Settlement and Bearing Capacity of a Strip Footing. Nonlinear Analyses

Settlement and Bearing Capacity of a Strip Footing Nonlinear Analyses

Outline 1 Description 2 Nonlinear Drained Analysis 2.1 Overview 2.2 Properties 2.3 Loads 2.4 Analysis Commands 2.5 Results 3 Nonlinear Undrained - Drained Analysis 3.1 Overview 3.2 Properties 3.3 Loads 3.4 Analysis Commands 3.5 Results 4 Nonlinear Effective Stress Undrained Analysis 4.1 Overview 4.2 Analytical solution 4.3 Boundary Conditions 4.4 Loads 4.5 Analysis Commands 4.6 Results 5 Nonlinear Total Stress Undrained Analysis 5.1 Overview 5.2 Analytical solution 5.3 Calculation of undrained soil parameters 5.4 Properties 5.5 Analysis Commands 5.6 Results Appendix A Additional Information Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 2/71

1 Description This tutorial is a continuation of the tutorial Settlement Analysis of a Strip Footing which presents a linear analysis of a strip footing. The present tutorial will use the same geometry for the concrete strip footing and soil layer, the same boundary constraints and the same mesh properties and thus, the presentation of these steps of the model will not be repeated here. Hence, this tutorial will focus on the different steps required for each analysis regarding mostly material properties, analysis procedures and results. This tutorial presents four nonlinear analyses which differ from the first tutorial and from each other in the following ways: Nonlinear Drained Analysis [ 2] Perform a nonlinear drained analysis to determine the settlement of the footing. The self-weight load is considered in a phased analysis and a Mohr-Coulomb model is used for the soil. Nonlinear Undrained-Drained Analysis [ 3] Perform a nonlinear undrained-drained analysis to determine the settlement of the footing. The drainage condition is changed and a third phase is added. Nonlinear Effective Stress Undrained Analysis [ 4] Perform a short term stability analysis using nonlinear effective undrained analysis to determine bearing capacity and compare with analytical results. Nonlinear Total Stress Undrained Analysis [ 5] Perform a short term stability analysis using nonlinear total undrained analysis to determine bearing capacity and compare with analytical results. Figure 1: Strip footing model Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 3/71

The material models used are linear elastic isotropic for the concrete and Mohr-Coulomb plasticity for the soil, with the properties described in Table 1. Table 1: Material properties Mohr-Coulomb plasticity Analyses [ 2][ 3][ 4] Analysis [ 5] Soil Young s modulus E Poisson s ratio ν 1.0E+4 0.33 1.128E+4 0.495 kn/m 2 Dry density ρ 1.631 1.8 T/m 3 Saturated density ρ Reduced density ρ 1.835 0.835 1.8 - T/m 3 Cohesion c 10 9.06 kn/m 2 Friction angle φ 25 0 [Deg] Dilatancy angle ψ K 0 0 0.577 0 0.812 [Deg] Linear elasticity Concrete Analyses [ 2][ 3][ 4][ 5] Young s modulus E Poisson s ratio ν 3.5E+7 0.15 kn/m 2 Mass density ρ 2.44748 T/m 3 Analyses: [ 2] Nonlinear Drained Analysis [ 3] Nonlinear Undrained - Drained Analysis [ 4] Nonlinear Effective Stress Undrained Analysis [ 5] Nonlinear Total Stress Undrained Analysis Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 4/71

2 Nonlinear Drained Analysis 2.1 Overview In this case the analysis is performed with drained condition. The analysis has two phases that account for the following sequence of construction stages: Phase 1: Soil is active with self-weight and ground supports Analysis is performed for drained condition Phase 2: Soil and footing are both active Footing support is activated along with ground support Pressured load on footing is added Analysis is performed for drained condition Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 5/71

2.2 Properties The soil has a new material model for Mohr-Coulomb and Drucker-Prager plasticity including the aspects of initial stress and drainage. So we create a new material model for the soil and assign it to the soil shape. Main Menu Geometry Analysis Property assignments [Fig. 2] Shape assignment Add new material [Fig. 3] [Fig. 7] Figure 2: Assign soil properties Figure 3: Soil - Add new material Figure 4: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 6/71

Figure 5: Soil - Material properties Figure 6: Soil - Material properties Figure 7: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 7/71

2.3 Loads In addition to the Pressure Load we consider the dead weight as a global load. So, automatically there are two load combinations defined [Fig. 9] [Fig. 10]: 1. Load combination 1: with load case 1 (self weight) 2. Load combination 2: with load case 2 (pressure load) With this changes the model is complete and we can re-mesh. Main Menu Geometry Analysis Global load [Fig. 8] Main Menu Geometry Analysis Generate mesh Figure 8: Load - Self weight Figure 9: Geometry browser Figure 10: Load combinations Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 8/71

2.4 Analysis Commands We will perform a nonlinear analysis with two phases as described in Section 2.1. For that we add a new analysis and a phased command. Main Menu Analysis New Analysis [Fig. 11] <Rename Analysis 1 to Nonlinear Phased Analysis > Analysis browser Add command Phased [Fig. 12] [Fig. 13] Figure 11: Analysis browser Figure 12: Add command Figure 13: Analysis tree Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 9/71

We edit the shapes and the geometry support sets that are active in the first phase: Soil and Ground Support. Analysis browser Nonlinear Phased Analysis Phased Edit properties [Fig. 14] Figure 14: Edit analysis phases Note: You can see the correspondent active element and support sets if you select the option Show mesh related items. You can edit the analysis phases at the geometry level before meshing and they will then be automatically connected to the correspondent element and support sets after meshing. As we did this phase edition after meshing, any changes that occur at the geometry level will require the re-meshing of the model and re-definition of the characteristics of the phased analysis. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 10/71

We add a command for structural nonlinear analysis and a start step execution block. We need to remove the default loads execute steps. Analysis browser Nonlinear Phased Analysis Add command Structural nonlinear [Fig. 15] [Fig. 16] Analysis browser Nonlinear Phased Analysis Structural nonlinear Add Execute steps - Start steps Analysis browser Nonlinear Phased Analysis Structural nonlinear new execute block 2 Remove [Fig. 17] Figure 15: Add command Figure 16: Analysis browser Figure 17: Analysis tree Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 11/71

In the start steps we define the reference load for the initialization of the newly active elements. All the active loads up to the current stage are on the right hand side of the equilibrium equation (accumulated loads). Stress initialization in soil is achieved by balancing the internal stress state in the soil with the external load (the self weight, i.e. load combination 1). We tick off the Liquefaction and we tick on the option for Suppress result superposition. Analysis browser Nonlinear Phased Analysis Structural nonlinear new execute block 2 Start steps Edit properties [Fig. 19] Property panel Physic nonlinear options [Fig. 20] Figure 18: Analysis browser Figure 19: Properties of start steps Figure 20: Properties of execute block Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 12/71

In the equilibrium iteration properties we choose the displacement convergence norm. Analysis browser Nonlinear Phased Analysis Structural nonlinear new execute block 2 Equilibrium iteration [Fig. 19] Figure 21: Analysis browser Figure 22: Properties of equilibrium iteration Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 13/71

In the output properties we set the name of the results file as Phase 1 and we select the desired results for output: displacements, total and plastic strains and total and effective stresses. Analysis browser Nonlinear Phased Analysis Structural nonlinear Output [Fig. 23] Properties - OUTPUT Output Device Properties [Fig. 24] Properties - OUTPUT Output Results User selection Modify [Fig. 25] Figure 23: Output properties Figure 24: Output basename Figure 25: Results selection Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 14/71

We will now set up the second phase of the analysis where all the model is active. For that we add a new phase and we activate the remaining shapes and geometry support sets. Analysis browser Add command Phased [Fig.??] Analysis browser Nonlinear Phased Analysis Phased 1 Edit properties [Fig. 27] Figure 26: Analysis browser Figure 27: Edit analysis phases Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 15/71

Similarly what we did in Phase 1, we add a structural nonlinear analysis and a start step execute block. We move the start step before the default load step. Analysis browser Nonlinear Phased Analysis Add command Structural nonlinear [Fig. 28] Analysis browser Nonlinear Phased Analysis Structural nonlinear 1 Add Execute steps - Start steps Analysis browser Nonlinear Phased Analysis Structural nonlinear 1 new execute block 2 Move up [Fig. 29] Figure 28: Analysis browser Figure 29: Edit analysis phases Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 16/71

We define the start step as previously done in Phase 1 [Fig. 19] and we also tick off the Liquefaction option and tick on the Suppress result superposition option [Fig. 20]. In the load steps, we apply the pressure load (Load combination 2) in 10 equal steps (i.e., 10 steps with a factor of 0.1) [Fig. 31]. We consider default parameters for the equilibrium iteration. In the output properties we set the name of the results file as Phase 2 and we select the desired results for output in the same manner we did for the first phase [Fig. 25]. Finally we run the analysis. Analysis browser Nonlinear Phased Analysis Structural nonlinear 1 new execute block Load steps [Fig. 31] Analysis browser Nonlinear Phased Analysis Structural nonlinear 1 Output Properties - OUTPUT Output Device Properties [Fig. 32] Properties - OUTPUT Output Results User selection Modify <Same as [Fig. 23]> Analysis browser Nonlinear Phased Analysis Run analysis Figure 30: Analysis browser Figure 31: Edit analysis phases Figure 32: Edit analysis phases Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 17/71

2.5 Results We first create a contour plot of the initial vertical effective stresses in the soil. Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 33] [Fig. 34] Figure 33: Results browser Figure 34: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 18/71

We now create a contour plot of the final vertical displacement. Results browser Output Nodal results Displacements TDtY [Fig. 35] [Fig. 36] Figure 35: Results browser Figure 36: Displacements TDtY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 19/71

We select the nodes at the interface between the soil and the footing. We tabulate the vertical displacements for these nodes. We observe that the displacements in this area are approximately 9.03 mm. Main menu Viewer Node selection [Fig. 38] <Drag the mouse around the nodes at the connection between Soil and Footing> Results browser Output linear static analysis Nodal results Displacements DtY Show table [Fig. 37] [Fig. 39] Figure 37: Results browser Figure 38: Select nodes Figure 39: Table of results Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 20/71

We can compare these results with the results presented in the tutorial Settlement Analysis of a Strip Footing for the linear elastic analysis and the theoretical solution: Theoretical solution for linear elastic case: 7.92 Linear elastic solution: 7.77 mm mm Nonlinear drained solution: 9.03 mm 16.2% greather then linear elastic solution 14.01% greater than theoretical solution for linear elastic case Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 21/71

We will now create a contour plot of the final vertical effective stresses in the soil. For that, in the properties we specify the minimum and maximum values of the contour plot to -100 and 5, respectively, with 15 equidistant contour levels. Results browser Output Nodal results Displacements TDtY [Fig. 35] [Fig. 36] Properties panel Result Contour plot settings [Fig. 41] Figure 40: Contour plot properties Figure 41: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 22/71

We create a contour plot of the final principal plastic strain in the soil. Results browser Output Element results Plastic strains Ep1 [Fig. 42] [Fig. 43] Figure 42: Results browser Figure 43: Principal plastic strain Ep1 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 23/71

3 Nonlinear Undrained - Drained Analysis 3.1 Overview In this case the drainage condition will be changed and a third phase will be added. The analysis has three phases that account for the following sequence of construction stages: Phase 1: Soil is active with self-weight and ground supports Analysis is performed for drained condition Phase 2: Soil and footing are both active Footing support is activated along with ground support Pressured load on footing is added Analysis is performed for undrained condition Phase 3: Analysis is performed with drained condition to dissipate excess pore pressure All other parameters remain unchanged from previous stage Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 24/71

3.2 Properties Th only change in the properties is the drainage condition in the Mohr-Coulomb soil model. Instead of considering the drained behaviour as in Section 2, we now choose the option of Undrained with penalty factor for the Drained/Undrained behaviour. Geometry browser Materials Soil MC Edit properties [Fig. 44] [Fig. 45] Figure 44: Geometry browser Figure 45: Edit material Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 25/71

3.3 Loads We need to add a load combination to consider the self-weight and pressure load simultaneously. Finally we need to re-mesh the model. Geometry browser Loads Combinations Open geometry load combination table [Fig. 46] [Fig. 47] Main Menu Geometry Analysis Generate mesh Figure 46: Geometry browser Figure 47: Edit load combinations Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 26/71

3.4 Analysis Commands We will perform a nonlinear analysis with three phases as described in Section 3.1. The first two phases are defined in the same manner as in the Nonlinear Drained Analysis, so we repeat the steps presented in Section 2.4. In order to distinguish the analyses, we rename it as Nonlinear Phased Analysis 2. The only difference is that in the results of the second phase we ask for an additional result to be output, the pore pressure in the element center. Analysis browser Nonlinear Phased Analysis 2 Structural nonlinear 1 Output <Same as [Fig. 23]> Properties - OUTPUT Output Results User selection Modify [Fig. 50] Results Selection PRESSU TOTAL Properties Location [Fig. 51] Figure 48: Analysis browser Figure 49: Output properties Figure 50: Results selection Figure 51: Results selection Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 27/71

We now continue with the definition of the third phase. We add a new phase and a command for a structural nonlinear analysis. We add a start step execution block and remove the default load execute steps. In the start steps we define the reference load as load combination 3 (self-weight and pressure load). In the physic nonlinear options we tick on the Drained behaviour. Analysis browser Add command Phased Analysis browser Nonlinear Phased Analysis 2 Add command Structural nonlinear Analysis browser Nonlinear Phased Analysis 2 Structural nonlinear 2 Add Execute steps - Start steps Analysis browser Nonlinear Phased Analysis 2 Structural nonlinear 2 new execute block 2 Remove [Fig. 52] Analysis browser Nonlinear Phased Analysis 2 Structural nonlinear 2 new execute block 2 Start steps Edit properties [Fig. 53] Property panel Physic nonlinear options [Fig. 54] Figure 52: Analysis browser Figure 53: Properties of start steps Figure 54: Properties of execute block Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 28/71

In the equilibrium iteration properties we choose the displacement convergence norm. Analysis browser Nonlinear Phased Analysis Structural nonlinear new execute block 2 Equilibrium iteration [Fig. 56] Figure 55: Analysis browser Figure 56: Properties of equilibrium iteration Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 29/71

In the output properties we set the name of the results file as Phase 3 and we select the desired results for output: displacements, total and plastic strains, total and effective stresses and pore pressure in the element center. Finally we run the analysis. Analysis browser Nonlinear Phased Analysis Structural nonlinear Output [Fig. 57] Properties - OUTPUT Output Device Properties [Fig. 58] Properties - OUTPUT Output Results User selection Modify [Fig. 59] Analysis browser Nonlinear Phased Analysis Run analysis Figure 57: Output properties Figure 58: Output basename Figure 59: Results selection Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 30/71

3.5 Results We first create a contour plot of the initial vertical effective stresses in the soil, correspondent to Phase 1. Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 60] [Fig. 61] Figure 60: Results browser Figure 61: Effective stresses SEYY The initial effective stress S i yy may be calculated by subtracting the hydrostatic stress from the vertical soil stress S v yy: S i yy = S v yy hρg = 5 1.835 9.81 5 1.0 9.81 = 40.96 kn/m 2. This value of 40.96 kn/m 2 can be seen in the legend of the contour plot corresponding to the initial effective vertical stress. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 31/71

We now show the vertical effective stresses at the end of Phase 2 (undrained). Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 62] [Fig. 63] Figure 62: Results browser Figure 63: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 32/71

And at the end of Phase 3 (drained). Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 64] [Fig. 65] Figure 64: Results browser Figure 65: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 33/71

We now create a contour plot of the pore pressure at the start of the Phase 2. Results browser Output Element results Pore pressure PR [Fig. 66] [Fig. 67] Figure 66: Results browser Figure 67: Pore pressure PR Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 34/71

And at the end of the Phase 2. Results browser Output Element results Pore pressure PR [Fig. 68] [Fig. 69] Figure 68: Results browser Figure 69: Pore pressure PR Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 35/71

Now we show the contour plot of vertical displacements at the end of Phase 2 (drained condition). Results browser Output Nodal results Displacements TDtY [Fig. 70] [Fig. 71] Figure 70: Results browser Figure 71: Displacements TDtY Giving a short term vertical displacement beneath the footing of approximately 7 mm. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 36/71

And at the end of Phase 3 (undrained condition). Results browser Output Nodal results Displacements TDtY [Fig. 72] [Fig. 73] Figure 72: Results browser Figure 73: Displacements TDtY Giving a long term vertical displacement beneath the footing of approximately 12 mm. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 37/71

Finally we show the principal plastic strain at the end of Phase 3. Results browser Output Element results Plastic strains Ep1 [Fig. 74] [Fig. 75] Figure 74: Results browser Figure 75: Plastic strains Ep1 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 38/71

4 Nonlinear Effective Stress Undrained Analysis 4.1 Overview In this analysis we will assess the short-term stability of the footing using nonlinear effective undrained analysis, in order to determine the bearing capacity and compare with an analytical solution. The analysis has two phases that account for the following sequence of construction stages: Phase 1: Soil is active with self-weight and ground supports Analysis is performed for drained condition Phase 2: Soil and footing are both active Footing support is activated along with ground support Prescribed displacement applied on top of the footing Analysis is performed for effective undrained condition The properties of the model are equal to the ones used in the previous analysis Nonlinear Undrained - Drained Analysis [ 3]. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 39/71

4.2 Analytical solution The analytical solution for bearing capacity of infinitely long (strip / continuous) smooth rigid footing on finite undrained soil with constant undrained cohesion is given by Giroud (1972) 1 as follows: Q f = C 0 N 0 CZ + γd = 9.06 5.55 = 50.28 kp a Q f = Bearing capacity C 0 = Undrained cohesion at ground surface = C u0 = c cos φ + 0.5(σ ini hor + σ ini ver ) sin φ = 9.06 kp a C 2 = C 0 N 0 CZ = P arameter dependent on C 2 C 0 = 1 and H B γ = T otal unit weight D = Depth of foundation = 0 = 2.5 = 5.55 1 Giroud, Settlement of rectangular foundation on soil layer, 1972 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 40/71

4.3 Boundary Conditions We need to attach a new load for prescribed deformation that we will use to determine the capacity of the footing. As we are applying a prescribed deformation we first need to define the supports to which the deformation will be attached. So, we will add this support to the existing ones previously defined. Main Menu Geometry Analysis Attach support [Fig. 76] [Fig. 77] Figure 76: Add top supports to the footing Figure 77: View of the model - Supports Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 41/71

4.4 Loads We now define the prescribed deformation load. Main Menu Geometry Analysis Attach load [Fig. 78] [Fig. 79] Figure 78: Add prescribed deformation Figure 79: View of the model - Prescribed deformation Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 42/71

We also need to change the load combinations. In this case we will have the default load combinations, i.e., one for each load case: self weight is load combination 1, pressure load is load combination 2 and prescribed deformation is load combination 3. Finally we need to re-mesh the model. Geometry browser Loads Combinations Open geometry load combination table [Fig. 80] [Fig. 81] Main Menu Geometry Analysis Generate mesh Figure 80: Geometry browser Figure 81: Load combinations Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 43/71

4.5 Analysis Commands We will perform a nonlinear analysis with two phases as described in Section 4.1. The analysis commands are similar to the ones defined for the Nonlinear Drained Analysis [ 2.4]. We rename this analysis as Nonlinear Phased Analysis 3. We will only refer here to the differences. The shapes and the geometry support sets that are active in the first phase are Soil and Ground Support as in Section 4.1 but now we have the new top support inactive in this phase [Fig. 83]. Analysis browser Nonlinear Phased Analysis 3 Phased Edit properties [Fig. 82] [Fig. 83] Figure 82: Analysis browser Figure 83: Edit analysis phases Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 44/71

In the start steps we define the reference load for the initialization of the newly active elements correspondent to the self weight (load combination 1). We tick off the Liquefaction and we tick on the options for Suppress result superposition and Drained behavior. Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear new execute block 2 Start steps Edit properties [Fig. 84] [Fig. 85] Property panel Physic nonlinear options [Fig. 86] Figure 84: Analysis browser Figure 85: Properties of start steps Figure 86: Properties of execute block Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 45/71

In the equilibrium iteration properties we choose the displacement convergence norm and the same output as in Section 2.4: displacements, total and plastic strains and total and effective stresses. Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear new execute block 2 Equilibrium iteration [Fig. 87] Analysis browser Nonlinear Phased Analysis Structural nonlinear Output [Fig. 88] Figure 87: Properties of equilibrium iteration Figure 88: Output properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 46/71

We now set up the second phase of the analysis where all the model is active. For that we add a new phase and we activate the remaining shapes and geometry support sets. Analysis browser Add command Phased [Fig. 89] Analysis browser Nonlinear Phased Analysis 3 Phased 1 Edit properties [Fig. 90] Figure 89: Analysis browser Figure 90: Edit analysis phases Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 47/71

As in Section 2.4 we add a structural nonlinear analysis and a start step execute block and we move the start step before the default load step. In the start step, we set the self-weight as reference load for the initialization of active elements (load combination 1). We also tick off the Liquefaction option and tick on the Supress result superposition option. Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear new execute block 2 Start steps Edit properties [Fig. 91] [Fig. 92] Property panel Physic nonlinear options [Fig. 93] Figure 91: Analysis browser Figure 92: Edit analysis phases Figure 93: Properties of the execute block Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 48/71

In the load execute block we apply the prescribed deformation load (load combination 3) in 25 equal steps [Fig. 95]. We choose the force convergence of the equilibrium iteration for both execute blocks of the second phase [Fig. 96]. We define the desired results for output: displacements, total and plastic strains, total and effective stresses, pore pressure in the element center and reaction forces [Fig. 97]. We can now run the analysis. Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear new execute block Load steps Edit properties [Fig. 94] [Fig. 95] Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear <Do the same for the equilibrium iteration of the new execute block> new execute block 2 Equilibrium iteration Edit properties [Fig. 94] [Fig. 96] Analysis browser Nonlinear Phased Analysis 3 Structural nonlinear Output Edit properties [Fig. 96] Analysis browser Nonlinear Phased Analysis 3 Run analysis Figure 94: Analysis browser Figure 95: Execute load steps Figure 96: Equilibrium iteration properties Figure 97: Output properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 49/71

4.6 Results We first create a contour plot of the initial vertical effective stresses in the soil, corresponding to Phase 1. Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 98] [Fig. 99] Figure 98: Results browser Figure 99: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 50/71

And at the end of Phase 2 (undrained). Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 100] [Fig. 101] Figure 100: Results browser Figure 101: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 51/71

We show the contour plot of pore pressure at the end of Phase 2. Results browser Output Element results Pore pressure PR [Fig. 102] [Fig. 103] Figure 102: Results browser Figure 103: Pore pressure PR Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 52/71

And the principal plastic strain. Results browser Output Element results Plastic strains Ep1 [Fig. 104] [Fig. 105] Figure 104: Results browser Figure 105: Plastic strain Ep1 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 53/71

In order to determine the bearing capacity of the footing and to compare with the analytical solution described in Section 4.2 we need to sum the reaction forces at the nodes of the top of the footing. We first show the vector plot of the reaction forces. Results browser Output Nodal results Reaction Forces FBY Show vectors [Fig. 106] [Fig. 107] <Show only the footing in the mesh browser> Figure 106: Results browser Figure 107: Reaction forces FBY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 54/71

We select the nodes by dragging the mouse around the nodes at the top of the footing. We tabulate the selected nodes, we sum the reaction forces and we need to multiply by a factor of two in order to get the reaction force of the complete footing. Main Menu Viewer Node selection Results browser Output Nodal results Reaction Forces FBY Show table [Fig. 108] Analytical solution F analytical = Q f S = 50.28 2 1 = 100.56 kn DIANA FEM solution = 105.9 kn The difference is 5% Figure 108: Table of results Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 55/71

5 Nonlinear Total Stress Undrained Analysis 5.1 Overview In this analysis we will assess the short term stability of the footing using nonlinear total stress undrained analysis, in order to determine the bearing capacity and compare with an analytical solution. The material properties of the soil will differ from the previous analysis as described in Table 1. The calculation of the undrained strength and stifness parameters of the soil are presented in Section 5.3. The difference between the present analysis and the previous one, Nonlinear Effective Stress Undrained Analysis [ 4], is that the cohesion varies linearly with depth instead of being constant. In the same manner, this analysis has two phases that account for the following sequence of construction stages: Phase 1: Soil is active with self-weight and ground supports Analysis is performed for drained condition Phase 2: Soil and footing are both active Footing support is activated along with ground support Prescribed displacement applied on top of the footing Analysis is performed for effective undrained condition Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 56/71

5.2 Analytical solution The analytical solution for bearing capacity of infinitely long (strip / continuous) smooth rigid footing on finite undrained soil with linearly (with depth) varying undrained cohesion is given by Giroud (1972) 2 as follows: Q f = C 0 N 0 CZ + γd = 9.06 6.84 = 61.97 kp a Q f = Bearing capacity C 0 = Undrained cohesion at ground surface = C u0 = c cos φ + 0.5(σ ini hor + σ ini ver ) sin φ = 9.06 kp a C 2 = c cos φ + 0.5(σ ini hor + σ ini ver ) sin φ = 22.38 kp a N 0 CZ = P arameter dependent on C 2 C 0 = 2.5 and H B γ = T otal unit weight D = Depthoffoundation = 0 = 2.5 = 6.84 2 Giroud, Settlement of rectangular foundation on soil layer, 1972 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 57/71

5.3 Calculation of undrained soil parameters The cohesion of the soil is assumed to increase with the soil depth at a rate of 2.66 kn/m 2 : Increment of cohesion = C 2 C 0 H = 22.38 9.06 5 = 2.66 Reference height = Ground surface = 5 m The value of the undrained elastic modulus E u and undrained stress ratio K 0 may be calculated as follows: E u = 3G u = 3E 2(1+ν ) = 3 1000 2(1 + 0.33) 11280 kp a K 0 = K 0 γ + γ w γ tot = 0.812 γ may be calculated from the difference between the total (saturated) unit weight of soil and the bulk unit weight of water as: γ = γ tot γ w = 18 10 = 8.0 kn/m 3 The undrained Poisson s ratio is assumed to have a value of ν u = 0.495. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 58/71

5.4 Properties The soil has new properties including the variable cohesion. So first we define the function which will be used to define the cohesion gradient. After that we change the properties of the material Soil MC that is assigned to the soil shape. As we made the changes in the geometry we need to re-mesh. Main Menu Geometry Functions Add [Fig. 109] Geometry Browser Materials Soil MC Edit material [Fig. 111][Fig. 112] Main Menu Geometry Analysis Generate mesh Figure 109: Function Figure 110: Geometry browser - materials Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 59/71

Figure 111: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 60/71

Figure 112: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 61/71

Figure 113: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 62/71

Figure 114: Soil - Material properties Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 63/71

5.5 Analysis Commands The setting up of the analysis is equal to the Nonlinear Effective Stress Undrained Analysis described in Section 4.5. We call this analysis Nonlinear Phased Analysis 4. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 64/71

5.6 Results We first create a contour plot of the initial vertical effective stresses in the soil, corresponding to Phase 1. Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 115] [Fig. 116] Figure 115: Results browser Figure 116: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 65/71

And at the end of Phase 2 (undrained). Results browser Output Element results Cauchy Effective Stresses SEYY [Fig. 117] [Fig. 118] Figure 117: Results browser Figure 118: Effective stresses SEYY Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 66/71

And also the principal plastic strain. Results browser Output Element results Plastic strains Ep1 [Fig. 119] [Fig. 120] Figure 119: Results browser Figure 120: Plastic strain Ep1 Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 67/71

In the same manner as we did for the previous analysis Nonlinear Effective Stress Undrained Analysis we sum the reaction forces at the nodes ot the top of the footing to compare with the analytical solution presented in Section 5.2. Results browser Output Nodal results Reaction Forces FBY Show vectors [Fig. 121] [Fig. 122] <Show only the footing in the mesh browser> Figure 121: Results browser Figure 122: Reaction forces Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 68/71

Main Menu Viewer Node selection Results browser Output Nodal results Reaction Forces FBY Show table [Fig. 123] Analytical solution F analytical = Q f S = 61.97 2 1 = 123.94 kn DIANA FEM solution = 126.97 kn The difference is 1.0% Figure 123: Table of results Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 69/71

Appendix A Additional Information Folder: Tutorials/SettleFootingNL Number of elements 694 Keywords: analys: nonlin phase physic. constr: suppor. elemen: ct12e pstrai. load: deform edge elemen force weight. materi: elasti functi harden isotro mohrco plasti porosi soil strain undrai. option: direct newton nonsym penal regula units. post: binary ndiana. pre: dianai. result: cauchy displa effect force green plasti pressu princi reacti strain stress total. References: [1] J.P. Giroud. Settlement of rectangular foundation on soil layer. Journal of the Soil Mechanics and Goundations Division, 98(1):149 154, 1972. Settlement and Bearing Capacity of a Strip Footing (Nonlinear Analyses) https://dianafea.com 70/71

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