LEARNING OBJECTIVES By the end of this chapter, you should be able to:1.  State and define: – The following properties of materials – Strength, Stiffness, Ductility, Brittleness and Elasticity.2.  State:  (a)  – The factors affecting the strength of a material.(b)  – Characteristics of brittle materials.(c)  – The factors which determine the amount of deformation of a material.3. (a)  State: – Hooke’s law. (b)  Describe: – Experiments to verify Hooke’s law using– A Nuffield spring and– A copper wire.4. Define: – Stress and Strain.5. (a) State:  – The factors for stability and safety of structures. ecolebooks.com (b) Define:  – Ties and strut. (c) Identify:  – Ties and struts in a structure.6. (a)  Explain: –  The effects of compression and tension forces on beams. (b)  State:-  The uses of beams.7. (a)  Describe:-  How concrete is made. (b)  State: –  The uses of concrete. (c)  Describe:-  How concrete is reinforced.

13.1  Materials

Materials are used in construction of structures like buildings, communication masts,

bridges, dams, tanks, motorcars etc. Before these materials are put to use, their

mechanical properties are tested. Some of the properties tested are: strength, stiffness

(toughness) ductility, brittleness and elasticity.

1.  Strength

Strength
is the ability of material to withstand stress.

It is the measure of how great an applied force a material can withstand before

breaking. Thus strength is a measure of the size of breaking stress a sample of a

material has.

(a)  Breaking Stress

Breaking strength is the force needed to break a piece of material of cross-sectional

area of 1m2.

It is given by the formula:  Breaking stress = N/m2 (Pascal)

The greatest stress (breaking stress), the material can withstand before it can break is its ultimate tensile strength.

(b)  Factors affecting the strength of a material

(i)  Cross-sectional area (diameter)

For samples of the same substance, it requires a larger force to brake

one of larger cross-sectional area.

(ii)  Nature of the material

It is for example harder to break a steel rod than to break a sisal rod

(iii)  Force applied

The strength of a material may depend on how a force is applied on it. For example concrete is very strong when compressed but very weak when stretched.

2.  Stiffness (toughness)

Stiffness is the ability of a material to resist bending. Stiffness tells us about

opposition a material sets up when being deformed i.e. having its size or shape

changed.

Stiff materials resist forces which try to change their shapes or sizes. They can

withstand large compressional forces but stretch by little amount when pulled apart.

3.  Ductility

Ductility is the ability of a material to be drawn into wires or worked into any shape without breaking. Examples of ductile materials include all metals except mercury. Metals can be hammered, rolled, cut or stretched into any useful shapes.

4.  Brittle materials

Brittle materials are substances which bend very little, then suddenly crack without any warning. Examples of brittle materials are glass, brass, bronze and other alloys.

Characteristics of brittleness

(i)  Undergo very little deformation before finally breaking

(ii)  Can withstand fairly large compressional forces but break when subjected to

tension

(iii)  Cracks form easily and are rapidly transmitted through the material once formed

(iv)  When broken, the broken pieces can be fitted together accurately.

5.  Elasticity

Elasticity is the ability of a substance to recover its original shape and size after

deformation.

Examples of materials which have this property called elasticity are said to elastic.

E.g. all metals except mercury and rubber. While materials which do not have

elasticity are said to inelastic. E.g. Wood, paper, etc.

The amount of deformation depends on

(i)  The nature of the material

(ii)  The magnitude (Strength) of the distorting force.

There is a limit to the elasticity of the substance as it will break if the force applied is strong. The elasticity of a material depends on the force of attraction between its molecules. When a substance is stretched, its molecules are pulled apart as the distorting force overcomes the force of attraction between the molecules. On removing the distorting force, the substance will return to its original shape if the elastic limit is not exceeded. The relation between force and extension was investigated by Robert Hooke and stated it in his law called Hooke’s law.

13.2  Hooke’s Law

Hooke’s Law states that:

The extension is directly proportional to the load provided the elastic limit is not exceeded.

Robert Hooke illustrated his law by carrying out four different experiments.

• Loading a horizontal beam fixed at one end (deformation and depression of the free end).
• Tightening a watch spring (deformation = angular rotation)

Experiment 13.1  To verify Hooke’s Law, Using a spring

(a)  Arrange the apparatus as shown in figure 13. below.

(b)  Attach a pointer to a spring and suspend the spring from a support clamped up

using a retort stand.

(c)  Read and record the position, Po, of the pointer on the meter rule.

(d)  Suspend a mass hunger of 100 g from the spring.

(e)  Read and record the new position, P1, of the pointer.

(f)  Determine the extension, e1, of the spring.

(g)  Repeat the procedures (d) to (f) for values of slotted masses of 200 g, 300 g,

400 g, 500 g and 600 g.

(h)  Remove the masses in steps of 100 g and each time record the position, P2, of

the pointer until the spring returns to its original length.

(i)  Determine the extension, e2, of the spring.

(j)  Enter your results in the table 13.1 below.

Table of results

 Increasing load Decreasing Load Averageextensione = M/kg Load/N P1/cm e1 = (P1 – P0)cm P2/cm e2= (P2 – P0)/cm 0.1 0.2 0.3 0.4 0.5

Table 13.1

(k)  Plot a graph of load (N) against extension, e.

If the experiment is carefully performed, and the graph is accurately drawn, the shape of the graph below is obtained.

The graph of load against extension

Observation:  When the load is doubled, the extension is also doubled.

Conclusion:  Hooke’s law which states that:

The extension of an elastic material is directly proportional to the load provided the

elastic limit is not exceeded, is verified.

Experiment 13.2

To verify Hooke’s Law, Using a Copper wire

Apparatus

A smooth pulley, G-clamp, a mass hanger and slotted masses, a table, a pointer (made

of paper) and a millimeter scale.

Procedure

(a) Set up the apparatus as in figure 13.3 below.

(b)  Add small masses one at a time to the weight hunger and observe the scale reading.

(c)  Determine the extension in millimeters.

(d)  Repeat the procedure (b) to (c) by adding more masses until the wire breaks.

(e)  Enter your results in table 13.2 below.

Figure 13.2

(f)  Plot a graph of load/force against extension for the wire.

If the experiment is carefully performed, and the graph is accurately drawn, the

shape of the graph resembles that of a spring except the extension is much smaller

for a given force.

Note:  The extension for a given load depends on:

• The metal used,
• The length of the wire and
• The diameter or cross sectional area of the wire.

The graph of load against extension for a wire

Explanation of the shape of the graph

Line OP:  Line OP is a straight line passing through the origin. It shows that the extension is directly proportional to the load. And therefore indicates that Hooke’s law is obeyed. When the load is removed from the wire within this range of load, the wire returns to its original unloaded length.

Point E:  Point E is called elastic limit.

After the elastic limit, the wire does not return to its original length when the load is removed. Some of the extension remains and the wire is deformed permanently giving a residual extension, OR, on the diagram.

Point Y:  Just after the elastic limit is point Y. It is called yield point. Beyond this point, an increase on the load causes the wire to ‘flow’ thus producing a larger extension than the previous one. After point Y, plastic deformation occurs.

Point M:  Point M
indicates the maximum load the wire can take.

Reducing the load after reaching point M has little effect; the wire forms a waist at its weakest point and breaks on reaching point B called breaking point.

Worked Examples

1.  A force of 20 N extends a spring by 5 cm. Calculate the force required to extend the

same wire by 20 cm.

Solution:  Let:  F1 = 20 N,  e1
= 5 cm, e2
= 20 cm, F2 = ?

20 N extends the spring by 5 cm,

1cm is extended by a force of N

The force that can extend the spring by 20cm =
= 80 N

2.  A force of 10 N extends a wire by 2 cm. Find the extension by a force of 50 N.

Solution:  Let:  F1 = 10 N, e1 = 2 cm, e2 =? cm, F2 = 50 N

10 N extends the by 2 cm,

1 N extends by 10/2

Then the extension by 50 N  = 2 x = 10 cm

13.3  Stress and strain

(a)  Stress

Stress is defined as force per unit area of cross-section.

i.e.  Stress =

The SI unit of stress is Newton per metre squared, Nm-2
= Pascal (Pa).

(b)  Strain

Strain is defined as the change in length per unit length.

Strain =

Strain has no units because it is a ratio of the same quantity.

13.4  Structures

Structures are generally designed and built to carry or support a load (i.e. to with

stand a force). A well designed structure must be able to stay in position without

collapsing for a long period of time.

13.41  Factors for stability and safety of structures

The stability and safety of structures depend on the following.

(i)  The nature of the material.

(ii)  Shapes of the materials.

(iii)  The way of arrangement of the materials.

The behaviour of structures under stress or strain can be investigated by using models

The straws are arranged to give structures of different shapes and a heavy load is then

attached to one end of each structure as shown in the diagrams in figure 13.3 below.

Experimental results show that the triangular structure figure 13. (b), provides better

support than (a) and the rectangular structure (c) for the same load. However, the

rectangular structures can be strengthened by fixing more girders diagonally. Notable

examples are seen in door and window frames.

13.42  Beams and Girders

(a)  Beams:

A beam refers to a large, straight piece of material with uniform cross section whose width and thickness are small compared to the length. Beams are used as major structural items in a building or structure.

(b)  Girders:

These are the smaller pieces of materials used to strengthen a structure.

In any structure, some girders are under tension while others are under

compression. Girders under compression bend or buckle and those under tension become thin and may break.

13.43  (a)  Ties and Struts

Girders and beams are used to make the frame works of structures such as water tank supports, radio masts, roof trusses, bridges, cranes, electricity

pylons etc. In these structures, materials ties and others are struts.

(i)  Ties:   A tie is a girder under tension.

(ii)  Struts: A strut is a girder under compression.

(b)  Identification of Ties and struts in a structure

To identify ties and struts in a structure can be identified by replacing them with

strings. If the string is pulled tightly, the girder is in tension and therefore it is a tie. And if it becomes slacked, it is under compression and therefore it is a strut.

Worked Examples

1.  The diagram in figure 13.4 (a) and (b) shows a structure supporting a heavy load

hanging from a nail at A. Identify ties and struts in the diagrams below.

Answer: (a)  P, Q, S and R are ties (b)  Tie:  W

Only T is a strut.  Struts:  X, Y and Z

(c)  Applications of struts and Ties

Ties and struts are applied in:

(i)  Roof supports (ii)  Water tank/reservoir supports

(iii)  Girder bridge (iv)  Communication masts

13.5  Effect of Compression and Tension forces in beams

Beams are used to support loads in buildings and bridges.

(a)  Effect of Compression forces in a beam

If a beam is acted on by two compression forces (two opposing forces that act toward each other) as shown in figure 13. (a) below, the particles in the body are pushed closer together. The body becomes shorter compared to the original length and the body is said to be in a state of compression.

(b)  Effect of Tension forces in a beam

If the beam is acted on by two tension forces (two opposing forces that act

away from each other) as shown in figure 13. (b) above, the particles in the

body are moved apart. The body becomes slightly longer than the original

length and the body is said to be in a state of tension.

(c)  Bending of beams

When a beam is subjected to a bending force, as shown in figure 13. below,

three things happen to it.

Diagrams showing the behaviour of a beam under a load

1. The particles in the upper region marked P become closer and shorter. The part region is under compression.

(ii)  In the lower region, R, the particles become further apart and the region becomes longer. It is under tension.

(iii)  Near the middle point Q, the distance between the particles remains the same. That is the region is neither under compression nor under tension. The length of this surface has not changed. This region is called the neutral plane or neutral surface.

13.51  (a)  Stiffness of three wooden beams

Three wooden beams can be arranged in five possible ways to give different shapes as shown in figure 13.7 below. One end of each beam is then subjected to a heavy load.

Experimental results show that the I-shaped beam is the most stiff i.e. the strongest. That is why many beams in structures such as bridges, buildings are I-shaped.

I-shaped beams can be made lighter by removing material along the neutral axis.

(b)  Effect of shear force on a body

When a body is acted upon by sheer forces (two equal and opposite forces whose line of action are parallel to each other) the body becomes twisted and deformed. See the diagrams in figure 13.8 below.

(c)  Uses of beams

(i)  I-Shaped beam:  I-Shaped beam resists bending and sagging under load. They are used in building constructions such as buildings, bridges, masts, cranes etc.

(ii)  Joist.  A joist is a wooden beam which helps to support a floor or ceiling. The joists are laid on their edges since they are then stiffer and bend less.

13.52  Other materials used in construction of structures are the following:

1.  Natural stone:  Stone is an inorganic mineral or soil concretion of the earth. The commonly used types of stones in building, civil engineering, manufacturing and art are: sedimentary, igneous, or metamorphic origin. Some of the building stones are basalt, flint, granite, limestone, marble, porphyry, sandstone, slate, and flagstone.

2.  Bricks:  Brick
is block of clay or other ceramic used for construction and decorative facing. They are made by mixing together clay and water. The mixture is molded into different shapes and baked in a kiln (i.e. fired at high temperature) or may be dried in the sun. Well known places for making bricks is Lweza Clays at Kajjansi in Wakiso district.

3.  Mortar:  Mortar is a mixture of sand and cement made into paste by adding water. When dried it produces a hard stony material used for bonding bricks. They resist dampness and heat, and can last longer than stone. Some bricks are made of special fireclays for use in fireplaces or ovens.

4.  Concrete:  Concrete is made by mixing gravel or small stones, sand, cement and water in right proportions. The gravel or stones make the concrete very strong; the sand fills up the spaces between the stones.

(a)  Uses of concrete

It is used where heavy loads have to be supported, e.g. in

• the foundations of tall buildings,
• airport runways,
• dams,
• the piers of bridges
• etc.

Concrete has great compressive strength but little tensile strength. Its tensile strength is increased by reinforcing it.

(b)  Reinforcement of concrete

Concrete used in most construction work is reinforced with steel. Steel is embedded in the concrete in the form of a mesh. A bond forms between the steel and the concrete. When reinforced concrete is subjected to extreme tensile stresses, the steel supplies the necessary strength and stresses can be transferred between both components.

13.6  Notches

A notch is an indentation or incision on an edge or surface.

Notches and cracks spread more easily when brittle materials such as glass and concrete are under tension. This effect is applied in cutting glass where a notch is made on the surface of the glass and the glass is subjected to bending from below.

Self-Check 13.0

1.  The strength of a material depends on the:

(i) Nature of the material (ii) Diameter of the material (iii) Length of the material

A. (i) only B. (i) and (ii) only  C. (ii) and (iii) only  D. (i), (ii) and (iii)

2.  A mass of 0.2 kg produces an extension of 8 cm in a spring. The force required to

produce an extension of 6cm is

A. 0.75 N B. 1.50 N C. 2.70 N D. 24.00 N

3.  A ductile material is that which

A. Is fragile B. Is not elastic

C. Can be molded into any shape D. Easily breaks under compression.

4.  In a wire supporting a load, stress is given by

A. B. force x area  C. D.

5.  The diagram in figure below shows a framework of a bridge.

Which of the girders are ties?

Q

A. XQ, QY, PX, YR   B. PQ, QR, XY  C. XQ, QY D. PX, YR

6.  Which of the following are all brittle materials?

A. Leather, rubber, thread.  B. Clay, glass, wood.

C. Glass, cast iron, stone.  D. Rubber, polyester, copper wire.

7.  The diagram below shows a structure of wooden beams P, Q, R, S and T supporting a

heavy road L. Which of the beams can be replaced by ropes if the same shape is to be maintained?

A. P, R, S and T

B. P, Q, S and T

C. Q, R, S and T

D. P, Q, R and S

8.  A rod of cross-sectional area 40cm2 needs a tensile force of 2N to break it. What is its

breaking stress?

A. 0.005 Nm-2 B. 0.05 Nm-2 C. 5 Nm-2 D. 500 Nm-2

9.  A mass of 0.5kg causes a spiral spring to extend by 4cm. The force that would cause an extension of 6cm would be

A. 2.0 N B. 3.3 N C. 4.8 N D. 7.5 N

10.  If a load 1N extends a spring by 5cm, what extension will a load of 0.6 N produce?

A.1.2 cm B. 3.0 cm C. 8.3 cm D.30.0 cm

11.  Which of the following are brittle substances?

A. Dry clay, steel, chalk, wood B. Chalk, steel, plastic, glass.

C. Glass, chalk, concrete and steel D. Dry clay, glass, chalk and concrete.

12.  Reinforced concrete is stronger that ordinary concrete because concrete and the steel are

A. Both brittle materials. B. Strong in tension and compression respectively.

C. Both ductile materials. D. Strong in compression and tension respectively.

13.  The graph above represents the extension of a wire with increasing load. Where does the yield point occur?

A. between point P and Q B. between point Q and R

C. between point R and S D. at point S

14.  A load of 4 N stretches a spring by 0.5 cm. Calculate the extension when a load of 8 N

is applied.

A. 0.25 cm B. 1.0 cm C. 2.0 cm D. 4.0 cm

15.  A notch on a material spreads more rapidly when the metal is

A. in tension B. in compression  C. pre-stressed  D. reinforced

SECTION B

1.  (a)  With the aid of a diagram, describe the effect of a shear force on a body.

(b)  (i)  What is meant by strength as applied on a material?

(ii)  State the factors on which strength on a material depends.

(c)  (i)  Describe a simple experiment to describe Hooke’s law using a spring.

(ii)  State any three characteristics of concrete which make it a desirable building material.

2.  (a)  Define the following terms

(i)  Strain. (ii)  Stress.

(b)

The curve in the figure shows the force versus extension graph for a copper wire.

Describe what is happening between points A and B.

3.  (a)  Explain with the aid of a sketch diagram, how a notch weakens a beam of a brittle

material.

(b)  Sate two ways in which concrete may be made stronger.

(c)  A force of 100 N stretches an elastic spring by 2 cm. what force would stretch the same spring by 3.5 cm?

(d)  The diagram in the above figure shows a simple bridge on a stream.

(i)  Mark the neutral axis

(ii)  Label the part that will be in tension

(iii)  Indicate on the diagram how the bridge can be strengthened.

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