Reinforced Concrete Sections under Moment and Axial Load

Concrete International, Oct 2007 by Alaoui, Sanaa S, Klingner, Richard E

A spreadsheet-based program for producing moment-curvature and moment-axial force diagrams

Moment-curvature plots and moment-axial force interaction diagrams are essential tools for understanding the load-deformation behavior of structural elements. Moment-curvature plots readily illustrate stiffness, strength, and cross-sectional ductility, and allow the calculation of deflections after materials become nonlinear. Moment-axial force diagrams show the behavior of a section under combined axial load and bending moment, and whether that behavior is controlled by tension in the reinforcement or by compression in the concrete.

To make the use of these tools simpler, we created a user-friendly spreadsheet-based program that can generate moment-curvature plots and moment-axial force interaction diagrams for reinforced concrete sections.1 These plots can include the effects of confinement provided by transverse reinforcement on the stress-strain relationship of concrete. The Microsoft Excel� spreadsheet contains a series of macros written in Visual Basic for Applications (VBA) and has been named RECONASANCE (REinforced CONcrete Analysis Spreadsheet, enhANCEd). It allows the user access to equations used in calculations and can be downloaded for free with the electronic version of this article at www.concreteinternational.com .

BACKGROUND

The strain distribution in the cross section is assumed to vary linearly from a specified value at the extreme compression fiber to zero at the selected neutral axis location a distance c from the extreme compression fiber. The default steel stress-strain relationship used in the spreadsheet is shown in Fig. 1. For the concrete, the user can select between the following three stress-strain relationships:

* The Hognestad2 model;

* The Park and Kent3 model; and

* The Scott, Park, and Priestley4 model for low strain rates.

These relationships are illustrated for 4000 psi (27.6 MPa) concrete in Fig. 2. The last two models address confined as well as unconfined concrete.

At large curvatures, analysis of reinforced concrete cross sections can be difficult. Because the tensile reinforcement can be strain hardening while the concrete is on the descending branch of the stress-strain curve, multiple solutions can exist, making convergence unreliable or even unachievable.

To avoid this potential difficulty, the program doesn't iterate. For every maximum concrete strain and position of the neutral axis, a corresponding axial force and moment are calculated. Using the resulting table of equilibrium configurations, the equilibrium configuration corresponding to a particular axial load is found by linear interpolation between the two closest values.

PROGRAM INPUT

The "Input" sheet of the spreadsheet is used to enter material properties and cross-sectional geometry. All values must be entered in the units displayed.

Concrete properties

To define the concrete stress-strain relationship, the user inputs the compressive strength, ultimate strain, and model for the stress-strain relationship. Two of the models require information about confinement, so the user must input the spacing of hoops or spirals as well as the volumetric shear reinforcement ratio (defined as the ratio of the volume of shear reinforcement to the total volume of the core confined by and measured out-to-out of the shear reinforcement).

Some of this input is used for other calculations as well. For example, the spacing of the spirals or hoops is used in shear calculations. If the user chooses to use the third material model, Hognestad's relationship, the spacing of transverse reinforcement must still be input if shear calculations are desired.

Steel properties

To model the stress-strain relationship for the longitudinal reinforcement, the user must input yield strength, modulus of elasticity, strain at onset of strain hardening, initial strain hardening modulus, ultimate strain, and ultimate strength. For transverse reinforcement, only yield strength is required. The user may choose to use default values for Grade 60 steel by simply clicking on a button.

Geometry and reinforcement

Several steps are required to define the geometry of the initial and confined cross sections. First, under "Cross-Section Geometry," the user must enter the number of layers used to define the initial and confined cross sections. This number cannot exceed 20 and can be different for the two sections. The user must then enter the number of layers of longitudinal reinforcement (assumed to be equal for both sections), the depth of the initial section, and the depth of the confined section.

To define the dimensions of the initial and confined cross sections, the user must define each layer's width, thickness, and the distance y from the extreme compression fiber to the start of the layer. Finally, layers of reinforcement are described by an area of steel and a distance y measured from the extreme compression fiber of the initial cross section.

Shear calculations

The first input value required for shear calculations is the effective length of the member, used to calculate a moment on the section corresponding to the shear capacity. The other inputs include the width of the web or diameter of a circular section, the ratio of tension reinforcement to effective area, the cross-sectional area of transverse reinforcement within the spacing input under "Concrete Material Properties," and the effective depth of the cross section. In the example problem shown in this article, the effective depth was approximated as 0.8 times the total depth of the cross section.