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### Fundamental principles of engineering

BADI Year 1

Fundamental principles

Suppose we design a chair. We need to know how much weight the chair must support.

We could safely make the chair from a solid block of steel, or mahogany. However these would be expensive and rather heavy! (also not very green!)

For elegance and economy we usually choose to use the minimum amount of material. This must be based on an understanding of the stresses imposed on each part of the chair, and the ability of the materials used to withstand these stresses.

Fundamental principles 1: Newton’s laws (1642 -1727)

- A body will continue in a state of rest or uniform straight-line motion unless acted upon by a force
- The acceleration of a body is dependent on its mass and the total force acting on it F = m a
- Every action has an equal and opposite reaction

Mass : the amount of matter in a body

We experience mass in two main ways:

- Through the effort required to lift the body against the earth’s gravity, and
- Through its resistance to acceleration, i.e. its inertia. Try picking up a dumbell and shaking it rapidly. You are attempting to accelerate the dumbell and this takes a lot of force.

NB Mass is not the same as weight! We measure mass in kg but your dumbell (10kg) would weigh nothing in space – however it would have the same amount of matter, and be just as hard to accelerate!

Force

Cant be seen, we experience force

- By its effects on our body and the way in which we need to oppose it to maintain our balance.
- By observing the effect it has on objects around us.

Force is measured in Newtons (N) and 1 Newton is roughly the force exerted by an apple in the earth’s gravitation. (Actually 1kg m s-2: 9.81N is the force exerted by 1kg)

Forces in equilibrium

- A man of mass m = 60kg experiences a force of F = 60 * 9.81 N = 588N due to gravity. (2)
- If he was able to move freely in any direction (eg jump off roof) he would fall with an acceleration of a = F / M = 60kg / 60 * 9.81N = 9.81m s-2
- If he is stationary, standing seated or lying down he is not accelerating. a=0; hence F/m = 0 so F = 0.
- This shows that the downforce acting on him is exactly balanced by the floor pushing him upwards!

Forces in equilibrium

The earth pushes us up by the same amount as gravity attracts us down!

What is making this balancing force and how does it work?

F = 588N

Fr = 588N

Restoring force

When a force is applied to a body the body will be deformed by the force. The body may remain deformed afterwards (like clay) or spring back (like a rubber band).

Deformation is what produces the force that stops us from sinking into the earth. The material we walk on can be elastic (like floorboards) or plastic (like sand).

The work done in deforming the material is

4. Work = Force * distance.

Elastic and plastic

- Concrete is very hard and very elastic. When we walk on it we don’t deform it very much, and it springs back when we move on. You can see the effect better if you walk on a trampoline. Hard elastic materials are easy to walk on because the energy used in deforming them is given back when you lift your foot.
- Sand or mud are very plastic and you can see the change you have made by stepping on it. Wet sand is harder than dry sand and deforms less – so each step uses less work, and makes it easier to walk on then dry sand.

Hooke (1635 – 1703) noted that the length of an elastic* solid changed in exact relation to the force applied.Hooke’s law as applied to a wire under tension

Original length Lo

Wire of cross-sectional area A

Change L in length due to applied force

Force F

Stress

We define stress as the internal force at a point within the body. There are different ways in which a body can be stressed (see later)5: Stress s = force per unit area s= F / A

Strain

We define strain as the relative change in dimension of a body that results from a stress.6: Strain e = change in length / original length Strain e = L / Lo

Hooke’s law

Hookes Law can be expressed as strain stress.

The constant of proportionality is the elastic modulus.

In this example of a body under tension it is

Young’s modulus

7: Y = s / e

When a force acts at a distance from a point we call this a moment. The moment of the force F at point P is Fx and acts in the same direction as the force.

We will use this notion later in combining the effect of forces.

P

Moment and torqueForce F

Distance x

Torque

When we use a spanner to turn a nut the force we apply to the handle is opposed by an equal and opposite force (reaction) at the nut.

This pair of forces is called a couple and produces a turning motion - we call this torque.

If point P is fixed in space but free to turn the turning force T is T = Fx and here acts in a clockwise direction.

Force F

Force -F

Distance x

Notice that moment is a vector, having magnitude and direction, while torque has only magnitude and sense

Summary: concepts and equations

- Mass kg
- Force N ( F = m * a )
- Equilibrium
- Work (Force * distance) * Newton metres or Joules
- Elastic and plastic
- Stress (s= F / A) Newtons per square meter
- Strain (e = L / Lo)
- Youngs modulus (Y = s / e)
- Moment of a force
- Couple
- Torque (T = Fx)

*Strictly work = force * displacement IN THE DIRECTION OF THE FORCE

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