Elastic Analysis of a Deep Beam with Web Opening
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1 Elastic Analysis of a Deep Beam with Web Opening
2 Outline 1 Description 2 Finite Element Model 2.1 Units 2.2 Geometry definition 2.3 Properties 2.4 Boundary conditions Constraints Vertical load 2.5 Meshing 3 Structural linear static analysis 3.1 Analysis commands 4 Results 4.1 Reaction forces FBY 4.2 Contour plot Total displacements DtXYZ 4.3 Contour plot Stress field SXX 4.4 Contour plot Stress field SYY 4.5 Contour plot Stress field SXY 4.6 In-plane principal components Stress field SXX 4.7 Diagram Stress field SXX 4.8 Diagram Stress fields SYY and SXY Appendix A Additional Information Elastic Analysis of a Deep Beam with Web Opening 2/30
3 1 Description We analyze a concrete continuous deep beam with web openings presented in (Ashour and Rishi) 1 and here shown in [Fig. 1]. The beam is 3 meters long, 625 mm height and 120 mm thick. The web opening is of 250 x 250 mm. Due to symmetry we model one half of the beam. Supports and loading plate are made of steel. The material properties are listed in [Table 1]. P Concrete Steel Young s modulus Poisson s ratio N/mm 2 Table 1: Concrete and steel properties The nodal force P is equal to 4e+05 N. 250 We will study the elastic response of the beam by performing a linear elastic static analysis. y z x Figure 1: Schematic representation of the model (units are in millimeters). 1 Tests of Reinforced Concrete Continuous Deep Beams with Web Openings, A. F. Ashour and G. Rishi, 2000, ACI Structural Journal Elastic Analysis of a Deep Beam with Web Opening 3/30
4 2 Finite Element Model For the modeling session we start a new project for structural analysis [Fig. 2]. The dimensions of the domain for the 2D model are set equal to 10 m. We will use quadratic quadrilateral finite elements. Main menu File New [Fig. 2] Figure 2: New project dialog Elastic Analysis of a Deep Beam with Web Opening 4/30
5 2.1 Units We choose millimeter for the unit Length, ton for Mass, Newton for Force. Geometry browser Reference system Units [Fig. 3] Property Panel [Fig. 4] Figure 3: Geometry browser Figure 4: Property Panel - Units Elastic Analysis of a Deep Beam with Web Opening 5/30
6 2.2 Geometry definition We create a sheet for the beam. We will change the viewpoint to a top view, fit the shape in the Workspace window and hide the working plane. Main menu Geometry Create Add polygon sheet [Fig. 5] Main menu Viewer Viewpoints Top view Main menu Viewer Fit all Figure 5: Add Beam polygon sheet Figure 6: Top view of the Beam Elastic Analysis of a Deep Beam with Web Opening 6/30
7 Similarly, we create sheets for the left and right supports and the loading block. For the left support an additional vertex is created at (160, -40, 0) mm to account for the constrain we will apply along the Y -direction. Similary, we add a vertex at (820, 665, 0) mm to the loading block where to apply the nodal vertical force. Main menu Geometry Create Add polygon sheet [Fig. 7] [Fig. 8] [Fig. 9] Main menu Viewer Fit all Figure 7: Add Left support polygon sheet Figure 8: Add right support polygon sheet Figure 9: Add Loading block polygon sheet Elastic Analysis of a Deep Beam with Web Opening 7/30
8 Figure 10: Top view of the model Elastic Analysis of a Deep Beam with Web Opening 8/30
9 We create a sheet for the web opening. Main menu Geometry Create Add polygon sheet [Fig. 11] Figure 11: Add Opening polygon sheet Figure 12: Top view of the model Elastic Analysis of a Deep Beam with Web Opening 9/30
10 We subtract the sheet Opening from the sheet Beam. Main menu Geometry Modify Subtract shapes [Fig. 13] Figure 13: Subtract shapes Figure 14: Top view of the model Elastic Analysis of a Deep Beam with Web Opening 10/30
11 2.3 Properties We assign the element class and the material and geometrical properties to the beam. We use regular plane stress finite elements. The material is linear elastic. Regarding the geometry, we need to specify the thickness of the beam (120 mm). Main menu Geometry Assign Properties [Fig. 15] Properties Material Add material [Fig. 16] Edit material [Fig. 17] Properties Geometry Add new geometry [Fig. 18] Figure 15: Property assignments Figure 16: Add new material Figure 17: Material properties Figure 18: Geometrical properties Elastic Analysis of a Deep Beam with Web Opening 11/30
12 We assign the element class and the material and geometrical properties also to the supports and the loading block. We also use regular plane stress finite elements. The material is linear elastic. The thickness is equal to that of the beam (120 mm), thus we do not need to specify a new Geometry set. Main menu Geometry Assign Properties [Fig. 19] Properties Material Add material [Fig. 20] Edit material [Fig. 21] Figure 19: Property assignments Figure 20: Add new material Figure 21: Material properties Elastic Analysis of a Deep Beam with Web Opening 12/30
13 2.4 Boundary conditions Constraints We support the right edge (i.e., the centerline) of the beam in X-direction to get the symmetry condition. Main menu Geometry Assign Supports [Fig. 22] [Fig. 23] Figure 22: Apply symmetry boundary conditions Figure 23: Symmetry boundary conditions Elastic Analysis of a Deep Beam with Web Opening 13/30
14 We support the center of both support blocks in Y -direction. Due to the symmetry boundary condition (the center bottom point of the Right support has already been supported in X-direction), we now have a roller under the center of the left support and we pinned the center of the right support. Main menu Geometry Assign Supports [Fig. 24] [Fig. 25] Figure 24: Apply vertical (Y -direction) boundary conditions Figure 25: Vertical (Y -direction) boundary conditions Elastic Analysis of a Deep Beam with Web Opening 14/30
15 2.4.2 Vertical load We apply a nodal force of -4e+05 N to the center of the loading block. Main menu Geometry Assign Loads [Fig. 26] [Fig. 27] Figure 26: Apply vertical nodal force Figure 27: Vertical nodal force Elastic Analysis of a Deep Beam with Web Opening 15/30
16 2.5 Meshing We set the mesh properties with an element size of 50 mm and we generate the mesh. Main menu Geometry Assign Mesh properties [Fig. 28] Main menu Geometry Generate mesh [Fig. 29] Figure 28: Mesh properties Figure 29: Finite element mesh Elastic Analysis of a Deep Beam with Web Opening 16/30
17 3 Structural linear static analysis 3.1 Analysis commands We will perform a linear structural analysis. Main menu Analysis Add analysis Analysis browser Analysis1 Rename LinSta [Fig. 30] Analysis browser LinSta Add command Structural linear static [Fig. 31] [Fig. 32] Main menu Analysis Run selected analysis Figure 30: Analysis window Figure 31: Add command Figure 32: Analysis tree Elastic Analysis of a Deep Beam with Web Opening 17/30
18 4 Results 4.1 Reaction forces FBY To validate the results we check that the reaction forces are in equilibrium with the applied load. Since the load has only vertical component, we check only the reaction forces along the Y -axis (FBY). Results browser LinSta Output linear static analysis Nodal results Reaction Forces FBY Show table [Fig. 33] [Fig. 34] Figure 33: Results browser - show table The summation of all nodal reaction forces along the Y -axis is equal to 400 kn, in equilibrium with the total applied load P = 400 kn. Figure 34: Nodal reaction forces FBY Elastic Analysis of a Deep Beam with Web Opening 18/30
19 4.2 Contour plot Total displacements DtXYZ We make a contour plot of the total displacements DtXYZ [Fig. 36]. Results browser LinSta Output linear static analysis Nodal results Displacements DtXYZ [Fig. 35] [Fig. 36] Figure 35: Results browser Figure 36: Total displacement (DtXYZ) As expected, the highest total displacements DtXYZ are located near the applied nodal force and in the proximity of the opening [Fig. 36]. Elastic Analysis of a Deep Beam with Web Opening 19/30
20 In order to have a smooth contour plot, we choose a continuous color scale [Fig. 38]. Main menu Show view settings Property Panel Result Contour plot settings [Fig. 37] [Fig. 38] Figure 37: Output settings Figure 38: Total displacement (DtXYZ) Elastic Analysis of a Deep Beam with Web Opening 20/30
21 4.3 Contour plot Stress field SXX We create a contour plot for the stress SXX [Fig. 42]. We only display the Element set Beam. We change the colorscale to specified values with minimum value equal to -12 N/mm 2 and maximum value 4 N/mm 2 (this provides a better understanding of the stress field in the beam). Mesh browser Mesh Shapes Beam Show only [Fig. 39] Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXX [Fig. 40] Main menu Show view settings Property Panel Result Contour plot settings [Fig. 41] Figure 39: Show only Beam Figure 40: Results browser Figure 41: Output settings Elastic Analysis of a Deep Beam with Web Opening 21/30
22 Figure 42: Stress field SXX Due to the bending of the beam, the stress component SXX takes the highest values (in tension) at the bottom part of the beam between the two supports and at the top-right (the center of the full beam). High values of SXX are also noticed at the bottom-left and top-right corner of the opening. On the contrary, negative values (compression) of SXX are observed at the other two corners. Elastic Analysis of a Deep Beam with Web Opening 22/30
23 4.4 Contour plot Stress field SYY We create a contour plot for the stress SYY [Fig. 44]. Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SYY [Fig. 43] Figure 43: Results browser Figure 44: Stress field SYY The highest absolute values of SYY are located close to the supports and the loading block. Highest concentration of stresses are also noticeable at the opening s corners as observed for the case of SXX. Elastic Analysis of a Deep Beam with Web Opening 23/30
24 4.5 Contour plot Stress field SXY We create a contour plot for the stress SXY [Fig. 47]. The colorscale limits are changed to the minimum and maximum values of SXY that correspond to -4 N/mm 2 and 4 N/mm 2, respectively. Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXY [Fig. 45] Main menu Show view settings Property Panel Result Contour plot settings [Fig. 46] [Fig. 47] Figure 45: Results browser Figure 46: Output settings Figure 47: Stress field SXY The contour plot shows that the highest absolute values of SXY are in two regions with an inclination of 45 starting from the loading block. It is interesting to observe how the stresses redistribute around the opening. Elastic Analysis of a Deep Beam with Web Opening 24/30
25 4.6 In-plane principal components Stress field SXX We create a vector plot of the in-plane principal components of the SXX stresses. Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXX Show in-plane principal components [Fig. 48] [Fig. 49] Figure 48: Results browser Figure 49: In-plane components of SXX Elastic Analysis of a Deep Beam with Web Opening 25/30
26 4.7 Diagram Stress field SXX We create diagrams of the stress field along a vertical cut at X = 1050 mm (450 mm from the symmetry line). Therefore, we create a probe-curve. Property Panel Result Probing curve setting Add... Curve [Fig. 50] Property Panel Result Probing curve setting probe-curve Number of intervals between points 20 [Fig. 51] Property Panel Result Probing curve setting probe-curve Add... Point coordinates [Fig. 52] Property Panel Result Probing curve setting probe-curve Point coordinates [Fig. 53] Figure 50: Add curve Figure 51: Rename curve and interval number Figure 52: Add coordinates Figure 53: Point coordinates Elastic Analysis of a Deep Beam with Web Opening 26/30
27 We now create a diagram of SXX along the vertical cut at X = 1050 mm [Fig. 55]. Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXX Show contours Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXX Show contour probe [Fig. 54] [Fig. 55] Figure 54: Show contour probe Figure 55: Stress SXX along the cut at X = 1050 mm The diagram clearly shows that at the bottom of the beam we have positive stresses (tension) that become negative (compression) while moving toward the top. Nevertheless, close to the top surface (near the loading plate) the stress becomes again positive. Elastic Analysis of a Deep Beam with Web Opening 27/30
28 4.8 Diagram Stress fields SYY and SXY Similarly, we create a diagram of SYY and SXY along the vertical cut at x = 1050 mm [Fig. 56] [Fig. 57]. Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SYY Show contour probe [Fig. 56] Results browser LinSta Output linear static analysis Element results Cauchy Total Stresses SXY Show contour probe [Fig. 57] Figure 56: Stress SYY along the cut at X = 1050 mm Figure 57: Stress SXY along the cut at X = 1050 mm Elastic Analysis of a Deep Beam with Web Opening 28/30
29 Appendix A Additional Information Folder: Tutorials/DeepBeamWithWebOpening Number of elements 350 Keywords: analys: linear static. constr: suppor. elemen: cq16m pstres. load: force node. materi: elasti isotro. option: direct units. post: binary ndiana. pre: dianai. result: cauchy displa extern force green reacti strain stress total. References: Elastic Analysis of a Deep Beam with Web Opening 29/30
30 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) DIANA FEA BV Disclaimer: The aim of this technical tutorial is to illustrate various tools, modelling techniques and analysis workflows in DIANA. DIANA FEA BV does not accept any responsibility regarding the presented cases, used parameters, and presented results.
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