

AxialForceMomentCurvature Relationships for RC Sections 

Table of Contents 



Welcome to the help page for Java Module B. This program allows users to investigate the nonlinear axialforcemomentcurvature relationships for rectangular reinforced concrete (RC) beamcolumn crosssections. The module determines the expected behavior of a userdefined crosssection 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 crosssections to be investigated in the laboratory session are specified. These include the material properties for concrete and reinforcing steel, crosssection dimensions, and reinforcement amount and location. The module displays warning messages for user input that does not conform to ACI318 requirements.
Axial Forces: Up to five compressive axial forces may be specified to generate momentcurvature relationships for each crosssection. 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 PM Interaction: The PM interaction diagrams in Window 3 correspond to a userspecified maximum concrete strain in each crosssection. 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 crosssection number to be displayed in Window 1 is selected.
Window 2 Control: The crosssection number to be displayed in Window 2 is selected.
Window 3 Control: The crosssection number(s) to be displayed in Window 3 are selected.
Window 4 Control: The crosssection 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 14.
A demonstration of the use of
Module B and its application on three crosssections (S1S3) 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 PM interaction (default= concrete crushing strain, ).
Currently, the entire crosssection is assumed to be unconfined. The compression stressstrain relationship of the unconfined concrete is determined using a method developed by Mander et al. [3].
In this method, the concrete
stress, , is given as a
function of the strain, , as:
(1) 
in MPa or psi,
where
(2) 
and
(3) 
In equations (2) & (3) above, is the strain at peak stress ( ) and E_{c} is the tangent modulus of elasticity of
concrete calculated as:
(4) MPa 
or
(5) psi 
E_{sec}, the secant modulus of elasticity, is the slope of the line connecting the origin and peak stress on the compressive stressstrain 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.
· Nonlinear BendingMomentCurvature Relationship:
Window 2 generates the momentcurvature relationships of the userdefined crosssections. This is an iterative process, in which the basic equilibrium requirement P= CT is used to find the neutral axis location, c, for a particular maximum concrete compressive strain, , where P= userspecified 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 crosssection is in compression.
The entire process can be summarized for a
crosssection 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 A_{s1}
and A_{s2} 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:
(6) 
The internal lever arms, z_{c} and z_{ct},
from the concrete compressive and tension stress resultants to the
"instant" centroid of the crosssection is
calculated, after which
(7) 
The instant centroid
is described in the calculations for the Sample CrossSection 2 Below.
The calculated moment M and curvature define a point on the momentcurvature relationship of the
crosssection. The entire moment curvature relationship is constructed by
repeating the above procedure for values between 0 and .
· AxialForceBendingMoment Interaction Diagrams:
Window 3 generates PM interaction diagrams
for the userdefined crosssections by determining the axial load and moment
pairs of each section for a userspecified maximum concrete compression
strain, (see "strain condition for PM
interaction"). The PM interaction diagram for each crosssection 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
userspecified 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 A_{s1}
and A_{s2} 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, z_{c} and z_{ct},
from the concrete compressive and tension stress resultants to the instant centroid of the crosssection is calculated, after which
(5) 
The derivation of the instant centroid is shown in detail in the calculations for Sample CrossSection 2 below.
A sample laboratory session to demonstrate Java Module B can be found here. Calculations for one of the crosssections from this session (Sample CrossSection 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 13 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. CMS9874872 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.
ACI318, "Building code requirements for
structural concrete (31899) and commentary (318R99)", American Concrete
Institute,
Nilson, A., "Design of concrete structures", McGraw Hill, Twelfth edition, 1997, 779 pp.
Mander, J.B., Priestley, M.J.N., and Park, R. "Theoretical stressstrain model for confined concrete," Journal of Structural Engineering, American Society of Civil Engineers, Vol.114, No. 8, 1988, pp. 18041825.