Chapter 3 Analysis of Original Steel Post

Transcription

1 Chapter 3. Analysis of original steel post 35 Chapter 3 Analysis of Original Steel Post This type of post is a real functioning structure. It is in service throughout the rail network of Spain as part of the via general infrastructure. For it to have reached actual service, this post would have been designed beforehand. Thus, this chapter does not aim to design the post but only to analyse it under the loading conditions provided in the project outline. The chapter outlines the problem data in terms of geometry, materials, and boundary conditions. The behavioural response of the structure is analysed using the Finite Element Method (FEM). Finally, the validity of the results from the numerical model is assessed analytically using the Resistance of Materials model. 3.1 Problem Data Geometry The post is a lattice-type structure and its geometry is shown in figure 1. The particular structure analysed is registered as type X3B of the poste de via general of Adif. It is composed of two U-section beams (UPN) fixed together by plates. The post is an entire height of 8 m of which 7 m is above the fixed support conditions. The post s width tapers towards the top of the structure from a maximum of approximately mm at the base to 200 mm at the highest point. The UPN section consists of a web of 140 mm, a flange of 60 mm and a thickness of 7 mm. The configuration of the beam and plate elements in the structure creates a closed cross-section type profile for the post. While the loading conditions do not suggest it, this type of cross-section is preferred in the presence of torsional affects as all the parts at the crosssection s periphery are connected thereby having a higher torsional stiffness than that of an open cross-section profile. Figure 3.1 shows the geometry of the steel post structure.

2 Chapter 3. Analysis of original steel post 36 Figure 3.1: Geometry of structure The plates between the two UPNs are connected by weld. The type of connection implemented between the plate and the UPN is of butt-weld type and is shown in figure 3.2. More specifically, the thickness of the plate (8 mm) is welded to the edge thickness of the flange of the UPN (7 mm) where the penetration of the weld through the thickness of the materials is complete.

3 Chapter 3. Analysis of original steel post 37 Figure 3.2: Butt-weld connection of UPN and plate with complete penetration Load Development and Boundary Conditions The base of the post is situated in a foundation of concrete at a depth of 1000 mm of the entire height of the structure, thereby, impeding the movement of the post completely at the base. Figure 3.3 shows the typical foundation of the structure. Figure 3.3: Fixed boundary condition at post base There are two load cases provided in the project outline. These include two moments that are induced by wind loading with a maximum velocity of 120km/h. They occur in directions parallel

4 Chapter 3. Analysis of original steel post 38 (case 1) and perpendicular (case 2) to the rail line which represent, in global coordinates, moment about the z and y axes, respectively. Both moments are schematically shown below. Figure 3.4: Orientation of induced moments for both load cases Both moment magnitudes are presented in table 3.1. These two loading types will be defined throughout the project by their case numbers. Perpendicular to line (Case 1) Parallel to line (Case 2) (Nm) (Nm) Table 3.1: Moment at the post s base (120km/h) Figure 3.5 shows a simplified scheme of the structure, with the appropriate boundary conditions, rotated 90 o simulating a cantilever beam-type structure. Figure 3.5: Post structure modelled as cantilever beam with udl The moments above can be translated simply to a uniformly distributed load (udl) along the length of the beam using the following expression:

5 Chapter 3. Analysis of original steel post 39 2 wl M (3.1) 2 Where M is the moment, w is the uniformly distributed load and l is the free length of the cantilever beam. The resultant load R is represented by the area formed by the intensity w (force per unit length of beam) and the length l over which the force is distributed. R wl (3.2) The wind loading applied on the structure is more efficiently represented as a pressure on the post s surface. The pressure P is expressed as the following. P R A (3.3) Where A is the area occupied by the vertical face of the post in directions parallel and perpendicular to the rail line which represent each of the moment cases in table 1. This pressure is directly applied as a boundary condition in the FEM subroutine. Numerically, the expressions for w, R and P are shown, for both load cases, in table 2. Perpendicular to line (Case 1) Parallel to line (Case 2) w (N/m) R (N) P (N/m 2 ) Table 3.2: Udl w, resultant R and pressure P for cases 1 and 2 It is worth noting at this stage that the pressures exerted on the post surface are not equal in both directions, i.e. the perpendicular pressure is approximately 40% greater than that of the pressure for Case 2 (parallel to line). The difference in magnitude between both pressures is considered to be a result of additional loading requirements associated with the perpendicular case such as loading from the catenary cantilever and its assembly which consist of support wires, droppers and contact wires. The additional loading effect of such components is considered as a separate analysis in the following numerical analysis of the composite structure in Section 4.7.

6 Chapter 3. Analysis of original steel post Materials The material used in this model is a carbon steel S275JR of density between kg/m 3. The steel is of structural type (S) with an elastic limit or yield strength of 275 N/mm 2. The principal mechanical properties of this type of steel are given in table 3.3. Mod. of Elasticity E (GPa) 205 Shear Modulus G (GPa) 80 Poisson s Ratio v 0.3 Table 3.3: Mechanical properties of carbon steel S275JR The material is isotropic which considers the elastic or mechanical properties to be equivalent in all directions and as a consequence, the model applied in the numerical program ANSYS is of structural type, lineal elastic and isotropic. 3.2 Finite Element Model The following section relates to the most efficient approach, as regarded by the author, to create an accurate representation of the post structure. The following approach considers the most suitable element types, meshing requirements and application of boundary conditions Element Type In this project, ANSYS is the preferred finite element program to be used. Two different types of elements are employed in this model: PLANE42 element and BEAM188 element. The first of the two elements, PLANE42, creates the plane cross-sections of the structure which in turn largely defines the contour of the structure. In this analysis the post is treated as a beam-type structure which is implemented by the second of the two elements, BEAM 188. The beam element is used to create a mathematical one-dimensional idealization of the 3-D structure. In comparison to other ANSYS beams, BEAM188 provides significant improvements in cross section analysis and visualization. As an overview of the structure s development, each cross section is defined by a section ID number. All sections are custom created, i.e. they are not common sections recognized by ANSYS and have therefore been developed by the user. Custom cross sections are required in this model for two main reasons: the post is of varying width, i.e. it tapers towards the top; and secondly, as stated previously, the beam fixation through welded plates creates a discontinuous section throughout the post s length. The post is therefore defined by two types of sections of varying plate width which are shown in figure 3.7. They consist of firstly, the section where the plate is connected to the flanges of either UPN beam (closed section) and secondly, where there is no plate connection (open section).

7 Chapter 3. Analysis of original steel post 41 The section mesh is also user defined and is stored in the section ID. As a result of the variable section areas a linear-beam tapering command is carried out between respective sections. The length of taper is defined by the length between the two respective sections. By maintaining a constant number of key points for each section each of the sections keypoints are connected by the tapered line thereby forming visually a complete meshed structure. The tapered beam between two respective sections can also be further separated into divisions. PLANE42 Element PLANE42 is a linearly interpolated element used in 2-D modeling of solid structures. The element is defined by four nodes all of which have two degrees of freedom at each node: translation in x and y directions. The element PLANE42, its nodes and degrees of freedom are shown in figure 3.6. The model was constructed over a number of custom sections dictated by the previously described complexity of the structure s geometry. These sections are defined by the element PLANE42. Figure 3.6: PLANE42, 2-D element In total, 50 cross sections were created in order to consider the change in width and those sections that contain a plate connection (closed section) and those that do not contain a plate connection (open section) between the UPNs, and are both depicted by general sketches below. The flanges and web of the beam are subdivided into the 2-D elements which are defined by their keypoints (KP) and lines (L). Figure 3.7: Sketches of a general closed and open section

8 Chapter 3. Analysis of original steel post 42 BEAM188 Element BEAM188 is a 3-D linear finite strain beam element. It is recommended for analysis of slender to moderately thick beams. The post under analysis complies with the slenderness ratio recommended for this beam type. The applicability of the element is given by the following criterion: GAl 2 > 30 (3.4) EI Where G is the Bulk Modulus, A is the cross-sectional area, l is the length and EI is the bending stiffness or flexural rigidity. As the cross section of the beam tapers towards the top of the structure the cross-sectional data which includes I and A subsequently vary throughout the structure. The cross-sectional data with most critical criterion is therefore used to satisfy the slender ratio criterion. This most critical criterion data is found in the largest cross section, i.e. the closed section at the bottom of the post (sec1). This element is based on the Timoshenko beam theory where the cross sections remain plane and undistorted after deformation. The element is defined by the nodes i and j in the global coordinate system in which the orientation of the element x-axis is defined node i toward node j which is shown in figure 3.8. Each node contains six degrees of freedom. The degrees of freedom include translations and rotations in the x, y and z directions. The KEYOPT command which is common to all element types permits the user to determine different value settings for that element. For example, a seventh degree of freedom found in the quadratic beam element which is a warping magnitude can be defined in the analysis however, with respect to the present model, this seventh degree of freedom is not considered (KEYOPT(1) 0). BEAM188 is set as a first order, linear polynomial beam element which uses one point of integration along the length (KEYOPT(3) 0) [8]. Figure 3.8: BEAM188, 1-D line element

9 Chapter 3. Analysis of original steel post 43 BEAM188 allows for the analysis of built-up beams, i.e. beams fabricated from two or more sections joined together to form a single, solid beam. The sections are assumed to be perfectly bonded with the beam thereby behaving as a single member. As already briefly discussed, BEAM188 element is utilized to connect the previously-defined custom sections of the element PLANE42, where the elements of the beam are one-dimensional linear elements in space and the section chosen is associated with the beam element by specifying the ID number of that section. The method including commands, to construct the entire beam with custom defined sections are explained in the following Model Development Method The sections are constructed from keypoints (KP), lines (L) and areas (AL). The global location in Cartesian coordinates of the points for each section was done in an Excel code in which took into account the change in width of the structure and the type of section in question (i.e. open or closed section). The area segment is composed of four keypoints resulting in perfectly straight rectangular areas. The next step, before meshing, is to specify the divisions within each line. This is achieved by selecting the appropriate lines (LSEL) that are to be divided and by choosing the number of divisions required in each of these lines (LESIZE). The process of choosing the number of divisions depends on the size of the element in question and the continuity of the section of the structure. These conditions are highlighted effectively in the meshed model in figure 3.9. The figure shows the step type geometry between the plate and the UPN due to the difference in thickness of each component creating a corner or discontinuity in the sectional geometry between the two components. The UPN has a thickness of 7 mm while the plate has one of 8 mm, hence the size of the divisions coincide with the thickness difference of 1 mm between the two components.

10 Chapter 3. Analysis of original steel post 44 Figure 3.9: Meshing requirements for step in geometry between plate and UPN beam components After each of the 50 cross sections is meshed, they are saved in separate files (SECWRITE) that contain their nodal and elemental data. Other data contained in each file include sectional properties such as the centroide, shear centre, origin and inertia. The sections are introduced in the program algorithm by the command SECREAD. The section s geometry and properties can be displayed through the SECPLOT command. Figure 3.10 shows an example of both the closed and open sections plotted in ANSYS. Figure 3.10: Examples of closed and open sections plotted in ANSYS (section 26 & 27)

11 Chapter 3. Analysis of original steel post 45 The next part of the model development is to define the BEAM188 element and its mechanical properties. Between the most extreme sections created (i.e. the base and top section) of the model there is a notable difference in width. That is to say, the post is of variable section which reduces in width along its length from a maximum at the base to a minimum at the top of the structure. For the element BEAM188, it is possible to define specific beams that contain variable sections by introducing the command TAPER. The section varies linearly between two points or as in this case, between two specified sections. The linear tapered section analysis evaluates the cross-sectional properties at each Gauss point, thereby making the analysis more accurate but computationally intense (KEYOPT(12) 0). The difference in location of the two specified cross sections is related directly to the length of the beam element BEAM188 which, in this model, only considers a change in length according to the z-axis (length of beam). In order to construct a linearly tapered beam segment in the model, the cross section at each end of the tapered length must be defined (SECTYPE) and their appropriate data files previously stored must be read (SECREAD). Two SECDATA commands are required to define the tapered length of the beam which in this case, coincides with two consecutive keypoints of the beam. The line is constructed between these two keypoints, selected (LSEL) and its material attribute defines (LATT). Finally the line is subdivided (LESIZE) into the number of elements desired at the meshing stage. Creating the plate is achieved by defining two beam segments with their appropriate consecutive cross sections. This requires, at the boundary between both segments, consecutive open and closed cross sections to be defined at this same location but of distinctly defined types of taper (SECTYPE). Figure 8 is an example of two cross sections (closed and open) that are at the same location defining two different tapered beam segments and subsequently the plate connection in the post structure. The first of the images in figure 3.11 is a simple example of testing a beam creation using the TAPER command where the two most extreme sections (base and top sections) of the model are used. From here, it is possible to carry out the same command between each of the consecutive sections, which are 50 in total. The second image in figure 3.11 represents the completed post through the repetition of the previously described command for the consecutive sections. The change between the open and closed cross sections of PLANE42 elements combined with the beam elements BEAM188 creates the desired structural effect of the plates connecting the two UPN beams.

12 Chapter 3. Analysis of original steel post 46 Figure 3.11: Meshed beam of variable section in preliminary test form (left) and true form (right) Boundary Conditions and Loads The boundary conditions have been summarised previously in Section The conditions include the analytical development of the loading from moments given in the project outline. For load cases parallel and perpendicular to the rail line, a uniform distributed load w (N/m) is applied as a static load onto the BEAM188 elements over a distance of 7 m thereby effectively simulating wind loading on the free surface of the post. The bottom metre contained in the concrete foundation is completely fixed preventing translations and rotations about all axes. 3.3 Results The most significant results of the model are shown in tables 3.4 and 3.5 which include the maximum stress (Von Mises) due to bending, maximum point displacement, and the reactions due to wind loading applied for both types of load cases. As a result of the section s shape and hence the second moment of inertia I, the maximum stress due to bending ( MPa) and the maximum displacement ( mm) are found in the analysis of load Case 2. That is to say, the moment inertia about the y-axis I YY is less than that about the z-axis. This maximum stress occurs at the section directly above the post s foundation. This section corresponds to the section with an identification number equal to 7 in the FEM model and is shown subsequently in detail in figure The maximum reactions include a horizontal force of N and a moment of Nm and occur in case 1 where the moment inertia here is the greater of the two, thereby reducing the deflection, and consequently increasing the resisting moment. The results of lesser magnitude shown in table 3.5 are negligible (NG) and are treated as numerically zero.

13 Chapter 3. Analysis of original steel post 47 Load Direction Stresses: Von Mises (MPa) Displacement (mm) Perpendicular to line (Case 1) Parallel to line (Case 2) Table 3.4: Results of maximum stresses (Von Mises) and displacements Load Force F X Force F Y Force F Z Moment M X Moment M Y Moment M Z Direction (N) (N) (N) (Nm) (Nm) (Nm) Case 1 NG NG 0.05 NG Case 2 NG NG NG Table 3.5: Reactions (Forces and Moments) Figure 3.12 and 3.13 show the stress distribution equivalent to Von Mises for Case 1 and Case 2, respectively. The figures focus on points localised around the most critical areas of the structure. The first of the figures shows the stress concentrations occurring at the post s thickness (beam web) where there is a slight stress increase towards the centre of the thickness as a result of localized bending in the web of the UPN beam. The second figure shows a stress concentration in the width extremities (flanges of UPNs) due to load Case 2. Focussing along the thickness of the structure, an attenuation of the stress can be seen towards the thickness centre and as a result of the symmetry of the structure, an increase in stress is observed once again towards the other thickness extremity. As a result of this symmetry and the specific cases of applied load direction, the maximum stress produced by bending is the equal on both faces (extremities) of the structure but are opposite in sense (tension and compression). Taking the maximum stress value due to bending as MPa, and knowing the elastic limit of the material (σ e 275 MPa), the Factor of Security (FoS) of the structure in steel is calculated to be approximately equal to 3.9. An overall analysis of the FoS for the models is given in Section 6.1.

14 Chapter 3. Analysis of original steel post 48 Figure 3.12: Stresses (Von Mises) for load case 1 (y direction) Figure 3.13: Stresses (Von Mises) for load case 2 (z direction)

15 Chapter 3. Analysis of original steel post Validation of Numerical Model The validation of the numerical model is carried out using the method of Resistance of Materials. As indicated previously, the most critical point of the structure is the section directly above the fixed end (post foundation) of the structure which corresponds approximately to section 7 of the model. This most critical section and its properties are shown in figure Figure 3.14: Most critical section of structure (ID section 7) By knowing the second moment of area of the section it is possible to determine its resistance to bending and the maximum displacement. The general expression for the second moment of area of a section is given in equation (1) where B is the width, D is the depth, A is the area of the local section and h is the distance from the centroide of the local section to the neutral axis of the entire section I BD + Ah 2 (3.5) Taking into account the sectional diagram the figure 3.15, the second moment of area with respect to the y-axis, I YY is calculated as the following:

16 Chapter 3. Analysis of original steel post 50 Figure 3.15: Cross section at fixed boundary condition with inertia calculated about the y-axis I yy ((120)(7 )) + ((120)(7) )( 66.5 ) 2 + ((14)(126 )) I yy x10 mm Taking into account the sectional diagram the figure 3.16, the second moment of area with respect to the z-axis, I zz is: Figure 3.16: Cross section at fixed boundary condition with inertia calculated about the z-axis I zz ((140)(7 )) + ((140)(7) )( ) 2 + ((14)(53 )) + ((14)(53) )( ) I xx x10 mm

17 Chapter 3. Analysis of original steel post 51 As it was predicted, the values of I calculated by hand are equal to the values calculated in the program model shown in the figure of section 7. The maximum stress due to bending produced in the critical section of the structure is given by the classic expression in the equation (3.5) where M is the moment with respect to the neutral axis and y is the perpendicular distance to the neutral axis. My σ I (3.5) Then, the maximum stress due to bending with respect to the neutral axis of y is: σ (8.649x10 6 yy x10 )(70) N mm 2 σ yy MPa And the maximum stress due to bending with respect to the neutral axis of x is: σ (10.54 x10 6 xx x10 )( ) N mm 2 σ xx MPa The calculated results of stresses through the use of the model of resistance of materials are approximately equal to the maximum stresses calculated in the numerical model. The slight difference between both sets of results is proposed in two areas. The first of the two areas deals with the site of the most critical section of the structure. It has been highlighted that the most critical section lies just above the fixed point at the post s base, however the applied section 7 is not exactly where the maximum stress lies. That is to say, section 7 is not the most critical section in the structure but is gives a good approximation of the size of the actual critical section, and therefore, the second moment of area. In reality, the most critical section is found a small distance above section 7 and is visually evident from the stress distribution in figure 3.13 where it can be seen the increase of stress away from the closed section, i.e. above the welded plate. The second area concerns the type of stress that is being evaluated in the model. The method of resistance of materials calculates the principal stresses with respect to the x and y axis independently. The stresses calculated by the ANSYS model are equivalent to Von Mises where the principal stresses are not calculated independently but with the following expression.

18 Chapter 3. Analysis of original steel post 52 σ VM 2 2 ( σ σ ) + ( σ σ ) + ( σ σ ) x y y 2 z z x 2 (3.6) The deflection of the cantilever beam with a uniform distributed load is given by the expression in (3.7). It is necessary to take into account that the deflection calculated by this expression is an approximated value as the beam is composed of a variable section. The second moment of inertia of the open section is applied in the expression below as approximately 80% of the post s cross section is composed of this open section type. As a consequence the maximum displacement calculated is more conservative. δ max 4 wl 8EI (3.7) δ max 4 ( )( 7000 ) 3 7 ( x10 )( 0.977x10 ) δ max 64.46mm