willam and Warnke (1974) model to define the failure of the concrete. EX is the modulus of elasticity of the concrete (E ), and PRXY is the Poisson’s ratio (c ν ). The
modulus was based on the equation,
' 57000 c c E = f (3.1) with a value of ' c
f equal to 4,800 psi. Poisson’s ratio was assumed to be 0.3. The
compressive uniaxial stress-strain relationship for the concrete model was obtained using the following equations to compute the multilinear isotropic stress-strain curve for the concrete (MacGregor 1992) 2 0 1 c E f ε ε ε = + (3.2) ' 0 2 c c f E ε = (3.3) c f E ε = (3.4) where:
f = stress at any strain ε , psi ε = strain at stress f
0
ε = strain at the ultimate compressive strength ' c
f
The multilinear isotropic stress-strain implemented requires the first point of the curve to be defined by the user. It must satisfy Hooke’s Law;
E σ ε
= (3.5)
The multilinear curve is used to help with convergence of the nonlinear solution algorithm. 0 1000 2000 3000 4000 5000 6000 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 Strain (in./in.) S tr ess ( p si)
Figure 3.8 – Uniaxial Stress-Strain Curve
Figure 3.8 shows the stress-strain relationship used for this study and is based on work done by Kachlakev, et al. (2001). Point 1, defined as 0.30 '
c
f , is calculated in the
linear range (Equation 3.4). Points 2, 3, and 4 are calculated from Equation 3.2 with ε0 obtained from Equation 3.3. Strains were selected and the stress was calculated for each
0
ε
'
0.30
f
cStrain at Ultimate Strength
c
E
Ultimate Compressive Strength
' c
f
5 4 3 2 1strain. Point 5 is defined at ' c
f and ε0 =0.003in.in. indicating traditional crushing strain for unconfined concrete.
Implementation of the Willam and Warnke (1974) material model in ANSYS requires that different constants be defined. These 9 constants are: (SAS 2003)
1. Shear transfer coefficients for an open crack; 2. Shear transfer coefficients for a closed crack; 3. Uniaxial tensile cracking stress;
4. Uniaxial crushing stress (positive); 5. Biaxial crushing stress (positive);
6. Ambient hydrostatic stress state for use with constants 7 and 8;
7. Biaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);
8. Uniaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);
9. Stiffness multiplier for cracked tensile condition.
Typical shear transfer coefficients range from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer). The shear transfer coefficients for open and closed cracks were
determined using the work of Kachlakev, et al. (2001) as a basis. Convergence problems occurred when the shear transfer coefficient for the open crack dropped below 0.2. No deviation of the response occurs with the change of the coefficient. Therefore, the
coefficient for the open crack was set to 0.3 (Table 3.4). The uniaxial cracking stress was based upon the modulus of rupture. This value is determined using,
'
7.5
r c
f = f (3.6)
The uniaxial crushing stress in this model was based on the uniaxial unconfined compressive strength ( '
c
f ) and is denoted as f . It was entered as -1 to turn off the t
crushing capability of the concrete element as suggested by past researchers (Kachlakev, et al. 2001). Convergence problems have been repeated when the crushing capability was turned on.
The biaxial crushing stress refers to the ultimate biaxial compressive strength ( '
cb
f ). The ambient hydrostatic stress state is denoted as σh. This stress state is defined as: 1 ( ) 3 h xp yp zp σ = σ +σ +σ (3.7)
where σxp, σyp, and σzp are the principal stresses in the principal directions. The biaxial crushing stress under the ambient hydrostatic stress state refers to the ultimate
compressive strength for a state of biaxial compression superimposed on the hydrostatic stress state ( f ). The uniaxial crushing stress under the ambient hydrostatic stress state 1
refers to the ultimate compressive strength for a state of uniaxial compression
superimposed on the hydrostatic stress state ( f ). The failure surface can be defined 2
with a minimum of two constants, f and t ' c
f . The remainder of the variables in the
concrete model are left to default based on these equations: (SAS 2003) ' 1.2 ' cb c f = f (3.8) ' 1 1.45 c f = f (3.9) ' 2 1.725 c f = f (3.10)
These stress states are only valid for stress states satisfying the condition '
3 h fc
σ ≤ (3.11)
Material Model Number 2 refers to the Solid45 element. The Solid45 element is being used for the steel plates at loading points and supports on the beam. Therefore, this element is modeled as a linear isotropic element with a modulus of elasticity for the steel (Es), and poisson’s ratio (0.3).
Material Model Number 3 refers to the Link8 element. The Link8 element is being used for all the steel reinforcement in the beam and it is assumed to be bilinear isotropic. Bilinear isotropic material is also based on the von Mises failure criteria. The bilinear model requires the yield stress ( fy), as well as the hardening modulus of the steel to be defined. The yield stress was defined as 60,000 psi, and the hardening modulus was 2900 psi.
Note that the density for the concrete was not added in the material model. For the control beam in Buckhouse (1997), the LVDT’s used to measure deflection at mid-span were put on the beam after it was set in the test fixture. Deflections were taken relative to a zero deflection point after the self-weight was introduced. Therefore, the self-weight was not introduced in this calibration model.
3.2.4 Modeling
The beam, plates, and supports were modeled as volumes. Since a quarter of the beam is being modeled, the model is 93 in. long, with a cross-section of 5 in. x 18 in. The
dimensions for the concrete volume are shown in Table 3.6. The zero values for the Z-coordinates coincide with the center of the cross-section for the concrete beam.
Table 3.6 – Dimensions for Concrete, Steel Plate, and Steel Support Volumes
ANSYS Concrete (in.) Steel Plate (in.) Steel Support (in.)
X1,X2 X-coordinates 0 93 60 66 1.5 4.5
Y1,Y2 Y-coordinates 0 18 18 19 0 -1
Z1,Z2 Z-coordinates 0 5 0 5 0 5
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
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
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 Displacement
Calculate 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
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
Beginning
Time Time at End of Loadstep Load Step Sub Step
Load Increment (lbs.) 0 5210 1 1 5210 5210 5220 2 10 10 5220 5300 3 16 5 5300 5400 4 20 5 5400 10000 5 92 50 10000 13000 6 30 100 13000 14000 7 10 100 14000 14500 8 50 10 14500 14700 9 20 10 14700 14800 10 20 5 14800 14900 11 100 1 14900 15000 12 10 10 15000 15100 13 10 10 15100 15200 14 50 2 15200 15300 15 20 5 15300 15600 16 150 2 15600 15900 17 150 2 15900 16200 18 150 2 16200 16300 19 50 2 16300 16382 20 41 2
The time at the end of each load step corresponds to the loading applied. For the first load step the time at the end of the load step is 5210 referring to a load of, P, of 5210 lbs applied at the steel plate.
The two convergence criteria used for the analysis were Force and Displacement. These criteria were left at the default values up to 5210 lbs. However, when the beam began cracking, convergence for the non-linear analysis was impossible with the default values. The displacements converged, but the forces did not. Therefore, the convergence criteria for force was dropped and the reference value for the Displacement criteria was changed to 5. This value is then multiplied by the tolerance value of 0.05 to produce a criterion of 0.25 during the nonlinear solution for convergence. A small criterion must be used to capture correct response. This criteria was used for the remainder of the analysis. As shown in Table 3.12, the steps taken to the initial cracking of the beam can be decresed to one load increment to model/capture initial cracking. Once initial cracking of the beam has been passed (5220 lbs), the load increments increased slightly until
subsequent cracking of the beam (14,000 lbs) as seen in Table 3.12. Once the yielding of the reinforcing steel is reached, the load increments must be decreased again. Yielding of the steel occurs at load step 13,400; therefore, the load increment sizes begin decreasing further because displacements are increasing more rapidly. Eventually, the load
increment size is decreased to 2 lb. to capture the failure of the beam. Failure of the beam occurs when convergence fails, with this very small load increment. The load deformation trace produced by the analysis confirmed the failure load.
3.3 Results
The goal of the comparison of the FE model and the beam from Buckhouse (1997) is to ensure that the elements, material properties, real constants and convergence criteria are adequate to model the response of the member. Figure 3.4 shows the different
components that were analyzed for comparison: the linear region, initial cracking, the nonlinear region, the yielding of steel, and failure.
3.3.1 Behavior at First Cracking
The analysis of the linear region can be based on the design for flexure given in MacGregor (1992) for a reinforced concrete beam. Comparisons were made in this region to ensure deflections and stresses were consistent with the FE model and the beam before cracking occurred. Once cracking occurs, deflections and stresses become more difficult to predict. The stresses in the concrete and steel immediately preceding initial cracking were analyzed. The load at step 5210 was analyzed and it coincides with a load of 5210 lbs. applied on the steel plate.
Calculations to obtain the concrete stress, steel stress and deflection of the beam at a load of 5210 lbs. can be seen in Appendix A. A comparison of values obtained from the FE model and Appendix A can be seen in Table 3.13. The maximums exist in the constant-moment region of the beam during load application. This is where we expect the maximums to occur. The results in Table 3.13 indicate that the FE analysis of the beam prior to cracking is acceptable.
Table 3.13 – Deflection and Stress Comparisons At First Cracking
Model Extreme Tension Fiber Stress (psi)
Reinforcing Steel Stress (psi) Centerline Deflection (in.) Load at First Cracking (lbs.) Hand-Calculations 530 3024 0.0529 5118 ANSYS 536 2840 0.0534 5216
3.3.2 Behavior at Initial Cracking
The cracking pattern(s) in the beam can be obtained using the Crack/Crushing plot option in ANSYS (SAS 2003). Vector Mode plots must be turned on to view the cracking in the model.
The initial cracking of the beam in the FE model corresponds to a load of 5216 lbs that creates stress just beyond the modulus of rupture of the concrete (520 psi) as shown in Table 3.13. This compares well with the load of 5118 lbs calculated in Appendix A. The stress increases up to 537 psi at the centerline when the first crack occurs. The first crack can be seen in Figure 3.15. This first crack occurs in the constant moment region, and is a flexural crack. Buckhouse (1997) reported first cracking at a load, P, of 4500 lbs using visual detection.
3.3.3 Behavior Beyond First Cracking
In the non-linear region of the response, subsequent cracking occurs as more load is applied to the beam. Cracking increases in the constant moment region, and the beam begins cracking out towards the supports at a load of 8,000 lbs.
Figure 3.15 – 1st Crack of the Concrete Model
Significant flexural cracking occurs in the beam at 12,000 lbs. Also, diagonal tension cracks are beginning to form in the model. This cracking can be seen in Figure 3.16.
3.3.4 Behavior of Reinforcement Yielding and Beyond
Yielding of steel reinforcement occurs when a force of 13,400 lbs. is applied. At this point in the response, the displacements of the beam begin to increase at a higher rate as more load is applied (Figure 3.4). The cracked moment of inertia, yielding steel, and nonlinear concrete material, now defines the flexural rigidity of the member. The ability of the beam to distribute load throughout the cross-section has diminished greatly. Therefore, greater deflections occur at the beam centerline.
1st Crack in Concrete Beam
Figure 3.16 – Cracking at 8,000 and 12,000 lbs.
Figure 3.17 shows successive cracking of the concrete beam after yielding of the steel occurs. At 15,000 lbs., the beam has increasing flexural cracks, and diagonal tension cracks. Also, more cracks have now formed in the constant moment region. At 16,000 lbs., cracking has reached the top of the beam, and failure is soon to follow.
Flexural Cracks
Figure 3.17 – Increased Cracking After Yielding of Reinforcement
3.3.5 Strength Limit State
At load 16,382 lbs., the beam no longer can support additional load as indicated by an insurmountable convergence failure. Severe cracking throughout the entire constant moment region occurs (see Figure 3.18). The deflections at the analytical failure load of the control beam were compared with the finite element model as shown in Table 3.14.
Multiple cracks occurring
Increasing Diagonal Tension Cracks
Figure 3.18 – Failure of the Concrete Beam
Table 3.14 – Deflections of Control Beam (Buckhouse 1997) vs. Finite Element Model At Ultimate Load
Beam Load (lb.) Centerline Deflection (in.) C1 16310 3.65 ANSYS 16310 3.586
The deflection of the finite element model was within 2% of the control beam at the same load at which the control beam failed.
Excessive cracking and of the beam in the constant moment region
Diagonal Tension Cracking
3.3.6 Load-Deformation Response
The full nonlinear load-deformation response can be seen in Figure 3.19. This response was calibrated by setting the tolerances so that the load-deformation curve fits to the curve from Buckhouse (1997). The response calculated using FEA is plotted upon the experimental response from Buckhouse (1997). The entire load-deformation response of the model produced compares well with the response from Buckhouse (1997). This gave confidence in the use of ANSYS (SAS 2003) and the model developed. The approach was then utilized to analyze a prestressed concrete beam.
0 2000 4000 6000 8000 10000 12000 14000 16000 18000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Avg. Centerline Deflection (in.)
A vg . Lo ad , P (l bs .)
Figure 3.19 – Load vs. Deflection Curve Comparison of ANSYS and Buckhouse (1997)
Buckhouse (1997) FEA
C1 theoretical ultimate load (14,600 lbs.)
CHAPTER 4
PRESTRESSED CONCRETE BEAM MODEL
4.0 Introduction
This chapter discusses the finite element modeling of a prestressed concrete beam. The prestressed beam has the same dimensions as the experimental beam modeled by Buckhouse (1997). In this case, the reinforcement in the beam is different. The shear stirrups are included as discussed in the calibration model. However, no mild-steel flexural reinforcement was used. The beam was prestressed using ½ in. diameter 270 ksi 7-wire strands as opposed to the #5 60 ksi mild-steel reinforcement. All the necessary steps taken to create the model and the analysis used for the prestressed beam are explained in detail.
4.1 Finite Element Model
The finite element model that was used for analysis of the prestressed concrete beam is very similar to the calibration model. Many of the steps taken to model the prestressed concrete beam were the same as the calibration model and can be found in chapter 3.
The rest of this chapter will discuss the steps taken that were different from those in the calibration model. These steps include definition of real constants, material properties, and the parameters for the nonlinear analysis.
4.1.1 Real Constants
The real constants for the concrete element were left untouched because the modeling of the concrete has not changed. Also, the constants for the Link8 element designated to the stirrups used in the model were left unchanged. However, the real constants for the Link8 element for the flexural reinforcement has changed. Since the beam is now using prestressing steel, the cross-sectional area of the steel has changed, and an initial strain due to prestressing is now added to the element. The change in area and strain is shown in Table 4.1.
Table 4.1 – Real Constants for Prestressed Beam
Real Constant Set Element Type Constants
Cross-sectional Area (in.2) 0.153 2 Link8 Initial Strain (in./in.) 0.001903 Cross-sectional Area (in.2) 0.0765 3 Link8 Initial Strain (in./in.) 0.001903
The cross-sectional area for real constant set 2 is the area of an equivalent ½ in. diameter 7-wire strand. The cross-sectional area for real constant set 3 is half of real constant set 2, because it is at a point of symmetry on the model.
The initial strains for each real constant set were determined from the effective prestress ( fpe) and the modulus of elasticity (Eps). An example of this can be seen in Appendix B. This prestress level is too low and not suitable for practical use. It was
utilized to prevent any convergence problems occurring from bursting and significant cracking near the support as the prestrain is applied.
4.1.2 Material Properties
The material properties for the shear reinforcement steel, steel plates at loading points, and steel support plates are the same. Density was added to the concrete material property so the self-weight of the concrete beam could be taken into account.
The material properties for the prestressing steel have been changed from bilinear isotropic to multilinear isotropic following the von Mises failure criteria. The
prestressing steel was modeled using a multilinear stress-strain curve developed using the following equations, 0.008 : ps ε ≤ 28,000fps = εps (ksi) (4.1) >0.008: ps ε 268 0.075 <0.98 ( ) 0.0065 ps pu ps f f ksi ε = − − (4.2)
The values entered into ANSYS (SAS 2003) for the stress-strain curve are given in Table 4.2 and Figure 4.1 shows the stress-strain behavior of the prestressing steel.
4.1.3 Solution
Most of the parameters used to control the solution are the same as those used for the calibration model. The only changes made were as follows. First, no point loads are applied in the first load step, due to the fact that the initial prestrain is applied. Second, the time at the end of the prestrain application load step is 1. Third, the self-weight was added in a load step. The addition of the self-weight is done by applying gravitational
acceleration of 386.4 in/s2 in the global y-direction. A mass density was entered in as a function of the gravitational acceleration (Density/g).
Table 4.2 – Values for Multilinear Isotropic Stress-Strain Curve
Strain Stress Strain Stress Strain Stress Strain Stress
0 0 0.0101 247.1667 0.0123 255.069 0.0145 258.625 0.008 224 0.0103 248.2632 0.0125 255.5 0.0147 258.8537 0.0083 226.3333 0.0105 249.25 0.0127 255.9032 0.0149 259.0714 0.0085 230.5 0.0107 250.1429 0.0129 256.2813 0.0151 259.2791 0.0087 233.9091 0.0109 250.9545 0.0131 256.6364 0.0171 260.9245 0.0089 236.75 0.0111 251.6957 0.0133 256.9706 0.0189 261.9516 0.0091 239.1538 0.0113 252.375 0.0135 257.2857 0.0215 263 0.0093 241.2143 0.0115 253 0.0137 257.5833 0.0259 264.134 0.0095 243 0.0117 253.5769 0.0139 257.8649 0.0301 264.822 0.0097 244.5625 0.0119 254.1111 0.0141 258.1316 0.0099 245.9412 0.0121 254.6071 0.0143 258.3846 0 50 100 150 200 250 300 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Strain (in./in.) St ress ( ksi)
4.1.4 Analysis Process for the Finite Element Model
Analysis for the prestressed concrete beam was very similar to the calibration model. However, different load step and restart points were used. The first load step taken was to produce the camber in the concrete beam due to the prestress. The second load step was the addition of the self-weight. From that point on, the concrete was modeled to the point of cracking, yielding of the prestress steel ( 0.85fpu), and eventual failure. A listing of the load steps, sub steps, and loads applied per restart file are shown in Table 4.3.
Table 4.3 – Load Increments for the Analysis of the Prestressed Beam Model
Beginning Time
Time at End of Loadstep
Load Step Sub Step Increment Load (lbs.) 0 1 1 1 Prestress 1 2 2 1 386.4 2 7840 3 1 7840 7840 7900 4 30 2 7900 8500 5 12 50 8500 10000 6 15 100 10000 20000 7 100 100 20000 25000 8 100 50 25000 27000 9 200 10 27000 28000 10 100 10 28000 28823 11 412 2
When the analysis reaches the point of initial cracking, the force convergence criteria is dropped, and the reference value of the displacement criteria is 5. From this point, the load increments are decreased to 100 lbs. to capture the initial cracking of the beam. When yielding of the steel occurs, the load increments are decreased to 10 lb. Finally, load increments are decreased to 2 lb. until unresolvable convergence failure of the non-linear algorithm occurs.
4.2 Results
The analysis yielded results for seven distinct points on the load-deflection application of the prestressed concrete beam and the full nonlinear load-deformation response. As seen in Figure 4.2, these distinct points are: effective prestress, addition of self-weight, zero deflection, decompression, initial cracking, steel yielding ( 0.85fpu), and failure.
4.2.1 Application of Effective Prestress
Calculation of the effective prestress for the beam can be found in Appendix B. The deflections computed using hand calculations and FE analysis due to the prestress force is shown in Table 4.4.
Table 4.4 – Analytical Results
ANSYS CalculationsHand Centerline Deflection at
Effective Prestress (in.) -0.0323 -0.0341
Top Fiber Stress at
Effective Prestress (psi) 156.05 163.04
Deflection at Application
of Self-Weight (in.) -0.0211 -0.0229
Top Fiber Stress at
Self-Weight (psi) 36.97 45.86
Zero Deflection Load
(lbs.) 2055 2126
Decompression Load
(lbs.) 2817 2858
Initial Cracking Load
(lbs.) 7850 7538
Prestress at Failure Load
(psi) 264,822 254,520
-5000 0 5000 10000 15000 20000 25000 30000 35000 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
Centerline Deflection (in.)
Lo ad , P ( lbs .) 0 1000 2000 3000 4000 5000 6000 7000 8000 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10
Centerline Deflection (in.)
Loa d, P ( lbs .)
Figure 4.2 – Load vs. Deflection Curve for Prestressed Concrete Model
Initial Cracking Steel Yielding Failure Zero Deflection Decompression Self-Weight Initial Cracking Linear Region Secondary Linear Region
SEE FIGURE BELOW
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.)
