Shrinkage Effects on a Concrete Slab on Ground

Transcription

1 Shrinkage Effects on a Concrete Slab on Ground

2 Outline 1 Description 2 Finite Element Model 2.1 Units 2.2 Geometry Definition 2.3 Properties Concrete slab Grid reinforcement Soil interface 2.4 Loads 2.5 Meshing 3 Nonlinear structural analysis 3.1 Results 4 Random Fields for Cracking Estimation 4.1 Concrete Properties 4.2 Strain Load 4.3 Nonlinear structural analysis 4.4 Results 5 Design Methodology: Shrinkage Strain as Equivalent Temperature Variation 5.1 Material properties 5.2 Temperature load 5.3 Linear analysis Appendix A Additional information Shrinkage Effects on a Concrete Slab on Ground 2/53

3 1 Description This tutorial presents the simulation of a reinforced concrete (RC) slab on soil and studies the risk of cracking due to shrinkage. The case study is a concrete floor of a large workshop and storage building with steel structural elements, casted in 2008 and located in Go teborg, Sweden, and analysed in the Master Thesis of Narin and Wiklund (2012) from Chalmers University of Technology1. The floor area is approximately 1650 m2 made with concrete class C20/25 (Eurocode EN ) casted in four sections separated by dilatation joints as presented in [Fig. 1], [Fig. 2]. The slab is 220 mm thick and has equal top and bottom reinforcement with a grid of bars with 10 mm diameter spaced of 125 mm in the two directions. The steel type is NPS500 / B500T (fyk = 500 MPa). Concrete cover is 25 mm at the top and 50 mm at the bottom. Some parts of the floor rest on the soil and some parts are supported by piles. The curing was made with water for a period of 7 days. Some cracks were observed in the floor as presented in [Fig. 3]. Figure 1: Geometry of the floor and location of measured cracks [1] 1 Figure 2: Floor under construction [1] Figure 3: Crack in the floor[1] Narin, F. and O. Wiklund (2012), Design of slabs-on-ground regarding shrinkage cracking Master of Science Thesis, Department of Civil and Environmental Engineering, Division of Structural Engineering, Concrete Structures, Chalmers University of Technology, Goteborg, Sweden (Master Thesis 2012:59). Shrinkage Effects on a Concrete Slab on Ground 3/53

4 The following considerations were taken into account to simulate this slab : We use the models of Eurocode EN (EC2) available in DIANA for the mechanical properties of concrete, including shrinkage and creep effects in time. We model only one panel of the slab, correspondent to the area between two dilatation joints (15m x 28.23m) [Fig. 4]. We model the full geometry of this panel (and not take advantage of symmetry conditions) because, in an extra section of this tutorial, we will use Random Fields for the spatial distribution of mechanical properties in concrete to see the crack patterns. At the end of the tutorial, we will present the general design procedure to account for the shrinkage strains as an equivalent temperature variation in the model and compare with the previous analysis. We use a 3D finite element model with solid elements for the slab and embedded grid reinforcements [Fig. 5]. We consider that the floor is on the ground and simulate this restriction though a boundary interface with nonlinear elastic friction material, with normal stiffness 1000 times the elasticity modulus of the concrete and shear stiffness 100 times the elasticity modulus of concrete, cohesion is N/m 2 and friction angle is 35 degrees. Figure 4: Section considered for modeling Figure 5: 3D FE model of the slab In this tutorial we will study the shrinkage effects on a RC slab and the risk of cracking by considering three different approaches: 1. Nonlinear structural analysis with time-steps, shrinkage development with time and mechanical properties considered through the equations of EC2 available in DIANA; 2. Nonlinear structural analysis with load steps, shrinkage strain imposed as an external load and spatial variation of mechanical properties of concrete with Random Fields, using the Joint Committee on Structural Safety Probabilistic Model; 3. Linear analysis with shrinkage strain imposed as an equivalent temperature variation. Shrinkage Effects on a Concrete Slab on Ground 4/53

5 2 Finite Element Model For the modeling session we start a new project. We will use quadratic hexagonal elements. Main menu File New project [Fig. 6] Figure 6: New project dialog Shrinkage Effects on a Concrete Slab on Ground 5/53

6 2.1 Units We use SI units (N, m, s) with the exception of temperature that we change to Celsius. Despite the fact that it would be handy to have the time unit in days we keep it in seconds (SI) to have a consistent unit set [Fig. 8]. Model Window Reference system Units [Fig. 7] Property Panel [Fig. 8] Figure 7: Model window Figure 8: Property Panel - Units Shrinkage Effects on a Concrete Slab on Ground 6/53

7 2.2 Geometry Definition We start the model by making a sheet with the dimensions of the slab and duplicate it twice to make the bottom and top reinforcements and finally extrude it to make the slab. Main Menu Geometry Create Add polygon sheet [Fig. 9] [Fig. 10] Figure 9: Geometry - Add polygon sheet Figure 10: Geometry - Sheet Shrinkage Effects on a Concrete Slab on Ground 7/53

8 We now duplicate and rename the sheets. Geometry browser Geometry Shapes Slab Duplicate (2x) [Fig. 11] Geometry browser Geometry Shapes Slab1 Rename (or use F2) <Rename to Grid bot> Geometry browser Geometry Shapes Slab2 Rename (or use F2) <Rename to Grid top> [Fig. 12] Figure 11: Geometry browser - duplicated sheets Figure 12: Geometry browser - renamed sheets Shrinkage Effects on a Concrete Slab on Ground 8/53

9 We move the sheets in the Z-directions for the positions of the top and bottom grid reinforcements. The thickness of the slab is 0.22 m and the concrete cover is 0.03 m at the top and m at the bottom. Main Menu Geometry Modify Move shape (2x, one for each grid) [Fig. 13] [Fig. 14] Figure 13: Move sheet for grid bottom Figure 14: Move sheet for grid top Figure 15: Geometry - Sheets Shrinkage Effects on a Concrete Slab on Ground 9/53

10 Finally we extrude the sheet of the slab in the Z-direction with the thickness of 0.22 m. And the geometry of the model is completed. Main Menu Geometry Modify Extrude shape [Fig. 16] Figure 16: Geometry - Add polygon sheet Figure 17: View of the mode: slab Shrinkage Effects on a Concrete Slab on Ground 10/53

11 2.3 Properties Concrete slab To model the concrete we use the Concrete design codes class available in Diana. We use the Eurocode 2 EN with the concrete class C20/25 and we include creep and shrinkage effects 2. We use Structural solids and we don t need to define geometry and data. Main Menu Geometry Analysis Property assignments [Fig. 18] Shape property assignment assignment Add new material [Fig. 19] [Fig. 20] Figure 18: Assign slab properties Figure 19: Add new material - Concrete C20/25 2 Note: At this point we will not include the Total Strain crack model in this material because it would significantly increase the complexity of the analysis. We will study the crack patterns latter in this tutorial by using Random Fields for the probabilistic spatial distribution of the mechanical properties of concrete. Shrinkage Effects on a Concrete Slab on Ground 11/53

12 Ambient temperature is 20 o C, notional size of member (h = 2A c /u) (only the top part of the slab is exposed to the environment) is 0.44 and the relative humidity is 40%. We choose a non-aging creep curve type: concrete age at loading is 28 days (2.4192e+6s) and the concrete age at end of curing period is 7 days (604800s). Figure 20: Concrete C20/25 material properties Shrinkage Effects on a Concrete Slab on Ground 12/53

13 2.3.2 Grid reinforcement We now define the properties of the reinforcement. We will use embedded grid reinforcement. We define a new material for the reinforcement using no-hardening Von Mises plasticity with elasticity modulus of 2e+11 N/m 2 (200 MPa) and yielding strength of 5e+8 N/m 2 (500 MPa). Main Menu Geometry Analysis Reinforcement property assignments [Fig. 21] Reinforcement property assignments Add new material [Fig. 22] [Fig. 23] Figure 21: Assign reinforcement properties Figure 22: Add new material - Steel Figure 23: Material properties - Steel Shrinkage Effects on a Concrete Slab on Ground 13/53

14 The grid of reinforcement is made of bars of 10 mm diameter spaced of m in the two orthogonal directions. Reinforcement property assignments Add new geometry [Fig. 24] Figure 24: Edit new geometry - Grid Shrinkage Effects on a Concrete Slab on Ground 14/53

15 2.3.3 Soil interface We simulate the interaction between the slab and the soil through an interface. We use a nonlinear elastic friction material with normal stiffness 1000 times the elasticity modulus of the concrete and shear stiffness 100 times the elasticity modulus of concrete, cohesion is N/m 2 and friction angle is 35 degrees. Main Menu Geometry Analysis Connection property assignments [Fig. 25] Connection property assignment Add new material [Fig. 26] [Fig. 27] Figure 25: Assign connection properties Figure 26: Interface - Add new material Figure 27: Interface - Material properties Shrinkage Effects on a Concrete Slab on Ground 15/53

16 We assign supports to this interface by fixing the translation in the three directions at the bottom surface of the interface. The geometry of the model is complete. Connection property assignment Attach support [Fig. 28] [Fig. 29] Figure 28: Interface - Add supports Figure 29: Boundary interface Shrinkage Effects on a Concrete Slab on Ground 16/53

17 2.4 Loads For the loads we only need to define the self weight because the shrinkage strains are automatically accounted in the material model for the transient nonlinear analysis. We set a dependency factor curve to the Self Weight to be considered during all the analysis: time from 0 to 1.5e+9 s with a factor of 1. Main Menu Geometry Analysis Global load [Fig. 30] Geometry browser Loads Cases SW Edit time dependency factors [Fig. 31] [Fig. 32] Figure 30: Self weight load Figure 31: Geometry browser - Loads Figure 32: Time dependendent factors for self weight Shrinkage Effects on a Concrete Slab on Ground 17/53

18 2.5 Meshing We set the element size as 0.5 m in the shapes and divide the slab edges into 3 elements in the thickness. Finally we generate the mesh. Main Menu Geometry Analysis Set mesh properties (2x) [Fig. 33] [Fig. 34] Main Menu Geometry Generate mesh [Fig. 35] Figure 33: Mesh properties Figure 34: Mesh properties - slab edges Figure 35: Model view - Mesh Shrinkage Effects on a Concrete Slab on Ground 18/53

19 3 Nonlinear structural analysis We perform a structural nonlinear analysis with time steps. Analysis browser New analysis [Fig. 36] <Rename (F2 or ) Analysis1 to ShrinkageAnalysis > Analysis browser ShrinkageAnalysis Add command Structural nonlinear [Fig. 37] Analysis browser ShrinkageAnalysis Structural nonlinear Add Execute steps - Time steps [Fig. 38] <Rename (F2 or ) the execute block as Shrinkage > Figure 36: Analysis window Figure 37: Add command Figure 38: Analysis tree NOTE: As the load execute block is the default, we first need to remove it, and then add a new execute block with time steps. Shrinkage Effects on a Concrete Slab on Ground 19/53

20 We choose User specified sizes for the time steps. We use smaller time steps in the beginning of the analysis (because is when shrinkage deformations are more relevant). We then use larger time steps until aproximatly 50 years. The considered time steps are (in seconds): (15) (24) e+06(12) e+06(6) e e+07(5) e+08(4) s That correspond to: 1 day(15) 7.6 days(24) 15 days(12) 1 month(6) 6 months(1) 1 year(5) 10 years(4) Analysis browser ShrinkageAnalysis Structural nonlinear Shrinkage Time steps Edit properties [Fig. 39] Properties - TIME User specified sizes <Type the time steps> [Fig. 40] Figure 39: Edit time steps Figure 40: User specified sizes for time steps Shrinkage Effects on a Concrete Slab on Ground 20/53

21 For the equilibrium iteration we increase the maximum number of iterations to 20 to ensure convergence along the analysis. We use the default displacement and force norms and we choose to continue the analysis if convergence is not achieved. Analysis browser ShrinkageAnalysis Structural nonlinear Shrinkage Equilibrium iteration Edit properties [Fig. 41] [Fig. 42] Figure 41: Equilibrium iteration properties - Shrinkage Figure 42: Continue if no convergence Shrinkage Effects on a Concrete Slab on Ground 21/53

22 With the output user selection we choose the results of displacements, strains and stresses. Finally we run the analysis. Analysis browser ShrinkageAnalysis Structural nonlinear Output Edit properties [Fig. 43] Properties - OUTPUT Result User selection Modify [Fig. 44] [Fig. 45] Analysis browser ShrinkageAnalysis Run analysis Figure 43: Edit output properties Figure 44: Selection of results Figure 45: Output user selection Shrinkage Effects on a Concrete Slab on Ground 22/53

23 3.1 Results We start the presentation of results with the contour plots of maximum principal stresses for different stages of the analysis: Time-step 16 (22 days), Time-step 37 (0.5 years), Time-step 63 ( approximately 7 years) and Time-step 67 (approximately 50 years ). In order to better see the results we hide the Ground interface from the mesh. Mesh browser Mesh Element sets Slab Show only [Fig. 46] Results browser Output Element results Cauchy Total Stresses S1 <In Case select the desired time-steps> [Fig. 47] Main Menu Viewer Viewpoints Isometric view 1 Main Menu Viewer Fit all Figure 46: Show only slab mesh Figure 47: Results tree Shrinkage Effects on a Concrete Slab on Ground 23/53

24 We can observe that the increment of stresses occurs mainly during the first month, with a small difference from half year to 50 years. There is a high risk of cracking since the first days as the stresses are very close to the concrete tensile strength (f ct = 2.2 MPa for concrete C20/C25). These principal stresses are very similar to the stresses SYY which is the direction normal to the appearance of the cracks (see [Fig. 1]). Figure 48: Principal stresses S1-22 days Figure 50: Principal stresses S1-7 years Figure 49: Principal stresses S1-0.5 years Figure 51: Principal stresses S1-50 years Shrinkage Effects on a Concrete Slab on Ground 24/53

25 To better see the development of strains and stresses in concrete due to the shrinkage effects, we show the graphs of maximum principal strains and stresses with time for a node located at the middle of the slab. Model Element selection <Select the element located at the middle of the slab> [Fig. 52] Result browser Output Element results Total Strains E1 Show table [Fig. 53] Figure 52: Selected element Figure 53: Show table Shrinkage Effects on a Concrete Slab on Ground 25/53

26 In the same graph we can select various results in the y-axis. In addition to the maximum principal strains E1 we also ask for the maximum principal stresses S1 to be displayed in the same graph. We observe the development of shrinkage strains in the concrete and the generated stresses due to the restriction of the ground. We see the large increment of strains during the first days and the stresses reaching the plateau of approximately 2.15e+6 N/m 2 at around 3 years of time. Figure 54: Principal strains and stresses vs. time Shrinkage Effects on a Concrete Slab on Ground 26/53

27 4 Random Fields for Cracking Estimation In reality, as the concrete properties are not uniform, cracks will appear and propagate through the weakest points of the slab. We can consider this probabilistic spatial distribution of the mechanical properties of concrete through the Random Fields available in DIANA. We will use the material model of the Joint Committee on Structural Safety (JCSS) Probabilistic Modal Code 3 to study the crack pattern of this slab. For this we will need to define a new material and assign it to the slab. With this material model we will need to apply an explicit strain load to the slab and perform a nonlinear structural analysis with load steps. 3 Technical University of Denmark, Joint Committee on Structural Safety, Probabilistic model. http : // ublications/p robabilistic Model code Shrinkage Effects on a Concrete Slab on Ground 27/53

28 4.1 Concrete Properties We add a new material model using the Concrete design codes class and choosing the JCSS Probabilistic Model Code. We include the JCSS Random field. We select concrete grade C15 because it is similar to the C20/25 from Eurocode 2. We define the parameters of the Random field in order to reproduce a realistic distribution of mechanical properties. Main Menu Geometry Materials Add material [Fig. 55]-[Fig. 57] Figure 55: Add material - JCSS Random fields Figure 56: Edit material - C15 Random field Figure 57: Edit material - C15 Random Field Note: the parameters of the random field have to be defined for each particular case by trial-and-error method until a realistic distribution of the mechanical properties is achieved in the structure. We see this distribution through the status output of the mechanical properties (elasticity modulus, tensile strength, compressive strength). Shrinkage Effects on a Concrete Slab on Ground 28/53

29 We now assign this material to the slab. We can do so through the geometry browser by changing the material properties of the slab from material C20/25 to material C15 Random Field. Geometry browser Geometry Slab Properties Material <Change material C20/25 to material C15 Random Field > [Fig. 58] Figure 58: Assign material C15 Random Field to the slab Shrinkage Effects on a Concrete Slab on Ground 29/53

30 4.2 Strain Load We add a strain load to the slab with strains of -5E-4 in the three orthogonal directions (which is proximately the long-term shrinkage strain of this material determined through the expressions of EC2) and -1E-8 in the transversal directions. As this new settings were defined in geometry, we generate the mesh again to pass the new data to the mesh. Main Menu Geometry Analysis Attach load [Fig. 58] Main Menu Geometry Generate mesh [Fig. 35] Figure 59: Shrinkage strain load Shrinkage Effects on a Concrete Slab on Ground 30/53

31 4.3 Nonlinear structural analysis We add a second analysis for the Random Fields. We will perform a nonlinear structural analysis with load steps. We need two execute blocks, the first for the Self Weight and the second for the shrinkage strain load. Analysis browser New analysis <Rename (F2 or ) Analysis1 to RandomFields > [Fig. 60] Analysis browser RandomFields Add command Structural nonlinear [Fig. 61] Analysis browser RandomFields Structural nonlinear Add Execute steps - Load steps <Rename (F2 or ) the first execute block as SW > <Rename (F2 or ) the second execute block as Shrinkage > [Fig. 62] Figure 60: Analysis window Figure 61: Add command Figure 62: Analysis tree Shrinkage Effects on a Concrete Slab on Ground 31/53

32 For the Self Weight (SW) execute block we choose the load set SW and apply it in one step. We keep the equilibrium iteration settings as default. Analysis browser RandomFields Structural nonlinear SW Load steps Edit properties [Fig. 63] [Fig. 64] Analysis browser RandomFields Structural nonlinear SW Equilibrium iteration Edit properties [Fig. 65] Figure 63: Edit load steps - Self Weight Figure 64: Edit load steps - Self Weight Figure 65: Equilibrium iteration properties - Self Weight Shrinkage Effects on a Concrete Slab on Ground 32/53

33 For the Shrinkage execute block we select the load set Shrinkage strain and the User specified sizes for the load steps. We apply the total load in 10 equal steps with a factor of 0.1. Analysis browser RandomFields Structural nonlinear Shrinkage Load steps Edit properties [Fig. 66] [Fig. 67] Figure 66: Edit load steps - Shrinkage Figure 67: Edit load steps - Shrinkage Shrinkage Effects on a Concrete Slab on Ground 33/53

34 We expect some convergence problems due to the cracking in multiple points at the time. For this reason, in the equilibrium iteration settings we increase the maximum number of iterations. We use the method Secant (Quase-Newton). We use the energy convergence norm and choose to continue the analysis if convergence is not achieved. Analysis browser RandomFields Structural nonlinear SW Equilibrium iteration Edit properties [Fig. 68] [Fig. 69] Figure 68: Equilibrium iteration properties - Shrinkage Figure 69: Continue if no convergence Shrinkage Effects on a Concrete Slab on Ground 34/53

35 With the output user selection we choose the results of displacements, strains and stresses in global and principal directions. Analysis browser RandomFields Structural nonlinear Output Edit properties [Fig. 70] Properties - OUTPUT Result User selection Modify [Fig. 71] [Fig. 72] Figure 70: Edit output parameters Figure 71: Output properties for Shrinkage Analysis Figure 72: Analysis browser view Shrinkage Effects on a Concrete Slab on Ground 35/53

36 We also add a new output block for the parameters of concrete (elasticity modulus, tensile strength, compressive strength) computed by the Random Fields to be output in the first step. Finally we run the analysis. Analysis browser RandomFields Structural nonlinear Add - Output - User <Rename (F2 or ) this Output block as Output - Random Fields > [Fig. 73] Analysis browser RandomFields Structural nonlinear Output - Random Fields Edit properties [Fig. 74] [Fig. 75] Analysis browser RandomFields Run analysis Figure 73: Edit output parameters Figure 74: Output properties for Shrinkage Analysis Figure 75: Analysis browser view Shrinkage Effects on a Concrete Slab on Ground 36/53

37 4.4 Results We first see the distribution of the parameters elasticity modulus and tensile strength calculated by the Random Fields. In order to better see the results we hide the Ground interface from the mesh. Mesh browser Mesh Element sets Slab Show only [Fig. 76] Results browser Analysis: RandomFields Output - Random Fields Element results Elastic parameters YOUNG [Fig. 77], [Fig. 78] Results browser Analysis: RandomFields Output - Random Fields Element results Total strain parameters TENSTR [Fig. 77], [Fig. 79] Figure 76: Show only slab mesh Figure 77: Results tree - Random Fields Shrinkage Effects on a Concrete Slab on Ground 37/53

38 We confirm that the distribution of the elasticity modulus and tensile strength in concrete is consistent with what is expected in reality in the slab. Figure 78: Random field - Elasticity modulus Figure 79: Random field - Tensile strength Note: please remember that these results will not be the exactly the same when running the computation again. Shrinkage Effects on a Concrete Slab on Ground 38/53

39 We now present the crack widths in the principal directions for different load steps. Results browser Analysis: RandomFields Output - Random Fields Element results Crack-widths Ecw1 [Fig. 80], [Fig. 81]-[Fig. 84] <In Case select the desired time-steps> Figure 80: Results tree - Crack widths Shrinkage Effects on a Concrete Slab on Ground 39/53

40 We choose to represent the following steps of the analysis: Load-factor 0.3 (1.5E-4 shrinkage strain), Load-factor 0.4 (2.0E-4 shrinkage strain), Load-factor 0.6 (3.0E-4 shrinkage strain) and Load-Factor 0.7 (3.5E-4 shrinkage strain). These strain load levels correspond proximately to the shrinkage strains in this type of concrete for 1, 1.5, 2 and 4 years respectively, calculated through the EC2 expressions. Due to the cracking in various points at the same point convergence is not achieved for steps after cracking. Even tough we can see the initiation and propagation of cracking that is consistent with the in-situ observations (see [Fig. 1]). Figure 81: Crack widths Ecw1-1.5E-4 shrinkage strain Figure 83: Crack widths Ecw1-3.0E-4 shrinkage strain Figure 82: Crack widths Ecw1-2.0E-4 shrinkage strain Figure 84: Crack widths Ecw1-3.5E-4 shrinkage strain Note: please remember that these results will not be the exactly the same when running the computation again. Shrinkage Effects on a Concrete Slab on Ground 40/53

41 We try to get more converged results after the start of cracking, by continuing the analysis with smaller load steps while the cracks propagate. For that we duplicate the RandomFields analysis [Fig. 85]. In this new analysis we duplicate the Shrinkage execute block. Analysis browser RandomFields Duplicate [Fig. 85] Analysis browser RandomFields - Copy 1 Shrinkage Duplicate [Fig. 86] Figure 85: Results browser view Figure 86: Duplicate execute block Shrinkage Effects on a Concrete Slab on Ground 41/53

42 In the first Shrinkage execute block we do the first 3 load steps with explicit load steps until the start of cracking [Fig. 87]. In the second Shrinkage execute block Shrinkage - copy we use automatic load steps for the shrinkage strain [Fig. 88] [Fig. 89]. Analysis browser RandomFields - Copy 1 Shrinkage Load steps Edit properties [Fig. 87] Analysis browser RandomFields - Copy 1 Shrinkage - copy Load steps Edit properties [Fig. 88] Properties - LOAD Automatic step sizes settings [Fig. 89] Figure 87: Edit shrinkage load steps Figure 88: Edit shrinkage automatic load steps Figure 89: Automatic load steps Shrinkage Effects on a Concrete Slab on Ground 42/53

43 In this execute block we increase further the maximum number of iterations to 50 and we decrease the energy convergence tolerance to This convergence norm is acceptable in this high complex problem with multiple formation of cracks at the same time. We finally run the analysis. Analysis browser RandomFields - Copy 1 Shrinkage - copy Equilibrium iteration Edit properties [Fig. 90] [Fig. 91] Analysis browser RandomFields Run analysis Figure 90: Equilibrium iteration properties Figure 91: Converge norm Shrinkage Effects on a Concrete Slab on Ground 43/53

44 We can t go so far in the analysis, but we can see some converged results after start of the first cracks. This difficulty in convergence is due to the complexity of the problem of restrained strains and propagation of cracks, when cracks at different locations appear at the same time. Results browser RandomFields - Copy 1 Output Element results Crack-widths Ecw1 [Fig. 92], [Fig. 93] <In Case select the desired time-steps> Figure 92: Equilibrium iteration properties Figure 93: Crack widths Ecw1-1.35E-4 shrinkage strain Shrinkage Effects on a Concrete Slab on Ground 44/53

45 5 Design Methodology: Shrinkage Strain as Equivalent Temperature Variation In practice the use of nonlinear analysis (transient or static) is not often applied for the design of concrete slabs. Instead, the usually design methodology is based on a linear static analysis with a temperature variation applied to the slab, generating a strain that is equivalent to the shrinkage effects. We will exemplify how this can also be done in DIANA and compare with the results of the nonlinear transient analysis. We need to include thermal effects in the material properties. For the concrete we need to add a new material model to include this aspect. For the reinforcement we can include the thermal aspect directly in the material model for steel previously defined. We need to define a temperature load and apply it to the slab. Finally we run a linear static analysis and check the crack risk based on the level of principal stresses in the slab. Shrinkage Effects on a Concrete Slab on Ground 45/53

46 5.1 Material properties We add a new linear elastic isotropic material for concrete with the same properties of the concrete class C20/25 and assign it to the slab through the geometry browser. Main Menu Geometry Materials Add material [Fig. 94] [Fig. 95] Geometry browser Geometry Slab Properties Material <Change material C15 Random Field to material Concrete > [Fig. 96] Figure 94: Add material - Linear elastic isotropic Figure 95: Edit material - Concrete Figure 96: Assign material Concrete to the slab Shrinkage Effects on a Concrete Slab on Ground 46/53

47 We include thermal effects in the materials steel and ground interface. Geometry browser Materials Steel Edit material [Fig. 97] [Fig. 98] Geometry browser Materials Ground interface Edit material [Fig. 99] Figure 97: Edit material - Steel Figure 98: Edit material - Steel Figure 99: Edit material - Ground interface Shrinkage Effects on a Concrete Slab on Ground 47/53

48 5.2 Temperature load We define an incremental temperature load equivalent to the shrinkage strains considered in the Random Field analysis (-5E-4). T = ɛ sh /α = -5x10 4 / 1x10 5 = -50 degrees The equivalent temperature gradient is calculated as: As this new settings were defined in geometry, we generate the mesh again. Main Menu Geometry Analysis Attach load [Fig. 100] Main Menu Geometry Generate mesh Figure 100: Incremental temperature load Shrinkage Effects on a Concrete Slab on Ground 48/53

49 5.3 Linear analysis We add a new analysis to perform a structural linear static calculation. Analysis browser New analysis <Rename (F2 or ) Analysis4 to Temperature > [Fig. 101] Analysis browser Temperature Add command Structural linear static [Fig. 102] Analysis browser Temperature Run analysis Results browser Analysis: Temperature Analysis: Temperature Run analysis Figure 101: Analysis window Figure 102: Add command Shrinkage Effects on a Concrete Slab on Ground 49/53

50 We see the results of stresses in concrete in X and Y directions due to the temperature variation [Fig. 105] [Fig. 107]. As already known we get similar results for the linear analysis considering the Shrinkage Strain load case [Fig. 106] [Fig. 108]. Results browser Temperature Temperature Output linear static analysis Element results Cauchy Total Stresses SXX and SYY [Fig. 103], [Fig. 105 Results browser Temperature Shrinkage strains Output linear static analysis Element results Cauchy Total Stresses SXX and SYY [Fig. 104], [Fig Figure 103: Result browser - Temperature case Figure 104: Result browser - Shrinkage strain case Shrinkage Effects on a Concrete Slab on Ground 50/53

51 We observe that the tensile stresses are high (around 1.9 MPa) that can lead to cracks in the weakest points of the structure (see probabilistic distribution of the tensile strength of concrete in [Fig. 79]). Figure 105: Stresses SXX - Temperature variation Figure 107: Stresses SYY - Temperature variation Figure 106: Sresses SXX - Shrinkage strain Figure 108: Stresses SYY - Shrinkage Shrinkage Effects on a Concrete Slab on Ground 51/53

52 Appendix A Additional information Folder: Tutorials/SlabOnGround Number of elements 6800 Keywords: analys: nonlin physic. constr: suppor. elemen: chx60 cq48i grid interf reinfo solid struct taper. load: time weight. materi: concre creep elasti en1992 fricti harden isotro maxwel nonlin plasti shrink strain viscoe vonmis. option: direct newton regula units. pre: dianai. Shrinkage Effects on a Concrete Slab on Ground 52/53

53 DIANA FEA BV Delftechpark 19a 2628 XJ Delft The Netherlands T +31 (0) F +31 (0) DIANA FEA BV Vlamoven TN Arnhem The Netherlands T +31 (0) F +31 (0)