Chapter 4 Design and Analysis of the Proposed Composite...

Chapter 4. Design and analysis of composite post and related structures 53

Chapter 4

Design and Analysis of the Proposed Composite

Post and Related Structural Elements

In the previous section, the analysis of the steel post was presented. The present section differs

greatly from the previous in that the proposed composite post must be designed before any

type of analysis treatment can be done. In relation to this project, the original steel post has a

particular geometry, material, and boundary conditions which are unchangeable, that is to say

this post has been designed previously and all that is required in this project is an analysis of the

structure. This analysis brings the project up to the present point where the determined

behavioural response is utilised in the initial design of the proposed composite post.

The chapter defines the geometry of the composite post, its structural elements, and a number

of laminate combination sub-models to be analysed. The analysis is carried out using the FEM

with the behavioural response magnitudes of the original post, in particular displacement,

being implemented as the constraints imposed on the proposed post model. The authenticity

of the FEM model is confirmed by semi-analytic analyses which include CLT and a smeared

approach. Additional loading scenarios are also analysed in the chapter while although

conservative, they represent loading issues specific to the proposed structure’s service

environment. Finally, a design and analysis of a steel resistant moment base is realised. The

objective of this related structure is to provide the constrained boundary conditions necessary

to enable the post to execute its service requirements.

4.1 Problem Data

4.1.1 Geometry

While the overall geometrical constraints are maintained, the geometry of the proposed post

differs slightly from the original steel geometrical configuration so as to incorporate general

fabrication methods of structures made from composite materials. The original steel structure

is composed of two U-section beams (UPN) fixed together by plates. The plates are connected

by butt-welding. This specific geometry creates problems if it is to be directly applied in

composite materials. A more efficient geometry in terms of fabrication and joining was

Chapter 4. Design and analysis of composite post and related structures

therefore examined and implemented in the composite model.

geometrical configurations presented in this project. The only difference geometrically between

the two is the inclusion of hole

from this, the overall dimensions are the same in both models.

Within each model there lies a second variation. Each model is split into two sub

are a result of distinct lamin

contains a ply mix of woven fibres (weave) and unidirectional fibres (tape) while the second

sub-model (B) in each model contains plies of unidirectional fibres only. These two different ply

types are described more extensively in Section 4.1.

schematic representation of the two different types of models and their subsequent variations

within due to different laminate composition. In total there are four post designs and will be

denoted by their number/letter comb

composed of laminates containing unidirectional fibre

Figure 4.1: Four different designs of proposed composite post

The new structure consists of four separate structural components

U-profile beams and two identic

through adhesion. The beam flanges are fixed to the inside of the plate surface and are aligned

along the plate edges so as to converg

the second model type are contained in the plate component (mechanized post

incorporated in the free surface

the UPNs. Figure 4.2 shows firstly the four components (including holes in plates of model 2)

separately but in their correct orientation

form the complete post. The diameters of the holes are contained within the area o

not fixed to the beam flanges and appropriately reduce along the leng

the top.

2 Model Types

of composite post and related structures

therefore examined and implemented in the composite model. In all, there are two types of


the two is the inclusion of holes or cut-outs in the width of the post in one of the models. Apart

from this, the overall dimensions are the same in both models.

Within each model there lies a second variation. Each model is split into two sub

are a result of distinct laminate configurations for both. The first sub-


model (B) in each model contains plies of unidirectional fibres only. These two different ply

re extensively in Section 4.1.3 of Materials. Below



denoted by their number/letter combination, e.g. 1B is the post structure

of laminates containing unidirectional fibres only.

: Four different designs of proposed composite post

of four separate structural components which include two identical

profile beams and two identical plates of varying width. The components


along the plate edges so as to converge towards the top of structure. The holes which define

the second model type are contained in the plate component (mechanized post

incorporated in the free surface of the plate, i.e. the plate surface area not directly bonded to

firstly the four components (including holes in plates of model 2)

separately but in their correct orientation and secondly the components joined together to

. The diameters of the holes are contained within the area o

not fixed to the beam flanges and appropriately reduce along the length of the post towards

1) No Holes

1A) Weave + Tape

1B) Tape

2) Holes

2A) Weave +Tape

2B) Tape

54

In all, there are two types of


outs in the width of the post in one of the models. Apart

Within each model there lies a second variation. Each model is split into two sub-models which

-model (A) of the two


model (B) in each model contains plies of unidirectional fibres only. These two different ply

3 of Materials. Below in figure 4.1 is a



st structure with no holes

: Four different designs of proposed composite post

which include two identical

components are fixed together


s the top of structure. The holes which define

the second model type are contained in the plate component (mechanized post-curing) and are

of the plate, i.e. the plate surface area not directly bonded to

firstly the four components (including holes in plates of model 2)

and secondly the components joined together to

. The diameters of the holes are contained within the area of the plate

th of the post towards

1A) Weave + Tape

1B) Tape

2A) Weave +Tape

2B) Tape


A box-type section for the structure was decided upon instead of using an open-type (I-type)

section as its sectional configuration have a more effective resistance to torsion than the open-

type section as all its parts at the periphery of the box section are connected to one another.

Secondly, while no torsional conditions are applied in the project data, the box section is a

design variable that is maintained from the original steel post structure.

Figure 4.2: Separate substructures and complete post

A detailed geometric representation of the structure is shown in figure 4.3. The thicknesses of

the UPNs and plate elements have been omitted as they vary with different types of layer

configuration of the laminate. In this report, there are three distinct types of composite

configurations examined. As with the original steel structure, the geometric sectional

boundaries are identical so as to incorporate the cantilever and the catenary elements without

making any additional adjustments at their connections. The structure stands at a height of 7 m

which is actually 1 m shorter than its steel counterpart. This is due to the distinct boundary

conditions of the two which are described in detail later. The structure’s width varies from 460

mm at its base to 200 mm at the top. The thickness of the post is 140 mm.


Figure 4.3: Dimensions of composite post structure

4.1.2 Boundary Conditions and Loads

The two moment cases of the original steel post outlined in Section 3.1.2 are generated by the

wind pressure exerted onto the post’s surface in directions perpendicular and parallel to the

rail line. The pressures calculated from this section for both load cases provide an approximate

representation of the wind loading that is also applied onto the free surface of the composite

post structure. The pressures for both cases are again outlined below in table 4.1, recalling that

the difference in magnitudes between both cases is assumed to be due to the additional

loading exerted by the catenary assembly in Case 1.


Pressure Load Perpendicular to line

(Case 1)

Parallel to line

(Case 2)

P (N/m2) 3072.886 2188.072

Table 4.1: Pressure P for cases 1 and 2

As outlined in the geometry, the proposed composite post differs from the original steel post in

terms of its method of fixation. While the original post of overall length of 8 m is embedded in a

concrete foundation of a depth of 1 m, the proposed post is instead fixed onto the top of this

same type of concrete pad by a steel moment resistant base as shown in figure 4.4. The overall

height of the proposed post is therefore reduced to 7 m with the bottom 200 mm fixed in the

moment resistant base. For simplification purposes of the composite model this bottom section

of 200 mm is treated as completely fixed, i.e. translations and rotations impeded. A separate

analysis of the combined base and post is carried out later in Section 4.8.

Figure 4.4: Composite post (blue) fixed to steel moment resistant base (grey)

As well as being the overall lighter structure of the two, the composite post with holes

experiences a reduced resultant force R for load Case 2, in comparison to the composite post

with no holes, due to its reduced surface area. However, this reduced surface area

consequently reduces the moment inertia (about the y-axis) and therefore its mechanical

resistance is negatively affected. This effect is analysed in more detail in the second part of

Section 4.1.4. As previously stated in Section 3.1.2, the additional loading effect due to the

catenary assembly is considered as a separate numerical analysis in Section 4.7.2.


4.1.3 Materials

The post is a composite based structure that is composed of two principal components, the

fibre and the matrix. The fibre component consists of a carbon based material. There are two

processes in which to fabricate the carbon fibre, they include the PAN (polyacrylonitrile) and

pitch process. The principal objective of the fibre component is to support approximately 70%

to 90% of the applied loading, and to provide mechanical resistance and stiffness to the

material. The matrix component consists of a polymeric resin which in this case is epoxy. The

objective of the epoxy is to give cohesion and maintain the fibre’s principal direction(s),

transmit loads to the fibres, protect the fibres from environmental degradation and provide

shear resistance between plies. There are two types of carbon/epoxy composite plies utilised in

the analysis depending on the model type, they are defined, with their thickness included, in

table 4.2 below.

Ply type Thickness (mm)

1 Woven fibres (weave fabric) 0.280

2 Unidirectional fibres (tape) 0.184

Table 4.2: Ply types and their thicknesses incorporated in laminates

The first is a carbon fibre weave which contains the same quantity of fibres in orthogonal

directions i.e. 0o and 90

o angles. The weave thereby gives equal mechanical resistance in the

orthogonal angles but consequently the maximum resistance that can be achieved is decreased.

The second type of composite utilised in the model a unidirectional fibre ply composite. The

unidirectional fibre provides a high mechanical resistance in one direction only of the ply or

lamina and an extremely limited resistance in the direction perpendicular to the fibre.

As previously outlined, sub-model A of both models is composed of a combination of woven

and unidirectional plies with the bulk of the structure to be composed principally of the weave

composite while the unidirectional composite is to be incorporated in areas/directions of high

tension and large displacements. Sub-model B of both models is composed solely of

unidirectional plies. To account for the limited transversal mechanical resistance of the

unidirectional ply, the laminate configuration requires off-axis and transversal orientation of

the ply to be included.

In the both types of composites utilised the resin introduction during ply fabrication is the

same, which is that the resin component is impregnated into the lamina in a fresh/uncured

state and therefore requires a system of curing to gain consistency [3], [9].


4.1.3.1 Elastic Properties

The material properties for both composites to be inputted in the model are orthotropic. The

state of plane stress in the ply causes the stress associated with the transversal direction to be

equal to zero ( 031233 === σσσ ) thereby reducing the associated properties in the

compliance matrix S. The relevant elastic properties for both composite orthotropic materials

are listed below in table 4.3. These values are typical elastic constants of carbon/epoxy-based

composites utilised in aeronautical applications and have been taken from an experimental and

numerical analysis of a mixed laminate (weave and tape) cylindrical stiffened composite panel

tested at the TEAMS facility in conjunction with the School of Engineering, University of Seville,

Spain. The elastic properties include: Young’s Modulus in the both in-plane orthogonal

directions (E11 and E22); the shear modulus (G12); and Poisson’s ratio (v12). Both the longitudinal

and transversal elastic properties influence the total stiffness of the composite. The elastic

properties for the two types of composites utilised in the project are as follows:

Property Weave Fabric

Composite Ply

Unidirectional Tape

Composite Ply

E11 (GPa) 61 131

E22 (GPa) 61 9.75

G12 (GPa) 4.9 4.65

G13 (GPa) 4.9 4.65

G23 (GPa) 3 3

v12 0.05 0.3

Table 4.3: Elastic properties of weave and unidirectional composites

4.1.3.2 Mechanical Strength Properties

The strength of the composite takes into account longitudinal, transversal and shear strengths.

These mechanical strength properties include tensile resistance in the direction of the fibres XT,

resistance to compression in the direction of the fibres XC, tensile resistance in the transverse

direction to the fibres YT, and resistance to compression in the transverse direction to the fibres

YC and shear resistance S. These mechanical strength properties, supported by a selected failure

criterion, will allow for the prediction of failure in the structure. There are three predefined

failure criteria in the ANSYS program which include Maximum Strain, Maximum Stress and Tsai-

Wu Failure Criterion. Additional types of failure criteria will be later proposed by the author

through the application of a macro-written code applicable to the ANSYS program. Table 4.4

gives the mechanical strength values for ply types.


Properties Weave Fabric

Composite

Unidirectional Fibre

Composite

MPa MPa

XT 460 2220

XC 435 1300

YT 460 60

YC 435 240

S 155 108

Table 4.4: Mechanical strength properties of weave and unidirectional composites

4.1.4 Ply Orientation and Laminate Design

Unlike materials with isotropic characteristics, composites can include an additional design

variable which is ply orientation. The elastic and mechanical properties outlined above clearly

indicate that the stiffness and strength of a composite depend on the orientation of the fibres.

A laminate’s mechanical capabilities can be varied to account for direction of loading by

changing the fibre orientation of the plies while not affecting the overall thickness of the

laminate. With the provision of using stiffeners in a structure, a higher modulus of elasticity can

be applied to the flange part of the stiffener by concentrating the fibre direction of the plies

along its longitude. It is therefore possible to tailor a laminate to the specific conditions applied

to the problem.

As it has been stated before, the project presents the results of two general types of laminates:

The first is a combination of woven and unidirectional plies; and the second is a laminate

containing unidirectional plies only. The difference in structural behaviour between the two

laminate types is directly associated to the type of ply used, quantity and its orientation.

Both of the types of composites utilised have their own application purposes. The woven fabric

allows easy processing as it is produced in prepreg laminas and can be more readily applied and

wrapped into relatively complex shapes than the unidirectional composite. The matrix of the

unidirectional composite tends to separate from the fibre in its weak transversal direction when

applied to irregular shaped surfaces. The main advantage of the unidirectional fibre composite

is that it is utilised in the directions of large stresses and displacements allowing for maximum

manipulation of strength in the most critically anticipated directions. A number of plies are

orientated off the global axes; these include orientations of 45o and -45

o which increases the

shear strength of the laminate and reduces torsion within the structure.

The configurations of each plate and UPN component for the two models are shown

respectively below in table 4.5 and table 4.6. Within the laminate configuration, each ply’s

orientation is described by the angle (in degrees) written in normal size font. Angles that are


underlined define the weave ply while the remaining angles not underlined define the tape ply.

The subscript number relates to the multiple of the number of plies defined within the bracket

pair. Finally, subscript s implies that the configuration is symmetric.

It is worth noting at this point that all laminates shown are symmetric which implies a reduction

in the number of independent elastic constants from 21 which characterises anisotropic

materials to 13 independent constants which is termed as a monoclinic laminate material. From

viewing the configurations of the laminates in sub-models A (weave + tape) for both models, it

is evident that no 90o plies have been defined in the configuration. Mechanical resistance in the

transversal direction is gained through inclusion of the weave ply as it gives equal mechanical

resistance in the orthogonal angles so by only defining the 0o, it is implied that equal resistance

is attained in its orthogonal angle, which in this instance is 90o. Angled plies of +45

o and -45

o are

defined in equal quantities so as to maintain equivalent proportions of each angle in the

laminate.

On a final observation of the configurations, it is worth highlighting that only the unidirectional

ply is orientated at +/-45o in all laminates, this is due to a fabrication constraint imposed on the

weave ply. During lamination, the post’s substantial length of 7 m causes an issue of ply join-up.

At an angle of 45o, a number of plies are required to be laid-up together at their edges in order

to completely cover the laminate. Plies cannot be jointly laid-up together end to end through

the direction of the fibre as this creates a discontinuity and stress are not transferred through

the entire layer. For this particular reason, the weave ply cannot be jointly laid-up as one of its

orthogonal axes of mechanical resistance would be rendered discontinuous. See Section 5.2 on

the lay-up process for additional information.

Model 1 Substructure Ply Configuration

A Weave + Tape Plate [(0/0)2/(0/45/0/-45)3/(0)2]s

Beam [(0/0)2/(0/45/0/-45)4/(0/0)2]s

B Tape Plate [(0/45/90/-45/(0)4)2]s

Beam [(0/45/90/-45/(0)4/(0/45/90/-45)2/(0)6]s

Table 4.5: Laminate configuration for post with no holes (Model 1)

Model 2 Substructure Ply Configuration

A Weave + Tape Plate [(0/0)4/(0/45/0/-45)3]s

Beam [(0/0)4/(0/45/0/-45)4/(0/0)2]s

B Tape Plate [(0/45/90/-45/(04))2/(0)2]s

Beam [(0/45/90/-45/(04))2/(0/45/90/-45)2/(0)8]s

Table 4.6: Laminate configuration for post with holes (Model 2)


Tables 4.7 and 4.8 presents a summarised view of the design considerations for the laminate

configurations including overall percentage thickness of both the plate and beam components

and percentage of fibres orientated off-axis, or at +/-45o. In the cases of combined weave and

tape laminates, the percentage off-axis laminates is not considered by the number of plies but

rather by the thickness of such orientated fibres. Other data presented include the number of

plies and thickness for each laminate.

The objective of the laminate configuration design is to maintain equivalent proportions (%

thickness and % +/-45o) between both sub-models (A and B) for each model (1 and 2) while also

maintaining equivalent magnitudes of the critical design control, which in this project is the

maximum displacement permitted. While the variation in the quantity of plies/thickness

between sub-models A and B is a result of the different mechanical resistance provoked by

distinct ply combinations, proportionally both sub-models are maintained approximately the

same. In all models, the overall percentage thickness of the plate and the beam are

approximately 42-43% and 57-58%, respectively. A slight difference in the proportion of +/-45o

occurs between both models but is maintained relatively equal for the sub-models A and B. The

slight difference in proportions arises due to the requirement of maintaining +45o and -45

o at

equal quantities while the total number of plies in the laminate varies between both models.

Model 1 Substructure

No. of

plies

Thickness

(mm)

Overall %

Thickness % +/-45o

A Weave + Tape Plate 36 8.352 43 27

Beam 48 11.136 57 26.4

B Tape Plate 32 5.888 42 25

Beam 44 8.096 58 27.3

Table 4.7: Laminate design of post in Model 1

Model 2 Substructure

No. of

plies

Thickness

(mm)

Overall %

Thickness % +/-45o

A Weave + Tape Plate 40 9.28 42 23.8

Beam 56 12.992 58 22.6

B Tape Plate 36 6.624 43 22.2

Beam 48 8.832 57 25

Table 4.8: Laminate design of post in Model 2


4.2 Finite Element Model

The following section relates to the most efficient approach, as regarded by the author, to

create an accurate representation of the newly designed post structure in composite materials.

The following approach considers the most suitable element type, meshing requirements and

application of boundary conditions. Due to the orthotropic nature of the composite materials

analysed in this model and the complexity of the geometry, issues encountered with ply

orientation are emphasised extensively in the following.

4.2.1 Element Type

As in the case of the original structure, ANSYS is the preferred finite element program to be

used for modelling. One type of element is employed in this composite model, namely

SHELL181. Shell elements are designed to efficiently model thin structures. Structures

fabricated from composite materials by stacking methods of plies to create laminates are well

defined by shell elements. The SHELL181 element is a 4-node finite strain shell which is suitable

for analyzing thin to moderately thick shell structures. The element contains six degrees of

freedom which include translations in the x, y and z directions, and rotations about the x, y and

z-axes. Figure 4.5 shows the quadrilateral element with its coordinate direction with respect to

the configuration of the nodes (i, j, k, l). The coordinate system is directly related to the

configuration of the nodes where the z-axes is positive and transversal in an element when the

nodes are defined in an anti-clockwise manner, commencing at node i and moving out from the

x-axes towards the subsequent node j [8].

Figure 4.5: 4-node SHELL181 element

A number of elemental key options settings utilised in this model include applying a full

integration analysis (KEYOPT(3) = 2). This option is recommended as it is highly accurate (even

with coarse meshes) in that its application is well suited to cantilever-type problems which are

dominated by in-plane bending. The second key option setting utilised for this element stores


data for top and bottom for all layers in the laminate (KEYOPT(8) = 1). This permits the user to

obtain more accurate interlaminar results (e.g. stresses and strains) caused by different ply

types and orientations. Data for a specific layer is attained by calling on that layer through the

LAYER command. In relation to this project and briefly outlined in its objectives, this key option

setting enables the author to apply any type of desired failure criteria created in macro-style

parametric language (APDL), contained in the ANSYS program. This process is described in

detail later in Section 4.4. A typical laminate section with distinct interlaminar stress results (σx)

is shown in figure 4.6.

Figure 4.6: Interlaminar stress distribution through laminate section

4.2.2 Model Development Method

The complete post structure is divided essentially into four elemental sub-structures which

include two U-profile beams and two plates joined to the flanges of the beams as shown

previously in figure 1. The model is further subdivided into planar areas as shown in figure 4.7

which are themselves defined by keypoints (KP), lines (L) and areas (AL). For ease of analysis,

the plate area and to the flange area of the beam which are considered adhesively joined

together are considered unique and as one area. As a result, the laminate configuration at this

new unified area includes the laminate configuration of the plate plus the laminate

configuration of the flange.

Laminate Thickness


Figure 4.7: Post segment showing separated planar areas for lamination

The local coordinate system (LOCAL) defines the ply orientation of the laminate in question

with its relevant capabilities shown in figure 4.9. The origin of the local coordinate system is

defined by three points on the global Cartesian coordinate system and its orientation through

rotation of Euler angles. This permits each area to orientate the direction of the composite

fibres independently. Each area is meshed individually or collectively depending on whether or

not two or more areas share (1) the same configuration and (2) orientation. Before meshing of

the area, elemental attributes need to be assigned to that area and are done so by the AATT

command. From here the section information is associated with a section ID number through

the SECTYPE command which also defines the type of element (SHELL) and the name to be

given to that particular section, e.g. the first constructed area is ‘AREA1’. The geometry data

describing this section type is defined by the SECDATA command. It describes the lamina

configuration of the section by the thickness of each shell layer, material type and the angle (in

degrees) of the already-defined local coordinate system of the layer element.

A simplified sketch of the cross section of the structure is shown below figure 4.8. In total, there

are 8 planar areas that make up the outer boundary of the structure which highlighted in red in

the figure below. Their dimensions are equivalent to those of the original steel post, thereby

maintaining the same profile as has been outlined as a project constraint. From this, lamination

of all areas occurs from the areas surface towards the centroide of the cross section. As a

result, each section’s defined shell layers or configuration originates on the plane marked in red

and advances or stacks up in accordance to the sections own local coordinate system. This

method of lay-up is achieved by offsetting each section to the origin of their local z-axis through

the SECOFFSET command. Note that the SECOFFSET command is determined by the node

orientation of the element (i, j, k, l).


Figure 4.8: Section of structure with individual sections according to lamina lay-up direction

Orientation issues arise when attempting to represent laminates in a 3-D model. An example of

this is incurred with the orientation of a single layer about the UPN beam section. As described

in Section 5.5.2 of fabrication, the plies of the beam are laminated onto and around a suitable

mould profile. Three planar surface areas of the mould are to be covered by the layer. Figure

4.9 shows the layer and its fibre orientation before and after lamination. Before lamination and

in the first image, one local coordinate system defines the complet layer. In order to maintain

fibre orientation throughout lamination of a single layer around the three planar surfaces of the

mould the layer needs to be subdived into same number of local axes as the amount of

distinctly orientated planar axes, and in this case are three axes which are shown in the second

image of figure 4.9. Each of the three local coordinate systems are different and are defined as

so through the LOCAL command. While attention is required in defining the local coordinates,

the ply configuration or stacking sequence for the three laminates remain the same.

A complication arises in the stacking sequence due to the simplification of the unique surface

defined earlier between the plate area and beam flange adhered together. Taking the example

in bottom figure again, the top local coordinate system (x1, y1, z1) equivalent to the coordinate

system for the combined top flange/plate area defines this laminate configuration adequately.

However, the combined bottom flange/plate area becomes problematic in that the flange

configuration is adequately defined by the local coordinate system (x3, y3, z3) while the plate

configuration is not. The orientation of the fibres in top and bottom areas of the third image

describes issue. According to both sets of local coordinates in these areas (1 and 3), if an

abitrary layer is defined the same for both plate areas the fibre orientation would be opposite

to each other on the global axes as clearly evident in the final image of figure 7. To overcome

this event, the plate stacking of off-axis plies must be altered by 90o. For example, if a layer in

both the top and bottom plate is orientated 45o on the global axes the top layer, with local

coordinate system (x1, y1, z1), is defined at 45o while the bottom layer is defined as -45

o in

relation to its appropriate local coordinate system (x3, y3, z3).


Figure 4.9: Fibre orientation in laminated UPN beam

In relation to meshing of the structure, both U-profile beams are regular in shape and their

areas that make up the sub-structure are mapped meshed (forming straight-sided elements)

with a specific number of divisions made on the selected lines. In relation plate structures in

Model 2 where holes are incorporated in the plates, meshing requirements are more

specialised. As shown previously in figure 4.7 and below in figure 4.8, the plate components are

divided into 3 areas: 2 of which coincide with the area of the beam flange (blue), and the third

is the remaining area between the two beam flanges (green). This third area shares the same

line number with the two bordering flange/plate areas and thus is divided into the same

number of elements in its longitude so that boundary nodes between the 3 plate areas


coincide. The centre section however, is free meshed so as to incorporate the holes in the

section. Figure 4.10 shows a plane view of a segment of the plate meshed in accordance to the

three described areas.

Figure 4.10: Plate segment with mapped and free mesh

4.2.3 Boundary Conditions and Loads

The boundary conditions and loading applied in the numerical model are representations of the

physical conditions imposed on the structure which are outlined previously in Section 4.1.2.

These boundary conditions include a moment resistant base at a height of 200 mm connecting

the composite post to the concrete foundation pad (figure 4.4) and wind loading for Cases 1

and 2 (table 1). For simplification purposes of the analysis of the post structure, the moment

resistant base is not considered and is replaced by complete limitation of movement of the post

in its equivalent region, i.e. the bottom 200 mm of the post. As previously mentioned, a

separate analysis of the combined base and post is carried out later in Section 3.8. This

simplification is deemed satisfactory and is proven justifiable by the minimal displacement

incurred at the base in the subsequent analysis of the combined base-post. The wind loading is

thus, applied on the remaining free surface length of the post (6.8 m) for both load cases

separately. These conditions are applied as FEM boundary conditions and not as Solid Model

boundary conditions. That is, the conditions are applied directly to the nodes of the relevant

elements whereas in the Solid Model the loads are applied to the structure’s surface, i.e.

keypoints, lines and areas. Applying FEM and Solid Model boundary conditions together at the

solution stage of the analysis is not recommended as Solid Model conditions when being

transferred to nodes or elements may overwrite directly applied loads.

Coinciding

Nodes

Plate areas connected to beam

flange (mapped mesh)

Plate areas with

holes (free mesh)

Holes in plate


4.3 Failure Criteria

Due to the intrinsic anisotropy of composite materials and the existence of multiple failure

modes, i.e. failure of the material at a micromechanical, ply and laminate level, there is great

difficulty in the development of a comprehensive failure theory in which envelopes all cases.

There are numerous criteria developed for the failure analysis of composite materials. These

criteria fall primarily into three categories.

The first contains failure criteria not directly associated with failure modes. These include

criteria of full quadratic interaction such as Tsai-Hill, Tsai-Wu and Modified Tsai-Wu. The second

category contains failure criteria that are associated with failure modes, these include the more

traditional lamina approaches such as Maximum Stress, Maximum Strain, which are non-

interactive limit criteria, and also criteria that attempt to distinguish between fibre and matrix

failures. The most popular of these theories include the partially interactive criteria of Hashin-

Rotem (73), Yamada and Sun, and Hashin (80). There also includes in this category the criteria

of Puck, of which was determined to be the most accurate in the World Wide Failure exercises

[10], [11]. However, Puck’s method requires additional material data which often is relatively

difficult to obtain, and if not available, can be approximated [12]. A more recent theory has

been developed by NASA named the LaRC02 approach which is gaining popularity. Overall, the

data required to execute the non-associative criteria are essentially the same as that of the

associative failure mode criteria. The third category of composite strength prediction is

laminate approaches. This approach does not attempt to define the stress/strain state at

lamina level, but instead for the laminate. This method requires a different set of material data

and will not be examined in this project.

A number of these failure criteria are selected and applied in this analysis so as to ensure an

extensive analysis is made for possible failure modes in composite laminates, and also to

provide an adequate comprehension of the theory behind each of the selected criteria, their

assumptions and issues arising from them, and finally a numerical comparison of their results.

In this project, there are three criteria applied in the post-processing stage of the analysis.

These include the

• Tsai-Wu

• Maximum Stress

• Puck

These three criteria were chosen for a number of reasons. Firstly the three criteria represent a

number of fundamental concepts and assumptions that have been associated with the

evolution and development of composite failure criteria over the past 50 years. As previously

mentioned, they are classified principally into two categories: criteria that distinguish between


fibre failure and matrix failure (direct failure mode) and those that do

failure criteria directly associated with failure modes can be sub

categories: interactive and non

types is the existence (or non

and the influence of the transversal stress component for

describes the formation of the discussed categories and given are the criteria associated with

each type.

Figure 4.11: Categorisation of failure criteria applied in analysis

The second for choosing the above criteria is due to the presence of two types of composite

plies utilised in the design of the struc

(tape) composites. Issues arise with determining an appropriate failure criterion for each of the

two ply types. For instance, the direct

the application to unidirectional composites and not for woven

may therefore be called into question with the application of this criterion for the analysis of

the mixed ply laminates of M

for the application to the wove

criterion of Tsai-Wu [13].

The final reason for choosing these particular criteria is down to the discretion of the author.

The high accuracy of the Puck criterion in the World Wide Exercise a

Maximum Stress criterion in FEM commercial programs and industry contributed to these

criteria inclusion.

In this post-processing analysis of the project, the above criteria are executed by two different

methods. They include analysis through

• Predefined criteria in ANSYS

• Macro-style parametric language in ANSYS (APDL)

Failure Mode

(Criterion)


fibre failure and matrix failure (direct failure mode) and those that do

failure criteria directly associated with failure modes can be sub-divided further into two

categories: interactive and non-interactive limit criteria. The difference between these two

types is the existence (or non-existence) of association between in-plane stress components

and the influence of the transversal stress component for a particular failure mode. Figure 4.11


: Categorisation of failure criteria applied in analysis


plies utilised in the design of the structure, namely woven fibre (weave) and unidirectional fibre

pe) composites. Issues arise with determining an appropriate failure criterion for each of the

two ply types. For instance, the direct-interactive criterion of Puck was originally developed for

the application to unidirectional composites and not for woven composites. Accuracy of results


odels 1B and 2B. Of the three presented, the criterion most suited

for the application to the woven fibre (in-plane stress) is the non-direct, fully inter


The high accuracy of the Puck criterion in the World Wide Exercise and the extensive use of the


processing analysis of the project, the above criteria are executed by two different

alysis through

Predefined criteria in ANSYS

style parametric language in ANSYS (APDL)

Failure Mode

Non-Direct

(Tsai-Wu)

Direct

Non-Interactive

(Max. Stress)

Interactive

(Puck)

70

fibre failure and matrix failure (direct failure mode) and those that do not (non-direct). The

divided further into two

interactive limit criteria. The difference between these two

plane stress components

a particular failure mode. Figure 4.11


: Categorisation of failure criteria applied in analysis


) and unidirectional fibre

pe) composites. Issues arise with determining an appropriate failure criterion for each of the

interactive criterion of Puck was originally developed for

composites. Accuracy of results


odels 1B and 2B. Of the three presented, the criterion most suited

direct, fully interactive


nd the extensive use of the


processing analysis of the project, the above criteria are executed by two different

Interactive

(Max. Stress)

Interactive

(Puck)


Of the three failure criteria to be applied in the analysis only the Maximum Stress and Tsai-Wu

criteria are readily available in the ANSYS FEM program. As a result, the third criterion (Puck) is

written as a macro file in APDL (ANSYS Parametric Design Language) and is executed in the

post-processing stage of the analysis. As stated previously, this parametric script coding is

described in detail in Section 4.4. For validity purposes of the macro-written criteria, the

Maximum Stress and Tsai-Wu criteria are also written in APDL and compared with failure

results of the predefined equivalent criteria in ANSYS.

All failure criteria results are presented as failure index values where a value greater or equal to

one signifies failure.

4.3.1 Tsai-Wu

The theory behind the Tsai-Wu criterion (1971) is a simplification of the Tsai-Hill criterion for

generalized failure theory of anisotropic materials. There are two forms of the Tsai-Wu failure

criterion presented in the predefined ANSYS failure analysis [8], [5]. The first form of the

criterion is the ‘strength index’ or TWSI which expressed as the following.

BAIF += (4.1)

The second form presented is the inverse of the ‘strength ratio’ (TWSR) given as

12

122

1−

+

+−==AA

B

A

B

RIT

(4.2)

where, in the 2-D case of plane stress, A is

( ) ( ) ( )

CTCTCTCT YYXXc

SYYXXA 21

12

212

22

21 σσσσσ +++= (4.3)

where c1 is a coupling coefficient of the Tsai-Wu theory which by default is taken to be -1. The B

term is defined as

222122

1111 σσ

−+

−=

CTCT YYXXB (4.4)


4.3.2 Maximum Stress

As previously mentioned, the maximum stress criterion does not consider any interaction

between stress/strains acting in the lamina and dictates failure to occur when the stress in any

direction exceeds the strength in that direction. As a result, this type of criterion under-predicts

the strength when combined in-plane stresses are acting on the composite. This type of

criterion is a simple, straightforward way to predict failure of composites. This more traditional

criterion predicts no material fracture, for a state of tension, occurs if:

TX<11σ ( )011≥σ (4.5a)

TY<22σ ( )022 ≥σ (4.6b)

S<12σ (4.6c)

And for a state of compression if:

CX<11σ ( )011<σ (4.7a)

CY<22σ ( )022<σ (4.7b)

Where one or more of the inequalities are not met fracture occurs in the material according to

the mechanism associated to that equation in which the inequality has not been met [5].

4.3.3 Puck

The Puck failure criterion is one type of criterion that is associated with failure modes. The

criterion distinguishes between fibre failure (FF) and inter-fibre failure (IFF). In relation IFF or

failure of the matrix, the in-plane failure parallel to the fibres is governed by the three stress

vector components associated with that plane which include the normal stress acting on the

plane, and two planar stresses, one acting parallel and the other perpendicular to the fibre. The

criterion proposes that the two shear stresses always promote fracture, while the normal stress

only promotes fracture if it is in a traction state and has the adverse effect in compression, i.e.

it impedes fracture [12].

IFF contains three distinct modes of failure: mode A is when perpendicular transversal cracks

appear in the ply under transversal tensile stress with or without in-plane shear stress; Mode B

also implies transversal cracks, but are a caused by in-plane shear stress with small transverse

compression stress; and finally Mode C denotes oblique cracks onset (typically of angle 53o in


carbon/epoxy composites) when the material is under significant transversal compression. The

FF yields one failure index which assumes that fibre failure only depends on longitudinal

tension [14]. It is defined as

TX<11σ ( )011≥σ (4.8a)

CX<11σ

( )011<σ (4.8b)

The three failure modes in the IFF imply that it yields three separate index failures. The

particular IFF mode to be activated for analysis depends on the stress state of the lamina in

question. For instance, mode A is activated if the transversal stress is positive. It is expressed as

11 212

2

2

1212 =+

−+

Sp

YS

Yp

S TT

T

T σσσ

( )022 ≥σ (4.9)

where T

p12 is the slope of the failure curve for 022≥σ at the point 022=σ , also known as a

fitting parameter. Without experimental values, the parameter is assumed to be 0.3 which is

representative of carbon based composites. In relation to negative transverse stress 022<σ ,

either Mode B or Mode C is activated, depending on the relationship between in-plane shear

stress and transversal stress. The selection of either mode is defined by the limit of the relation

YA/SA where

−+= 121

2 1212 S

Yp

p

SY C

A C

C

(4.10)

C

pSS A 221+= (4.11)

and

A

A

S

Ypp

CC 122 = (4.12)

where C

p12 is another fitting parameter but which corresponds to the interlaminar shear

strength S. A value of 0.2 is selected to represent this parameter [14]. The failure index and its

limitation YA/SA for Mode B are defined as


( ) 11

2122

2122

12 =

++ σσσ

CCpp

S

≤

<

A

A

S

Y

12

2

22 0

σσ

σ

(4.13)

and for Mode C

( ) 112

2

2

2

2

12

2

=

+

+−

C

C

YSp

Y

C

σσσ

≥

<

A

A

S

Y

12

2

22 0

σσ

σ

(4.14)

4.4 Macro Modelling: ANSYS Parametric Design Language (APDL)

APDL is a scripting language in ANSYS that permits the user to automate common tasks and

even build models in terms of parameters or variables. APDL includes a wide range of features

including repeating commands, inclusion of separately constructed macros, if-then-else

branching, do-loops, and scalar, vector and matrix operations. In the presented model, APDL is

utilised in the post processing phase. It is used to determine the mode of failure within the

composite laminate, concentrating on the most critical areas of the structure, by employing the

failure criteria outlined previously in Section 4.3. In this case, APDL allows the user a more

detailed and comprehensive view of the possible failure incurred in the model than that given

in the program’s own failure criteria as a step by step analysis can be performed from the

constructed APDL code by the user. Also it is possible to determine the most critical failure

mode of the criteria. An overview of the code is made subsequently.

The areas considered most critical in the structure are chosen for failure analysis. These include

the plate areas containing the mechanised holes. It is well known that the most critical stress in

a cantilever beam subjected to a uniformly distributed load over its free length occurs at the

proximities of the fixed end of the structure and more so in areas of change in geometry where

stresses tend to accumulate i.e. stress concentration around the mechanised holes. Hence, the

most critical areas were selected and examined accordingly. This reduction also would reduce

the calculation time involved as a lesser amount of data would be associated in the post

processing analysis.

As with the failure criteria predefined in ANSYS, nodal result data is also utilised in the APDL

failure analysis. The APDL macro analysis is based upon selecting specific components (i.e.


areas) for time-reduced, localised failure analysis. The failure analysis is carried out by using

vector functions and *DO loops of nodal data for each layer of the laminate. Figure 4.12 shows

the manner in which the nodal results are stored in the matrix. Extrapolation of elemental

results rather than nodal results is more facilitated to this type of written parametric analysis

due to the fact that elements of the selected area would have maintained its consecutive

numeric order during merging while the nodal numeric order of the selected area(s) can vary

due to the merging of the boundary nodes at adjoining areas. However, initial APDL macro

analysis which utilised elemental results was deemed to be outside a tolerable difference for

comparison purposes with those same criteria results yielded in the predefined ANSYS failure

analysis which uses nodal data. To avoid such differences, nodal result data would have to be

obtained and utilised in the failure criteria written in APDL. The following is the process

developed by the author to obtain nodal result data from the analysis, store it and call it for the

APDL macro.

As a result of the inconsecutive numerical order of the nodes in the selected areas caused by

merging, the required nodal result data is obtained and coherently recalled for the failure

analysis through the following steps. Information from the model is retrieved through the *GET

command. *VGET has a similar functionality, but it act upon an array rather than a single

parameter which is a more rapid form of obtaining information than by looping the single

parameter command. Because *VGET acts upon a vector, *VMASK typically is used in

conjunction with *VGET. *VMASK is a masking array which tells ANSYS to perform vector

operations only on certain items in the array, which in this case are the nodes of the selected

areas. The desired areas are selected by the ASEL command and the nodes associated with

these areas (NSLA) are then selected. The objective of this part of the macro is to store the

required nodal results for all layers in a matrix array. The matrix dimensions are determined by

retrieving the number of nodes in the structure (rows) and defining the number of layers in the

equivalent laminate in the selected area(s) (columns). An extra column is defined in the matrix

which is occupied by the node number for that particular row. After the masking vector has

been applied at the stage of retrieving the nodal results, the node number will be retrieved also

thereby insuring the data for each row of the matrix are defined by their appropriate nodal

number. This coherent method of defining the rows of data by their equivalent node number is

vital for retrieving information such as the location most critical in the particular failure

analysis. Figure 4.12 shows the manner in which the nodal results are stored and their

equivalent number.


Figure 4.12: Matrix layout for nodal stress results of each layer

Below is a segment of the macro that defines the number of layers (36) in the laminate, the

maximum amount of nodes in the structure, the relative areas to be selected and their

associated nodes.

*SET,NUMPLY,36 !36 LAMINAS *SET,NUMCOL,37

*GET,NMAX,NODE,,NUM,MAX

!SELECCIONAR AREAS: PLATES ASEL,S,AREA,,12 ASEL,A,AREA,,16

NSLA,S,1

At this point, the nodal stresses are to be retrieved from the model’s results database. This is

done, as previously mentioned through the *VGET command. To use *VGET with masking, the

following steps need to be performed:

1. Define the masking vector by its dimensions with the *DIM command

2. Define the regular array to hold results of interest with the *DIM command

3. Fill the masking vector with the selected nodes *VGET,,NODE,1,NSEL

4. Activate the masking array with *VMASK

5. Fill the regular array with the nodes selected using *VGET

The following is a continuation of the macro above where the previously outlined steps are

executed to obtain the nodal stress data in the x-direction of the model’s global axes. The script

below combines vector function and *DO loop capabilities where the function commands

*VMASK and *VGET are repeated for each layer of the laminate through the LAYER command

thereby only obtaining data for each layer of the selected areas and leaving the rest a null. This

segment of the macro is repeated for obtaining stresses in the y and xy directions.


*DEL,SXMASK *DEL,SXARRAY

*DIM,SXMASK,ARRAY,NMAX *DIM,SXARRAY,ARRAY,NMAX,NUMCOL

*VFILL,SXARRAY(1,1),RAMP,1,1

*VGET,SXMASK(1),NODE,1,NSEL !GET SELECTED NODES

*DO,j,1,NUMPLY,1

LAYER,j *VMASK,SXMASK(1) *VGET,SXARRAY(1,j+1),NODE,1,S,X

*ENDDO

The results are directed to a file by the /OUTPUT command. The masking command is once

again applied here so as to preserve the file for only information of the selected nodes.

*MWRITE writes the obtained results to the file in a formatted sequence. Below is the segment

which writes the stresses (x-direction) into a file. The /NOPR command suppresses the

expanded interpreted input data listing. This command reduces the file to the leave only the

matrix of results which is directly applicable to the failure criterion analysis part of the macro.

The precision of results is handled by the FORTRAN format F contained in the brackets.

/NOPR /OUTPUT,SXARRAY,FILE *VMASK,SXMASK(1) *MWRITE,SXARRAY(1,1),,,,JIK (37F12.6) /OUTPUT

The failure criteria analysis initiated by recalling the result data from the previously written files

through the *VREAD command. The array is defined by its dimensions with the *DIM

command. The process of reading a specific results file (σx) is described below.

*DEL,SX *DIM,SX,ARRAY,NODOS,NUMCOL *VREAD,SX(1,1),SXARRAY,FILE,,JIK,NUMCOL,NODOS (37F12.6)

Before the criterion can be applied, the stresses must be rotated to their principal directions. In

order to rotate the stresses from the orientated global coordinates to the principal axis (1,2)

the angle (in degrees) of each layer needs to be defined in the macro by using the *VFILL

command. The angles are then converted into radians. This is realised by using an APDL vector

operation command *VOPER where the vector of angles is multiplied by its radian equivalent.

Again, this vector operation reduces the calculation time that would be incurred if a do-loop


process was to be used instead. The transformation of orientated stresses to principal stresses

is performed by the following matrix expression.

−−

−=

xy

y

x

σ

σ

σ

θθθθθθ

θθθθ

θθθθ

σ

σ

σ

.

sincossin.cossin.cos

sin.cos2cossin

sin.cos2sincos

22

22

22

12

2

1

;

[ ]

=

xy

y

x

T

σ

σ

σ

σ

σ

σ

.

12

2

1

(4.15)

where T is the matrix of transformation. Resolving the transformation above, the stresses in the

principal axis in plane stress are.

θθσθσθσσ sin.cos2sincos 221 xyyx ++=

θθσθσθσσ sin.cos2cossin 222 xyyx −+=

( )θθσθθσθθσσ 2212 sincossin.cossin.cos −++−= xyyx

(4.16)

where sine and cosine of each angle is carried out by another vector operation (*VFUN). These

three equations above are incorporated in the macro in the form of a do-loop process which

incorporates the off-axis stresses for each node (i) of all layers (j) at their respective angle of

orientation θ (j).

At this stage of the macro the principal stresses have been calculated for each layer of each

element and the failure criterion can now be applied. Below is shown a segment of the

Maximum Stress criterion in which it determines if failure occurs due to breakage of the fibre

caused by traction. As this part of the criterion determines the possibility of failure caused by

traction, it therefore implies that nodal stress results need to be separated in terms of their

state (in compression/tension) before they can be applied to the criterion. The separation is

achieved by an if-else statement which simply defines the stress state as being negative or

positive and is then applied accordingly to the criterion in question. For example, in the

criterion of failure of the fibre in tension the if-else statement utilises the stresses greater than

zero i.e. tensile stresses, and set the compressive stresses as null.

All of the three criteria applied in the APDL macro have been written so as to locate whether

failure occurs in each of the mechanisms and in the event of failure occurring in two or more

mechanisms, the script would locate in which of the mechanisms failure would occur initially.

After determining the primary mechanism of failure the script then relays to the user the most

critical node and layer within the laminate.


!----------------------------------------------------------- ! MAXIMA TENSION: TRACCION EN DIRECCION DE LAS FIBRAS (Xt) !-----------------------------------------------------------

*DEL,S1_TRAC *DEL,VALORCRIT_TRAC1 *DIM,S1_TRAC,ARRAY,NODOS,NUMPLY

*DO,i,1,NODOS,1

*DO,j,1,NUMPLY,1 *IF,S1(i,j),GT,0,THEN S1_TRAC(i,j)=S1(i,j) *ELSE S1_TRAC(i,j)=0 *ENDIF *ENDDO

*ENDDO

VALORCRIT_TRAC1=S1_TRAC(1,1)/XT LAMINACRIT_TRAC1=1 NODOCRIT_TRAC1=1

*DIM,TRAC1,ARRAY,NODOS,NUMPLY

*DO,i,1,NODOS,1

*DO,j,1,NUMPLY,1 TRAC1(i,j)=S1_TRAC(i,j)/XT *IF,TRAC1(i,j),GE,VALORCRIT_TRAC1,THEN VALORCRIT_TRAC1=S1_TRAC(i,j)/XT LAMINACRIT_TRAC1=j NODOCRIT_TRAC1=SX(i,1) *ENDIF *ENDDO

*ENDDO


4.5 Results

Tabulated below are the relevant results of the behavioural response for both models. Table

4.9 and 4.10 present the nodal stress results in global coordinates for Model 1 and Model 2,

respectively. Nodal stress results include stress in the directions of the global axes (σx, σy, σz)

and shear stresses with respect to the two planes in which laminates of the structure lie (σxy,

σxz). Both tables are subdivided by the type of applied load case with also results shown for

both laminate composition sub-models A (weave + tape) and B (tape). Both traction (T) and

compression (C) stresses are tabulated for each stress component. The maximum displacement

experienced in each load case is tabulated with their respective displacement direction.

σx σy σz σxy σxz Displacement

Model 1 MPa MPa MPa MPa MPa mm

T C T C T C T C T C

Case 1: Wind load perpendicular to rail line

A (Weave +Tape) 5.952 5.922 1.168 1.167 0.975 1.079 0.154 0.003 0.074 0.075 6.432 (Uy)

B (Tape) 13.279 13.210 0.150 0.152 0.190 0.191 0.247 0.005 0.103 0.102 6.500 (Uy)

Case 2: Wind load parallel to rail line

A (Weave +Tape) 24.267 24.229 4.529 4.678 4.139 4.815 0.737 0.740 0.002 0.895 54.470 (Uz)

B (Tape) 52.39 52.678 0.959 0.848 0.6774 0.767 0.978 0.978 0.007 0.976 53.439 (Uz)

Table 4.9: Model 1 (no holes) nodal stress and displacement results (global coordinates) for Case 1 and 2

σx σy σz σxy σxz Displacement

Model 2 MPa MPa MPa MPa MPa mm

T C T C T C T C T C

Case 1: Wind load perpendicular to rail line

A (Weave +Tape) 5.2967 5.3872 1.896 1.647 0.763 0.812 1.464 0.498 0.109 0.104 5.482 (Uy)

B (Tape) 12.066 12.155 1.516 2.793 0.160 0.161 2.778 0.948 0.118 0.112 5.881 (Uy)

Case 2: Wind load parallel to rail line

A (Weave +Tape) 46.005 46.355 9.9549 11.949 8.382 7.745 6.640 5.507 0.000 1.313 57.368 (Uz)

B (Tape) 83.368 81.818 8.0318 6.319 0.867 0.856 12.381 8.617 0.001 1.185 58.202 (Uz)

Table 4.10: Model 2 (holes) nodal stress and displacement results (global coordinates) for Case 1 and 2

The laminate configurations of tables 4.5 and 4.6 and their design formats of tables 4.7 and 4.8

are directly associated with the results of the above tables. In terms of the design of the

laminates of each model, the principal constraint imposed was the magnitude of maximum

deflection permitted for each type of structure. These constraint values were taken from the

analysis of the original steel post where equivalent loading conditions were applied. As a result,

displacement correlation is the means by which the composite models are designed in this


project. The displacements shown in the above table of results confirm this type of correlation.

Values of displacement for the set of sub-models A and B, in their respective case types, are

approximately equal with the largest variation between a sub-model set being just over 1 mm.

However, while the displacements of sub-model sets are equal, differences emerge in terms of

their maximum stresses experienced. With regards to the stress maximums along the direction

of the beam caused by bending (σx), each sub-model set demonstrate differences in magnitude

of approximately 100% between the A and B laminate types. Such variations in magnitude are

caused by variations in

1. Laminate stiffness

2. Sectional inertia

Laminate sub-model A for each case is composed of plies of woven and unidirectional fibres

whereas sub-model B is entirely compose of plies of unidirectional fibres. The differences in the

elastic constants of both types of plies can be appreciated by recalling table 3. It must be noted

that the elastic modulus is not the same as stiffness. It is instead a property of the constituent

material whereas stiffness is a property of the laminate structure. That is to say, the modulus is

an intensive property of the material whereas stiffness is an extensive property which is

dependent on the material, ply orientation, shape and boundary conditions of the structure. As

a consequence, the higher stiffness of the laminate B in the direction of the beam length

induces an increased load transfer and higher stress magnitude than that of the mixed ply

laminate of A. In order to maintain a comparable maximum displacement magnitude in the

lower stiffness laminate A to that of B, two approaches need to be performed: the first is by

adding additional woven plies to the laminate thereby increasing the inertia of the section to a

certain extent and the second is orientating unidirectional fibres along the direction of the

beam’s length.

Figures 4.14 and 4.15 show the stress variation in the x-direction σx around post section of

Model 1 for load Case 1 and load Case 2, respectively. Results for both sub-models A and B are

together for each load case. The section shown is equal to the section 40 mm above the fixed

end of the beam. Figure 4.13 depicts a sketch of the equivalent section with the dimensional

path related to the graphical figures defined also.


Figure 4.13: Beam section dimension for stress variation path

The figure below represents the loading perpendicular to the rail line in which stress maximums

in the x-direction are experienced over the thickness segments of the post structure with one

side in a state of tension and the opposing side of equal magnitude in compression. Between

these, the stress varies approximately linearly along the width of the structure with stresses

equal to zero occurring for both laminate A and B at the centre of the post’s width.

Figure 4.14: Stress Variation in post section at 40 mm above constraint boundary conditions

(Case 1)

0 200 400 600 800 1000-15

-10

-5

0

5

10

15

Length (mm)

Str

ess

(MP

a)

Stress Variation in Post Section for Case 1

1A:Weave & Tape

1B: Tape


Figure 4.15 represents the loading parallel to the rail line in which stress maximums in the x-

direction are experienced over the width segments of the post structure. For both sides of the

structure which are in opposing states of stress, a maximum value is found for both at the

corners of the structure which reduce towards the centre of the width. This variation in stress

across the width represents the different load carrying capabilities of the structural elements

with the beam flange sections being the greatest of all elements. It is worth recalling that the

flange elements were determined as strong design drivers in the project outset.

Figure 4.15: Stress Variation in post section at 40 mm above fixed end (Case 2)

Figure 4.16 shows the stress distribution (σx) over the bottom region of the post Model 1A

respectively, under loading conditions of Case 2.

Figure 4.16: Stress distribution (σx) of Model 1A for load Case 2

0 200 400 600 800 1000-50

-40

-30

-20

-10

0

10

20

30

40

50

Length (mm)

Str

ess

(MP

a)

Stress Variation in Post Section for Case 2

1A:Weave & Tape

1B: Tape


Large variations are presented between the maximum stresses between Model 1 and Model 2

for load Case 2. The cause for such differences is the presence of holes in the plate elements of

the structure. These cut-outs provoke a stress concentration factor at the edge of the laminate.

As describe in Section 2, the stress concentration is caused by the combination of the fibre

direction and the load direction in which the material is subjected. The complexity of stress

analysis around the hole’s edge increases with the introduction of additional plies in the

laminate with various orientations. The stress concentrations occurring in such regions are

areas of concern and are therefore analysed using various strength criteria for failure analysis

as outlined previously in Section 4.3. Figure 4.18 shows the variation of nodal stress in the x-

direction around the hole edge for laminate sub-models A and B. The sketch in figure 4.17

depicts where the starting point and direction of the graphical stress distribution in figure 4.18.

Figure 4.17: Sketch of circumferential path for stress variation around hole

Figure 4.18: Stress variation in laminate around bottom hole for Model 2A and 2B, load case 2

0 100 200 300 400 500 600 700 800

0

10

20

30

40

50

60

70

80

90

Circumference (mm)

Str

ess

(MP

a)

Stress (sigmax) Distribution Around Bottom Plate Hole

2A) Weave + Tape

2B) Tape


Figure 4.19 shows the stress distribution (σx) over the bottom region of the post Model 2A

respectively, under loading conditions of Case 2.

Figure 4.19: Stress distribution (σx) of Model 2A for load Case 2

Tables 4.11 to 4.16 present the failure criteria results for both the predefined ANSYS failure

criteria and for the APDL macro failure criteria. Criteria include Tsai-Wu and Maximum Stress

for the predefined FEM program, and Tsai-Wu, Maximum Stress and Puck for the macro. Only

the most critical failure criteria results of the two models are presented below so as to reduce

the quantity of result information displayed. These critical failure values occur in Model 2

where the cut-outs/holes in the plate components creates stress concentrations in the laminate

bordering these regions as can be seen in the stress distribution of figure 4.17.

The failure criteria results are presented as failure index values where a value of one or above

signifies failure in one of the laminate’s layers. Both sets of results are obtained from nodal

stress data of each layer. Regions of high stress concentrations are analysed through one of the

defined failure criteria firstly by the APDL macro and secondly, by the predefined criteria in

ANSYS. By initially carrying out the failure analysis by the criteria written in the macro,

additional information such as the failure mode type and particular layer of failure are

attainable. The equivalent criterion analysis is then carried out in the ANSYS program in which

the user can define the desired layer for analysis, i.e. the layer at which failure has occurred in

the macro analysis. Failure index values and their node location are obtained and compared to

those of the macro.

Table 4.11 and 4.12 both present the most critical failure index values for the Tsai-Wu and

Maximum Stress, respectively, for laminate sub-model A analysed by both the ANSYS program

and the macro, with their % difference calculated and presented in the final column. Also

included are the node and layer number at which the critical index value occurs. Additional


information can be obtained from the APDL macro for the Maximum Stress criterion that

includes the mode in which the critical index value occurs. Table 4.13 presents the failure

analysis results for the Puck criterion defined in the APDL macro. Information including critical

node and layer number are obtained from the analysis. The macro also outputs the failure type

(FF or IFF) and in the case of the IFF, its mode number (1, 2, 3) for the most critical index value.

Tsai-Wu: Model 2A ANSYS Macro (APDL) % Difference

FPF Index Value 0.13800 0.13816 0.1%

Node 15322 15322

Layer 34 34

Table 4.11: Tsai-Wu Index failures for Model 2A in FEM (ANSYS) and macro program (APDL)

Max Stress: Model 2A ANSYS Macro (APDL) % Difference

FPF Index Value 0.12216 0.12223 (Yt) 0.05%

Node 15322 15322

Layer 34 34

Table 4.12: Maximum Stress Index failures for Model 2A in FEM (ANSYS) and macro program (APDL)

Puck: Model 2A Macro (APDL)

FPF Index Value 0.12303

Failure Mode 1 (IFF)

Node 15322

Layer 34

Table 4.13: Puck Index failure for Model 2A in macro program (APDL)

Tables 4.14 to 4.15 below present the equivalent result format as for the set of tables above,

however this time the results presented are for the model composed of laminate sub-model B.

Again, the Tsai-Wu and Maximum Stress criteria are given in the first two tables for both the

ANSYS and APDL analysis. The third table contains the results of the Puck criterion for the APDL

analysis only.

Tsai-Wu: Model 2B ANSYS Macro (APDL) % Difference

FPF Index Value 0.11787 0.11770 0.14%

Node 12728 12728

Layer 1 1

Table 4.14: Tsai-Wu Index failures for Model 2B in FEM (ANSYS) and macro program (APDL)


Max Stress: Model 2B ANSYS Macro (APDL) % Difference

FPF Index Value 0.10917 0.10902 0.13%

Node 12728 12728

Layer 1 1

Table 4.15: Maximum Stress Index failures for Model 2B in FEM (ANSYS) and macro program (APDL)

Puck: Model 2B Macro (APDL)

FPF Index Value 0.10902

Mode FF

Node 12728

Layer 1

Table 4.16: Puck Index failure for Model 1B in macro program (APDL)

All of the above criteria results demonstrate that failure does not occur in any of the structures

(i.e. index failure < 1). However, it is worth noting that the more critical index failures occur for

the sub-model A (weave + tape) even though the stress concentrations around the holes shown

in figure 4.19 are greater in sub-model B (tape).

The index result values for the predefined ANSYS criteria and APDL macro criteria are identical

for each criterion with only minute differences between the results occurring due to variation

of precisions.


4.6 Validation of Numerical Model

Two procedures have been implemented to verify the adequacy of the FEM model. These

include

1. Classical Lamination Theory (CLT) for Narrow Beams [15], [16], [17]

2. Smeared Approach

Both procedure types compare stress results for a given section of the post with those obtained

in the FEM model. The accuracy of both the CLT and Smeared Approach validation methods are

analysed in terms of their axial stiffness, curvature and bending stiffness results with the

effectiveness and restrictions of each of the procedures considered.

4.6.1 Classical Lamination Theory for Narrow Beam

The response of the beam is dependent on the width to height ratio of the beam’s section. A

beam of a small ratio is defined as being ‘narrow’ while that of a large section ratio is defined as

a ‘wide’ beam. The difference in response for both beam types is related to the significance of

the induced lateral curvature caused by the beam subjected to axial bending. For the narrow

beam the axial strain distribution gives rise to a significant amount of deformation of the cross-

section which is induced due to the Poisson effect. Figure 4.20 shows a simplified

representation of the response of a narrow beam. As a result, loading Nx and moment Mx acting

on the axial directions are considered while lateral moment is concluded as being equal to zero.

In summary, the initial conditions of a narrow beam,

0==== xyyxyy MMNN

0≠yK (4.17)

Figure 4.20: Narrow composite beam deformed cross-section


The Classical Laminate Theory (CLT) of Chapter Two is applied

validity of the numeric model of the FEM.

stiffness, bending stiffness and stres

figure 4.21. The process includes the CLT described in Chapter Two up to determination of the

extension, coupling and bending stiffness. From that point, the narrow beam conditions are

implemented and the responses are subsequently determined in the order shown in the

flowchart. The CLT validation is written using MATLAB parametric language.

Figure 4.21: Classical Laminate Theory

The theory described below relates t

of the flowchart in figure 4.21.

load-deformation relation can be summarized as

or as its inverse

Material Data + Structural

Configuration

Axial Stiffness (EA) Bending Stiffness (EI)

Centroid (ZC)


The Classical Laminate Theory (CLT) of Chapter Two is applied in this section

validity of the numeric model of the FEM. Comparable response results such as the axial

stiffness, bending stiffness and stresses are calculated using the process shown as a flowchart in



d and the responses are subsequently determined in the order shown in the


: Classical Laminate Theory (CLT) process for narrow beam

The theory described below relates to the Narrow Beam Theory step and the subsequent steps

of the flowchart in figure 4.21. Recalling the constitutive equations in (2.50) and (2.51), the

can be summarized as

=

kDB

BA

M

N oε.

=

M

N

db

ba

k T

o

.ε

Compliance Matrix [S]

Reduced Stiffness [Q]

Narrow Beam Theory

Inverse Compliance

Matrix

NX, MX, MXY

εX, εY, εXY

kX, kY, kXY

σ = Q.ε

89

in this section to verify the

Comparable response results such as the axial

ses are calculated using the process shown as a flowchart in



d and the responses are subsequently determined in the order shown in the


(CLT) process for narrow beam

o the Narrow Beam Theory step and the subsequent steps

Recalling the constitutive equations in (2.50) and (2.51), the

(4.18)

(4.19)

Transformed Stiffness [Q]

Extension [A] Coupling [B] Bending [D]


Applying the initial conditions of the narrow beam, where Ny= Nxy= My= Mxy = 0, the relation is

reduced to the following equations

xxox MbNa 1111 +=ε

xxx MdNbk 1111 +=

(4.20)

or its matrix form as

=

x

x

x

ox

M

N

db

ba

k.

1111

1111ε

(4.21)

And inverting back the matrix as the load-deformation relation

=

−

x

ox

x

x

kdb

ba

M

N ε.

1

1111

1111 =>

=

x

ox

x

x

kDB

BA

M

N ε.

'

''

11

11

(4.22)

where

11

211

11

1

1'

d

ba

A−

=

(4.23)

11

111111

1

1'

b

dab

B−

=

(4.24)

11

211

11

1

1'

a

bd

D−

=

(4.25)

The load-deformation relation of equation (4.22) for the plate and flange laminates in the xy-

plane (i = 1, 2, 3, 4) can be expressed by the following two equations.

ixio

ixiix kBAN ,,1,,1, '' += ε

ixio

ixiix kDBM ,,1,,1, '' += ε

(4.26)

However, for the web laminate in the xz plane (i = 5), no radius of curvature exists.


05, =xk (4.27)

which implies that the force and moment in the x-direction for the web are reduced to

oxx AN 5,5,15, ' ε=

oxx BM 5,5,15, ' ε=

(4.28)

Proceeding from here, the structural elements are reconfigured along the axis at which bending

is considered. Figures 4.22 and 4.23 represent the actual and reconfigured elements,

respectively. In figure 4.23 the elements are defined by numbers which are required in the

following theoretical development.

Figure 4.22: Sketch of actual section

Figure 4.23: Sketch of reconfigured section about y-axis which structural elements numbered


Centroide (ZC)

In relation to figure 4.23, the centroide of the structure is defined as the average location of the

forces acting on each part of the cross-section. The y-coordinate is located at the bottom of the

section and the z-coordinate, due to symmetry, is located at the centre of the section passing

through the centre of the web thickness. The net force acting on the centroide is given by

∑=

=5

1,

iiixicx zNbzN (4.29)

where bi is the breadth of the elements, zi is the distance from the y-axis to the centre of the

element, and xN is

∑=

=5

1,

iixix NbN (4.30)

Recalling the first equation in (4.26), the centroide can be expressed as

( )( )∑

= ++

=5

1 ,,1,,1

,,1,,1

''

''

i ixio

ixii

iixio

ixiic kBAb

zkBAbz

εε

(4.31)

As the strain along the x-axis of the structure is the same for all elements, it can be deduced

that cx

oix εε =, (4.32)

where cxε is the strain at the centroide in the x-direction. As well as that, with only load

application at the centroide of the section along the x-axis, no radius of curvature exists.

0, =ixk (4.33)

Which implies the expression for determining the centroide location is simplified to

∑=

=5

1 ,1

,1

'

'

i ii

iiic Ab

zAbz

(4.34)

In relation to the composite post sectional detail of the project, it is evident that the

configuration is symmetric about the centre of the web element in the y-direction. This implies

that the centroide is located at this point.


Axial Stiffness (EA)

In relation to calculating the axial stiffness, an axial force is applied at the centroide of the

entire section. The entire force in the x-direction can be written as

cxx AEN ε= (4.35)

where AE is the equivalent axial stiffness of the entire section. Substituting the equations

(4.26) and (4.28) into (4.30), the total force is

( ) ( )ox

iixi

oixiix AbkBAbN 5,5,15

4

1,,1,,1 ''' εε +

+= ∑=

(4.36)

and recalling the conditions of the axial force in (4.32) and (4.33), the total force can be written

as

( ) cx

iiix AbN ε

= ∑

=

5

1,1' (4.37)

where the total axial stiffness is

( )ii

i AbAE ,1

5

1

'∑=

=

(4.38)

Bending Stiffness (EI)

The axial bending stiffness can be determined by a moment applied at the centroide which is

equal to the bending stiffness multiplied by the curvature experienced at the centroide and is

given in the following expression.

cx

cxx kDM = (4.39)

The total moment is made up of moment and force components, in the x-direction, of the five

sectional elements.

( ) 5,

4

1,,, x

iiixiixiix MzNbMbMC

++=∑=

(4.40)

where the moment at the web (i = 5) is


( )∫

−

=2

2

5,5,

5

5

d

d

xx dzzNM

(4.41)

Thus

( )( )∫∑

−=

++=2

2

5,

4

1,,

5

5

d

d

xi

iixixix dzzNzNMbMC

(4.42)

Recalling the constitutive equations in (4.26), and considering that the curvature is equal for all

sectional elements

cxix kk =, (4.43)

And axial strain at the centroide is equal to zero

0=cxε (4.44)

The moment xM for the flange and plate elements (i = 1, 2, 3, 4) can be expressed as

( ) cxiiiiiiiixix kDzBzAbzNM

CCC ,1,12

,1,, ''2' ++=+ (4.45)

and the force component for the web element is

( )cx

cx

oxx zkAAN 5,5,15,5,15, '' +== εε (4.46)

Applying conditions (4.43) and (4.44),

cxx zkAN 5,15, '= (4.47)

The moment xM for the web is

dzzkAMd

d

cxx ∫

−

=2

2

25,15,

5

5

'

(4.48)

With the curvature cxk common to all elements, the total moment in (4.40) can be expressed as

( ) cx

d

diiiiiiix kdzzADzBzAbM

CC

+++= ∫∑−=

2

2

25,1

4

1,1,1

2,1

5

5

'''2'

(4.49)

Where the bending stiffness is

( ) dzzADzBzAbDd

diiiiiii

cx CC ∫∑

−=

+++=2

2

25,1

4

1,1,1

2,1

5

5

'''2'

(4.50)


Knowing the bending stiffness and total moment, the curvature can be found by manipulating

the above equation into the form below.

cx

xcx

D

Mk =

(4.51)

The stresses and strains are calculated using equations in (2.44) and (2.45) of Chapter Two, and

are recalled as

+

=

ixy

iy

ix

ik

oixy

oiy

oix

kixy

iy

ix

k

k

k

z

,

,

,

,

,

,

,

,

,

,

γ

ε

ε

ε

ε

ε

(4.52)

[ ] [ ] [ ]kikki Q εσ = (4.53)

The relation of the laminate force and moments to strains and curvatures, which is expressed in

the equations of (4.26), is used to determine the force and moments at a specific laminate. The

bending stiffness conditions of (4.43) and (4.44) are implemented in the equations of (4.26).

The strains and curvatures are calculated by the inverse constitute relation in (4.19). The

transformed stiffness ijQ is calculated for each layer of the laminate and subsequently

multiplied by the strain of the corresponding layer. Note that the curvature component of the

strain in (4.52) is dependent on the distance of the particular layer of interest from the

centroide (zk). This process is carried out for each laminate element in the section. Using the

bending stiffness approach, the steps in determining the stresses and strains are summarised as

follows.

1. Calculate the bending stiffnesscxD

2. Knowing the moment xM , calculate the curvature cxk from equation (4.51)

3. Applying bending stiffness conditions in (4.43) and (4.44), find the force Nx and

moments Mx and Mxy using equations in (4.26)

4. Determine strains and curvatures from the deformation-load relation in (4.19)

5. Multiply the transformed stiffness matrix ijQ of a particular layer by its corresponding

strains and curvatures to determine the stresses in that layer.


4.6.2 The Smeared Approach

The smeared approach involves determining the equivalent axial Elastic Modulus Ex for the

composite section. This approach is based on the assumption that the equivalent Elastic

Modulus is constant throughout the laminate. While it is a more simplistic method of

determining responses, it does not consider the effect of the stacking sequence of the laminate.

Knowing the elastic constants values in principal directions (1, 2) and orientation of the fibres

for a given layer, the equivalent axial Elastic Modulus Ex for off-axis fibres can be determined

from the relation of the transformed compliances ijS in (2.32).

ki

ki

ki

ki

x EEGEEθθθνθ 4

22

22

11

12

12

4

11

sin1

sincos21

cos11 +

−+= (4.54)

where i is the laminate element (plate, flange, or web) and k is the particular layer in the

laminate. The equivalent axial Elastic Modulus is calculated by multiplying the transformed

stiffness in (4.54) in each layer by its corresponding layer thickness and dividing by the overall

thickness of the laminate.

∑=

=5

1

,,

iki

ki

kix

ixt

tEE (4.55)

The total equivalent axial stiffness is gotten by summing the equivalent stiffness of each

element (i = 1, 2, 3, 4, 5).

∑=

=5

1,

iiix AEAE (4.56)

where Ai is the sectional area. Similarly, the bending stiffness is calculated by summing the

individual sectional elements.

∑=

=5

1,

iiix IEIE (4.57)

As this approach is based on the assumption that the equivalent Elastic Modulus is constant

throughout the laminate, the centroide zi is always located at the mid-plane of each laminate.

The centroide of the entire section is found by the summation of all the laminate stiffnesses

multiplied by their corresponding centroide, and dividing them by the total stiffness equivalent.

( )∑

=

=5

1 ,

,

i iix

iiixc AE

zAEz (4.58)


At this point, the maximum stresses for a given point in the section are determined. To

incorporate the variation of equivalent stiffness between the sectional elements, the equivalent

width technique is used. From equation (4.55), the equivalent axial Elastic Modulus for the

plate element (sections 1 and 4) is determined to be the stiffer than the beam elements

(sections 2, 3, and 5). That implies that to maintain an equivalent stiffness throughout the

entire section, the geometric sectional properties of the plate elements must be altered by a

factor of n where

beam

plate

E

En = (4.59)

Consider the two sectional drawings in figures 4.23 and 4.24, the first figure shows the original

section with elements of varying stiffness (Ex,plate, Ex,beam) whereas the second figure depicts the

equivalent section in which all elements have the same stiffness value (Ex,beam) and also shows

the necessary geometric transformation to maintain the mechanical properties of the original

section. Note that due to symmetry, the section elements below the plane y-y are considered

to take on the same transformation of their symmetric counterparts above. This also implies

that the centroide zc remains at the centre of the entire section.

Figure 4.24: Original section configuration, elements of varying stiffness


Figure 4.25: Transformed section configuration, elements of equivalent stiffness

The inertia Iyy of the entire section is recalculated, implementing this time the transformation in

figure 4.24. The stress at the plate extremity is calculated using the beam bending relationship

multiplied by the transformation factor n.

=

I

yMn i

i

.σ (4.60)

4.6.3 Results Comparison

One composite model type is used for the results comparison. The model type chosen is the

composite structure without holes and laminates of unidirectional fibres only, i.e. Model 1B. As

the structure is of variable section along its length, the section dimensions above the fixed end

are chosen. This is equivalent to the section at 200 mm above the bottom of the post. The

loading is of Load Case 2, i.e. wind loading parallel to the rail line which has an equivalent total

moment M of 17790 Nm about the y-axis.

Table 4.17 presents the results for the both the CLT and Smeared Approach validation models.

The results include the centroide location, axial stiffness, curvature, and bending stiffness. A

percentage difference between the two sets of results is made also. Note that the FEM results

are not included in this table as the structure’s variable section along its length affects the axial

stiffness, curvature, and bending results.


Units CLT for Narrow

Beams

Smeared

Approach

% Difference

Centroide (zc) mm 70 70 0.0%

Axial Stiffness (EA)x N 8.3999e8 8.8244e8 5.1%

Curvature (kc)x mm

-1 5.7853e-6 5.3755e-6 7.1%

Bending Stiffness (EI)x Nmm2

3.0578e12 3.2908e12 7.6%

Table 4.17: Behavioural response result comparison for CLT for narrow beams and the smeared approach

As both the entire structural section and each laminate element are symmetric about the y-

plane, the centroide distances are the same for both the CLT and Smeared Approach analyses.

Variation between both validation models occurs for the axial stiffness, curvature and bending

stiffness. This occurs due to the presence of extension, coupling and bending parameters (A’i,

B’i, D’j) in the CLT model while the Smeared Approach is an analogous method of elastic

equivalents (Ex) and sectional properties (area and moment inertia) used to determine intrinsic

laminate response. As all the laminates are symmetric about one plane (y-plane) they can be

classified as monoclinic, which causes the coupling terms B’I to be reduced to zero meaning that

only extension and bending terms remain. The difference between the two models increases if

the ply stacking sequence is altered to an un-symmetric configuration where coupling returns

to the load-deformation relation.

The maximum stresses of the plate element in a state of tension are tabulated below in table

4.18. They include results for the FEM, CLT, and Smeared Approach. Both the FEM and CLT

results are presented at a layer level while the Smeared Approach stress result is of a laminate

level. Both the FEM and CLT results are the maximum tensile stresses experienced and occur in

the top layer (0o) of the top laminate, i.e. the layer furthest from the centroide of the section.

The higher stiffness caused by the ply orientation and its distance from the centroide confirms

that maximum stresses would develop in this zone of the cross-section. The minute difference

between both results (1.24%) would indicate that the approximation made in the FEM model of

the composite post is valid. The large variation between the Smeared Approach and FEM/CLT

models indicates the limitations incurred by assuming equivalent elastic properties for a

laminate as opposed to a layer analysis made in the CLT for Narrow Beams. The Smeared

Approach does not account the for the stress variations between layers and consequently

underestimates the maximum stress incurred in the section.

Location σx (N/mm2)

FEM Top Plate (1) 0o Ply 52.390

CLT for Narrow Beams Top Plate (1) 0o Ply 53.046

Smeared Approach Top Plate (Laminate) 38.637

Table 4.18: Stress result comparison for FEM, CLT for narrow beams, and the smeared approach


4.7 Analysis Under Other Various Loading Types

The primary objective of the design-analysis for the proposed post is to ensure behavioural

response compliance between it and the original post, i.e. both the original and proposed post

have similar maximum deflections and factor of safety magnitudes for stresses (σmax/ σult). This

objective was undertaken using equivalent loading conditions for both structures. After

determining an adequate design for the proposed structure, a subsequent or secondary load

analysis can be considered. This secondary load analysis composes of

1. Dynamic loading

2. Loading hypothesis

Both of these additional types of loading account for conditions that are experienced by the

post structure at instantaneous, temporary, and permanent levels. The loading types are

caused by the post’s proximity to moving trains (dynamic loading), by related structural

components joined to the post (load hypothesis) and additional loading related to seasonal

changes (loading hypothesis).

4.7.1 Dynamic Loading

4.7.1.1 Background and Development

It is well known that relatively-high to high velocity trains can exert aerodynamic loading on

objects in their proximity. An isolated train passing through the air induces a complicated flow

field that are best simulated in computational fluid mechanics (CDF) simulations and finite

volume numerical methods. However, in order to simulate a typical aerodynamic pressure

exerted on the post structure a number of papers related to this phenomenon [18], [19], [20]

were analysed and an appropriate pressure loading consistent with the conditions of the

project was applied dynamically in ANSYS.

The overall resistance R to movement (on level track) that needs to be overcome by the

traction effort of the locomotive is a result of rolling friction between wheel and rail, bearing

resistance, train dynamic loses and air resistance, and are presented in the following empirical

expression.

A

A

S

Ypp

CC 122 = (4.61)

where A and B.V are the mechanical resistances and C.V2 is the aerodynamic resistance. By

specifically analysing this external aerodynamic resistance part of the expression, it can be

further expressed in terms of the coefficient C:


DAA CSVVC ...21. 22 ρ=

(4.18)

where ρ is the air density, S is the frontal area of the train, VA is the train velocity relative to the

air, and CD is the drag coefficient. The value of CD is influenced by the train’s geometric profile

where values of CD for S on the order of 10 m2 and L on the order of 300 m can range from

approximately 1, or less, for highly streamlined trains to 10 to 15 for freight trains. For analysis

based on data correlations, the drag coefficient, CD, is broken down as:

( ) 5.0AllCCC LTTBDLD −++= λ

(4.19)

where CDL is the drag coefficient of the leading car or locomotive, CB is the base drag at the

train’s tail, λT is the friction along the train, which includes the bogies, wheels, interference and

underbelly effects, and lT and lL are the lengths of the total train and the leading car,

respectively, [21].

When the train closes onto the stationary object and in this case the post, it applies a pulse of

pressure on the object. Figure 4.27 shows graphically the pressure history subjected on a

proximate structure due to a passing train and is taken from experimental data in [20]. It is felt

that this figure and the results provided are suitable to be interpreted in the dynamic analysis

of this project. The experimental results were collected by using a prototype high-speed

locomotive namely the JetTrain built by Bombardier. This passenger train was built in an

attempt to make European-style high-speed service more financially appealing to passenger

railways in North America. The validity of the experimental results is due to the similarity in the

train’s profile to a number of locomotives/leading cars in service in Spain which include, for

example, Alstom S-100 AVE, Siemens S-103 AVE and Talgo-Bombardier S-130 Alivia. Figure 4.26

shows similarities between the profiles of the Bombardier JetTrain (left) and the Alstom S-100

AVE (right). An additional validation for the use of the experimental results in the numerical

model is the velocity at which the train carried out the tests. The maximum velocity reached by

the Bombardier JetTrain was 130 mph or approximately 210 kph which is in line with the

velocities of the train fleet of the ‘High-Velocity Variable-Width’ section of the Spanish railway

system and it is in this section which the modelled structure would be designated to perform.


Figure 4.26: Bombardier JetTrain and Alstom S-100 AVE

In relation to the graph in figure 4.27, above the zero mark represents the positive pressure

exerted on an adjacent structure and below zero represents the negative pressure or vacuum

created. Maximum values on the graph range from approximately 0.06 psig to 0.09 psig or

413.68 Pa and 620.58 Pa, respectively. The concluding information gathered by the reports

allowed the author to make a more conservative representation of the pressure exerted on the

post as train velocities. It is worth highlighting that this proposed post would not be designated

for such rail High-Velocity only corridors, can reach up to 350 kph (Siemens S-103 AVE) but for

the general rail lines (via general) of reduce velocity capacities.

Aerodynamic pressures, which include a Factor of Safety of 2.0, utilised in the numerical

program include a maximum pressure value of 1.2 MPa at the point at which the nose of the

leading car reaches the structure and, in a relatively instantaneous moment, a pulse of vacuum

of 1.4 MPa which has the opposite directional loading effect and occurs when the nose of the

train immediately passes the structure. As a result, the numerical model intends to simulate an

equivalent pressure change applied on the structure. Figure 4.28 is a representation of the load

history inputted into the transient analysis of the numerical model. The pressure history below

also highlights an interesting phenomenon that consists of a number of relatively large pressure

variations contained within the passing of the nose and tail of the train. The figure shows two

pressure spikes sustained in the negative portion of the pressure curve. These variations are

directly related to the discontinuities of the train, i.e. the inter-car gaps of the passing train.


Figure 4.27: Pressure history of passing train [20]

Figure 4.28: Transient history analysis of passing train inputted in ANSYS

0 0.5 1 1.5 2 2.5 3-1500

-1000

-500

0

500

1000

1500

Time (sec)

Load

(N

/m2 )

Pressure History of Passing Train


4.7.1.2 Modal Analysis

Modal analysis is used to determine the vibration characteristics (natural frequencies and mode

shapes) of a structure. It also can be a starting point for another, more detailed, dynamic

analysis, such as a transient dynamic analysis, a harmonic response analysis, or a spectrum

analysis. Modal analysis uses the structure’s overall mass and stiffness to determine the various

periods that it will naturally resonate at. This issue is important in the design stage of the

structure as dynamic loading such as wind and aerodynamic pulse loading from passing trains

can cause the structure to vibrate and if this excitation coincides with the natural frequency of

the structure resonance will occur which can be detrimental to the structure’s performance.

Following the modal analysis, a transient analysis of the post structure is carried out. Tables

4.19 and 4.20 show the first three natural frequencies of the composite structure calculated in

ANSYS (ANTYPE, MODAL) and also below are the mode shapes of their natural frequencies for

Model 1A [8].

Mode 1A 1B

(Hz) (rad/s) (Hz) (rad/s)

1 6.335 39.805 8.768 55.089

2 19.617 123.257 15.355 96.478

3 158.415 995.349 176.572 1109.433

Table 4.19: Modal analysis, first three natural frequencies of post in Model 1

Mode 2A 2B

(Hz) (rad/s) (Hz) (rad/s)

1 4.249 26.694 5.151 32.364

2 12.941 81.311 15.375 96.607

3 45.721 287.272 64.859 407.522

Table 4.20: Modal analysis, first three natural frequencies of post in Model 2

Figure 4.29: Mode 1 shape for 1B




4.7.3 Transient Analysis

Transient dynamic analysis (sometimes called time-history analysis) is a technique used to

determine the dynamic response of a structure under the action of any general time-dependent

loads. The basic equation of motion solved by the transient analysis is:

[ ][ ] [ ][ ] [ ][ ] )(tFuKuCuM =++ &&&

(4.20)

where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u is the nodal

displacement vector, u& is the nodal velocity vector, and u&& is the nodal acceleration vector.

In this project, the transient analysis is used to determine the time-varying stresses and

displacements in the structure as it responds to a prescribed variable load history. The

experimental load-history curve shown in figure 4.27 is applied in the transient analysis in

ANSYS. Figure 4.28 shows the approximated, more conservative load history converted from

psig to N/m2. The transient analysis involves loads which are functions of time, and thus are

divided into load steps. For each load step, the load and time values are specified with the

option to specify whether the load is applied as a ramp function (KBC,0) or as a step function


(KBC,1). The time value is specified after the load is applied via the TIME command. The load-

history curve is solved by employing time integration of the equations of motion with the

integration time step (ITS) used as the time increment in this process. The size of time

increment step determines the accuracy of the transient solution where the smaller the

increment the more accurate the solution [22]. There are three methods of transient analysis

available which include Full, Reduced and Mode Superposition Method and are implemented

through the TRNOPT command. The one used in this analysis is the Full Method as it permits all

types of non-linearities however it is much more CPU intensive than the other two methods as

the full system matrices are used [23].

In most structures, damping is present in some form. The problem of dissipating the energy in

structures is an important feature in mechanical design in terms of vibration control, noise and

fatigue endurance. The amplitude and frequency of vibration in a structure is controlled by the

applied excitation and the response of the structure to that particular excitation. Damping

becomes important in the structure when the response is close to the natural/resonant

frequency or if the vibration response is short term. Energy dissipation mechanisms are

separated into classes: the first is damping in the structure due to friction at joints. While this

source is usually dominant in metal structures it does not contribute significantly to composite

structures as joints are tended to be kept to a minimum or are adhesively bonded. The second

class and more predominant of the two is inherent material damping [24]. For fibre reinforced

composite materials, material damping is dependent on two mechanism types: at a microscale

level, the energy dissipation is induced by viscoelastic behaviour of the matrix, damping at the

fibre-matrix interface, and damping due to damage. The second type is at the laminate level

where energy dissipation is dependent on the constituent properties, the ply orientation,

interlaminar effects, and stacking sequence [25].

In relation of the finite element model, damping is provided by Rayleigh damping equation

which is represented by the expression

KMC βα +=

(4.21)

where C is the damping matrix, M is the mass matrix and K is the stiffness matrix. The

parameters α and β are inputted into the program through the ALPHAD (mass damping) and

BETAD (material damping) commands, respectively. The parameters α and β are unknowns but

are related to the modal damping ratio ξ. The modal damping ratio is the ratio of actual

damping to the critical damping for a particular mode of vibration. If ωi is the natural frequency

(as calculated previously in the modal analysis) of mode i, then α and β satisfy the relation


22

i

ii

βωωαξ +=

(4.22)

Figure 4.32 below describes the influence of each of the parameters over a range of natural

frequencies. For damping of at low frequencies, α plays a more dominant role whereas higher

frequencies are damped more by the β parameter and to a lesser extent by α-damping.

Figure 4.32: Influence of α and β over a range of natural frequencies

To specify both α and β for a given damping ratio, it can be assumed that the sum of the two

damping functions is nearly constant over a range of frequencies. Therefore, given a modal

damping ratio ξ and a frequency range ωi and ωj, two simultaneous equations can be solved for

α and β [8].

22

i

i

βωωαξ +=

22j

j

βωωαξ +=

(4.23)

In relation to the transient analysis presented in the project, only one model type and material

configuration was analysed. This is predominantly due to the intensive computational demands

required to carry out a full transient analysis on models with a large quantity of elements such

is the case in this project. The structure without holes composed of only laminates with

unidirectional material (1B) was analysed. To specify both α and β for a given damping ratio,

the first two modal natural frequencies (ω1 and ω2) of the model were taken from table 4.19

and solved in the above simultaneous equations. The damping ratio ξ of carbon fibre

composites is known to approximately 0.01 [26]. The values yielded from the above

simultaneous equations include 0.7 and 0.00132 for the α-damping and β-damping,

respectively.


It must be stressed that the relevant data for damping of a particular composite laminate

structure requires a certain degree of vibration damping experiments at a laminate and possibly

a structural level. The above damping parameter values accumulated are approximations and

their response results obtained in the transient analysis are viewed with caution. A number of

additional transient analyses were carried out with varying α and β values so as to show the

influence of both types of damping on the structure. Figure 4.33 shows the vibration response

of Model 1B caused by the given excitation in figure 4.28 for different values of the α and β

parameters. Figure 4.34 shows the stress variation in the x-direction at the bottom of the

structure (node 4553), close to the fixed end of the beam caused by the same load history.

The behaviour responses of the figures below indicate the importance of dynamic response

analysis. The initial two loadings applied to the structure occur in opposite directions, which

simulates a positive-negative pressure condition caused by the passing train. This load pair

induces an initial displacement of 32.14 mm and a subsequent displacement in the opposite

direction of 79.2 mm from its stationary position. The range of displacement of the pair is equal

to 111.34 mm and occurs over a time interval of 0.11 s whereas the maximum displacement for

this model in the static analysis was 53.439 mm. Dynamic ranges of this magnitude over a large

number of cycles can merit the need for a fatigue analysis. As this structure is intended for only

temporary substitution of a fail post structure, its service life duration is relatively low and for

that reason a fatigue analysis is not considered.

In terms of the effect α and β damping values, the analysis demonstrates that both damping

parameters influence the overall damping of the structure. The vibration response below shows

that in the case of reduced α and β damping values, the damping is significantly less than that

of the two cases where alternately one of the damping values is reduced. In both these cases,

the damping response is quite similar.


Figure 4.33: Vibration response for different values of the α and β parameters (Model 1B)

Figure 4.34: Stress (x-direction) response for different values of the α and β parameters (Model 1B)

0 1 2 3 4 5 6 7

-0.05

0

0.05

Time (sec)

Dis

plac

emen

t (m

)Displacement Variation of Node 14495 for Load History

ALPHAD=3.0 BETAD=0.00132ALPHAD=0.7 BETAD=0.000132

ALPHAD=0.7 BETAD=0.00132ALPHAD=3.0 BETAD=0.000132

0 1 2 3 4 5 6 7-5

-4

-3

-2

-1

0

1

2

3

x 107

Time (sec)

Str

ess

(N/m

2 )

Stress Variation at Node 4553 for Load History

ALPHAD=3 BETAD=0.002ALPHAD=0.7 BETAD=0.000132

ALPHAD=0.7 BETAD=0.00132ALPHAD=3.0 BETAD=0.00-132


As part of the post’s dynamic design-analysis, the frequency response generated by the

excitation is determined and compared with the natural modal frequencies of that structure.

The objective at this stage of the design-analysis is to ensure that the response frequency does

not coincide with the natural frequencies of the structure, as resonance would occur and the

behavioural response maximums of the structure would be amplified.

The frequency is determined by Fast Fourier Transform (FFT). A sample of vibration or discrete

signal is taken from the response analysis in figure 4.33. In this case, the sample taken extends

over a time domain of 4.5 s that begins at the moment the post is free to vibrate, i.e. after the

application of the transient load history of figure 4.28. Figure 4.35 and 4.36 shows the extent of

the sample used in the FFT taken from the original vibration response. Of the four amplitude-

varying responses in figure 4.33, the response with the damping originally calculated from the

above simultaneous equations (4.23) is analysed in the FFT, recalling that the α-damping and β-

damping values are equal to 0.7 and 0.00132, respectively.

Figure 4.35: Extent of sampling frequency taken from the dynamic response (Model 1B)

0 1 2 3 4 5 6 7

-0.05

0

0.05

Time (sec)

Dis

plac

emen

t (m

)

Displacement Variation of Node 14495 for Load History

ALPHAD=0.7 BETAD=0.000132


Figure 4.36: Sampling frequency from the dynamic response (Model 1B)

The FFT is a faster version of the Discrete Fourier Transform (DFT) which utilizes alternative

algorithms to carry out the same operation as DFT but in a substantially reduced time period.

The FFT function in MATLAB is an effective tool for computing the discrete Fourier transform of

a signal and is therefore used to carry out the discretisation of the sample taken here where the

transform is given by the following expression.

( ) ( ) nk tjwn

N

nk etxX −

−

=∑=

1

0ω

(4.24)

where x(tn) is the input signal amplitude (real or complex), tn is sampling (sec), X(ωk) spectrum

of x at frequency ωk, and N is the number of time samples. Additional variables of the algorithm

include the sampling interval T (sec) and its inverse, the sampling rate fs (Hz) [27].

Only the positive frequency spectrum is considered in the analysis. Figure 4.37 presents the

discrete frequency domain representation of the sampled signal. Note that windowing, filtering

among other techniques were not applied in this analysis. These techniques are relevant for

analysis of more complex discrete signals [28].

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04Dynamic Response Post-Load History

time (seconds)

y(t)

ALPHAD=0.7 BETAD=0.000132


Figure 4.37: Discrete frequency spectrum from vibration response of Model 1B in transient analysis

From initial examination of the dynamic response sample in figure 4.36, it is quite evident that

the periodic response composes of approximately six cycles per second, or 6 Hz. This

observation is confirmed by the FFT analysis of the sample where figure 4.37 above shows the

discrete frequency components with maximum amplitude occurring at 5.9867 Hz. While they

do not coincide, both the response frequency and first natural mode frequency ω1 lie within

close range of each other resulting in possible amplification of the behavioural response.

However, other relative structural components such as the attached cantilever assembly and

electrification equipment are not considered in this dynamic analysis. Such components would

impede the structure from suffering from detrimental resonance.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7x 10

-3

X= 5.9867Y= 0.0060523

Freq (Hz)

Am

plitu

dePositive frequency for FFT


4.7.2 Load Hypothesis

The objective of the load hypothesis is to examine the mechanical behaviour of the structure

considering various combinations of loading associated to the structure. These combinations

are presented as cases. As it has already been stated, the only loading condition provided in the

project outline is a moment at the base of the structure which is a result of wind loading. Other

loading types that the post is subjected to, which are modelled in the following loading

hypothesis, include:

• Permanent loading

o Catenary cantilever support

o Catenary assembly

• Temporary loading

o Wind loading

o Ice loading

Other types of loading, but not considered in this analysis, include the transversal loading effect

due to posts located at rail curves, stresses induced longitudinally in the line as a result of

temperature change, construction and maintenance loading.

4.7.2.1 Permanent Loading

There are numerous types of patented designs of the cantilever support which can be broadly

classified into two groups: lattice and tubular. The main difference between both groups is that

the tubular support is in direct contact with the high voltage catenary assembly while the lattice

support is not. In the lattice support, insulators are situated between the catenary assembly

and the support itself while, in the tubular support assembly, insulators between the post and

support prevent the transfer of voltage to the post from the tubular support. The main

parameters that dictate the cantilever’s profile and connection points on the post are the

height of the contact wire and its offset distance, the height of the support wire, the span

between the posts, and the distance from the post to the centre of the track.

In this analysis, the lattice cantilever support and catenary assembly similar to that in figure 1

are to be considered with the objective of proposing an appropriate equivalent loading and

reaction diagram. The lattice support is made up of two steel UPNs (80x45x5 mm) welded

together to form an I-section profile. The support is made of steel (7900 kg/m3) and its self

weight is calculated and shown schematically ahead in figure 4.38.


Figure 4.38: Post structure complete with cantilever and catenary assembly

The catenary assembly is the overhead lines used to transmit electrical energy to the train

through a pantograph connected to the locomotive.

600, 750, 1500 and 3000 V in direct current (DC) and 25 kV in alternate current (AC).

catenary assembly consists of a mechanical support wire and an electrical contact wire which

are connected together by a series of droppers which are all supported

support and contact wire are kept in a state of tension independent of temperature. The

spacing between the droppers is 4.50 m with the distance between two consecutive posts

known as the span. The span can vary from between 30 m to

of curvature of the rail line. Figure

components of the assembly include electrical insulators

Figure

The permanent loading of the cantilever assembly consists of the total weight of the bronze

support wire (8300 kg/m3) of sectional area 65 mm

wire (107 mm2) both of which are made from copper (8960

distributed load W of the contact wire (kg/m) and the length of two adjacent spans,


: Post structure complete with cantilever and catenary assembly


through a pantograph connected to the locomotive. The power systems employed in Spain are

750, 1500 and 3000 V in direct current (DC) and 25 kV in alternate current (AC).


are connected together by a series of droppers which are all supported by the posts. Both the



known as the span. The span can vary from between 30 m to 70 m and depends on the degree

. Figure 4.39 shows a simplistic view of the catenary assembly. Other

components of the assembly include electrical insulators, stabiliser and bracing arm.

Figure 4.39: Simplistic scheme of catenary assembly

of the cantilever assembly consists of the total weight of the bronze

) of sectional area 65 mm2, the droppers (38.5 mm

) both of which are made from copper (8960 kg/m3). Knowing the uniformly

of the contact wire (kg/m) and the length of two adjacent spans,

114

: Post structure complete with cantilever and catenary assembly


The power systems employed in Spain are

750, 1500 and 3000 V in direct current (DC) and 25 kV in alternate current (AC). The typical


by the posts. Both the



and depends on the degree

shows a simplistic view of the catenary assembly. Other

and bracing arm.

of the cantilever assembly consists of the total weight of the bronze

, the droppers (38.5 mm2) and the contact

). Knowing the uniformly

of the contact wire (kg/m) and the length of two adjacent spans, li and li+1,


the total weight supported by one cantilever corresponds to half of that of the two adjacent

spans.

( )2

. 1++= iicontactcontact

llWP (4.24)

The weight of the support wire is similarly calculated to that in equation (4.24). Because the

wire is not linear but parabolic in shape, an additional 5% of the span is added to the support

wire’s length so as to account for this non linear shape.

( ) ( )05.1.2

. 1supsup

++= iiport

port

llWP (4.25)

The final element of the catenary assembly load is the dropper. The number of droppers

supported by each cantilever is calculated simply in equation (4.26). A span of 45 m is chosen

for this analysis and with a dropper spacing of 4.50 m, the number of droppers supported by

the cantilever is 11.

1+=spacing dropper

spann (4.26)

The total weight contributed by the droppers is calculated by multiplying the dropper linear

load (kg/m) by the average dropper length, and subsequently by the number of droppers per

span n.

averagedropperdropper lWnP ..= (4.27)

These three load parts of the catenary assembly are summed together to form a total vertical

load applied on the cantilever as depicted in figure 4.41.

4.7.2.2 Temporary Loading

As already defined, the temporary loading accounted for in this analysis includes wind and ice

loading. The wind loading consists of two parts: the first part, which has been dealt with in the

preliminary analysis, is the horizontal loading on the post in both directions; the second part is

the horizontal loading of the catenary assembly, transversal to the post. The ice loading also

consists of two parts: the first is ice loading on the vertical surface of the post due to blizzard-

type conditions; and the second is ice loading on the catenary wires. In relation to the wind

loading, the horizontal wind pressure acting on the surface of one of the cylindrical wire

elements of the catenary assembly and its resulting force component are shown in figure 4.40.


Figure

The wind pressure is calculated by the following generic formula

where Cd is the drag coefficient where cables and wires have a value between 1.0 and 1.3 with

a value of 1.2 chosen in this instance,

(m/s) which has a maximum of 120 km/hr in the project outline. The horizontal force vector is

given by

where A is

The total horizontal wind load is divided into two components as dictated to the boundary

conditions of the catenary assembly. The wires are supported by an insulator above the

horizontal part of the cantilever and by a stabiliser/insulator midway up the a

cantilever. These components are shown in figure 4

Ice loading on the vertical post surface is a phenomenon due to precipitation driven

horizontally by wind and is defined as ice accretion

structure dictates the surface area on which the snow collects: winds of low velocity tend to

have accumulation on the leading surfaces while higher velocities cause accumulation on the

trailing surface due to wake turbulence. Persistent freezing precipitation and c

can cause this snow to compact and form ice. This part of the loading analysis is quite

conservative as a sheet of ice (960 kg/m

vertical surface for cases in either direction. This lo

of the sectional axes and thereby affects the corresponding mom


Figure 4.40: Wind loading on catenary wire

The wind pressure is calculated by the following generic formula

2...21

vCP airdwind ρ=

is the drag coefficient where cables and wires have a value between 1.0 and 1.3 with

his instance, ρair is the air density (kg/m3), and


APF windwind .=

+= +

2. 1ii ll

DA



horizontal part of the cantilever and by a stabiliser/insulator midway up the a

cantilever. These components are shown in figure 4.41.


and is defined as ice accretion. The winds velocity and profile of the

dictates the surface area on which the snow collects: winds of low velocity tend to


trailing surface due to wake turbulence. Persistent freezing precipitation and c


conservative as a sheet of ice (960 kg/m3) of 40 mm depth is adopted to be frozen to the post’s

vertical surface for cases in either direction. This loading type affects the symmetry about one

of the sectional axes and thereby affects the corresponding moment inertia. In relation the

116

(4.28)

is the drag coefficient where cables and wires have a value between 1.0 and 1.3 with

), and v is the wind velocity


(4.29)

(4.30)



horizontal part of the cantilever and by a stabiliser/insulator midway up the angled part of


. The winds velocity and profile of the

dictates the surface area on which the snow collects: winds of low velocity tend to


trailing surface due to wake turbulence. Persistent freezing precipitation and cold temperatures


) of 40 mm depth is adopted to be frozen to the post’s

ading type affects the symmetry about one

ent inertia. In relation the


numerical analysis, the load is applied as its total (N) distributed vertically over the nodes that

form the relevant surface.

In terms of the vertical loading due to ice on the wires of the catenary assembly, the European

Standards EN 50119:2009 sets out a number of cases where the load magnitude depends on

altitude ranges or zones. The most critical of these is zone C of altitude superior to 1000 m

above sea level where the maximum loading due to ice (g/m) is 160. D where D is the diameter

of the wire (mm) [29]. This vertical loading type applies only to the horizontal cables such as the

support and the contact wires and not to the dropper.

All the above permanent and temporary loadings are shown schematically in figure 4.41 in

terms of the positioning and direction on the cantilever, and also shown is the horizontal

reaction of the strut T and the horizontal and vertical reactions RH and RV respectively, at the

point where the cantilever is connected to the post.

Figure 4.41: Loading and reaction system for cantilever and catenary assembly

The cantilever is fixed by an angle steel and bolt assembly as shown in figure 4.42. The

cantilever and catenary assembly, through the surface area of the angle steel in contact with

the post, applies a pressure onto the post surface. In relation to this structure, the pressure is

applied by the right-hand-side plate horizontally and vertically downwards. The strut is similarly

connected to the post, however as the strut is in tension, the pressure is applied onto the post

by the angle steel piece on the left-hand-side.


Figure 4.42: Components of post-cantilever connection

In terms of the FEM model, the horizontal loading is applied as a pressure over the equivalent

sized areas through elements of the meshed model so as to represent the boundary conditions

more accurately thereby avoiding the possible development of local stress spikes often

associated with force application to nodes. Vertical loading is distributed into vertical force

vectors on the nodes of the corresponding elements.

4.7.2.3 Cases of Load Combinations

Both the permanent and temporary loadings described above are combined with the previously

analysed post wind loading to form various possible loading scenarios that can affect the

entirety of the structure. The possible loading scenarios or cases are outlined in table 4.21.

There are five in total with the cantilever and catenary self weight being common to all of the

five. The cases are placed in order of increasing severity of loading culminating at Case E. The

wind loading cases of table 1 also contribute to the load hypothesis in this section. Case B and

Case D consist of the wind loading Case 1 while Case C and Case E contain wind loading Case 2

among additional loadings as describe above. The cases propose loading combinations that are

realistic for example, post wind loading perpendicular to the rail line (Case 1) is accompanied by

wind loading of the catenary assembly while post wind loading parallel to the rail line (Case 2)

does not include wind loading of the catenary assembly as the assembly runs parallel to the rail

line.


Cases Case Description

Case A Catenary assembly (self weight)

Case B Case 1 + Catenary assembly (self weight + wind)

Case C Case 2 + Catenary assembly (self weight)

Case D Case 1 + Ice (web of UPN) + Catenary assembly (self weight + wind + ice)

Case E Case 2 + Ice (plate of post) + Catenary assembly (self weight +ice)

Table 4.21: Description of cases of load combinations

4.7.2.4 Results

Table 4.22 and 4.23 shows the nodal results for the stresses (MPa) in global coordinates σx, σy,

σz, σxy and σxz, and the maximum displacement (mm) of the structure for the five cases of the

load hypothesis in Model 1 and Model 2, respectively. Both the tension (T) and compression (C)

maximum values are given for each stress component. Depending on the load case type, the

maximum displacement is given for one or two directions.

σx σy σz σxy σxz Uz Uy

Model 1 MPa MPa MPa MPa MPa mm mm

T C T C T C T C T C

A) Weave + Tape

Case A 5.107 10.374 5.742 10.577 13.400 14.251 0.800 0.217 0.585 0.592 NG 6.686

Case B 7.915 12.052 6.602 12.116 15.323 16.302 0.915 0.234 0.662 0.670 NG 13.837

Case C 25.544 25.750 6.405 11.867 15.103 16.089 0.909 0.734 0.660 0.910 54.768 7.563

Case D 10.786 21.551 11.862 21.803 27.597 29.357 1.648 0.433 1.198 1.212 NG 19.989

Case E 27.100 27.596 10.866 20.123 25.551 27.197 1.532 0.730 1.116 1.140 55.144 12.783

B) Tape

Case A 15.200 29.915 2.273 3.708 5.133 5.353 1.083 0.425 1.123 1.133 NG 6.876

Case B 17.490 34.558 2.618 4.254 5.872 6.126 1.237 0.480 1.269 1.282 NG 14.117

Case C 55.060 55.639 2.516 4.154 5.782 6.037 1.228 0.813 1.269 1.295 53.852 7.776

Case D 31.095 61.951 4.700 7.650 10.574 11.029 2.229 0.870 2.298 2.320 NG 20.442

Case E 58.450 59.678 4.274 7.047 9.783 10.209 2.072 0.813 2.143 2.179 54.315 13.143

Table 4.22: Stress and displacement results for Model 1


σx σy σz σxy σxz Uz Uy

Model 2 MPa MPa MPa MPa MPa mm mm

T C T C T C T C T C

A) Weave + Tape

Case A 4.379 8.432 4.201 8.089 10.538 11.335 0.850 0.393 0.462 0.465 NG 5.755

Case B 7.023 9.785 4.830 9.255 12.049 12.963 1.022 0.379 0.522 0.526 NG 12.083

Case C 41.885 46.247 12.168 12.592 11.876 12.592 2.918 3.436 0.520 1.365 61.596 6.509

Case D 9.290 17.505 8.678 16.673 21.701 23.347 1.801 0.739 0.945 0.953 NG 17.378

Case E 43.613 48.789 12.322 15.218 20.091 21.631 2.818 3.618 0.880 1.400 61.977 11.002

B) Tape

Case A 12.755 25.937 1.577 2.888 4.256 4.469 1.195 0.547 0.938 0.974 NG 6.149

Case B 15.840 29.921 1.812 3.313 4.868 5.113 1.437 0.530 0.020 0.020 NG 12.858

Case C 92.593 102.480 2.674 3.231 4.795 5.043 3.777 4.469 1.059 1.268 60.522 6.951

Case D 26.093 53.671 3.251 5.958 8.767 9.206 2.532 1.031 1.920 1.940 NG 18.516

Case E 96.663 108.540 2.951 5.483 8.113 8.525 3.557 4.723 1.790 1.822 60.574 11.749

Table 4.23: Stress and displacement results for Model 2

From observation of the above results, the largest displacement change experienced occurs for

in the y-direction (towards the rail line) caused by Case D. For both Model 1 and Model 2

subjected to load Case D, the displacement increases 3.14 times from that experienced for load

Case 1 of tables 4.9 and 4.10 However, this increase is still relatively small with only a

magnitude change of 13.942 mm and 12.636 mm for Models 1 and 2, respectively.

The introduction of the cantilever and contributing loads from the catenary assembly creates

new load conditions on the structure. Recalling the loading-reaction diagram of figure 4.35, the

resultant forces from the post-cantilever connection can be represented as those in the figure

below where T is the horizontal strut force and FH and FV represent the horizontal and vertical

force components at the pin joint of the cantilever. Both the horizontal forces T and FH are

applied as pressures over a region of elements that simulate the post-cantilever connection

shown in figure 4.43. Often such loads can be applied as forces directly to a node set however

local stress concentrations can occur as a result. On the other hand, the application of the

equivalent pressure over a set of elements reduces this undesired stress spiking effect. Finally,

the vertical force component FV is applied to the nodes that make up the element set on which

the FH force component is applied as a pressure.


Figure 4.43: Stress concentrations (σx) at post-cantilever connection

The application of loading combinations in directions parallel and perpendicular to the rail line

causes the post to suffer a twisting effect as can be observed from the stress distribution (σx)

diagram of the structure below in figure 4.44. Such loading combinations are found in Case C

and Case E where the wind pressure is parallel to the rail line and the self-weight of the

catenary assembly causes a moment about the z-axis at the top of the structure. The stress

variation across the post section is captured graphically in figure 4.45. As before, the section is

taken 40 mm above the fixed end making again the sketch of the section in figure 4.13

applicable in this instance. The stress path demonstrates the effects caused by the twisting in

the section where the stress variation is not symmetrical about the centre of the post’s width as

was observed for load Case 2 in figure 4.15.

T

FH

Fv


Figure 4.44: Stress distribution of Model 1 post for load Case C

Figure 4.45: Sectional stress variation of Model 1 post for load Case C

0 200 400 600 800 1000-60

-40

-20

0

20

40

60

Length (mm)

Str

ess

(MP

a)

Stress Variation in Post Section for Case C of Load Hypothesis

A) Weave & Tape

B) Tape


4.8 Design and Analysis of a Proposed Steel Moment Base

The proceeding section details the intermediate design-analysis of the moment resistant base

accompanying the composite post structure. This type of mechanical fixation was chosen above

other types for the following reasons

� New concrete foundation not required

� Rapid mounting and dismounting process of post

� Relatively light, can be carried manually

� Relatively cheap fabrication costs

� Reusable

� Can be designed and analysed with a high level of accuracy.

As briefly described in Section 4.1.2, the proposed composite post differs from the original steel

post in terms of its method of fixation which subsequently alters the overall height of the

structure and the boundary conditions imposed. The proposed structural fixation consists of

mounting the post onto the top of the original post’s concrete foundation by the steel moment

resistant base. The base itself is secured to the concrete pad and to the composite post by bolt

connections. The model brings together three different element types with the motivation for

their implentation and their description outlined below.

4.8.1 Proposed Base Description

The base’s preliminary design is based on the classical moment resistant base design for post

type structures where overturning of the structure is impeded by plate components fixed

transversally to the axis of overturning. Carbon steel S275JR similar to the original post, is the

material of choice because of its mechanical properties as outlined previously in table 3.3 of the

original steel post analysis. Steel was also chosen for the relative ease in which it can be

fabricated into complex mechanical structures including. Figure 4.46 shows an image of the

proposed base structure developed in the numerical model. Also included are the separate area

components which determine the required meshing characteristics in the latter stages.


Figure 4.46: Image of proposed steel base taken from the numerical model

The geometry of the structure is shown as CAD drawings below in figures 4.47 to 4.49. The

main features of the base include a flat plate in contact with the surface of the foundation,

vertical thin-walled section at a height of 200 mm and an inner section with the ‘equivalent’

outer dimensions of the post structure. The word equivalent is used with caution as the inner

dimension of the vertical walls of the base are fabricated to the same outer dimensions of the

post plus a tolerance. This extra-dimensional tolerance factors in discrepancies that come about

during fabrication, these include residual stresses and therefore deformations experienced in

the steel base caused during fabrication (welding and cutting), and deformations and

curvatures in the composite post which can occur during the curing process as a result of

unsymmetrical laminates, and fibre alignment discrepancies.

As already mentioned above, the base includes plate components connected at their thickness

to the vertical wall face and to the horizontal plate. The majority of moment resistance in the

structure is provided by these plates. In terms of dimensions, the plate component has a width

of 75 mm which tapers at one side inwards towards its height of 150 mm. These moment plates

are fabricated such that the taper does not reach its bottom or top. This ‘blunting’ of the taper

removes the likely occurrence, during loading, of stress concentrations of infinitesimal

magnitudes caused by severe geometric discontinuities. Bolt holes in the horizontal plate are

situated outside the section wall and between the moment plates. In total, there are ten bolt

holes in the horizontal plate. The bolt holes that connect the base and post are situated in the

base’s vertical walls spaced between the moment plates. Each of the longitudinal and

transversal walls consists of four and two bolts holes, respectively.


Figure 4.47: Longitudinal view of moment resistant base

Figure 4.48: Transversal view of moment resistant base

Figure 4.49: Plan view of moment resistant base


4.8.2 Finite Element Model

The steel base is considered as a solid structure. The SOLID45 element is therefore

implemented as it is appropriately used for the 3-D modelling of solid structures. This solid first-

order element is defined by eight nodes having three degrees of freedom at each node:

translations in the nodal x, y, and z directions. The elements attributes include plasticity, creep,

swelling, stress stiffening, large deflection, and large strain capabilities. Figure 4.50 depicts the

solid element with its local and surface coordinate systems included.

Figure 4.50: 8-node SOLID45 element

As can be seen in the above figure, the geometry of the element is defined by its eight nodes

with its coordinate system directly related to the configuration of the nodes. Accuracy of the

model is enhanced by solving the system by full integration (KEYOPT(2) = 0) with extra

displacement shapes (KEYOPT(1) = 0) although it is more CPU intensive. The full integration

analysis allows the capture of bending behaviour of single layer elements [8].

4.8.3 Model Development Method

Before the model can be analysed, a number of assumptions and limitations are defined at its

design stage. Complexity arises in attempting to best approximate the boundary conditions of

the structure, i.e. the base’s interaction with the foundation and post. The composite post

structure is therefore modelled in the analysis with its relevant boundary conditions included,

i.e. wind loading over its free surface. The inclusion of a second mechanical structure in the

design-analysis stage introduces a number of additional issues that require clarification. These

are dealt with by means of imposing assumptions and limitations onto the model. These include


• Post and base are completely adhered together

• Post width does not taper within the height of the base’s vertical walls

• Bolts on vertical wall (base-post connection) removed from analysis

• Fillets and welds are not accounted for in the base model geometry

• Foundation-base connection bolts not simulated as solid elements.

The problem of contact caused by bordering elements of distinct identities is omitted from the

numerical analysis. As a result, the associated non linearities are not considered and the overall

computational time is reduced. The elements of both the post (shell) and base (solid) in contact

within the base wall are assumed to be adhered with one another thereby sharing equivalent

transformations and rotations while retaining their own material properties within their

respective geometries. Note that contact in italics refers to the physical state and not to the

numerical problem. In order to maintain complete adherence between both sections, the

tapered width of the post is rendered vertical for its bottom 200 mm which is within the base’s

walls. Above this the post begins to taper towards its final width of 200 mm at the top.

However, this limitation creates a new section variation along the post’s length, thereby

changing its overall behavioural response from the analysis made for Cases 1 and 2 in tables 4.9

and 4.10. In continuation of the above assumptions, the bolts on the base’s vertical walls are

not considered. The connection between both structures is achieved by merging their

bordering nodes. Figure 4.51 shows the geometry of the numerical model before and after the

bolts are removed from the analysis.

Figure 4.51: Bolts on vertical walls removed due to merging of both solid and shell elements


As this design-analysis is considered to be at an intermediate stage, modelling details such as

fillet and weld components are not taken into account in the geometry due to their time-

consuming formation. The omission of such components creates stress concentration issues at

the areas of geometric discontinuity. Such behavioural responses occurring in the results stage

of the analysis are distinguished and recognised.

Finally, the issue of modelling the bolt assembly arises at the base-foundation connection.

Figure 4.52 shows two typical base connection and bolt assembly relevant to this design-

analysis.

Figure 4.52: Sectional view of bolt assembly for base-foundation connection

There are numerous methods to simulate the bolt assembly which include a complete solid

bolt, hybrid bolt, spider bolt, Rigid Body Element (RBE) bolt, coupled bolt, and no-bolt

simulation, with each method varying in their degree of accuracy. The most accurate approach

is to model the complete 3D solid bolt assembly however, with regards to the present analysis,

a large number of bolt assemblies are required (10 in total) and as a result, modelling of solid

bolts is impractical. Therefore, the analysis of the bolt assembly is carried out using line

elements (LINK10) and coupled nodes. This method reduces drastically the number of elements

required in the bolt assembly analysis with the LINK10 element adequately simulating the nut.

The typical bolt joint assembly and its numerical model equivalent are shown in figure 4.53.


Figure 4.53: Numerical model representation of bolt assembly

The figure above shows the bolt element (LINK10) passing through the bolt hole and coupled to

surface nodes that represent the contact area between the head of the bolt and the base plate.

The LINK10 element is a 3-D spar element with three degrees of freedom at each node

(translations x, y, and z directions). The element simulates the nut of the bolt assembly which

has uniaxial tension capabilities only. The head is represented by coupled nodes. Coupling is

carried out between the outermost node of the LINK10 element (master node) and the

circumferential nodes of the flange at the hole (slave nodes). These circumferential nodes are

created by defining separate circular areas around the holes during the model development

and are mapped meshed. The bottom node of the link element simulates the point at which the

bolt enters the foundation. The displacement at this node is impeded.

In order to perform the iterative process of design, a number of the design drivers were defined

parametrically in the numerical input code of the model. Design drivers of the base structure

include the thickness of the horizontal base plate, the walls, and the moment plates which are

all defined separately. Also included as parameters are the bolt hole size and the area of the

bolt head in contact with the base. After completing the iterative design process, it was

determined that for the analysis stage the thickness parameters of the horizontal base plate

and the walls/moment plate would be equal to 9 mm and 7 mm, respectively.

Master Node

Base surface in

contact with bolt

Link Element

Slave Nodes

Plate thickness


Taking into account that the steel base structure has two planes of symmetry, it is initially

modelled as one-quarter of its entire size. Due to the base’s isotropic properties, the REFLECT

command is adequately used in terms of overall time reduction in the geometric and meshing

development of the solid element. Figure 4.54 describes the progression of the model’s

development from one-quarter to complete structure using the REFLECT command.

Figure 4.54: One-quarter and complete structure

However, the REFLECT command is not utilised for the shell element structure (i.e. the

composite laminates of the post) due to its anisotropic nature. This command would create

conflicting lamina configurations at areas either side of the planes of symmetry. In order to

merge the nodes of both element types at the interface of the base/post, the elements of both

structures must be identical in the areas of contact. To achieve this, the complete base

developed and meshed, a copy of the keypoints at the base’s internal vertical surface (in

contact with composite laminates) is made at their same location. The shell areas and their

orientation are defined by these new keypoints and their sequence (i, j, k, l). The newly meshed

SHELL181 elements are dimensionally identical as to those of their corresponding SOLID45

elements. Figure 4.55 shows firstly the complete meshed base model (grey) and secondly the

coinciding meshed shell areas (green) of the post that adjoin the interior faces of the base’s

walls.


Figure 4.55: Complete meshed base model and coinciding meshed shell areas of the post

Figure 4.56 shows the meshed base and included in its interior is the section of the meshed

post that is merged with the base which effectively anchors the post. Also shown are the

stacked layers that create the UPN beam and plate laminate components of the post.

Figure 4.56: Meshed base plus meshed post section joined to base

Meshed shell areas of post


The remaining length of the composite post is created in the same manner as described

previously in the method statement of Section 4.2.2. As the base structure is of the most

importance in this analysis, the unconstrained post length is coarsely meshed so as to

reduce computational time. While the meshing maybe relatively coarse, the response

obtained remains at a high degree of accuracy in its approximation due to the full

integration option for the SHELL181 element used in the analysis (KEYOPT(3) = 2). Figure

4.51 shows a segment of the coarsely meshed unconstrained post (green) connected to the

finely meshed post segment contained within the base structure.

Figure 4.57: Base-post connection for numerical model

4.8.4 Boundary Conditions and loads

The loading applied in this analysis consists of the two distinct load-direction cases in table 1. As

the loading simulates wind, it is also applied over the relevant areas of the base structure as

well as the post. The boundary conditions at the base-foundation connection (plate component

at y-z plane) vary with the case type. The boundary conditions for each case are divided into

two situations which are defined by their location either side of the axis of rotation. The base

area on the side at which the load is originating from can be defined as the ‘lift’ zone as the

base plate in this area is simulating separation from the foundation while the other area is

pressing down onto the foundation. The base plate area segments of the ‘lift’ zone are not

constrained allowing them to translate and rotate as is the case in the actual base-foundation

connection. The area segments on the other side of the axis of rotation are constrained in


translation and rotation so as to prevent them ‘pushing’ down into the foundation. As

previously described, all the bolts are constrained by the bottom node of the LINK10 element.

4.8.5 Results

The most significant stress and displacement results are shown in table 4.24 below. They

include principal stress σ1, σ2, and σ3, and Von Mises equivalent stress.

Case σ1 σ2 σ3 Von Mises Displacement

MPa MPa MPa MPa mm

1 87.869 41.847 27.174 75.461 0.030

2 232.52 102.10 72.910 196.30 0.110

Table 4.24: Principal stresses maximums and equivalent Von Mises stresses

The final bolt assembly dimensions are shown in figure 4.58 and were determined from the

British Standard Whitworth (BSW) system. The bolt is of Grade A2/A4 - DIN 965 – DIN EN ISO

7046-2. The assembly includes also a lock nut which resists loosening under vibration by locking

the first nut in position. An alternative method to prevent loosening is to utilise only one nut

which has a nylon sleeve insert. As the nut is fed and tightened onto the bolt the thread cuts

into the nylon thereby holding the assembly together by means of friction.

Figure 4.58: Bolt assembly dimensions (Whitworth System)

Table 4.25 lists the reaction forces of the bolts for both load cases. A slight pretension is applied

to the bolts before the solution phase of the numerical model so as to simulate a tight bolt

assembly. The results are given as reaction force and tonnes.


Load Case 1 Case 2

Bolts Reaction Force Tonnes-Force Reaction Force Tonnes-Force

N T N T

1 33.922 0.00347 36.143 0.00369

2 33.736 0.00344 36.142 0.00369

3 34.473 0.00351 1086.3 0.11077

4 33.922 0.00346 5874.3 0.59901

5 33.736 0.00344 14832.0 1.51244

6 2348.0 0.23943 14832.0 1.51244

7 2413.6 0.24612 5872.7 0.59885

8 7649.3 0.78001 1085.9 0.11073

9 2414.5 0.24621 36.143 0.00369

10 2348.5 0.23948 36.142 0.00369

Table 4.25: Bolt traction force reactions and equivalent force in tonnes

The maximum reaction force is experienced in Case 2 its equivalent tonnes-force equal to 1.512

T. A Factor of Safety of 1.8 is applied to this maximum reaction force giving a value of 2.7216 T.

The maximum tensile force applicable to the bolt specified is 3.01 T implying that bolt election

satisfies the design-analysis.

In viewing the results in table 4.25, the most critical stresses are deemed to occur due to load

Case 2. The maximum stress values given are caused by stress concentrations occurring at the

bolt head connection and to a lesser magnitude, at the vertical wall corner and horizontal base

plate connection, both of which occur in the ‘lift’ zone of the structure.

Figure 4.59: Stress concentrations located at bolt head-base plate contact area


Figure 4.60: Stress concentrations located at wall-base plate connection

While it is accepted that both types of stress concentrations occur at these regions, they are

however viewed as being overestimations of the actual stresses incurred. The overestimation of

the stress magnitudes are a result of a number of the limitations outlined at the beginning of

the base design. Two of which are of most relevance are recalled as being

• Foundation-base connection bolts not simulated as solid elements

• Fillets and welds are not accounted for in the base model geometry.

The stress response for both limitations is shown above in figures 4.59 and 4.60. The method of

the nodal coupling used to represent the bolt assembly has its limitations. All translations and

rotations of the coupled nodes which represent the bolt head in contact with the base are

completely constrained. This in turn causes high local stresses in its proximate unconstrained

region. The severe change in geometry at the wall-base plate interface of the model is reduced

by the presence of welded connections between the two.

Figures 4.61 and 4.62 show the stress distribution and deformed shape of the base structure for

load Case 2. It must be highlighted that the displacement scale is amplified and the stress range

is reduced in order to appreciate the behavioural response. A number of interesting

observations can be made from figure 4.55 which include the sectional shape of the walls and

the response of the moment connections. The inward deformation of the bottom wall is

connected to the composite post width in compression while conversely the top wall is adhered

to the post area in tension. Consequently, the outer moment plates connected to the

longitudinal walls are forced into flexure while the centre moment plate, which coincides with


the symmetrical plane, does not contribute any significant behavioural response. This overall

sectional response coincides with the type of behavioural response experienced by narrow

beams where the axial strain distribution in the post gives rise to a significant amount of

deformation of the cross-section.

Figure 4.61: Behavioural response in load Case 2 (amplified displacement)

Figure 4.62: Behavioural response in load Case 2 (reduced VM stress range)


Figure 4.63 shows the behavioural response of the base structure for load Case 1 which again

contains an amplified displacement scale and a reduced stress range.

Figure 4.63: Behavioural response in load Case 1 (amplified displacement)

Chapter 4 Design and Analysis of the Proposed Composite...

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