Cross Section Analysis and Design - Worked Examples
Cross Section Analysis and Design - Worked Examples
Cross Section Analysis and Design - Worked Examples
Worked Examples
Worked examples
Contents
Introduction ............................................................................................................................. 3 Example 1 Reinforcement design of a beam section ............................................................... 4 Example 2 Reinforcement design of a beam section with rebar size constraints .................. 18 Example 3 Moment vs. Curvature Curve for specific axial force levels .................................. 23 Example 4 Moment vs. Curvature Curve for specific reinforcement ratios ........................... 33 Example 5 Control of concrete cracking: Estimation of the stress at reinforcement (Serviceability Limit State) ...................................................................................................... 41 Example 6 Control of concrete cracking: Estimate the stress at reinforcement (Serviceability Limit State) considering concrete tensile strength and using a parabolic concrete curve ...... 46 Example 7 Control of concrete cracking by reducing the maximum reinforcement stress ... 49 Example 8 Reinforcement design for both Ultimate and Serviceability Limit States ............. 53 Example 9 Check of user-defined reinforcement pattern in a composite cross section of a column ................................................................................................................................... 59 Example 10 Creation of interaction diagram for a composite cross section of a column ...... 76 Example 11 Cross sectional properties of a built-up steel section......................................... 86 Example 12 Creation of interaction diagram for a built-up steel section ............................ 100 Example 13 Estimation of maximum and minimum developed stresses on a built-up steel section under given external loads ....................................................................................... 108 Example 14 Check of a timber cross section ........................................................................ 114 Example 15 Steel jacketing for improvement of column strength and ductility .................. 126
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Worked examples
November 2012
Introduction
Thank you for using ENGISSOL products. Our experienced scientific team can assure you that you will be provided with a robust, powerful and reliable application of high standards and guaranteed accurate results, regarding the analysis and design of any cross section (generic, built up of one or more geometric entities, composite etc.). ENGISSOLs know-how with respect to engineering application development is incomparable, a fact that is proven by thousands of satisfied end-users around the world. Cross Section Analysis & Design performs all cross section calculation tasks. Among the programs capabilities are the calculation and illustration of Moment vs. Curvature graphs, interaction surfaces, stain distributions as well as stress contour plots. Moreover, reinforcement check or design options are also supported. The application fully complies with all major codes for reinforced concrete sections (AASHTO, UBC, AS 3600, IS 456, ACI 318, BS 8110, CSA A233, EC2, NZS 3101 and CP 65) but can also handle other user-defined materials (linear, bilinear, trilinear, parabolic or fully customized materials by providing a stress/strain curve). Cross Section Analysis & Design covers efficiently a design engineers everyday needs due to the programs simplicity and the wide range of supported cases. This document contains a series of worked examples for Cross Section Analysis & Design which can be found in the application using the File -> Open sample project menu item. We have tried to include characteristic examples from an extended range of common cases in the field of concrete, steel and composite structures. At ENGISSOL we are always looking for feedback on how to improve our products. You can contact us at engissol@engissol.com.
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Load cases
lc1: lc2: lc3: lc4: N=0 N=0 N = 15 kN N = -7 kN My = -250 kNm My = 20 kNm My = -45 kNm My = -65 kNm Mz = 0 Mz = 0 Mz = 0 Mz = 0
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Selection of ACI 318 regulation and definition of Stirrup type (Tied) and Frame type (Ordinary, Intermediate) Since the examined section does not belong to a column, we can select the User values option in the Reinforcement ratio limits box. We set a minimum ratio equal to 1.5 and keep the maximum value to 4%. Since the program is intended to handle generic cross sections, you should review the limits for reinforcement ratios, as they can differ depending on the frame elements which the examined section is assigned to. But in case of columns, you can keep the option Default by code for columns.
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Next, we are going to specify the material properties. Concrete can be specified by selecting Materials -> Concrete
The Default Concrete material can be overridden by changing its properties, as shown in the form below.
Value Concrete strength has been set equal to 25 MPa. The concrete is not assumed confined and the elasticity modulus is to be calculated according to ACI 318, so we do not
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need to change anything else. The concrete defined here is now accessible through the name C25. Alternatively we could click the Add new button in order to define a new concrete material. The reinforcement can respectively be defined from the Material -> Reinforcement menu item.
We modify the Default Reinforcement material by specifying a new name (S500) and a yield stress of 500 MPa.
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We are now ready to draw the geometry of the cross section, by clicking the Draw -> Rectangle using dimensions. The center point of the rectangular section can be inserted by entering its coordinates (0,0) and then clicking the button, or just by clicking on the point (0,0) with the mouse. Then the Length and Width values should be set to 0.25 and 0.60 respectively. Finally, the C25 concrete should be selected for the rectangle we are drawing.
After we have clicked the OK button, the rectangle is shown in the drawing area.
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Before drawing the rebars, we have to make sure that a proper rebar set is available for the design. This can be checked by selecting Project -> Available Rebars.
The rebars shown above are available in the project. This means that we can draw rebars of these diameters and additionally the program will only choose from these rebar sizes when performing a reinforcement design. So, we can now draw the reinforcement bars by clicking on the Draw -> Rebar Line menu item.
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We can enter a value (0.04 m) for the reinforcement cover to the rebar center in the field, which becomes active when inserting reinforcement bars, as shown below. The program automatically draws help lines at the specified distance from the concrete rectangle edges and enables mouse snapping at their intersections.
Start and end point coordinates have been automatically filled in the corresponding form. The remaining properties of the bottom reinforcement can be specified, by selecting a rebar count (4), a material (S500) and a reinforcement size ( 20).
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Note: The specified reinforcement (20) is not important at this point, since their diameter will be calculated by the program automatically and the required rebar will be assigned to them during the design procedure.
After clicking the OK button, the lower rebars are shown in the drawing area. We take the same steps in order to draw the top rebars. After clicking on the top points, we can define the corresponding properties, as below.
Note that the rebar count is now set to 3 and the reinforcement size is 12. Moreover, the option Constant size for design has been selected, so that the rebar sizes will not change
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during the design. After clicking the OK button, we can continue to draw the side reinforcement bars.
The rebar count is now set to 2 and both s1, s2 distances to 0.15 m. The option Constant size for design is still enabled since we do not want these rebars to be re -dimensioned.
In order to draw the side reinforcement bars on the right, we can take advantage of the replicate option.
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First of all, we click on the reinforcement line that has been defined before in order to select it.
The line appears now thicker with its related dimensions, which informs us that it is selected. In order to replicate it to the right side (16 cm to the right), we click on Edit -> Replicate menu item and on the Linear tab enter a dy distance equal to 0.17 m.
After clicking OK, the side reinforcement on the right is shown correctl y.
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Note that the rebars have automatically been numbered. When referring to specific rebars (for example when reporting the reinforcement forces), these numbers will be used.
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We need to specify the corresponding Analysis Parameters set by clicking Analysis -> Analysis Parameters. We will use the default ULS set. All parameters related to ACI 318 code can be found in this form. The concrete stress block is rectangular according to ACI 318, the tensile resistance of concrete is ignored and all remaining data have the default values according to the code. You can override these values by modifying any of the fields of the form.
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The load cases can be selected by clicking on the Analysis -> Reinforcement design -> Load cases menu item. The 4 load cases can be entered in the corresponding table. A new load case can be inserted by clicking the button. We make sure that the assigned Analysis Parameters set to each load case is ULS.
We just click Analysis -> Reinforcement design -> Analyze, to perform the reinforcement design procedure. The program asks if it should assign the calculated rebars to the existing ones. We click on Yes.
Results
The results can be obtained from the Analysis -> Reinforcement design -> Show results menu item. As we can see, the cross section is adequate and the bottom rebars have been chosen to be 18. Remaining rebars (top and side reinforcement) did not change (12), as we previously checked the Constant size for design option. Furthermore, the calculated rebars have been assigned to the reinforcement bars we provided in the beginning.
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Note: The program has calculated the required rebar sizes so that the provided reinforcement ratio lies between the limits specified in the Reinforced Concrete Regulation form.
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Rebars 7, 8, 9
Demonstration of area constraint between rebars Lets assume the following constraint: The area of each rebar 1, 2, 3 and 4 to the area of 7, 8 or 9 should be equal to 1.5. We suppose that side rebars are constant 12, as in Example 1.
Load cases
Same to these defined in Example 1.
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First of all we click on the File menu and select Open in order to open the file we created in Example 1.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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Next, we move the mouse over the reinforcement line at the top rebars (7, 8, 9) till the rebar line text appears.
After right-clicking, we select Properties from the menu, in order to view the properties of the top rebars.
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In the corresponding form, we make sure that the option Constant size for design is unchecked and the Grow factor value is set to 1 and finally click OK.
Proceeding respectively for the bottom rebars, we set the Grow factor value to 1.5.
Note that the selected Reinforcement values are not important, since the reinforcement bars will be designed by the program.
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We just click Analysis -> Reinforcement design -> Analyze, to perform the reinforcement design procedure, as in Example 1. The program asks if it should assign the calculated rebars to the existing ones. We click on Yes.
Results
The results can be obtained from the Analysis -> Reinforcement design -> Show results menu item. As we can see, the cross section is adequate and the bottom rebars have been chosen to be 18. Top rebars are now of size 14. But the calculated required area for reinforcement bars 1 to 4 and 7 to 9 conforms to the applied constraint, since 2.2966 cm 2 = 1.5 x 1.5311 cm2. Remaining rebars (side reinforcement) did not change (12), as their Constant size for design option has been checked (in Example 1). Note: The program has calculated the required rebar sizes so that the provided reinforcement ratio lies between the limits specified in the Reinforced Concrete Regulation form and also the applied constraint to be valid. Furthermore, it should be noticed, that the constraint may not be precisely valid for the placed rebar areas as well, since the program selects rebars from the available ones in project (Project -> Available Rebars menu item) under the condition that their area is greater than the required.
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Example 3 Moment vs. Curvature Curve for specific axial force levels
Data
The creation of Moments vs. Curvature curves using Cross Section Analysis & Design will be demonstrated in this example. The used cross section is similar to that defined in Example 2.
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First of all we click on the File menu and select Open in order to open the file we created in Example 1.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We click on Analysis -> Analysis Parameters menu item to view the existing Analysis Parameters sets. In the ULS set, the concrete stress distribution is chosen to be rectangular, which only applies to ultimate limit states and it has to be changed since the Moment vs. Curvature curve will be created for all possible stress states of the section, from the unstressed one to its failure. Thus, we choose a Parabolic-Linear stress strain - curve for concrete compressive parts. All remaining values (factors, strains etc.) are now set according to ACI 318 regulation. By clicking OK the form is closed.
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The load cases for Moment vs. Curvature analysis can be selected by clicking on the Analysis -> Moment Curvature -> Load cases menu item. By clicking the button we can add a new load case. The Moment curvature type is set to Moment about y, the Custom Angle (coordinate system rotation) is left to 0 and the Analysis Parameters item is changed to ULS so that we can use the parameters defined previously.
In order to use specific axial forces for the analysis, we click on Options button and in the form that shows, we select the option Alternate Axial Force.
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Next we enter the axial force values successively in the corresponding field at the top right corner of the form and click the Add button, so that all axial forces have been filled, as shown below.
We can also define the slope of the second branch of the equivalent bilinearized curve here, but we keep this value equal to 0. This means that the calculated bilinearized curve will be elastic-fully plastic. We click OK to close the form.
Carry out the analysis
We just click Analysis -> Moment Curvature -> Analyze, to perform the Moment Curvature analysis.
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The results can be obtained from the Analysis -> Moment Curvature -> Show results menu item.
The Moments vs. Curvature diagrams for the three specified axial force levels are shown in the form. We note that the equivalent bilinearized curves (same color, but dashed) have also been calculated. These can disappear by unchecking the Show equivalent bilinearized curve.
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The color of each curve can be controlled from the corresponding Color box after selecting the curve to modify by its axial force level.
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Section stress distribution for a specified point on Moments vs. Curvature curve
Assuming we would like to display the strain distribution and stress contour for a specific point on the diagram for the curve which refers to axial force level of -1000 kN. First of all we select this value from the related list as shown in the picture below.
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Point to show
By clicking on Show strain/stress distribution for current point, a form is shown, which represents the following data: Neutral axis location Strain distribution Equivalent force couple Stress contour on the cross section area Reinforcement forces Stress information per material
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First of all we click on the File menu and select Open in order to open the file we created in Example 3.
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Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We click on the Analysis -> Moment Curvature -> Load cases menu item and next on the button in order to insert a new load case lc 2. Load case lc 2 contains information regarding the reinforcement ratio levels used for the analysis. We change the Analysis Parameters to ULS and keep the remaining values. Afterwards, the different reinforcement ratio levels can be defined by clicking on the Options button of the lc 2 row.
On the form following that shows, we choose the Alternate Reinforcement Ratio option and provide the reinforcement ratio values on the list on the right.
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We just click Analysis -> Moment Curvature -> Analyze, to perform the Moment Curvature analysis.
Results
The results can be obtained from the Analysis -> Moment Curvature -> Show results menu item. On this form, we choose lc 2 in the Load case list at its top left corner in order to view the results of the load case defined previously.
The Moments vs. Curvature diagrams for the three specified reinforcement ratio levels are shown in the form. We note that the equivalent bilinearized curves (same color, but dashed) have also been calculated. These can disappear by unchecking the Show equivalent bilinearized curve box.
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The color of each curve can be controlled from the corresponding Color box after selecting the curve to modify by its reinforcement ratio level. Note: All reinforcement size constraints are respected when a reinforcement ratio is calculated. Thus, the side reinforcement bars are constant and bottom reinforcement area is 50% greater of the top ones (as they grow factor has been specified equal to 1.5).
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Section stress distribution for a specified point on Moments vs. Curvature curve
Assuming we would like to display the strain distribution and stress contour for a specific point on the diagram for the curve which refers to reinforcement ratio level of 0.015. First of all we select this value from the related list as shown in the picture below.
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Point to show
By clicking of Show strain/stress distribution for current point, a form is shown, which represents the following data: Neutral axis location Strain distribution Equivalent force couple Stress contour on the cross section area Reinforcement forces/stresses Stress information per material
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Example 5 Control of concrete cracking: Estimation of the stress at reinforcement (Serviceability Limit State)
Data
We will extend the file created in Example 2 in order to calculate the stress at each rebar under the loads defined below. As many regulations propose, the effect of cracking can be controlled by considering a reduced reinforcement stress which is specified by other parameters, such as allowable crack width, rebar diameters, rebar spacing etc. The concrete stress/strain curve is assumed to be linear.
Load cases
lc1: lc2: lc3: N=0 N=0 N = 20 kN My = -120 kNm My = 37 kNm My = -54 kNm Mz = 0 Mz = 0 Mz = 0
First of all we click on the File menu and select Open in order to open the file we created in Example 2.
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Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We need to specify the corresponding Analysis Parameters set by clicking Analysis -> Analysis Parameters. We will use the default SLS set. The concrete stress block is linear and the corresponding strain values are defined according to ACI 318. Strength reduction factors do not need to be changed since we are performing an analysis in Serviceability Limit State (stress calculation). The concrete tensile strength is ignored. You can override these values by modifying any of the fields of the form.
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We click on the Analysis -> Deformed configuration -> Load cases menu item and next on the button in order to insert a new load case under the name lc 1. Afterwards we fill the values for loads and make sure that the SLS Analysis Parameters set is enabled. We can create the remaining load cases lc 2 and lc 3 likewise.
We just click Analysis -> Deformed configuration -> Analyze, to perform the analysis.
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The results can be obtained from the Analysis -> Deformed configuration -> Show results menu item. On this form, we choose the lc 1 load case at the top left corner, in order to view the results related to the first defined load case. The following data can be found on this form. Neutral axis location Strain distribution Equivalent force couple Stress contour on the cross section area Reinforcement forces/stresses Stress information per material
As we can see, the maximum stress values at reinfordcement are reported as expected for rebars 1 to 4 (bottom layer reinforement) and have a value of 222.72 MPa. By the above mentioned way, we can obtain the results for the remaining load cases lc 2 and lc 3 by changing the selected load case item at the top left corner of the results form.
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Example 6 Control of concrete cracking: Estimate the stress at reinforcement (Serviceability Limit State) considering concrete tensile strength and using a parabolic concrete curve
Data
We will modify the file created in Example 5 in order to re-estimate the reinforcement stresses, but considering the concrete tensile strength and using a Parabolic-Constant concrete curve for compression.
Load cases
As defined in Example 5.
First of all we click on the File menu and select Open in order to open the file we created in Example 5.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We need to modify the previously created Analysis Parameters set SLS by clicking Analysis -> Analysis Parameters and selecting SLS from the list at the top left corner. Next, we select a Parabolic-Linear stress strain curve for compressive parts of concrete. Finally, in order to consider the tensile concrete strength, we select the Linear and Drop to Zero option in the tensile concrete data, as shown in the form below.
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We just click Analysis -> Deformed configuration -> Analyze, to perform the analysis.
Results
Similarly to Example 5, the results can be obtained from the Analysis -> Deformed configuration -> Show results menu item.
We notice that the stress at rebars 1 to 4 (bottom layer rebars) is now 208.39 MPa for load case lc 1.
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Load cases
As defined in Example 5.
First of all we click on the File menu and select Open in order to open the file we created in Example 6.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We need to modify the previously created Analysis Parameters set SLS by clicking Analysis -> Analysis Parameters and selecting SLS from the list at the top left corner. At the bottom right corner of the form, we activate the option Tensile stress limit and enter the value 180 MPa.
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We just click Analysis -> Deformed configuration -> Analyze, to perform the analysis.
Results
Similarly to Example 6, the results can be obtained from the Analysis -> Deformed configuration -> Show results menu item.
The maximum tensile stress at reinforcement was 208.39 MPa in Example 5, where no reinforcement limit was applied. For the same loads (lc 1), we can now see that the cross section is now inadequate.
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Cross section is adequate for the remaining load cases, where apparently the developed reinforcement stresses are smaller than the limit of 180 MPa.
Note: These limits are respected for other analysis types as well (such as Interaction diagram, Reinforcement design etc.) provided that such stress limits have been applied to the corresponding Analysis Parameters sets.
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Example 8 Reinforcement design for both Ultimate and Serviceability Limit States
Data
In this example, we will perform a reinforcement design for both Ultimate and Serviceability Limit States (ULS and SLS). Such procedure is very common when designing reinforced concrete beams, since both requirements, regarding ultimate strength and cracking due to service loads should be checked. Cross Section Analysis & Design can easily handle similar cases by modifying the default Analysis Parameters sets ULS and SLS. If more design situations should be considered, we can create new Analysis Parameters sets. We will start from the file created in Example 7. So materials and section geometry will not change. The additional design data are as follows: Concrete and Reinforcement data Compressive Stress/Strain part distribution of concrete Tensile part Reinforcement stress limits Ultimate Limit State (ULS) Rectangular according to ACI 318 Ignored (no tensile strength) No limits (use full strength) Serviceability Limit State (SLS) Parabolic-Constant Yes (consider tensile resistance) Reduce tensile strength to 220 MPa to limit crack width
Load cases
Moreover, the reinforcement design procedure will take place for the following loads: Ultimate loads (ULS) lc1-ULS lc2-ULS lc3-ULS N=0 N=0 N = -10 kN My = -140 kNm My = 55 kNm My = -44 kNm Mz = 0 Mz = 0 Mz = 0
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First of all we click on the File menu and select Open in order to open the file we created in Example 7.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We need to modify the Analysis Parameters sets ULS and SLS to match with the design data provided above. This can be done by clicking Analysis -> Analysis Parameters. First of all, we select ULS from the list at the top left corner.
We make sure that the compressive stress strain part of concr ete is set to Default rectangular by code for ULS. Moreover, the No Tension should be selected for the tensile part of concrete. The Stress/Strain limits for reinforcement are not changed as shown in picture above.
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Next, we select SLS from the list at the top left corner.
According to the given design data, we make sure that a Parabolic-Constant concrete curve has been selected. The tensile part should have the option Linear and Drop to Zero selected, in order to consider the tensile resistance of concrete. Finally, we have applied a tensile stress limit for reinforcement equal to 220 MPa. We notice that the strength reduction factors are in this case equal to 1.0, since the program automatically adjusts them to be compatible to current design situation SLS. On the other hand, the default values by ACI 318 were applied for the ULS case. We click ok to close this form.
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We click on the Analysis -> Reinforcement design -> Load cases menu item to enter the load cases.
First of all, we select all the rows and click the icon to delete them. Afterwards, we enter the load cases and assign the corresponding Analysis Parameters sets, as shown below.
We just click Analysis -> Reinforcement design -> Analyze, to perform the analysis.
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The results can be obtained from the Analysis -> Reinforcement design -> Show results menu item.
We note that the load case is adequate and we can see the placed reinforcement sizes. Notice that the most unfavourable load case is lc1-SLS, which refers to Serviceability Limit State. Although the loads for SLS are smaller than ULS, the reinforcement demand can apparently be greater, since the reinforcement stress was reduced to 220 MPa. Furthermore, the reinforcement size constraints applied in Example 2 are considered in this design as well. These can be verified for each load case. For instance for lc1-ULS, the needed reinforcement for rebars 1, 2, 3 and 4 is 0.99 cm2, whereas the reinforcement demand for rebars 5, 6 and 7 is 0.66 cm2. The size constraint is valid since 0.99 cm2 = 1.5 x 0.66 cm2.
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Regulation: Eurocode 2 Concrete (cylinder) strength: 20 MPa Yield stress of reinforcement steel: 500 MPa Reinforcement cover to rebar center: 4 cm Steel grade: Fe 360 Tensile concrete strength is ignored. Stirrup type: Tied All remaining data (safety factors, other coefficients etc.) to be taken according to Eurocode 2.
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Load cases
lc1: lc2: N = -1000 kN N = -770 kN My = 550 kNm My = 140 kNm Mz = 123 Mz = -365
Selection of Eurocode 2 regulation The reinforcement ratio limits are to be calculated according to Eurocode 2, so we keep the default option checked.
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Next, we are going to specify the material properties. Concrete can be specified by selecting Materials -> Concrete
The Default Concrete material can be overridden by changing its properties, as shown in the form below.
Value Concrete strength has been set equal to 20 MPa. The concrete is not assumed confined and the elasticity modulus is to be calculated according to Eurocode 2, so we do
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not need to change anything else. The concrete defined here is now accessible through the name C20. Alternatively we could click the Add new button in order to define a new concrete material. The reinforcement can respectively be defined from the Material -> Reinforcement menu item.
We modify the Default Reinforcement material by specifying a new name (S500) and a yield stress of 500 MPa.
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Finally we need to define the grade of the steel Fe 360. This steel grade can be imported from the library of the program by clicking on the Materials -> Import from Library menu item. From the top list we select Hot Rolled Steel European (no hardening) in order to view the materials belonging to this collection.
Then, we can select the first row (Fe 360) and click on Insert selected only button to use this material in our project.
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Fe 360 steel has been imported in our project. We can view/change its properties by clicking on it from the list of the main form, as shown in the picture below.
Alternatively, as Fe 360 steel is as bilinear material, it can be edited from the Materials -> Bilinear Materials menu item.
In this form, the stress/strain curve of Fe 360 hot rolled steel is represented. We do not need to change anything and click OK.
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We are now ready to draw the geometry of the cross section, by clicking the Draw -> Circle (2Points).
First of all we make sure that material C20 has been selected in the shown form. Then, we can specify the center point of the circle by entering its coordinates (0,0) and then clicking the button, or just by clicking on the point (0,0) with the mouse.
After that we need to define one point on the circle. This can be done by clicking at the point (0.30,0).
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So, the circle is now defined and we can see it in the drawing area.
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We can modify its properties by moving the mouse over it, right clicking and selecting Properties from the corresponding menu.
In this way, the Circle form is shown and w e can view/change its data.
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Afterwards, we can insert a hot rolled section from the library by clicking the Draw -> Standard Section menu item.
In this form, the collections of all sections of the library are represented. In order to insert a HEA-400 cross section, we select European steel shapes from the top list and then we choose the HE400A shape from the list below.
The insertion point of the shape is shown in red and can change by clicking on other possible points indicated in blue.
The insertion point of the shape is selected to coincide with its centroid.
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We should not forget to change the material for this section to Fe 360, as illustrated below.
We click OK and see that the steel shape is shown in the main drawing area.
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Furthermore, we need to draw the rebars, which will have a circular pattern. But first lets see if rebar sizes are defined. We select Project -> Available Rebars.
The rebars shown above are available in the project. This means that we can draw rebars of these diameters and additionally the program will only choose from these rebar sizes when performing a reinforcement design. Now we are ready to draw the reinforcement bars by clicking on the Draw -> Rebar Circle menu item.
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We can enter a value (0.05 m) for the reinforcement cover to the rebar center in the field, which becomes active when inserting reinforcement bars, as shown below. The program automatically draws help points at the specified distance from the concrete edges and enables mouse snapping.
To draw the rebars, we click on (0,0) point and then on one of the available green points.
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We change the rebar count to 8 and also check that the reinforcement material is S500 and choose a rebar size, 20. After clicking the OK button, the lower rebars are shown in the drawing area.
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We need to specify the corresponding Analysis Parameters set by clicking Analysis -> Analysis Parameters. We will use the default ULS set. All parameters related to Eurocode can be found in this form. The concrete stress block is rectangular according to Eurocode 2 parameters, the tensile resistance of concrete is ignored and all remaining data have the default values. You can override these values by modifying any of the field of the form.
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The load cases can be selected by clicking on the Analysis -> Reinforcement check -> Load cases menu item. The 2 load cases can be entered in the corresponding table. A new load case can be inserted by clicking the button. We make sure that the assigned Analysis Parameters set to each load case is ULS.
Reinforcement check
We just click Analysis -> Reinforcement check -> Analyze, to perform the reinforcement design procedure.
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The results can be obtained from the Analysis -> Reinforcement check -> Show results menu item. As we can see, the cross section is adequate for both load cases. This form reports the placed reinforcement ratio (9.42 ) which lies between the allowable limits (2 4%) calculated according to Eurocode 2 specifications.
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Case 3:
First of all we click on the File menu and select Open in order to open the file we created in Example 9.
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Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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We do not need to modify the previously created Analysis Parameters set ULS. We can view the assigned properties by clicking Analysis -> Analysis Parameters and selecting ULS from the list at the top left corner.
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The load cases can be selected by clicking on the Analysis -> Interaction (as per selected R/C code) -> Load cases menu item. Case 1 We click on the button to add a new load case. We change the name to Case 1 and select Moment about y Moment about z option. The assigned Analysis Parameters set should be ULS.
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Case 2 We click on the button to add the next load case and change the name to Case 2. We make sure that the option Moment about y Moment about z is selected and the Analysis Parameters set is ULS. Then we click on Options to specify the axial force levels for the interaction diagram.
On the form above, we select Alternate Axial Force in the Variations box on the left. Next, we enter at the top right corner the values -1000 and click on Add button. We repeat this step for the value 400.
In this way we request the interaction diagram to be produced for both specified axial force levels. We click OK to close the form.
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Case 3 As shown before, we click on the button to add the last load case Case 3. We make sure that the option Axial force - Moment about y is selected and the Analysis Parameters set is ULS. Then we click on Options to specify the reinforcement ratio values for the interaction diagram. We select the Alternate Reinforcement Ratio option and provide at the top right corner the values of 0.005, 0.015 and 0.025.
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We just click Analysis -> Interaction (as per selected R/C code) -> Analyze, to perform the analysis.
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The requested interaction diagrams can be obtained by clicking click Analysis -> Interaction (as per selected R/C code) -> Show results. Interaction diagram for Case 1: We choose Case 1 from the list at the top right corner of the form. The interaction diagram for Moment about Y vs. Moment about Z is shown below.
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Interaction diagram for Case 2: By choosing Case 2, appears the same interaction diagram, but for different axial force levels.
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Interaction diagram for Case 3: Respectively, by selecting Case 3, we can display t he interaction diagram in terms of Moment about Y vs. Axial force for three different reinforcement ratio values (0.5%, 1.5% and 2.5%).
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Since the cross section does not have a concrete part, we can optionally set the Reinforced Concrete code to None, in order to hide all data related to reinforced concrete analysis. We click on Project -> Reinforced Concrete code menu item and select None as below.
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Next, we are going to specify the material properties. The A36 Gr.36 steel grade can be imported from the library of the program by clicking on the Materials -> Import from Library menu item. From the top list we select Hot Rolled Steel US (no hardening) in order to view the materi als belonging to this collection.
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Then, we can select the first row (A36 Gr.36) and click on Insert selected only button to use this material in our project.
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A36 Gr.36 steel has been imported in our project. We can view/change its properties by clicking on it from the list of the main form, as shown in the picture below.
First we will insert a standard wide flange section from the library by clicking the Draw -> Standard Section menu item. We select the AISC13 library in the form below.
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After selecting AISC13 shape family, all sections belonging to it, appear in the list below.
The insertion point of the shape is shown in red and can change by clicking on other possible points indicated in blue.
The insertion point of the shape is selected to coincide with its centroid. We should not forget to check that the assigned material is A36 Gr.36.
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We click OK and see that the steel shape is shown in the main drawing area.
Next, we will insert a Tee section. We click again on the Draw -> Standard Section menu item and select the AISC13 library. We click on Tee section type, select WT155X26 and change the Place at point y value to 0.2 m.
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Now we need to rotate the Tee 90o clockwise. This can be done by right clicking on it and then selecting Rotate.
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The program asks for the center point of rotation, as the status bar shows.
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So we click on point (0.2,-0.15). Then, in order to enter the base point for rotation, we click on point (0.2,0.1). No we can move the mouse to rotate the cross section interactively.
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After aligning it horizontally, we can click anywhere with the mouse to fix the shape in this position. Following this action, we delete the initial section by right-clicking on it and selecting Delete.
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Now we are about to move the Tee section next to the W section. First, we uncheck the Snap to grid option, from the toolbar at the bottom of the screen.
In this way, we disable the Snap to nodes function during editing geometry. On the other hand, the Snap to points option should be checked in order to enable snapping to the remaining characteristic points of the section parts except for the grid points. In order to move the T section, we right-click on right and select Move.
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Then we click at the middle point of the Tee web and move it to the middle web point of the W section, as shown below.
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By left-clicking on that point, the T section is fixed in this position. Next we will mirror the T section about a vertical axis. This can be accomplished by right clicking on it and choosing Mirror. The program asks for the starting point of the mirror axis (see Status bar). We enable the snap option again by clicking on the Snap to grid item in the lower toolbar and successively click on points (0,-0.25) and (0,0.25). The geometry is now defined and the drawing screen should look like this.
We just click Analysis -> Inertia Data -> Analyze, to have the sectional properties computed.
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The calculated data can be found by clicking on Analysis -> Inertia Data -> Show results menu item.
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First of all we click on the File menu and select Open in order to open the file we created in Example 11.
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Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section.
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In order to review the Analysis Parameters, we click on the Analysis -> Analysis Parameters menu item. The reinforced concrete related parameters are hidden, since no concrete regulation has been selected. We can select a material from the list on the left and modify the existing preferences.
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We will use the advanced interaction analysis in this example. The other option for interaction analysis (as per selected R/C code) is available only when a reinforced concrete code has been specified. The load cases can be selected by clicking on the Analysis -> Interaction (advanced) -> Load cases menu item. Case 1 We click on the button to add a new load case. We change the name to Case 1 and select Axial force - Moment about y option. The assigned Analysis Parameters set should be ULS.
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Case 2 We click on the button to add the next load case and change the name to Case 2. We make sure that the option Moment about y Moment about z is selected and the Analysis Parameters set is ULS. Then we click on Options to specify the axial force levels for the interaction diagram.
On the above form, we select Alternate Axial Force in the Variations box on the left. Next, we enter at the top right corner the values -750 and click on Add button. We repeat this step for the value 900.
In this way we request the interaction diagram to be produced for both specified axial force levels. We click OK to close the form.
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We just click Analysis -> Interaction (advanced) -> Analyze, to perform the analysis.
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The requested interaction diagrams can be obtained by clicking click Analysis -> Interaction (advanced) -> Show results. Interaction diagram for Case 1: We choose Case 1 from the list at the top right corner of the form. The interaction diagram for Moment about Y vs. Axial force is shown.
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Interaction diagram for Case 2: By choosing Case 2, the corresponding interaction diagrams in terms of Moment about Y axis vs. Moment about Z are shown.
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Example 13 Estimation of maximum and minimum developed stresses on a built-up steel section under given external loads
Data
In this example we will calculate the stresses developed in the cross section discussed in Example 11 and create corresponding stress contours. The partial safety factor for steel is assumed to be 1.0.
Load cases
lc1: lc2: lc3: N = 500 kN N=0 N = -200 kN My = 150 kNm My = -250 kNm My = 0 kNm Mz = 100 kNm Mz = 50 kNm Mz = -147 kNm
First of all we click on the File menu and select Open in order to open the file we created in Example 11.
Unlock the model
Afterwards, if the model is locked, we click on Edit -> Unlock model, in order to modify the geometry of the cross section. In the popup window, we choose Yes to unlock the model.
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The partial safety factor of the steel A36 Gr.36 is set to 1, so we do not need to change anything in the form below. We click OK to close the form.
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First we will define the load cases by clicking on the Analysis -> Deformed configuration -> Load cases menu item. The load cases can be defined by successively clicking on the button. We have to make sure that the ULS Analysis Parameters set have been assigned to each load case.
We just click Analysis -> Deformed configuration -> Analyze, to perform the analysis.
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The requested interaction diagrams can be obtained by clicking click Analysis -> Interaction (advanced) -> Show results. Results for load case lc1: We choose lc1 from the list at the top right corner of the form. The corresponding strain distribution and stress contour are shown. Moreover, information about the maximum/minimum stresses developed on the section can be found on the right.
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Results for load case lc2: Respectively, by selecting lc2, appears the corresponding information for the second load case.
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Results for load case lc3: Respectively, by selecting lc3, appears the corresponding information for the second load case.
As we can see, the maximum stress is developed for load case lc2 and has a value of 248.21 MPa. The minimum stress is -248.21 MPa for lc2 as well. Since this stress is equal to the yield of A36 Gr.36, we can conclude that some parts of the cross section have entered the plastic region.
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Load cases
lc1: lc2: lc3: N = 5 kN N=0 N = -5 kN My = 14 kNm My = -15 kNm My = 4 Mz = 0 kNm Mz = 0 kNm Mz = 3 kNm
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Since the cross section does not have a concrete part, we can optionally set the Reinforced Concrete code to None, in order to hide all data related to reinforced concrete analysis. We click on Project -> Reinforced Concrete code menu item and select None as below.
Next, we are going to specify the material properties. The C20 solid wood grade can be imported from the library of the program by clicking on the Materials -> Import from Library menu item. From the top list we select Ti mber EN 338 Solid wood in order to view the materials belonging to this collection.
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Then, we can select the property C20 and click on Insert selected only button to use this material in our project.
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C20 wood has been imported in our project. We can view/change its properties by clicking on it from the list of the main form, as shown in the picture below.
First we will draw the web of the Tee section. To do this, we click on Draw -> Rectangle using dimensions from the top menu. The center point of the rectangular section can be inserted by entering its coordinates (0,0) and then clicking the button, or just by clicking on the point (0,0) with the mouse. Then the Length and Width values should be set to 0.08 and 0.15 respectively. Finally, the material C20 should be selected for the rectangle we are drawing.
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Next, we can draw the top flange by using the same command (Draw -> Rectangle using dimensions). On the forms that shows, we enter the coordinates of the top flange center (0,0.105) and click the button. Then we specify the values for length and width and click OK.
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After clicking OK, we can see the final geometry of the cross section in the drawing area.
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In order to review the Analysis Parameters, we click on the Analysis -> Analysis Parameters menu item. The reinforced concrete related parameters are hidden, since no concrete regulation has been selected. While the ULS set is active, we select the C20 wood grade from the list on the left and change the partial safety factor to 1.3 as shown in the picture below.
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We will first define the load cases by clicking on the Analysis -> Deformed configuration -> Load cases menu item. The load cases can be defined by successively clicking on the button. We have to make sure that the ULS Analysis Parameters set have been assigned to each load case.
We just click Analysis -> Deformed configuration -> Analyze, to perform the analysis.
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The calculated data can be found by clicking on Analysis -> Deformed configuration -> Show results menu item. Results for load case lc1: We choose lc1 from the list at the top right corner of the form. The corresponding strain distribution and stress contour are shown. Moreover, information about the maximum/minimum stresses developed on the section can be found on the right.
Design for load case lc1: adequate Maximum stress: Minimum stress: 6.95 MPa at point (0.1,0.135) -7.89 MPa at point (-0.075,-0.135)
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Results for load case lc2: Respectively, by selecting lc2, appears the corresponding information for the second load case.
Design for load case lc2: adequate Maximum stress: Minimum stress: 8.43 MPa at point (-0.075,-0.135) -7.31 MPa at point (0.1,0.135)
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Results for load case lc3: Respectively, by selecting lc3, appears the corresponding information for the second load case.
Design for load case lc3: adequate Maximum stress: Minimum stress: 6.56 MPa at point (0.1,0.135) -5.98 MPa at point (-0.075,-0.135)
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Next, we are going to specify the material properties. Concrete can be specified by selecting Materials -> Concrete
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We click on From library button to import an existing concrete material from the database of the program.
After choosing the European standard, we select the C16/20 material and click the Insert selected only button.
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Since the material has been defined, we click OK to close the form.
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The reinforcement can respectively be defined from the Material -> Reinforcement menu item. As before, we click the From library button to select from the exist ing predefined reinforcement materials.
We choose the S400 grade from the European standard and click on Insert selected only.
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We are now ready to draw the geometry of the cross section, by clicking the Draw -> Rectangle using dimensions. The center point of the rectangular section can be inserted by entering its coordinates (0,0) and then clicking the button, or just by clicking on the point (0,0) with the mouse. Then the Length and Width values should both be set to 0.40. Finally, the C16/20 concrete should be selected for the rectangle we are drawing.
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After we have clicked the OK button, the rectangle is shown in the drawing area.
Before drawing the rebars, we have to make sure that a proper rebar set is available for the design. This can be checked by selecting Project -> Available Rebars.
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The rebars that are shown above are available in the project. This means that we can draw rebars of these diameters and additionally the program will only choose from these rebar sizes when performing a reinforcement design. So, we can now draw the reinforcement bars by clicking on the Draw -> Rebar Rectangle menu item. We can enter a value (0.04 m) for the reinforcement cover to the rebar center in the field, which becomes active when inserting reinforcement bars, as shown below. The program automatically draws help lines at the specified distance from the concrete rectangle edges and enables mouse snapping at their intersections.
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To draw the bottom rebars, we click on the green points at the top left and at the bottom right successively.
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Next, as shown on the form below, we choose to use 20 and 16 rebar sizes for the corner and remaining rebars respectively. The rebar count (apart from the rebars at the corners) horizontally and vertically is set to 2.
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After clicking the OK button, the rebars are shown in the drawing area.
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Cross Section Analysis & Design Behavior assessment of the cross section
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First we are going to assess the behavior of the cross section by calculating a Moment vs. Curvature diagram for different axial force levels.
Review of Analysis Parameters
We need to specify the corresponding Analysis Parameters set by clicking Analysis -> Analysis Parameters. We will use the default ULS set.
The option Default rectangular by code for ULS for the compressive parts of concrete stress/strain curve cannot be used, since we are about to perform a Moment Curvature analysis. Default rectangular by code for ULS option is only valid for designing or checking the capacity of a cross section. So we choose the option Parabolic-Linear. The strain data have already been set by the program to their default values according to Eurocode 2. All parameters related to Eurocode 2 can be found in this form, including partial safety factors etc.
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The reinforcement bars are chosen to have a bilinear behavior. A hardening of 1.15 is assumed. Additionally, we check the Ignore compression part in order to ignore the contribution of the reinforcement to the sectional resistance, i.e. steel reinforcement will only be active if it is under tension. This assumption is common, especially when retrofitting damaged concrete elements, as the sparse stirrups will not provide a lateral support to vertical reinforcement bars. Consequently their compressive resistance cannot be used due to buckling. We click OK to close this form.
Definition of load cases
The load cases for Moment vs. Curvature analysis can be selected by clicking on the Analysis -> Moment Curvature -> Load cases menu item. By clicking the button we can add a new load case. The Moment curvature type is set to Moment about y, the Custom Angle (coordinate system rotation) remains 0 and the Analysis Parameters item is changed to ULS in order to use the parameters defined previously.
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In order to use specific axial forces for the analysis, we click on Options button and as shown on the form below, we select the option Alternate Axial Force.
Next we enter the axial force values successively in the corresponding field at the top right corner of the form and click the Add button, so that all axial forces have been filled, as shown below.
We can also define the slope of the second branch of the equivalent bilinearized curve here, but we keep this value equal to 0. This means that the calculated bilinearized curve will be elastic-fully plastic. We click OK to close the form.
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We just click Analysis -> Moment Curvature -> Analyze, to perform the reinforcement design procedure.
Results
The calculated Moment Curvature curves for the specified axial force levels can be found by clicking Analysis -> Moment Curvature -> Show results.
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Cross Section Analysis & Design Behavior assessment after the application of steel jacket Definition of a steel material
Worked examples
The steel grade of the jacket is assumed to be Fe 430. This steel grade can be imported from the library by clicking on the Materials -> Import from Library menu item. From the top list we select Hot Rolled Steel European (no hardening) in order view the materials belonging to this collection.
Then, we can select the corresponding row (Fe 430) and click on Insert selected only button to use this material in our project.
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The simplest way to draw the steel jacket geometry is to use the Draw -> Polygon from line offset.
In the form that appears, we select the third option in the box Method. This will enable the drawing of the steel jacketing by just providing a path of points which are the 4 corners of the cross section. The specified thickness is set to 5 mm (0.005m) and current material is Fe 430.
After successively clicking on the 4 corners of the cross section, we right-click at any point and choose Close polygon from the menu as shown above.
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The retrofitted column section has been now defined. We can continue by reviewing the used Analysis Parameters set ULS.
Informing the program that the concrete is confined
This can be easily carried out by clicking the Materials -> Concrete menu item and checking the Confined option for C16/20 concrete grade, as shown below.
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In this way, the program will handle the C16/20 concrete as defined, i.e. the confinement factors specified in Analysis Parameters will be applied to this material.
Review of Analysis Parameters
The Analysis Parameters set ULS will be modified in order to simulate the effect of steel jacketing on the behavior of the section. Since the existing concrete is assumed confined, we can define the related amplification factors for its strength, the strain that corresponds to its strength and its ultimate strain. These values can be calculated according to any regulation or theoretical method and can be applied in the program.
Example of confined and unconfined concrete stress/strain curves In this example we will assume that the confined concrete stress is 21 MPa, with a corresponding strain of -3 and its ultimate strain is -7. Consequently, the amplification factors can be calculated as follows: Concrete strength factor: ec2 factor: ecu factor: 21 MPa / 16 MPa 3 / 2 7 / 3.5 = = = 1.31 1.5 2
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All remaining data regarding concrete and reinforcement do not need to be modified.
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Furthermore, in tab Other Materials data, we can define a partial safety factor for the steel jacketing, which is taken as 1.15.
We just click Analysis -> Moment Curvature -> Analyze, to perform the reinforcement design procedure.
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The calculated Moment Curvature curves for the specified axial force levels can be found by clicking Analysis -> Moment Curvature -> Show results.
As expected, after the application of the steel jacket, the cross section behavior in terms of stiffness, resistance and ductility has dramatically been improved. By clicking the Export current graph data, we can export the points of selected diagram to a text file and use them in other applications, such as Excel. In the following graph, the Moment vs. Curvature curves before and after the retrofit are shown for the same axial force load (-500 kN).
Moment curvature diagrams 300 200
Moment (kNm)
100 0 -0.03 -0.02 -0.01 -100 -200 -300 Curvature (1/R) 0 0.01 0.02 0.03
N=-500 kN confined
N=-500 kN unconfined
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