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Section Analysis Functions(Moment Capacity - Moment Curvature - Interaction Diagram - Max Rebar Stress)

The functions below utilizes a non-linear strain-compatibility methodology. Non-linear material models are applied to concrete materials and reinforcing strands by associating curves with their material definitions. Essentially, these functions requires a section definition, orientation, concrete failure strain in compression, axial force, concrete failure strain in tension, resistance factor for compression, resistance factor for tension, refinement, a boolean value about using tendon initial strain, compression-controlled strain limit in the extreme tension steel inputs.

Here are the functions;

momentCapacity(section,orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain, εcl) → number : This function determines the bending moment capacity under a specific axial force and orientation while taking into account the failure strain of concrete in compression and tension. To obtain the negative flexural capacity of the section, the orientation should be set to PI.

momentCapacityNA(section,orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → number: This function determines the neutral axis location of the section under a specific axial force and orientation while taking into account the failure strain of concrete in compression and tension.

momentCapacityTension(section, orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → number: This function determines the tension force acting on the section under axial force and orientation.

momentCapacityTensionLoc(section, orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → number: This function determines the tension force location on the considered section.

momentCapacityCompression(section, orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → number: This function determines the compression force acting on the section under axial force and orientation.

momentCapacityCompressionLoc(section, orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement,useTendonInitStrain,εcl) → number: This function determines the compression force location on the considered section.

momentCapacityCurvature(section, orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → number: This function determines the Curvature value of the section under given loading.

momentCapacityStrainAndCondition(section,orientation,εcu, axial force,εtl,ϕ compression,ϕ tension, refinement, useTendonInitStrain,εcl) → [PhiFactor:number, MaxSteelStrain:number, StrainCondition:enumaration] : This function returns a list that includes the resistance factor (which takes the value of "εcl" for compression-controlled and "εtl" for tension-controlled and undergoes interpolation between etl and ecl for the transition zone), along with the maximum strain value and the strain condition of the extreme tension reinforcement. The strain condition is indicated as follows: 0 for Tension Controlled, 1 for Compression Controlled, and 2 for the Transition Zone.

maxRebarTensileStress(section, orientation, axial force, moment, εcu, εtl, φcompression, φtension, refinement, useTendonInitStrain, εcl) → number : This function determines the maximum rebar stress on the tension reinforcement under a given axial force and moment.

interactionDiagram(section, orientation, εcu, εtl, φcompression, φtension, refinement, useTendonInitStrain, axial force, εcl) → [[Mu1, Pu1],[Mu2, Pu2]….]: This function calculates the interaction diagram curve points based on given parameters and returns a list containing point coordinates where Mu1, Mu2… represents moment and Pu1,Pu2... represents axial force.

Here are explanations about the inputs;

section: ‘Section’

<O N="Rectangle" T="Section" Material="@Fc_4ksi\|Material" Ay="432.00000000000546" Az="432.00000000000546" Iy="20735.999999999978" Iz="11664.000000000027" Izy="8.526512829121202e-14" Cz="-1.8770992990421393e-15" Cy="-1.8812112361703803e-15" Perimeter="84" MinZ="-12" MaxZ="12" MinY="-9" MaxY="9">
 <P N="w" V="18" D="Width" Category="Rectangle Shape" UT="Length" UC="Property" />
 <P N="d" V="24" D="Depth" Category="Rectangle Shape" UT="Length" UC="Property" />
 <O T="Shape">
 <O T="Point" X="-w/2" Y="-d/2" />
 <O T="Point" X="w/2" Y="-d/2" />
 <O T="Point" X="w/2" Y="d/2" />
 <O T="Point" X="-w/2" Y="d/2" />
 </O>
 <O T="RebarLineLayout">
 <P N="UseReferenceEdge" V="1" />
 <P N="Material" V="A615_60" T="Material" />
 <P N="RebarProfile" V="ASTMA615No6" T="Rebar" />
 <P N="BarDiameter" V="0.75" />
 <P N="NumberOfRebars" V="4" />
 <P N="RowSpacing" V="0" />
 <P N="NumberOfRows" V="1" />
 <P N="ReferenceEdge" V="0" />
 <P N="DistanceToEdge" V="1" />
 <P N="RebarSpacing" V="3" />
 <P N="InnerRebarSpacing" V="3" />
 <O T="Point" X="(-(NumberOfRebars-1)*((NumberOfRebars-2)*RebarSpacing+InnerRebarSpacing)/(NumberOfRebars-1)/2)+0" Y="(sectionDepth(Rectangle)/2-(DistanceToEdge+RebarProfile.D/2+((NumberOfRows-1)*RowSpacing)/2))+0" />
 <O T="Point" X="((NumberOfRebars-1)*((NumberOfRebars-2)*RebarSpacing+InnerRebarSpacing)/(NumberOfRebars-1)/2)+0" Y="(sectionDepth(Rectangle)/2-(DistanceToEdge+RebarProfile.D/2+((NumberOfRows-1)*RowSpacing)/2))+0" />
 </O>
 <O T="TendonLayout" Tensioning="0" StrandCount="1">
 <P N="DistanceToEdge" V="2" />
 <P N="TendonSpacing" V="3" />
 <P N="Material" V="A416_270low_relaxation" T="Material" />
 <P N="StrandProfile" V="Grade250_5_16_No8" T="Strand" />
 <P N="NumberOfTendons" V="6" />
 <P N="UseReferenceEdge" V="1" />
 <P N="Force" V="100" />
 <P N="DebondingLength" V="0" />
 <P N="ReferenceEdge" V="1" />
 <P N="InnerTendonSpacing" V="3" />
 <O T="Point" X="(-(NumberOfTendons-1)*((NumberOfTendons-2)*TendonSpacing+InnerTendonSpacing)/(NumberOfTendons-1)/2)+0" Y="(-sectionDepth(Rectangle)/2+(DistanceToEdge))+0" />
 <O T="Point" X="((NumberOfTendons-1)*((NumberOfTendons-2)*TendonSpacing+InnerTendonSpacing)/(NumberOfTendons-1)/2)+0" Y="(-sectionDepth(Rectangle)/2+(DistanceToEdge))+0" />
 </O>
 </O>

| The provided ParamML code represents the definition of a rectangular section named "Rectangle." This section is described with its geometric properties, reinforcement, and other details. Here's a breakdown of what the code does:

  1. Section Properties: - The section is named "Rectangle." - It is defined as a "Section" type. - The material for the section is referenced as "@Fc_4ksi|Material," which indicates that the material properties for this section are taken from the material named "Fc_4ksi." - Various properties of the section are specified, such as area moments of inertia (Iy, Iz, Izy), centroid coordinates (Ay, Az), and more. 2. Geometry and Shape: - The width of the rectangle is specified using the parameter "w." - The depth of the rectangle is specified using the parameter "d." - The shape of the rectangle is defined using points in a clockwise order, starting from the bottom left corner and proceeding around the perimeter. 3. Reinforcement Layout: - The section includes a "RebarLineLayout" to define reinforcement details. - The layout specifies the use of a reference edge, material (A615_60), rebar profile (ASTMA615No6), bar diameter, number of rebars, and spacing between rebars. - The layout positions the rebars along the width of the rectangle, with a certain distance from the edges and a given spacing. 4. Tendon Layout: - The section also includes a "TendonLayout" for tendon (prestressing cable) details. - The layout specifies parameters like distance to edge, tendon spacing, material (A416_270low_relaxation), number of tendons, and force applied to the tendons. - The layout positions the tendons along the width of the rectangle, with a certain distance from the edges and a given spacing.

In summary, the code defines a rectangular section with specified geometric properties, reinforcement (rebars and tendons), and other relevant details for further analysis and calculations. The provided properties allow for the calculation of various structural characteristics, such as moment of inertia, centroid, and reinforcement layout, which are crucial for designing and analyzing the behavior of the rectangular section in structural applications.

To see further information about T=”Section” object, see Section Object . To view this example in the library, see (https://openbrim.org/app/?incubator=1&author=ParamML_Examples_OpenBrIM+Platform&obj=objid560qzp8sjvck01r5hxf6b9)

orientation : ‘Angle’ (in radians)

You can observe that the unaxial moment capacity of the section about x-axis and unaxial moment capacity of the section about y-axis are different and can be obtained by entering orientation as 0 for the flexural capacity of the section about x-axis and 90 degrees for the flexural capacity of the section about y-axis. The flexural capacity of a section subjected to biaxial flexure is calculated by considering the interaction between moments around both principal axes (X and Y axes) and the corresponding axial load.

The interaction between axial load and biaxial moment can be represented by an equation. This equation relates the axial load ratio, the biaxial moment ratio for both axes, and the column's capacity reduction factors. It takes the form:

→ φ(P/Pn) + φMx(Mu/Mnx) + φMy(Mu/Mny) ≤ 1.0

where φ is the capacity reduction factor, Mnx and Mny are the nominal moment capacities about the X and Y axes, and Mu is the applied biaxial moment.

When a section is subjected to bending moments around both the X-axis and the Y-axis, the biaxial moment value and its angle can be calculated using principles of structural mechanics.

The biaxial moment (Mλ) can be calculated using vector addition of the moments around the two axes:

→ Mλ = sqrt(Mx^2 + My^2)

The angle (θ) of the biaxial moment with respect to one of the axes can be determined using trigonometry:

→ θ = atan(My / Mx)

This angle is the orientation angle input for moment capacity functions that are explained in this page.

εcu : ‘Number’

εtl : ‘Number’

εcl : ‘Number’

failure strain of concrete in compression (in./in.)

tension-controlled strain limit in the extreme tension steel (in./in.)

compression-controlled strain limit in the extreme tension steel (in./in.)

refinement : 'Number' The section is defined by smaller segments based on the number entered for this parameter. This implies that the accuracy of the results pertaining to the section will get closer to correctness as the number of segments defining it increases. No matter the refinement value input, a segment is consistently formed at designated reinforcement and tendon locations, enhancing analysis result accuracy.

As illustrated in above images, the size of the segments are changing based on refinement value.

useTendonInitStrain : ‘Boolean' (1 for 'True', 0 for 'False’) A boolean is a data type in computer programming and mathematics that represents one of two possible values: true or false. If the tendon is not defined in the section, this variable remains unused. In cases where the tendon is defined, there exist two distinct methods for calculating the stress of each fiber with tendons: UseTendonInitStrain=1(True) → One approach involves setting the strain of the fiber with the tendon to be equal to the tendon stress(tendon force/tendon area) divided by the modulus of elasticity of the tendon. This approach can be activated by setting the parameter "useTendonInitStrain" to true. Using this method, the total force of the fiber with a tendon will consistently match the tendon force.

UseTendonInitStrain=0(False) → The other approach involves utilizing the overall strain, which is obtained by summing the initial tendon strain and the segment strain computed based on the strain compatibility of the fiber as shown in the figure. The total strain value is then input into the stress-strain diagram of the material in order to calculate the force at the location of the tendon fiber. This method is the default approach for computing the sectional capacity.

In the example below, the tendon force is 150 kips multiplied by 2, resulting in a total of 300 kips. As observed, when the initial tendon strain is set to true, strain compatibility calculations directly utilize the 300 kips without incorporating the strain of those fibers. However, when the initial tendon strain is set to false, the tendon force increases to 392 kips based on strain compatibility. This increase occurs because the total strain based on strain compatibility is greater than the tendon strain calculated using tendon force, cross-sectional area, and modulus of elasticity.

To see this section on the app, click https://openbrim.org/platform/?application=app&prj=objidn2nkqiu8ojl7895qkak74i&active=GraphicsView

Go to sections on DATA

Right click on the Rectange section. Select Section Editor.

Click 3 dots on the left top corner. Select Flexural Strength.

DocSectionAnalysis Object

With this object, section analysis and calculations, such as rebar stress, flexural capacity, and interaction diagrams, can be displayed in the document. The values that affect the results can be overridden in the document, and changes in the results along the section can be reviewed instantaneously.

This core object is a very useful tool for displaying the results of the functions described above and for observing the effect on the outcome by altering the inputs in real-time.

The following parameters need to be set properly to display desired results correctly.

AnalysisType = This parameter changes the report type to be displayed.

→ To display “Flexural Capacity”, set AnalysisType as 0.

→ To display “Moment Curvature”, set AnalysisType as 1.

→ To display “Interaction Diagram”, set AnalysisType as 2.

→ To display “Rebar Stress”, set AnalysisType as 3. <O T="DocSectionAnalysis" AnalysisType="0" />

<O T="DocSectionAnalysis" AnalysisType="1" />

<O T="DocSectionAnalysis" AnalysisType="2" />

<O T="DocSectionAnalysis" AnalysisType="3" />

Section = This parameter collects the section definition for analysis. <O T="DocSectionAnalysis" Section=" " />

Width/Height = These parameters define the size of the report to be displayed. <O T="DocSectionAnalysis" Width=" " Height =" "/>

AxialForce = This parameter collects the axial force that is applied to the section. This parameter does not apply to interaction diagram analysis(AnalysisType=”2”).

To view the document that has been overridden on the user interface, simply modify the parameters in the input boxes and then click the refresh button

next to report title. <O T="DocSectionAnalysis" AxialForce=" " />

Moment = This parameter collects the bending moment that is applied to the section. This parameter is only valid when AnalysisType=”3” (Rebar Stress).

To view the document that has been overridden on the user interface, simply modify the parameters in the input boxes and then click the refresh button

next to report title. <O T="DocSectionAnalysis" Moment=" " />

Orientation = This parameter collects the angular position of the section in radians. To view the document that has been overridden on the user interface, simply modify the parameters in the input boxes and then click the refresh button. <O T="DocSectionAnalysis" Orientation="pi/2" />

PhiComp = This parameter collects the resistance factor for compression based on AASHTO.

PhiTens = This parameter collects the resistance factor for tension based on AASHTO. <O T="DocSectionAnalysis" PhiComp=" " PhiTens=" " /> Ecu = Failure strain of concrete in compression (in./in.)

Ecl = Compression-controlled strain limit in the extreme tension steel (in./in.)

Etl = Tension-controlled strain limit in the extreme tension steel (in./in.) <O T="DocSectionAnalysis" Ecu="0.003" Ecl="0.002" Etl="0.005"/> Refinement = This parameter changes the precision of the diagrams. To override this parameter, simply change the value in degree on Refinement box then click refresh button

next to report title. <O T="DocSectionAnalysis" Refinement ="1000" />

Example:

The provided ParamML code includes pre-defined materials, an I-shaped section with tendons and reinforcement, and other necessary input parameters. It illustrates the utilization of the listed moment capacity functions and summarizes the results in a table. For further information about pre-defined objects in the code see Section Object Material ObjectDocument Objects To view this example in the library, see (https://openbrim.org/app/?incubator=1&author=ParamML_Examples_OpenBrIM+Platform&obj=objid10a0m4welkyvov6m2zcxd )

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