Axial-Force-Moment-Curvature Relationships for RC Sections
Table of Contents
Welcome to the help page for Java Module B. This program allows users to investigate the nonlinear axial-force-moment-curvature relationships for rectangular reinforced concrete (RC) beam-column cross-sections. The module determines the expected behavior of a user-defined cross-section by first dividing the section into a number of parallel concrete and steel "fibers". Then, the section forces and deformations are determined from the fiber strains and stresses using fundamental principles of equilibrium, strain compatibility, and constitutive relationships assuming that plane sections remain plane.
The module consists of several different input sections (illustrated in Fig. 1), which the user must complete:
Laboratory Setup: This part consists of defining the number of sections to be analyzed and the units used in the calculations.
Section Properties: The properties of the cross-sections to be investigated in the laboratory session are specified. These include the material properties for concrete and reinforcing steel, cross-section dimensions, and reinforcement amount and location. The module displays warning messages for user input that does not conform to ACI-318 requirements.
Axial Forces: Up to five compressive axial forces may be specified to generate moment-curvature relationships for each cross-section. One of the following three formats may be selected to input the axial forces for each section: (1) percentage of the balanced axial force; (2) percentage of the axial force capacity under concentric loading; and (3) numeric values of the axial forces. The module uses a default value of zero for the axial forces.
Strain Condition for P-M Interaction: The P-M interaction diagrams in Window 3 correspond to a user-specified maximum concrete strain in each cross-section. A strain value less than or equal to the concrete crushing strain may be used. The module uses the concrete crushing strain as the default value.
Window 1 Control: The cross-section number to be displayed in Window 1 is selected.
Window 2 Control: The cross-section number to be displayed in Window 2 is selected.
Window 3 Control: The cross-section number(s) to be displayed in Window 3 are selected.
Window 4 Control: The cross-section number(s) as well as analysis input and output parameters to be displayed in Window 4 are selected.
The user can obtain help on how to complete these sections by clicking on the question mark buttons throughout the module.
FIG. 1. Java Module B interface. Users will complete the input sections above and then click on the solve button to view the results in Windows 1-4.
A demonstration of the use of Module B and its application on three cross-sections (S1-S3) with different amounts of steel reinforcement can be found here. A brief overview of the analysis process used by the module is given below.
· User Input:
The following user input is required by the module: concrete and steel properties, section dimensions and reinforcement properties, axial forces (default= zero axial force), and strain condition for P-M interaction (default= concrete crushing strain, ).
Currently, the entire cross-section is assumed to be unconfined. The compression stress-strain relationship of the unconfined concrete is determined using a method developed by Mander et al. .
In this method, the concrete
stress, , is given as a
function of the strain, , as:
in MPa or psi,
In equations (2) & (3) above, is the strain at peak stress ( ) and Ec is the tangent modulus of elasticity of
concrete calculated as:
Esec, the secant modulus of elasticity, is the slope of the line connecting the origin and peak stress on the compressive stress-strain curve ( i.e., ). Crushing of the unconfined concrete is assumed to occur at . These are described in more detail in the sample laboratory session here.
Window 2 generates the moment-curvature relationships of the user-defined cross-sections. This is an iterative process, in which the basic equilibrium requirement P= C-T is used to find the neutral axis location, c, for a particular maximum concrete compressive strain, , where P= user-specified axial force; C= internal compression stress resultant; and T= internal tension stress resultant. The total concrete compressive stress resultant and the location of its centroid is determined by integrating numerically under the concrete stress distribution. A total of fifty "fibers" are used to model the concrete in compression. The bending moment is assumed to act such that the top surface of the cross-section is in compression.
The entire process can be summarized for a cross-section with two layers of reinforcing bars (Fig. 2) as follows:
FIG. 2. Section strains, stresses, and stress resultants.
1. Select any maximum concrete compression strain, , value less than .
2. Assume a value for the neutral axis depth, c, from the compression face at the top.
3. From the linear strain diagram (assuming plane sections remain plane), determine the steel strains and by using similar triangles.
4. Compute the steel stress resultants as , and , where As1 and As2 are the total reinforcing steel areas in each layer.
5. Determine the concrete compression and tension stress resultants and , respectively, by integrating numerically under the concrete stress distribution curve.
6. Check to see if within an acceptable tolerance. If not, the neutral axis location must be adjusted upward or downward, for the particular maximum concrete strain that was selected in Step 1, until equilibrium is satisfied.
This determines the correct value of c corresponding to.
Curvature can then be found from:
The internal lever arms, zc and zct, from the concrete compressive and tension stress resultants to the "instant" centroid of the cross-section is calculated, after which
The instant centroid is described in the calculations for the Sample Cross-Section 2 Below.
The calculated moment M and curvature define a point on the moment-curvature relationship of the cross-section. The entire moment curvature relationship is constructed by repeating the above procedure for values between 0 and .
Window 3 generates P-M interaction diagrams for the user-defined cross-sections by determining the axial load and moment pairs of each section for a user-specified maximum concrete compression strain, (see "strain condition for P-M interaction"). The P-M interaction diagram for each cross-section is generated by selecting successive choices of the neutral axis distance, c, from an initial small value that corresponds tp pure bending to a large one that corresponds to pure axial loading condition.
The axial load is assumed to be applied at the "instant centroid", which is determined from a uniform compression strain distribution over the section, equal to the user-specified maximum concrete strain, .
The entire process can be summarized as follows:
1. For a given user specified maximum concrete compression strain, and a neutral axis value, use the linear strain diagram (assuming plane sections remain plane) to compute the steel strains and .
2. Compute the steel stress resultants as , and , where As1 and As2 are the total reinforcing steel areas in each layer.
3. Determine the concrete compression and tension stress resultants and , respectively, by integrating numerically under the concrete stress distribution curve.
4.Then the axial load can easily be calculated as .
The internal lever arms, zc and zct, from the concrete compressive and tension stress resultants to the instant centroid of the cross-section is calculated, after which
The derivation of the instant centroid is shown in detail in the calculations for Sample Cross-Section 2 below.
A sample laboratory session to demonstrate Java Module B can be found here. Calculations for one of the cross-sections from this session (Sample Cross-Section 2) can be found by clicking on the link below. This PDF also contains examples of the different parameters that can be displayed in the control windows 1-3 of Module B.
Support for the development of the Virtual
Laboratories for Reinforced Concrete Education is provided by the National Science Foundation (NSF) under Grant No. CMS98-74872 as a
part of the CAREER Program and by the Portland Cement
Association (PCA). The support of
Dr. Shih C. Liu and Dr. Steven McCabe, NSF Program Directors, and Dr. David Fanella, PCA Manager on Buildings and Special Structures is
gratefully acknowledged. The authors also thank Prof. B. F. Spencer of the
Disclaimer: No responsibility is assumed by the authors, the University of Notre Dame, or the Portland Cement Association for any errors or misrepresentations in the laboratory modules, or that occur from the use of these modules.
Mander, J.B., Priestley, M.J.N., and Park, R. "Theoretical stress-strain model for confined concrete," Journal of Structural Engineering, American Society of Civil Engineers, Vol.114, No. 8, 1988, pp. 1804-1825.