structural basics-centre of mass ,centre of curvature , stable ,unstable,neutral equilibrium

Centre of mass and centre of curvature source:http://www.mace.manchester.ac.uk/project/teaching/civil/structuralconcepts/Statics/mass/mass_mod7.php chk this website guys a lot of useful stuff and very very useful

This demonstration shows the relationships between the three states of equilibrium and three relative locations of the centres of mass to the centres of curvature of bodies.

      
a.                                            b.
Figure 2.A1: The models

Figure 2.A1 shows four small aluminum axially symmetric objects that have different dimensions and different relative locations of their centres of mass to the centres of curvature (The centre of curvature of a curve at any point is the centre of the circle which is tangent line at the point on the curve. If a line is drawn perpendicular to the curve at the point, the intersection point of the line and the vertical axis of symmetry is the centre of curvature):

    a. A round ball: The centre of mass and the centre of curvature of the ball are at the same point, the centre of the ball.

    b. A circular solid cylinder attached to a small part of a solid sphere: The centre of mass of the object is lower than the centre of curvature at any point on the spherical surface.

    c. A circular solid cylinder attached to a half of a solid sphere: The centre of mass of the object is higher than the centre of curvature at any point on the spherical surface.

    d. A hollow circular cylinder attached to a small part of a solid sphere: The centre of mass of the object is lower than the centre of curvature at any point on the spherical surface.

An experiment may be conducted as follows:

    a. Applying a small lateral force on the ball causes it move from its original equilibrium position (Figure 2.A2a). It moves to a new position and a new state of equilibrium (Figure 2.A2b). The original state of equilibrium is a position of neutral equilibrium.

      
a. Initial position of the ball                       b. New position of the ball
Figure 2.A2: Neutral equilibrium

    b. Applying a lateral force on the top of the object (b) rotates the object as shown in Figure 2.A3a. Releasing the force, the object returns to its original equilibrium position (Figure 2A3b). The original state of equilibrium is a position of stable equilibrium.

      
a. Object (b) with applied lateral force                       b. Object (b) returns to its original position
Figure 2.A3: Stable equilibrium

    c. Allying a force on the top as above but object (c) of the third object and holding it to the position from its original equilibrium position as shown in Figure 2.A4a. When the finger is removed the object falls over as shown in Figure 2.A4b. This original state of equilibrium is a position of unstable equilibrium.

      
a. Object (c) with applied lateral force                       b. Object (b) Object (c) topples over
Figure 2.A4: Unstable equilibrium

    d. Applying a lateral force on the top of object (d) to the position shown in Figure 2.A5a. When the force is removed, the object moves back to its original position. The original state of a position of stable equilibrium.

      
a. Object (d) with applied lateral force                       b. Object (d) returns to its original position
Figure 2.A5: Stable equilibrium

It is observed from this demonstration that the states of equilibrium relate to the relative positions of the centres of mass to the centres of curvature of bodies:

• If the centre of mass and the centre of curvature of a body are at the same point, the body is in a state of neutral equilibrium.
• If the centre of mass is lower than the centre of curvature of a body, the body is in a state of stable equilibrium.
• If the centre of mass is higher than the centre of curvature of a body, the body is in a state of instable equilibrium

structure basics for civil and mechanical

Centreofmassandstability source:http://www.mace.manchester.ac.uk/project/teaching/civil/structuralconcepts/Statics/mass/mass_mod5.php

This demonstration shows how the stability of a body relates to the location of its centre of mass and the size of its base.

       (a)                                                                      (b)        (c)                                                                      (d) Fig. 2-9: Centre of mass and stability of three aluminum blocks

Fig. 2-9a shows three aluminum blocks with the same height of 150mm. The square sectioned block and the smaller pyramid have the same base area of 29mm x 29mm. The larger pyramid has a base area of 50mm x 50mm but has the same volume as that of the square sectioned block. The three blocks are placed on a board with metal stoppers provided to prevent sliding between the base of the blocks and the board when the board is inclined. As the board is inclined, its angle of inclination can be measured by the simple equipment shown in Fig. 2-9b. Basic data for the three blocks and the theoretical critical angles calculated using Eq. 2-7 are given in Table 2-1. Theory predicts that the largest critical angle occurs with the large pyramid and the smallest critical angle occurs with the square sectioned block.

The demonstration is as follows:

1. The blocks are placed on the board as shown in Fig. 2-9 in the order of increasing predicted critical angle.
2. The left hand end of the board is gradually lifted and the square sectioned block is the first to become unstable and topple over (Fig. 2-9b). The angle at which the block topples over is noted.
3. The board is inclined further and the pyramid with the smaller base is the next to topple (Fig. 2-9c). Although the height of the centre of mass of the two pyramids is the same, the smaller pyramid has a smaller base and the line of action of its weight lies outside the base at a lower inclination than is the case for the larger pyramid. The angle at which the smaller pyramid topples over is noted.
4. As the board is inclined further the larger pyramid will eventually topple over but its improved stability over the other two blocks is apparent (Fig. 2-9d). Once again the angle at which the block topples is noted.

The angles at which the three blocks toppled are shown in Table 2.1.

Table 2.1 Comparison of the calculated and measured critical angles

Model	Cuboid	Small Pyramid	Large Pyramid
Height of the model (mm)	150	150	150
Height of the centre of mass (mm)	75.0	37.5	37.5
Width of the base (mm)	29	29	50
Volume ( )
Theoretical Max Inclination (deg.)	10.9	21.9	33.7
Measured Max Inclination (deg.)	10.0	19.0	31.0

The results of the demonstration as given in Table 2.1 show that:

• The order in which the blocks topple is as predicted by Equation 2.7 in terms of the measured inclinations and confirms that the larger the base or the lower the centre of mass of a block, the larger the critical angle that is needed to cause the block to topple.
• All the measured critical angles are slightly smaller than those predicted by Eq. 2-7.

Repeating the experiment several times confirms the measurements and it can be observed that the bases of the blocks just leave the supporting surface immediately before they topple, which makes the centres of the masses move outwards. In the theory the bases of the blocks remain in contact with the support surface before they topple.

This demonstration shows that the larger the base and/or the lower the centre of gravity, the larger will be the critical angle needed to cause the block to topple and that this angle is slightly less than that predicted by theory.

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VISUALISING STRESS

Balloons on nails

This demonstration shows the effect of stress distribution.

      
(a)                                                                                  (b)
Fig. 6-1: Balloon on nails

Place a balloon on a single nail and hold a thin wooden plate above the balloon and position the balloon as shown in Fig. 6-1a. Gradually transfer the weight of the plate onto the balloon. Before the full weight of the plate rests on the balloon, it will burst. This happens because the balloon is in contact with the very small area of the single nail, resulting in very high stress and causing the balloon to burst.

Now place another balloon on a bed of nails instead of a single nail. Put the thin wooden plate on the balloon and add weights gradually onto the plate as shown in Fig. 6-1b. It will be seen that the balloon can carry a significant weight before it bursts. It is observed that the shape of the balloon changes but it does not burst. Due to its changed shape, the balloon and hence the weight on the balloon are supported by many nails. As the load is distributed over many nails the stress level caused is not high enough to cause the balloon to burst.

Fig. 6-2: A person lying on a nail bed

A similar example was observed at a science museum as shown in Fig. 6-2. A 6000-nail bed is controlled electronically allowing the nails to move up and down. When the nails move down below the smooth surface of the bed, a young person lies on the bed. Then the 6000 nails move up slowly and uniformly and lift the human body to the position shown in Fig. 6-2. If the body has a total uniform mass of 80kg and about two third of the nails support the body, each loaded nail only carries a force of 0.2 N or a mass of 20 grams. Thus the person is not hurt by the nails.

      
a) 49 plastic cups placed upside down                                   b) A person standing on the cups
Fig. 6-3: Uniform force distribution

A similar but simpler demonstration can be conducted following the observation of the nail beds. Fig. 6-3a shows 49 plastic cups which are placed upside down and side by side. Place two thin wooden boards on them and invite a person to stand on the boards. The boards spread the weight of the person, 650 N, over the 49 cups, each cup carrying about 13 N, which is less than the 19 N capacity of a cup. SOURCE:http://www.mace.manchester.ac.uk/project/teaching/civil/structuralconcepts/Statics/stress/stress_mod1.php

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Greek Symbol		Greek Letter Name	English Equivalent
Upper Case	Lower Case
Α	α	Alpha	a
Β	β	Beta	b
Γ	γ	Gamma	g
Δ	δ	Delta	d
Ε	ε	Epsilon	e
Ζ	ζ	Zeta	z
Η	η	Eta	h
Θ	θ	Thet a	th
Ι	ι	Iota	i
Κ	κ	Kappa	k
Λ	λ	Lambda	l
Μ	μ	Mu	m
Ν	ν	Nu	n
Ξ	ξ	Xi	x
Ο	ο	Omicron	o
Π	π	Pi	p
Ρ	ρ	Rho	r
Σ	σ	Sigma	s
Τ	τ	Tau	t
Υ	υ	Upsilon	u
Φ	φ	Phi	ph
Χ	χ	Chi	ch
Ψ	ψ	Psi	ps
Ω	ω	Omega	o

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