Part 9 - Estimation of magnitude of the unbalanced centrifugal forces driving tectonic movement pptx

Part 9 - Estimation of magnitude of the unbalanced

centrifugal forces driving tectonic movement

Based on the above observations, let us assume as a working hypothesis that the Earth can be

modelled as a rotating body where the centre of mass is offset from the principal axis of rotation.

For the purposes of this paper the author will consider the two principal approaches to determine

the circumferential forces associated with an unbalanced rotating body

Model 1. Rigid body dynamics

As discussed in Section 5, the Pacific plate has all the appearances of being in compression while

the almost diametrically opposed African plate appears to be subject to tensile forces

The simplest model is to consider the Earth as an eccentrically rotating solid body such as an

unbalanced flywheel. Although this model (shown in Figs 8 & 9 and enumerated in Appendix 1)

accounts for the compressive and tensile stresses developed in the outer rim, it does not describe

the unbalanced centripetal forces which the author believes to be linked to tectonic forces

resulting in plate movement.

Fig 8 below. Force/Vector diagram showing the centripetal force P and the circumferential or

tensile force F in the outer rim of a solid rotating body. Rotating Machines are designed to

ensure that the developed circumferential stress F, is less than the design hoop stress.

Fig 9 below. Force/Vector diagram showing the differential circumferential stress induced in the

outer rim of a solid rotating body, whose centre of mass is not co-incident with the principal axis

of rotation. Under these conditions the tensile forces are increased on the 'heavier' side and are

decreased on the 'lighter' side. Thus the hoop stress in the rim will be in tension on the 'heavier

side' and in compression on the 'lighter side'.

Model 2. Outer rim is able to slide relative to main body

In order to determine the magnitude and direction of the forces postulated as being responsible

for tectonic movement the model used in one in which the thin crust is able to slide relative to

the solid body at the crust /mantle interface. By way of illustration Fig. 10 shows that if an

unbalanced disc with an outer annular ring containing fluid is rotated about its principal axis, the

liquid will move to the ‘lighter’ side. Fig. 11 shows an analogous situation with the sliding

continental plates.

If we consider the crust as being able to move relative to the mantle, albeit it over a long

geological time scale, then a simple force diagram (Fig 12) can be constructed by making the

following assumptions: the crust is a thin shell that is able to slide relative to the mantle, the

forces due to eccentricity are superimposed on the stress caused by the general rotation and

gravity, and the stress that is of interest for the purposes of tectonic movement is the differential

stress due to this eccentricity.

By approaching the problem in terms of a thin shell moving relative to the mantle, it is possible

to consider what increments of the tensile force are responsible for putting the Pacific basin

under compression (note crumpled profile) and the African Plate under tension. (note the Rift

Valley). The calculations to derive the expression of the circumferential stress at the surface of

the earth, which are based on the consideration of the eccentrically induced loads on the thin

crust are detailed in Appendix 11. The term radius of eccentricity’ was introduced to denote the

distance between the centre of mass and the major axis of rotation.

From Appendix 2 the following relationship was derived:

Eq.2 shows that the magnitude of the circumferential forces or in this case the derived

circumferential stress is dependent on the distance between the geometric centre and the centre

of mass, i.e. the ‘radius of eccentricity’. In a limiting case, if the ‘radius of eccentricity’ is zero,

the rotating body will be balanced and the net force will be zero. Fig 13 shows this relationship

in between F (circumferential stress) and the E (radius of eccentricity).

Fig. 13 below. Relationship between the radius of eccentricity and the circumferential stress

Having derived an equation which relates the circumferential stress with the rotational velocity

and the centre of eccentricity it would seem appropriate to consider the possibility of tectonic

activity on the planet Venus. Due to the low peripheral velocity of Venus (1 Revolution in 243

days = 6.5 kmhr -1 ) as compared with 531.5 kmhr -1 on Earth, the centrifugal forces available as

compared to the similar sized planet Earth will be in the ratio of (42.5) 2

/ (531.5)2 =

1806.25/282492.25 = 0.006: 1. This would give a stress value of 3.9x10-3 Nmm-2 . (0.059 psig).

The unbalanced centripetal forces thus needed for tectonic activity are negligibly small.

In order to better understand the magnitude of the calculated circumferential stress in the

continental crust, it is helpful to relate the model to more familiar applications. (These are shown

pictorially in Cartoons 2, 3 &4) The stress value of 7.29x10-2 Nmm-2 if applied to a 1 tonne

braked motor vehicle with a rear surface area of 1000 mm x 1300 mm=1.3x106

mm2

will yield a

push force 94770 N. In imperial units this equates to a push of 21305 lbf (pound force) or 9.5

tonf (ton force). Rounded up and put more simply, this equates to the vehicle being pushed by

118 people each of whom weighs 180 pounds (81.8 kg) (see Cartoon 1). As the incline between

the height of the Andes (taken as 5 km) and distance between the Peru- Chile trench and the

Cordillera –Real (taken as c.1000 km) is c. 1:250, the vehicle can be considered to be on a level

surface for purposes of scaling. However a 3 tonne hoist will easily pull the vehicle up a 1:3

incline onto a pick-up truck It is also worth noting that an upward acting net force of 2.37 x 10-2

N/mm2 (3.5 psig) on a 60 metre long wing span of an aircraft is sufficient to keep a large 350

tonne aircraft flying.

A puff of wind with dynamic pressure as low as 0.135 x 10-2 N/mm2 (0.2 psig) acting on the

large surface area of a ship’s sail will cause a boat to move across water. Thus the unbalanced

centrifugal forces created by placing the centre of mass of the Earth 1 km off centre are large and

cannot be ignored. The calculated circumferential forces if applied to the cross sectional area of

the South American plate are more than sufficient to push it over the Nazca plate.

What is important to remember is longitudinal stresses are half as much

as the circumferential stresses. Therefore, we can say that longitudinal

strength is twice as strong as circumferential strength. This is only true

for illustration purposes

LECTURE 15

Members Subjected to Axisymmetric Loads

Pressurized thin walled cylinder:

Preamble : Pressure vessels are exceedingly important in industry. Normally two types of

pressure vessel are used in common practice such as cylindrical pressure vessel and spherical

pressure vessel.

In the analysis of this walled cylinders subjected to internal pressures it is assumed that the radial

plans remains radial and the wall thickness dose not change due to internal pressure. Although

the internal pressure acting on the wall causes a local compressive stresses (equal to pressure)

but its value is neglibly small as compared to other stresses & hence the sate of stress of an

element of a thin walled pressure is considered a biaxial one.

Further in the analysis of them walled cylinders, the weight of the fluid is considered neglible.

Let us consider a long cylinder of circular cross - section with an internal radius of R 2 and a

constant wall thickness‘t' as showing fig.

This cylinder is subjected to a difference of hydrostatic pressure of ‘p' between its inner and

outer surfaces. In many cases, ‘p' between gage pressure within the cylinder, taking outside

pressure to be ambient.

By thin walled cylinder we mean that the thickness‘t' is very much smaller than the radius Ri and

we may quantify this by stating than the ratio t / Ri of thickness of radius should be less than 0.1.

An appropriate co-ordinate system to be used to describe such a system is the cylindrical polar

one r, q , z shown, where z axis lies along the axis of the cylinder, r is radial to it and q is the

angular co-ordinate about the axis.

The small piece of the cylinder wall is shown in isolation, and stresses in respective direction

have also been shown.

Type of failure:

Such a component fails in since when subjected to an excessively high internal pressure. While it

might fail by bursting along a path following the circumference of the cylinder. Under normal

circumstance it fails by circumstances it fails by bursting along a path parallel to the axis. This

suggests that the hoop stress is significantly higher than the axial stress.

In order to derive the expressions for various stresses we make following

Applications :

Liquid storage tanks and containers, water pipes, boilers, submarine hulls, and certain air plane

components are common examples of thin walled cylinders and spheres, roof domes.

ANALYSIS : In order to analyse the thin walled cylinders, let us make the following

assumptions :

• There are no shear stresses acting in the wall.

• The longitudinal and hoop stresses do not vary through the wall.

• Radial stresses sr which acts normal to the curved plane of the isolated element

are neglibly small as compared to other two stresses especially when

The state of tress for an element of a thin walled pressure vessel is considered to be biaxial,

although the internal pressure acting normal to the wall causes a local compressive stress equal

to the internal pressure, Actually a state of tri-axial stress exists on the inside of the vessel.

However, for then walled pressure vessel the third stress is much smaller than the other two

stresses and for this reason in can be neglected.

Thin Cylinders Subjected to Internal Pressure:

When a thin – walled cylinder is subjected to internal pressure, three mutually perpendicular

principal stresses will be set up in the cylinder materials, namely

• Circumferential or hoop stress

• The radial stress

• Longitudinal stress

now let us define these stresses and determine the expressions for them

Hoop or circumferential stress:

This is the stress which is set up in resisting the bursting effect of the applied pressure and can be

most conveniently treated by considering the equilibrium of the cylinder.

In the figure we have shown a one half of the cylinder. This cylinder is subjected to an internal

pressure p.

i.e. p = internal pressure

d = inside diametre

L = Length of the cylinder

t = thickness of the wall

Total force on one half of the cylinder owing to the internal pressure 'p'

= p x Projected Area

= p x d x L

= p .d. L ------- (1)

The total resisting force owing to hoop stresses sH set up in the cylinder walls

= 2 .sH .L.t ---------(2)

Because s H.L.t. is the force in the one wall of the half cylinder.

the equations (1) & (2) we get

2 . sH . L . t = p . d . L

sH = (p . d) / 2t

Circumferential or hoop Stress (sH) =

(p .d)/ 2t

Longitudinal Stress:

Consider now again the same figure and the vessel could be considered to have closed ends and

contains a fluid under a gage pressure p.Then the walls of the cylinder will have a longitudinal

stress as well as a ciccumferential stress.

Total force on the end of the cylinder owing to internal pressure

= pressure x area

= p x p d2

/4

Area of metal resisting this force = pd.t. (approximately)

because pd is the circumference and this is multiplied by the wall thickness

Assessing the Biomechanical Impact of Medical Devices

Using MSC.Software at Texas A&M University

Contest

Winner

By Lucas H. Timmins, PhD Student, Texas A&M University, College Station, TX

The Vascular Biomechanics Group in the Department of Biomedical Engineering at Texas A&M

University investigates the role of biomechanics in the treatment of vascular disease. In

particular, we use MSC.Software to investigate the solid mechanical implications of implanting

vascular stents and aortic stent grafts (Figure 1).

Figure 1: A. Vascular stents on balloon catheters. B. Aortic stent grafts

Cardiovascular diseases are the principal cause of death in the United States, with total cost

associated with the disease in upwards of $450 billion USD. As a result, an extensive effort has

gone into developing biomedical devices to treat these diseases. As the placement of these

devices alters the biomechanical environment inside an artery, proper computational mechanical

analysis must be carried out to ensure their treatment efficacy, but also to examine the

mechanical loads that they place on the arterial wall. MSC.Software allows for proper analysis of

these inherently difficult (e.g. non-homogeneous, non-linear) contact mechanics problem.

MSC.Software allows for construction and simulation of vascular stent implantation to examine

the mechanical implications of varying specific design parameters on the arterial wall mechanics

(Figure 2). Through the modeling of biological tissues and the application of physiologic

boundary conditions, appropriate finite element models can be developed in MSC.Patran and

solved with MSC.Marc.

Figure 2: Stent Geometry: MSC.Patran models of vascular stent designs created by varying

specific design parameters (f,ρ,h). The designs on the right are approximately 5 and 20 mm

in outer diameter and length, respectively. Computational analyses of these models were

then carried out to assess their biomechanical impact on the arterial wall.

Figure 3: Stented Artery Model: An MSC.Software simulation illustrating the

circumferential stress distribution on the arterial wall after modeling stent implantation.

Patient specific computational models of abdominal aortic aneurysms (AAA) are also

investigated in our laboratory to assess treatment methods and/or device design. MSC.Patran

provides an excellent means of constructing finite element models from various medical imaging

techniques to examine the biomechanical environment after aortic stent graft implantation.

Figure 4: Aortic Stent Graft Model: A. Computational mesh of an AAA with different

biological tissue components B. Max principal strain field on outer vessel wall.

Through proper computational analysis, which MSC.Software provides, our laboratory is able to

examine the mechanical impact of implanted vascular medical devices. Such analysis, along with

clinical evidence of device success/failure, can provide information in optimizing biomedical

device designs and treatments.

For further information, please refer to the following publications:

Bedoya, J., Meyer, C.A., Timmins, L.H., Moreno, M.R., Moore, J.E. Effects of Stent Design

Parameters on Normal Artery Wall Mechanics. J Biomech Eng. 2006; 128:757-675.

Timmins, L.H., Moreno, M.R., Meyer, C.A., Criscione, J.C., Rachev, A., Moore, J.E. Stented

Artery Biomechanics and Device Design Optimization. Med Biol Eng Comput. 2007; 45:505-

513.

Thin Walled Cylinders Under Pressure.

The three principal Stresses in the Shell are the Circumferential or Hoop

Stress; the Longitudinal Stress; and the Radial Stress.

If the Cylinder walls are thin and the ratio of the thickness to the Internal

diameter is less than about 1/20 then it can be assumed that the hoop

and longitudinal stresses are constant across the thickness. It may also

be assumed that the radial stress is small and can be neglected. In point

of fact it must have a value equal to the pressure on the inside surface

and zero at the outside surface. These assumptions are within the

bounds of reasonable accuracy.