2. Chapter 2 - Example Applications _a4

8/8/2019 2. Chapter 2 - Example Applications _a4

http://slidepdf.com/reader/full/2-chapter-2-example-applications-a4 1/11

Chapter 2 Example Applications______________________________________________

14

CHAPTER 2

EXAMPLE APPLICATIONS

This chapter deals with some FEM applications in various fields of civil

engineering. Without any details regarding the modeling procedures, thegraphical representations of the virtual models and results are quite

suggestive, rendering evident the similarity with the real structures (or

domains) subjected to engineering analyses.

2.1 EXAMPLE 1

The first example shown in figure 2.1 is the finite element model of a 3D

frame structure, used for a removable stage roof. The structure is made of

steel beams and trusses, as braced circular arches. The envelope membrane,

made in light and flexible material, is attached to the frame structure in

several connecting points. The structure withstands its own weight, the

electrical equipment weight and the wind load (applied on the membrane).

The main goals of the finite elements analysis is to evaluate the structural

displacements and stresses due to various loading hypothesis. The results

are used to optimize the steel components cross sections and the assembling

details. The local and global structural stability is checked for different

assigned geometrical properties of components.

The reaction forces developed in the constrained base points are of utmost

importance for assessing the necessary ballast, in order to avoid the

structural lifting and sliding, due to the horizontal wind load.

An example of results representation is shown in figure 2.2. The axial forcediagram is drawn on each structural component, emphasizing the tension

and compression regions, as well as the associated force values.



______________________Basics of the Finite Element Method Applied in Civil Engineering

15

Fig. 2.1 Three dimensional finite element model for removable stage roof

Fig. 2.2 Axial force diagrams due to gravity and transversal wind load (N)




16

2.2 EXAMPLE 2

The second example refers to a dam-foundation interaction analysis, in the

design process of a gravity dam. The stress distribution over the dam body

and the structural stability is related to the dam’s cross section. The contact

stresses between the dam and the foundation layer are also significant for

design and in direct relationship with the dam’s geometry. As a massive

structure, the 3D effects should be taken into account.

Thus, the 3D model shown in figure 2.3 includes two main components: the

dam body, divided into separate blocks by joints with potential relative

movement, and the surrounding rock mass, following the geometry of the

valley. The different material properties of the dam body and the rock massare accordingly assigned.

Fig. 2.3 Three dimensional finite element model of a gravity dam and its

foundation rock mass

In figure 2.4 the horizontal displacement distribution is shown over the

deformed shape of the structure, due to water pressure applied on theupstream face of the dam. The displacement scale is larger then the

geometrical representation scale, in order to emphasize the joints behavior.




17

Fig. 2.4 Deformed shape of the dam and horizontal displacements due to upstream

water pressure (m)

Fig. 2.5 Principal stress field σ 3 due to water pressure and earthquake (pseudo-

static load) (KN/m2 )

Upstream view

Downstream view




18

The principal compression stress distribution σ 3 is represented on bothupstream and downstream faces of the dam in figure 2.5. These results are

the consequence of inertia loads due to the dam’s own mass and the

dynamic water pressure applied during an upstream-downstream

earthquake.

Internal stresses can be represented using cross sections throughout the

finite element model. The main cross section corresponding to block no. 9

(figure 2.6) shows the detail of the drainage outlet gallery. The figures in

the legend besides the drawing are values of the vertical stress σ z calculated

in the horizontal loading hypothesis (neglecting gravity loads). The span

between the minimum and maximum values is divided into 9 equal

intervals, each corresponding to a different color over the cross section.Sometimes, local extreme or abnormal values may be ignored after a careful

understanding of the discrete solution.

Fig. 2.6 Vertical stress field distribution σ z due to water pressure and earthquake

(pseudo-static load (KN/m2 )

2.3 EXAMPLE 3

Another type of analysis refers to seepage phenomena and water discharge

assessment, related to hydraulic structures. The results of such analyses mayexplain an unusual loss of reservoir storage and also suggest the best water

tightening solutions. The uplift pressures acting on the bottom of quasi




19

impervious structures, as well as the drainage effect can be evaluated. A 3Dmodel used for seepage analysis is shown in figure 2.7 (on purpose, only

half of the model is represented, to highlight the internal components of the

seepage domain). Each color corresponds to another material inside the

earth-fill dam or foundation layer, with a different permeability. Upstream

and downstream water levels are assigned as water head values. The

calculated distribution of water head inside the seepage domain (the

equipotential lines) is represented in the transversal cross sections shown in

figure 2.8. The difference between the two distributions is due to the grout

curtain effect. The other component of the flow net – the flow lines – may

be represented as oriented velocity vectors (which are always tangent to the

flow line). The elementary discharge (or flux), equivalent to the seepage

velocity, can be assessed and compared in various cross sections through themodel. By integrating these values over the areas, the total water discharge

is obtained. The assessment of uplift pressures (as difference between the

hydraulic head and the elevation), acting on the bottom of the hydraulic

structures is a preliminary step of any stability analyses.

Fig. 2.7 3D model of earth-fill dam and foundation used for seepage analysis(partial view)




20

a.

b.

Fig. 2.8 Hydraulic head distribution in main cross section with (a) and without (b)

grout curtain (m)

2.4 EXAMPLE 4

The following example represents probably the most common domain of

using the Finite Element Method in civil engineering: the building structuresdesign. The 3D model of a 5 storey block of flats, with a double basement

dedicated to parking places and a general foundation slab is represented in

figure 2.9. It is a typical reinforced concrete structure made of diaphragm

walls, columns, beams and horizontal floor plates. Static and dynamic

analyses are performed in order to calculate the maximum expected

displacements, on both vertical and horizontal directions. The envelope of

stresses, axial forces, shear forces and bending moments, acting on every

structural component are calculated. The finite element model can be easily

used in design optimization, because each geometric characteristic of the

model, such as beams and columns cross sections, floor or diaphragm

thicknesses, can be changed by simply assigning different numerical values.

For every set of geometrical characteristics, another structural response isattained. Some results as the vibration modes or displacements may even be

represented as animation pictures to emphasize in a suggestive way the




21

structural behavior. As examples of graphical representation, thelongitudinal (a) and transversal (b) displacements are shown in plan view in

figure 2.10. Both horizontal displacements are shown on the deformed

shape of the structure.

Fig. 2.9 3D model of a 5 storey building, double basement and foundation slab.Structural analysis.

a. b.

Fig. 2.10 Top view of lateral displacements due to seismic load (pseudo-static

analysis) applied on: (a) longitudinal direction; b) transversal direction (m).




22

2.5 EXAMPLE 5

The interaction between structural components and the surrounding rock or

earth mass is typical for underground structures, such as motorway, railroad

or hydraulic tunnels. The FEM represents an ideal tool for modeling the

excavation process, the stress and displacement field evolution inside the

rock mass and the tunnel lining for various loading hypotheses. The virtual

models are useful for design optimization, regarding the shape of the cross

section and the lining thickness. No other procedure allows taking into

consideration the rock mass heterogeneity, with its unsymmetrical effects

over the sustaining structure. Additionally, time-dependent response

dictated by the rheological properties of the ground may also be considered.

Modeling the initial stressing effects due to the anchorage system is also possible.

The last example refers to a 2D analysis associated to the monitoring system

of a hydropower plant cavern. The objective is to compare the recorded

displacements with the results of the numerical model, in order to draw out

conclusions regarding the fulfilling of safety requirements.

Fig. 2.11 Plain-strain analysis of a hydropower plant cavern




23

The 2D finite element model is represented in figure 2.11. The variousmaterial properties distribution is emphasized by different shadings. The

boundary condition symbols are represented around the contour of the

model. The horizontal pre-stressed anchors are also noticeable.

Fig. 2.12 Horizontal displacements due to the pre-stressed anchorage system (m)

Fig. 2.13 Initial horizontal stress distribution due to the anchorage system (KN/m2 )




24

The analysis objective is to determine the displacements of the concretevault, starting with the initial service conditions, by combining various

loading hypotheses due to rock mass relaxation, payload, rock failure zones,

thermal field, etc.

Some results concerning the peculiar load step assigned to the anchorage

system effect are represented in figures 2.12 and 2.13. Both horizontal

displacements and stresses are relative values, obtained for this load

combination only. In the results processing phase, they should be added to

the outcomes of other loading hypotheses, in order to draw out the envelope

of diverse effects.

Documents

2. Chapter 2 - Example Applications _a4