Conceptual sandwich-sandwich-steel joint design for light

weight rail vehicle

Tobias Magnusson

February 27, 2014

Abstract

In order to find a feasible solution for a joining method of a sandwich side-wall, a sandwichfloor and a steel underbody of a railway vehicle, conceptual joint designs have been devel-oped by using structural optimization software. It is shown that the joints are capable ofcarrying the loads assumed to act on the structure but that several improvements to theanalysis needs to be done to assure a safe design.

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Contents

1 Introduction 3

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Traditional joints in rail vehicles . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Sandwich joint design aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Method 3

2.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 FE-Models Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Steel FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Sandwich FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.1 Joint connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5.2 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6 Load case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.1 Transverse load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.6.2 Vertical load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.6.3 Floor payload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.4 Under body loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6.5 Longitudinal compression . . . . . . . . . . . . . . . . . . . . . . . . 112.6.6 Results Load case calculations . . . . . . . . . . . . . . . . . . . . . 122.6.7 Load case combinations . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7.1 Boundary condition floor . . . . . . . . . . . . . . . . . . . . . . . . 132.7.2 Boundary condition wall . . . . . . . . . . . . . . . . . . . . . . . . . 142.7.3 Boundary condition under body . . . . . . . . . . . . . . . . . . . . 142.7.4 Boundary condition compilation . . . . . . . . . . . . . . . . . . . . 14

2.8 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8.1 Load introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8.2 Topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8.3 Dimensioning by size optimization . . . . . . . . . . . . . . . . . . . 182.8.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Results 19

3.1 Results Topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Results Size optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Results Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Steel model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.2 Extruded joint with connection type A . . . . . . . . . . . . . . . . . 263.3.3 Extruded joint with connection type B . . . . . . . . . . . . . . . . . 273.3.4 Not extruded joint with connection type B . . . . . . . . . . . . . . 283.3.5 Longitudinal compression stress analysis . . . . . . . . . . . . . . . . 29

4 Conclusion 29

5 Discussion 30

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1 Introduction

1.1 BackgroundIn order to reduce the energy consumption as well as track and wheel wear for railway trans-portation it would be beneficial to reduce the weight of the car body. One way of doing this isby replacing the traditional load-carrying steel structure with sandwich panels. This could notonly lower the weight of the vehicle but also provide a more spacious cabin as well as reduceproduction and maintenance costs. A conceptual design of a vehicle car of this type has beenprepared by David Wennberg [1] in which both the floor, sidewalls and roof of a passenger carbody has been designed as sandwich panels. Since the joining properties of sandwich panelsdiffer considerably from those of steel structures this work aims to design a conceptual jointbetween the sandwich sidewall, sandwich floor and the steel underbody.

1.2 Traditional joints in rail vehiclesIn a traditional metal car body the wall, floor and underbody are usually joined together by thesole-bar. This is typically a steel or aluminium component that runs along the intersection ofwall and floor, which the floor-, underbody- and wall parts are either welded or bolted togetherwith. In a traditional car, the sole-bar or joint between floor and wall serves not only as aconnector between those two components but also carry the longitudinal load in the car body.Since the sandwich panels are designed to provide this function, the purpose of the joint betweenthe sandwich floor and wall should be to provide sufficient rigidity and strength during cornering,acceleration and deceleration.

1.3 Sandwich joint design aspectsWhen developing the different design concepts for the joint, some particular aspects must beconsidered. Since welding between a metal joint and the composite sandwich panels is notpossible the connection type need to be studied. The weak core of the sandwich panel also makeit difficult to bolt the panels and the joint without breaking the core. Besides basic structuralrequirements, manufacturing issues and material accessibility must also be taken into account.It is also an advantage if the joint is multifunctional meaning that it does more than just connectthe parts. For example, the joint could be used for attachment of the interior and necessaryinterior- and exterior equipment since this may prove difficult to do in the sandwich panels.The joint could also be designed so that it improves the aerodynamic properties of the traincar or provide housing for electric and hydraulic cabling. These properties are all desirable butnot considered to be a requirement. The main tasks when developing the joint is presented insection 2.1.

2 Method

A reduced FE-model of a section of a steel train car is used as a starting point to develop thedifferent design concepts. The goal is to analyze the same section of the sandwich train car. Todo this, a FE-model of the sandwich train is built (see section 2.3). The loads acting on this partof the car (calculated in section 2.6) are introduced into the model and an optimization software

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(a) Joint overlapping sandwich panels, calledtype A.

(b) Sandwich faces overlapping joint, calledtype B.

Figure 1: Main connection types

is used to determine the geometry of the joint. Details in how the FE-analysis i performed isshown in section 2.8.

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2.1 RequirementsThe main goal in this work is to design a joint that connects the floor, wall and underbodystructure without adding excessive weight. The joint should withstand the stresses that occursduring operation of the train. All design concepts should be developed in a way so that theyare possible to manufacture and mount.

2.2 Basic conceptsThe concepts evaluated consists of different aluminium profiles. Those can either be extrudedprofiles with constant cross section, free-shaped profiles with varying cross sections or combi-nations of those two. Two main concepts for the connections between the sandwich panels areused. Figure 1a shows the first solution where the joint overlaps the sandwich panels (from hereon referred to as type A). Figure 1b shows a connection where the joint replaces the edge partof the sandwich core (from here on referred to as type B). The type A joints main advantageis assumed to be that it allows for the sandwich panel to be produced as one plate with noalterations needed. The second solution (type B) is assumed to give better load introductionfrom joint to sandwich panel. Figure 1 also shows two different ways to connect the horizontal-and vertical part of the joint. Figure 1a shows how the flange used to bolt the underbody isleft open to ease assembly of the cart while Figure 1b shows a closed flange which would makeassembly more difficult, but probably add stiffness.

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Figure 2: FE-model of the steel structure section.

2.3 FE-Models OverviewFor the analysis, two basic models are used. They are both cross sections of one corner betweenthe wall and floor with a length of 1m. They also include a part of the underbody structure.One model is a typical steel structure cart, provided by Bombardier and is used for comparison.The other one is a model based on the steel model but with sandwich floor and wall. Mostmodeling is performed in Altair HyperMesh but some complex geometries are imported as .igesfiles from Siemens SolidEdge.

2.4 Steel FE-modelThe steel structure model, which is seen in Figure 2, consists of approximately 8000 elementsand 9000 nodes. All components but one are modeled with 2D shell elements with an elementsize above 30mm. The only component modeled by different element types is one part of thesole bar that is modeled with solid 3D elements. The structure modeled is a section from anexisting train car. The material data and properties of the components have not been alteredfrom its original values.

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Figure 3: FE-model of the sandwich structure section without joint.

2.5 Sandwich FE-modelTo design a joint suitable for the sandwich train car, an FE-model of the same section as thesteel model is built (see Figure 3). While the underbody structure remains the same as in thesteel model, the floor and wall are now modeled as sandwich panels. The outer surface of thesandwich wall has the profile of the outer wall in the steel model and the sandwich floor panelis on the same level as the top of the corrugated sheet in the metal floor. This is to make surethat the sandwich car does not add in size and to assure enough space between the floor andunder body structure to store necessary equipment. The sandwich model (without joint andconnectors) consists of 7 components (3 for the under body), 165564 elements and 157029 nodes.In this model, the sandwich cores are modeled with 3D solid elements in order to give a goodrepresentation of the shear stress in the core. The face sheets are modeled with HyperLaminate1

as 2D composite shell elements. The benefit of using this element type is that it allows differentstress distribution in different layers of the laminate. The adhesive joint between core and facesheets are assumed to give a perfect bond, hence the faces and cores share nodes. To savecomputational power but still give a high resolution solution close to the joint, the density ofthe mesh of the sandwich panels are set to give smaller element size close to the joint. In thefar ends of the panels the element size is roughly 40 mm and in the joint end roughly 10 mm.The mesh of the under body structure remains the same as that in the steel model.

2.5.1 Joint connections

In order to connect the different joint design concepts to the rest of the structure two differenttypes of connections are used. For the connection between joint and under body, rigid elementsare used to simulate a bolted connection. This means that the bolts are not dimensionedbut assumed to be sufficient. To connect the joint to the sandwich panels, an adhesive joint

1Function in Altair HyperMesh software to model composite laminates.

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is modeled as shown in Figure 4a. For this the function area connector2 is used. The areaconnector makes it possible to model the adhesive joint with 3D solid elements even though themesh of the components that are to be connected do not match. This allows for the stresses inthe adhesive joint to be analyzed. The elements representing the adhesive, are meshed with oneof the connected parts mesh as a base. The other end of the element is then connected to thesecond component with rigid elements (see Figure 4b). This type of modeling assumes that theadhesive bonds perfectly to the connected parts.

(a) Adhesive joint as modeled in between one of theconceptual joints and the sandwich floor.

(b) 3D solid element with mesh matching bottom sur-face and connected to the top surface (with unmatchedmesh) with rigid elements.

Figure 4: Adhesive joint between sandwich panel and joint modeled with area connectors.

2.5.2 Material data

The material data for the components used in the model are presented in Table 1. The lay upfor the sandwich panels are presented in table 2. The material data for the sandwich panels aregathered from [1]. Metal, and polymer data is found in [3].

Part Material PropertiesJoint Aluminum E = 70GPa, G = 27GPa, ⇢ = 2700kg/m3, �u = 300MPaFace CF epoxy E1 = 170GPa, E2 = 9GPa, G12 = 4.4GPa, ⇢ = 1600kg/m3

Core PMI Foam E = 75MPa, G = 24MPa, ⇢ = 52kg/m3

Under body Steel E = 200GPa, G = 77GPa, ⇢ = 7900kg/m3

Adhesive joint Epoxy E = 4000MPa, G = 1500MPa, �fracture = 30MPa

Table 1: Material data used in sandwich model.2Function in Altair HyperMesh software to model adhesive joints.

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Part Lay up CF skins (mm) Core thickness (mm)Roof 0�/± 45�/90� 0.48/2⇥ 0.32/0.91 102.3Floor 0�/± 45�/90� 0.81/2⇥ 0.26/2.05 116.6

Table 2: Sandwich panel lay up from [1].

2.6 Load caseSince a train car traditionally is a complex structure with multiple loads acting on it, a numberof simplified load cases must be defined for the analysis. The load cases are designed to simulatereal loading conditions that effect the joint and sandwich panels. Guidelines of how to calculatethe design loads in accordance with European norms have been summarized in [2] by DavidWennberg.

2.6.1 Transverse load

To make sure that the joint can withstand stress caused by inertia effects when the train cornersor due to movement of adjacent vehicles, a transverse force is calculated. To find a suitablemagnitude for this load the weight of the roof section per length of the car is determined. Tothis mass, the weight of the equipment stored in the roof section such as pantographs, air-conditioning units etc. are added3.The combined mass is then multiplied with the transverseacceleration constant from the design norms. In Figure 5, which shows an outward directedtransverse force, the design load is denoted Ft and the combined mass off the roof structure andequipment is denoted m

Figure 5: Transverse force acting on cross section of car body.

Since the analysis is performed on a reduced model which does not include the entire heightof the vehicle, the force need to be applied at a suiting position. To find a representative forceFt, the moment in point A is regarded.

yA =

ml

2gt ⇤ h1 (1)

3The weight per length used for calculations is the highest possible weight along the length of the car.

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Here gt is a transverse acceleration constant of ±1g which comes from the design norms, l isthe length on which the analysis is performed and m is the combined mass off the roofstructure and equipment per length. The mass m is divided by 2 since there are joints on bothsides of the train car. The force Ft should give rise to the same moment in A.

yA = Ft ⇤ h2 (2)

Combining equation (1) and (2) gives

Ft =mlgth1

2h2(3)

Since Ft acts in both directions (outwards and inwards) the Transverse force load case is splitinto two different cases with opposing directions of F .

2.6.2 Vertical load

This load case is designed to test the joints vertical load carrying ability. Figure 6 shows arepresentation of the Vertical load case.

Figure 6: Vertical force acting on cross section of car body.

The vertical design load Fv is calculated using

Fv =ml

2gv (4)

where m is the combined mass of the roof structure and equipment per length, l is the lengthon which the analysis is performed and gv is the vertical acceleration constant set to 3g inaccordance with design guidelines.

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2.6.3 Floor payload

This is a distributed load acting on the floor section. The weight used to calculate the forceis the combined weight of the payload and the interior attached to the floor. Figure 7 shows arepresentation of this load case.

Figure 7: Floor payload.

The design load F is calculated using

Fp = (mp +me)gp ⇤ 4A (5)

where mp is the maximum payload, me is the weight of the interior, gp is the accelerationconstant set to 1.3g in accordance with the design norms and 4A the fraction of the floor onwhich the analysis is performed.

2.6.4 Under body loads

Since the train carries a significant amount of equipment on the under body structure, and sincethe under body itself is relatively heavy, the forces transferred into the joint from this structureneeds to be included in the analysis. The acceleration constants used to calculate the designloads acting on the under body are the same as those used to calculate the loads on the sidewallstructure, i.e. ±1g for the transverse case and 3g for the vertical case. Figure 8 show the designforces Fut and Fuv.

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Figure 8: Under body loads.

2.6.5 Longitudinal compression

This load case investigates the joints ability to transfer the shear forces that arise duringlongitudinal compression of the train car. The force that the joint needs to be designed for is acompression force of 1500kN acting in the coupler area. Since the coupler area is not locatedon the neutral axis of the train car, it will cause the car to bend as can be seen in Figure 9.

Figure 9: Bending due to compressive loading.

In order to analyze the effects of this load in the joint and how well it transfers the load to thesandwich panels, the local displacements in a full scale FE model subjected to this load caseare studied. The full scale model is provided by David Wennberg and have been used whendesigning the sandwich panels for[1]. In this model the sandwich floor and wall (which aremodeled as 2D elements) are joined together by shared nodes and the loads from the underbody are introduced into the sandwich panels through rigid elements. The deformations in themodel when subjected to this load case are considered to be acceptable. This means that thejoint does not have to account for any extra strength or stiffness. However, the joint still mustbe able to transfer the loads without introducing any local stresses of high magnitude thatmay cause the sandwich panels or itself to fail. How the local displacements are introducedinto the reduced model is reported in Appendix B.

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2.6.6 Results Load case calculations

In this section the numerical values of the loads that are introduced into the reduced model arepresented. The reduced model is 1m long hence l = 1m. The results are presented in Table 3.

Load case Magnitude Direction Input data (mass, distances, areas, etc.)Transverse wall (Out) 5.9kN +y m = 526kg

m , h1 = 2.280m, h2 = 0.997m

Transverse wall (In) 5.9kN �y m = 526kgm , h1 = 2.280m, h2 = 0.997m

Vertical Load wall 7.7kN �z m = 526kgm

Floor Payload 3.7kN �z mp = 11638kg, me = 4289kg, 4A = 0.018Under body (Out) 4.9kN +y m = 1000kgUnder body (In) 4.9kN �y m = 1000kgUnder body Vert. 14.7kN �z m = 1000kg

Table 3: Numerical values of loads introduced into the reduced model.

2.6.7 Load case combinations

Since multiple load cases may act simultaneously, combinations of previously mentioned loadcases are simulated. For obvious reasons the transverse forces acting on the wall and underbody should have the same direction for all combinations. The different combinations of loadcases used are presented in Table 4.

Combination 1a/1b Combination 2T. Wall (Out/In) Wall Vertical

U.B (Out/In) U.B. Vertical

Table 4: Combinations of load cases.

2.7 Boundary conditionsSince the analysis is performed on a reduced model of the train car, boundary conditions areused to represent the continuation of the structure along the edges. To completely simulate thefull cart with the use of static boundary conditions is in this case not possible but it is importantthat the boundary conditions at least allow the right type of deformation in the part studied. Itis also desirable that the boundary conditions do not give raise to any local stresses in the joint.In this analysis, a number of different sets of boundary conditions are used for the different loadcases. Every set of boundary condition is designed to represent the deformation that wouldoccur in a full-scale model. Along some of the edges, the same boundary conditions are usedfor all load cases. Since the reduced model is supposed to represent an arbitrary section of thetrain car, symmetry boundary conditions are used along the edges perpendicular to the x-axis.Since the corner between the wall and roof is a fixed connection, capable of withstanding somemoment, the edge of the wall parallel to the x-axis is constrained in rotation around the x-axis.The common boundary conditions are shown in Figure 10.

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Figure 10: Common boundary conditions used for all load cases. The arrows shows in whatdegree of freedom the nodes along the edges are looked.

2.7.1 Boundary condition floor

Apart from the common boundary conditions described in section 2.7, the floor needs to beconstrained in additional ways. The longitudinal edge of the floor is located along the centeraxis of the cart and the obvious solution is to apply a symmetry boundary condition. This isdone by locking the displacement in y-direction and the rotation around the x-axis. This typeof boundary condition should be used for the load cases where the train section deforms in thesame way on both sides of the longitudinal axis, for example the vertical load acting on thewalls. For modeling reasons however, the model needs to be constrained in z-direction as well,in at least on node. In reality, the displacement in z-direction is limited by the contact betweenthe wheels and the rail. To get the right type of deformation in the joint, all nodes of the floorare locked in z-direction. This does not give the most accurate deformation of the entire modelbut it admits an opening moment in the joint without giving rise to local stresses. This methodis also used for combination of load cases where a constraint in z-direction is needed.

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Figure 11: Shape of deformation. 11a shows how the model would deform under inward andoutward wall bending with symmetry boundary conditions and 11b shows how the full cross-section is assumed to deform.

For the load case transverse bending, symmetry boundary conditions cannot be used. Thisbecause outward bending of the wall on one side corresponds to inward bending on the otherside as can be seen in Figure 11a. When analyzing the deformation with symmetry boundaryconditions one can see that outward bending raises an opening moment in the joint and elevatesthe floor. Inward bending instead gives a closing moment in the joint and forces the floordownwards. A combination of those gives an assumed total deformation of the floor sectionshown in Figure 11b. To achieve this deformation the longitudinal edge should be locked inz- and y-direction while the rotational degrees of freedom remains free. In order to attain thedesired deformation and to stop the model from only rotating around the x-axis, the modelalso must be constrained in z-direction in additional nodes. To achieve optimal shape of thedeformation, this zero-displacement constraint should be placed on the joint but since this maycause unwanted local stresses it is placed some distance away from the joint. This distance isdetermined iteratively so that it has little or no effect in the joint.

2.7.2 Boundary condition wall

For the load cases that include loads acting on the floor in z-direction, the floor cannot beconstrained in this direction. Instead the wall is locked in z-direction.

2.7.3 Boundary condition under body

For the load cases where only the wall is being subjected to forces, the edge of the under bodyis constrained in the same way as the floor edge. When vertical forces acts on the under bodystructure, the edge is constrained by symmetry boundary conditions and when transverse forcesacts, the edge is constrained only in rotation around the x-axis.

2.7.4 Boundary condition compilation

A visualization of all boundary conditions used for the different load cases can be found inAppendix A.

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Figure 12: Design process starting from design space leading up to dimensioned product.

2.8 AnalysisThe analysis part consists mainly of four steps. Those are; Load introduction, Topology Opti-mization, Dimensioning by size optimization, and Evaluation. The analysis is performed withthe Altair HyperWorks software package. HyperMesh is used as the pre-processor to set up theanalysis, RADIOSS/Optistruct is used as the solver and HyperView as post-processor to reviewthe results. Figure 12 shows a representation of the design method used. First a design spaceis assigned and loads are introduced. As a design tool, the topology optimization function inAltair Optistruct is used to find the basic geometry. The design is then fine-tuned by using thefunction Size optimization.

2.8.1 Load introduction

When introducing loads and boundary conditions in the models there are certain aspects thatneed to be considered. First of all there are numerous differences in how the loads are appliedto the sandwich structure model and to the steel model. Due to weakness of the foam core,loads are not applied directly to the core since this would raise unwanted local displacements.Instead loads are applied to the face sheets to a maximum extent possible. The loads are alsodistributed over as large areas as possible in both models to avoid local stress concentrations. Theboundary conditions used along the edges of the model are applied as single point constraints.This constrains every node along the edge in the degrees of freedom declared in section 2.7. Thedifference between the steel model and the sandwich model concerning the boundary conditionsalong the edges is that, in the steel model they are applied along a two dimensional edge and inthe sandwich model through the thickness of the sandwich plate. For the load cases requiringrotational freedom but zero displacement constraints along the longitudinal edge (Transverse

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(a) Zero displacement boundary conditions in y-and z-direction along the longitudinal edge thatwould stop rotation around the x-axis.

(b) Used boundary condition to the longitudi-nal edge for load case Transverse bending andcombination 1a/b.

Figure 13: Ways of constraining the longitudinal edge.

wall load case and combination 1 load case), the boundary conditions need to be applied ina special way to the sandwich model. This is since the sandwich panel is modeled as a 3dstructure which makes it impossible to constrain the displacement of all the nodes along theedge without also effecting the rotational freedom. Instead one node in the middle of the panelis constrained to zero displacement (in the y- and z-direction) but left free in rotation. Thisnode is then connected to all other nodes along the edge with rigid elements, meaning thatthe constraint is distributed along the edge but still allows the right type of deformations. Arepresentation of this is shown in Figure 13. Here Figure 13a shows how the longitudinal edgewould be constrained through the thickness with single point constraints. If the single pointconstraints prevents displacement in y- and z-direction the rotation around the x-axis would alsobe inhibited. Figure 13b instead shows how the edge actually is constrained. This technique isalso used for one section close to the edge which needs the same type of constraint.

2.8.2 Topology optimization

To find a suitable design for the joint, topology optimization is done for the different concepts.Topology optimization is a mathematical technique used to optimize the material distributionin a given design space. The type of design being used is a minimum compliance design whichactually means maximizing stiffness for a given volume of material. The design variable beingchanged during the optimization run is an element density function ⇢ ranging from 0 to 1. Bystudying the following equation for linear FE analysis;

K(⇢)u = F (6)

where K(⇢) is the stiffness matrix, u is the displacement and F is the force vector, one cansee that the stiffness is effected by the design variable ⇢. By combining equation 6 with thedefinition of the compliance bellow (equation 7);

c(⇢) = FTu (7)

the relationship becomes;c(⇢) = FTK(⇢)�1F (8)

Since the force vector F is constant one can see that minimizing the compliance is equivalentto minimizing the inverse of the stiffness matrix i.e maximizing stiffness. The response type

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(a) Design space for the joint overlappingthe sandwich panels.

(b) Design space for joint where sandwichfaces overlapping the joint.

Figure 14: Design space (magenta colored elements) used for topology optimization.

for the compliance in Optistruct is called weighted compliance. This is because it is possibleto define weight factors for the load cases that are used in the optimization4. In this case, allweight factors are set to 1, meaning that every load case is equally important when findingthe optimum shape. Since the goal with the topology optimization is to provide guidance inhow the joint should be designed, a maximum volume fraction of the design space is used asoptimization constraint. The design space defined for the topology optimization is built up by3d solid elements with a size of roughly 10 mm. The design space used for the different types ofconnections, shown in Figure 1, is presented in Figure 14. For every joint, two different types oftopology optimizations are preformed. One with a extrusion constraint and one without. Theextrusion constraint means that the optimized element density must remain constant along agiven path. The outcome of this is that the optimized result is a constant cross-section. Figure15 shows the node path defined for the extrusion constraint.

Figure 15: Node path for extrusion constraint.4Due to difficulties to set up boundary conditions, combination load cases are not included in optimization.

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Figure 16: Re-modeling of joint with design from Topology optimization. Figure shows resultsfrom extruded optimization of the joint overlapping the sandwich panels.

2.8.3 Dimensioning by size optimization

The goal with the size optimization is to find the thickness of the the different parts of the joint.The basic shape of the joint comes from the topology optimization. The joint is re-modeledwith 2d shell elements to allow thickness optimization. Figure 16 shows a representation of howthe geometry is extracted from topology optimization. What the size optimization basicallydoes, is automatically change the element thickness to find an optimal solution. The objectiveis to minimize the volume (or mass since the joint is made from a homogenous material) of thejoint. The design variable in this step is the element thickness. As optimization constraints,an interval for the element thickness and maximum stress constraints are used. The maximumstress constraint is set to 300MPa (von Mises) in the joint for all load cases. Since the jointis connected to the under body structure with rigid elements, the elements closest to the rigidconnection are deleted from the maximum stress constraint. This it to avoid that the entiresection becomes over-dimensioned. Figure 17 shows the elements excluded from the constraintfor one of the joints. To avoid that the optimized result is a joint where every element thickness isoptimized individually, leaving a joint impossible to manufacture, a relationship giving constantthickness for each section of the joint is used.

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Figure 17: Elements not included in maximum stress constraint.

2.8.4 Evaluation

The final step of the analysis is to evaluate the optimized joints. In this step the final weight ofthe designed joint is extracted as well as maximum stresses for the different load cases.

3 Results

In this section, the results from the analysis is presented. Topology optimization results arepresented for four joint concepts. This includes both extruded and not extruded optimizedsolutions for the joint overlapping the sandwich panel and the joint being overlapped by thecarbon fibre faces. Size optimization results are presented for three joint concepts. The conceptnot further investigated after the topology optimization is the not extruded joint that overlapsthe sandwich panel. In the last section Results Evaluation, the calculated weight for the threeconcepts modeled in Size optimization are presented. It also contains information of maximumstresses for the different load cases and where those occurred.

3.1 Results Topology optimizationThe number of iterations needed for the topology optimization is approximately 10-20. Theresults for the topology optimization is presented as figures of the geometry. The maximumvolume fraction used for the optimization runs are chosen to give the most representative figures.Figure 18 shows results for the extruded joint overlapping the sandwich panel.

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(a) Side view without material cut away. (b) Perspective view with material cut away.

Figure 18: Extruded joint that overlaps the sandwich panel. Volume fraction constraint used:Vf < 0.3.

Figure 19 shows topology optimization results for the not extruded joint that overlaps thesandwich panels.

(a) Side view with material cut away. (b) Perspective view with material cut away.

Figure 19: Not extruded joint that overlaps the sandwich panels. Volume fraction constraintused: Vf < 0.3.

Figure 20 shows the topology optimization results for the extruded joint overlapped by thesandwich panel faces.

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(a) Side view. (b) Perspective view with material cut away.

Figure 20: Extruded joint where the sandwich panel faces overlaps the joint. Volume fractionconstraint used: Vf < 0.2.

Figure 21 shows not extruded joint that is being overlapped by the sandwich faces.

(a) Side view with material cut away. (b) Perspective view with material cut away.

Figure 21: Not extruded joint where the sandwich panel faces overlaps the joint. Volume fractionconstraint used: Vf < 0.3.

3.2 Results Size optimizationIn this part, the thickness calculated from the size optimization is presented. First it is explainedhow the different 2d joints are divided into sections. The thicknesses are then presented for everysection of the joint. The thickness constraint used for all sections is 1mm t 5mm. Figure22 shows how the extruded joint type A is divided into sections.

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Figure 22: Extruded joint with connection type A section numbers.

Table 5 shows the calculated thicknesses for extruded joint type A .

Section 1 2 3 4 5 6 7 8 9Thickness [mm] 2.086 3.525 1.0 1.0 1.0 1.0 2.510 1.0 1.914

Table 5: Thickness calculations for extruded joint type A.

Figure 23 shows how the extruded joint type B is divided into sections. The part namedsection 1 is one single section, meaning the thickness of this part will be constant. This is tomake the manufacturing process easier.

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Figure 23: Extruded joint type B section numbers.

Table 6 shows the calculated thicknesses for extruded joint type B .

Section 1 2 3 4 5 6 7Thickness [mm] 1.0 2.781 1.603 1.367 1.0 3.508 1.361

Table 6: Thickness calculations for extruded joint type B.

The size optimization of the joint that is not an extruded profile is made in the same way asfor the extruded joints. However, different parts of the same sections may not be connected. Forexample, the two arms connecting the under body and the wall are designed in the same wayand share section number and therefore also thickness. Section numbers for the not extrudedjoint is found in Figure 24.

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Figure 24: Not extruded joint with connection type B numbers.

Table 7 shows the calculated thicknesses for the different sections for the not extruded jointwith connection type B .

Section 1 2 3 4 5 6 7 8Thickness [mm] 3.624 2.276 3.049 1.835 2.219 2.782 3.816 2.623

Table 7: Thickness calculations for extruded joint type B.

3.3 Results EvaluationIn this section, the total weights per length and stress levels of the joints are presented startingwith the original steel model. When evaluating stress levels, both von Mises stress and Principalstresses have been studied. The presented value is the highest of those, giving a conservativedesign. The values presented in tables 8 to 11 are the highest calculated stress levels in thejoint for each load case. It’s also declared in what part of the joint the highest stresses occur.Figure 25 to 28 shows visual representations of the load cases giving the highest stress levels inthe joints. The stress levels in the sandwich panels5 close to the joint have also been evaluated.

5Higher stresses may occur some distance away from the joint.

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3.3.1 Steel model results

Figure 25 shows the contour plot forthe steel model when subjected to loadcase Transverse wall out. The max-imum stress level for this load casewas as high as 500 MPa. However,the elements showing those stress lev-els are connected to- , or located inclose proximity to rigid elements. Thestress levels presented in Table 8 aretherefore taken at least two elementsaway from rigid connections.

Figure 25: Wall out Steel model.

Load case Max stress [MPa] Position in the jointTransverse Wall (Out/In) 176 Connection Wall/Floor beams

Vertical Load wall 89 Connection Wall/Floor beamsFloor Payload 60 Floor beams

Under body (Out/In) 85 Lower part of wall beamUnder body Vertical 156 Connection Wall/Floor beamsCombination 1(a/b) 185 Connection Wall/Floor beams

Combination 2 226 Connection Wall/Floor beams

Table 8: Highest calculated stress levels for each load case.

Total weight per length of the joint: 42 kg/m

Since the steel model does not have a defined joint, the weight presented is for the componentslocated between the floor and wall in the same area as the design space for the sandwich jointtype A.

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3.3.2 Extruded joint with connection type A

Figure 26 shows a close up of the stressdistribution for the joint and under-body connection when subjected toload case Under body vertical. Thestress levels presented in Table 9 aretaken at least two elements away fromrigid connections.

Figure 26: Load case Under body Vertical give highstress levels close to under body connection.

Load case Max stress [MPa] Position in the jointTransverse Wall (Out/In) 301 Intersection Inner wall flange.

Vertical Load wall 119 Lower part of section 9Floor Payload 9 Intersection Upper floor flange

Under body (Out/In) 295 Close to Under body connectionUnder body Vertical 320 Close to Under body connectionCombination 1(a/b) 301 Intersection Inner wall flange.

Combination 2 233 Intersection Lower floor flange

Table 9: Highest calculated stress levels for each load case for the extruded joint with connectiontype A .

Total weight per length of the joint: 6.688 kg/m

The joints influence on the rest of the structure can be summarized as;

• Maximum adhesive stress is 2.7 MPa

• Maximum composite stress is 55/2.7 MPa (In fibre direction/Transverse fibre direction)

• Maximum core shear stress is 0.3 MPa

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3.3.3 Extruded joint with connection type B

Figure 27 shows a close up of the stressdistribution for the joint and under-body connection when subjected toload case Combination 1. The stresslevels presented in Table 10 are takenat least two elements away from rigidconnections.

Figure 27: Load case Combination 1 give high stresslevels close to under body connection. The upperarrow shows region of high stress

Load case Max stress [MPa] Position in the jointTransverse Wall (Out/In) 332 Upper part of crossbeam 7.

Vertical Load wall 127 Upper part of crossbeam 7.Floor Payload 36 Intersection Lower floor flange

Under body (Out/In) 55 Connection section 1 and 2Under body Vertical 117 Inner/Lower corner of floor sectionCombination 1(a/b) 344 Close to Under body connection

Combination 2 175 Close to Under body connection

Table 10: Highest calculated stress levels for each load case for the extruded joint with connectiontype B .

Total weight per length of the joint: 5.941 kg/m

The joints influence on the rest of the structure can be summarized as;

• Maximum adhesive stress is 5.7 MPa

• Maximum composite stress is 112/2.5MPa (In fibre direction/Transverse fibre direction)

• Maximum core shear stress is 0.26 MPa

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3.3.4 Not extruded joint with connection type B

Figure 28 shows a close up of the stressdistribution for the joint whens sub-jected to Transverse wall load case.The stress levels presented in Table 11are taken at least two elements awayfrom rigid connections.

Figure 28: Load case Transverse wall give high stresslevels in the connections between the inner beams.

Load case Max stress [MPa] Position in the jointTransverse Wall (Out/In) 283 Connection between inner beams

Vertical Load wall 72 Connection between inner beamsFloor Payload 92 Connection between inner beams

Under body (Out/In) 161 Upper connection of beam 5Under body Vertical 148 Lower part of load carrying armsCombination 1(a/b) 281 Connection between inner beams

Combination 2 124 Intersection Lower floor flange

Table 11: Highest calculated stress levels for each load case for the extruded joint with connectiontype B .

Total weight per length of the joint: 9.597 kg/m

The joints influence on the rest of the structure can be summarized as;

• Maximum adhesive stress is 15 MPa

• Maximum composite stress is 112/18 MPa (In fibre direction/Transverse fibre direction)

• Maximum core shear stress is 0.17 MPa

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3.3.5 Longitudinal compression stress analysis

Longitudinal compression stress analysis is only performed for the extruded joint with connectiontype B due to modeling issues. Figure 29 shows models used for the Longitudinal compressionstress analysis.

(a) Area studied in 2d model. (b) Relative displacements im-ported into 3d model.

Figure 29: Figure a shows 2d model subjected to longitudinal compression and Figure b showsdisplacements for 3d model. Displacements in Figures are exaggerated.

The results for the Longitudinal compression stress analysis can be summarized as;

• Maximum joint stress 41 is MPa

• Maximum adhesive stress is 10MPa

• Maximum composite stress is 49/7.6MPa

• Maximum core shear stress is 0.17MPa

4 Conclusion

The steel model is originally not modeled with an analysis of this type in mind resulting ina model where the results are hard to interpret. Most components are connected with rigidelements and this in combination with a relatively large element size means that the actualstress levels for some areas can be hard to read out. However, the stress results presented isquite close to the assumed maximum stresses since this is a steel structure. The weight of thesteel model joint should not be directly compared to the other joints for two reasons. Firstof all, the geometry of this joint is totally different from the later design joints which makeit hard to decide which components that should be considered as parts of the joint. Second,the joint in the steel train is designed to account for other loads for example longitudinalcompression. The two extruded joints for the sandwich car does only differ marginally in totalweight. The first one, with connection type A shows lower stress levels for the load casesinvolving transverse wall bending but shows higher stress levels when subjected to vertical loadson the under body structure compared to the other extruded joint. The weight of the notextruded joint is considerably higher than that of the other two. As a result the average stresslevel is lower and maximum stresses appear more often in the cross beams and not in closeconnection with the under body. The highest stress measured (minimum two elements awayfrom rigid connectors) in any joint is 344 MPa so it should not be a problem to find aluminumcapable of withstanding this stress level. The influence on the surrounding structures seems tobe acceptable, however, the stress in the adhesive connection of the not extruded joint is very

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high. This occurs in the intersection of the inner beams and the part of the joint going intothe sandwich panel. Based on the results from this analysis, the extruded joint that is beingoverlapped by the sandwich faces is the preferred one. The weight of this is the lowest of thethree joints compared, the stress levels are within acceptable limits and the cross-section doeshave lower length to thickness ration of the individual sections than the other extruded joint(less prone to buckle). The open flange facing the underbody also means that a traditionalbolted connection could be used.

5 Discussion

The first thing that is conspicuous is that the weight of the not extruded part is heavier than theother two. This seems odd since the methodology used for all joints are the same but with extraconstraints for the extruded joints. The reason for this is probably that the re-design from thetopology optimization to 2d shell elements is not good enough. This highlights the main problemwith this analysis. Even though the method used seems reasonable, every step needs to be donewith greater accuracy if one should use this as a design method. Determination of load cases,boundary conditions and optimization constraints, all have immense impact on the calculatedthickness of the different sections and as a result also the stress levels. One thing that shouldalso be noticed is that the rigid connection between the joint and under body structure is alsoan important factor why the stress level results should be questioned. The stress levels in theelements connected to rigid elements are much higher than the presented values two elementsaway. However the stress levels one additional element away is usually much lower than thepresented value. To design a complete joint, a much more detailed connection between the jointand the under body should be modeled. One other thing that should be investigated furtheris buckling. Since the aluminum sheets in the joints are thin they would most likely buckle.This is not included in the optimization since this requires a totally different approach of theoptimization set up. This needs to be considered when comparing the weight results. To stopthe sections from buckling, thickness or stiffeners could be added. All those factors make it hardto decide if on one joint that is better than the other. The impression is that the weight savingthat possibly could be gained by not making an extruded part would not be enough to justify amore difficult manufacturing process. When choosing between the extruded ones the next stepwould be to investigate the manufacturing process of those to see what boundary conditions forthe thickness that should be used an run a new size optimization. In a new optimization thebolted connection should also be modeled with higher accuracy.

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Appendix A

This appendix shows the boundary conditions for all load cases.

Figure 30: Boundary conditions used for Transverse wall load case.

Figure 31: Boundary conditions used for Vertical wall, Underbody vertical and combination 2load case.

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Figure 32: Boundary conditions used for Under body transverse load case.

Figure 33: Boundary conditions used for combination load case 1a and 1b.

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Figure 34: Boundary conditions used for Payload load case.

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Appendix B In this appendix the method used to implement the forced displacements

used in load case Longitudinal compression is presented. The displacements originally comesfrom a 2d model of the full sandwich train car subjected to the compression load case shown inFigure 35. Those displacements are then transferred to the edges of the 3d model.

(a) Train car subjected to Longitudinal compressionload case. Deformations are exaggerated.

(b) Path from which the displacements are gathered.

Figure 35: 2d model subjected to load case to Longitudinal compression.

Since the element size of the 2d model does not match the 3d model, the relative dis-placements are plotted against the position along the path shown in Figure 35b. The relativedisplacements are plotted in x,y and z direction. The in MatLab built-in function Basic fit isused to find fitted equations for displacement along the different sections of the path in a leastsquare sense. The equations are then used to give interpolated values of the displacement in be-tween the nodes. The relative displacements are the imported into the 3d model. The downsideof the method is that it gives similar displacements through the thickness for the 3d model.

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References

[1] David Wennberg. Light-Weighting Methodology in Rail Vehicle Design through In-troduction of Load Carrying Sandwich Panels. Licentiate Thesis in Railway Tech-nology. Stockholm, Sweden 2011

[2] David Wennberg. A Light Weight Car Body for High-Speed Trains, Literaturestudy. Stockholm, Sweden 2009

[3] Norman E. Dowling. Mechanical Behavior of Materials, Third edition, PearsinEducation Ltd, London 2007

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