The object of studying structural mechanics is to enable architects and engineers to learn how to build structures with a view to stability, the

Transcription

1 1 The object of studying structural mechanics is to enable architects and engineers to learn how to build structures with a view to stability, the safety of human life and financial economy. Mechanics is a branch of knowledge covered by applied mathematics and deals with the motion of bodies, with the forces by which those motions are conditioned and with the balance of forces on a body at rest. The word 'Mechanics' is derived from the Greek Mechanikos meaning 'contrivance'. 1

2 2 Our body contains a number of these contrivances which through long ages have been adapted by nature to our needs. If we lift a weight or raise a foot from the ground, we are employing mechanisms which are admirably suited by nature for these purposes. Therefore, mechanics may be defined as the constituent part of applied mathematics where we study the conditions under which bodies around us move or remain at rest. In this course we are going to consider in particular structural mechanics, which is mainly concerned with forces-how they combine together, how they keep a body at rest, and in general with the effect they have on the stability of the part of the building or structure to which they are applied. Also, we will have to pay attention to the effect a force has on the size and shape of the material upon which it acts. The word 'body' is meant to include all things which have weight, and throughout this course it will enter into our discussions and into many of our calculations. 2

3 3 In the definition of mechanics two aspects have been mentioned: first, the conditions under which objects remain at rest; and second, the conditions under which objects move. These two aspects have brought about the division of the subject into two parts: Statics, which is concerned with solid bodies at rest, and Dynamics, which is concerned with solid bodies in motion. There is a further branch of the subject called Hydrostatics, which deals with the application to liquids and gases of those principles which have been mentioned above in relation to solid bodies. Hydrostatics is a study in its own right and is outside the scope of this course. 3

4 4 Take a small fairly heavy body such as a weight, suspend it by a piece of string and hold the free end of the string between the thumb and first finger. If you do this simple experiment yourself, you will be aware of the forces involved, and I would suggest that the way to get the most out of this experiment is to use different weights at the end of the string. 4

5 5 Now look at Fig 1. First, in order to keep the weight from falling to the ground you must exert an upward force by means of the muscles of your thumb, finger, wrist and arm. It is an upward force because, if you take the weight in your other hand, then your thumb, finger and arm--on being relieved of the weight--will shoot upwards. Second, this upward force is necessary to counterbalance a force which is acting vertically downwards on the suspended body, a force which we call the weight of the body. This downward force or weight is called the Force of Gravity. It is the attraction which the Earth exerts on all bodies and makes them fall to the ground. Now let us return to the experiment with the body suspended by a piece of string. Fig 2 explains the method which is employed to represent the forces involved. A represents the body and the straight lines show the vertical gravitational forces acting on it, the arrowheads indicating the direction of each of them. The forces are as follows: the weight acting vertically downwards is marked w and the balancing force acting vertically upwards exerted by the thumb and finger through the string is marked F. Now, so long as the pull upwards which you exert is equal to that of the weight w, the body will remain stationary, that is, at rest. If you increase the pull upwards, the body will move upwards; if you decrease the pull upwards, the body will move towards the ground. 5

6 6 Here is a summary of these principles. Thus an alteration in the forces acting on the body results in the body being at rest or in motion. So far the term 'Force' has been used without giving a definition of it. The experiment described earlier and the simple results gained from it have now made this possible. Force is that which tends to produce motion in a body, to change that motion or to keep the body at rest. The mental picture of a pull or push is for the moment a sufficient conception of force. 6

7 7 Familiar examples of forces are to be found in life around us. It is necessary to measure the amount of pull or push. This is called the measurement of the magnitude of a force and it is convenient to express this in terms of Newtons. In this course the measurement of a force will be in Newtons or Kilonewtons. The Newton (N) is the unit of force which acting on a mass of one kilogramme (kg) imparts to it an acceleration of one metre per second per second (1 m/s 2 ). thus 1 N = 1 kg x 1 m/s 2 = 1 kgm/s 2 weight = mass x acceleration due to gravity weight of 1 kg = 1 kg x 9.81 m/s 2 = 9.81 kgm/s 2 but 1 kgm/s 2 = 1 N Hence 1 kg = 9.81 N 7

8 8 Let us again refer to the experiment. The upward pull through the thumb and finger to maintain the weight at rest is apparent, but if the string is attached to a nail in a wall and the weight is left hanging downwards, the upward pull through the nail is not apparent. This phenomenon forms the basis for all static theory in mechanics. If we push against a wall we are conscious of an opposite force acting against our hands, and the harder we push, the greater will be the counter force. In other words, there is some hidden force balancing our pushing. Similarly, there must be a hidden counter force acting through the nail in the wall to balance the vertical force acting downward through the weight. 8

9 9 This hidden balancing force, or counter force, is known as the Reaction. This is a force which we seldom see at work, and because it is unseen, fail sometimes to appreciate fully. Stop this tape; then Go to the nearest solid wall and push against it with your hands and then with your shoulder. If you push against a wall with a force of 100 N, the reacting force back through the wall will be 100 N. If you push lightly against the wall through your thumb with a continuous pressure of 5 N, the wall will press back with a continuous reaction of 5 N. Now we have arrived at an important principle: Force and reaction are opposite in direction and equal in amount. You must never consider that reaction, because it is not apparent, is a purely imaginary force invented by mathematicians to help them to solve problems. 9

10 10 In the experiment in which a weight was suspended by a piece of string held by your finger and thumb no weight value was used. Let us now assume that the body is an actual weight of 10 N. The piece of string is now subject to a pulling or tensile force of 10 N. The vertical force acting downwards is 10 N, the reacting force upwards through your finger and thumb is 10 N and this must be so since there is no motion. The body is at rest. You may ask, very rightly, 'Why is the pulling or tensile force in the string not 20 N?', that is, 10 N downwards added to 10 N upwards. If we want to determine the actual force in the string, all we have to do is to insert a spring balance somewhere along the length of the string, as illustrated in the next slide. 10

11 11 If we carried out this experiment we would find that the spring balance would indicate a force of 10 N, no more and no Less. We can now further extend this experiment to determine the value of the reaction force which is taking place through the muscles of our f~irst finger and thumb. 11

12 12 Here is an experiment which I want you to carry out. Fix a smooth- running pulley to a wall or to a drawing board or to a piece of apparatus. Pass a piece of string over the pulley and on the vertical length fix the 10 N weight. Along the horizontal length of string insert a spring balance and hold the free end of the string again between your first finger and thumb. Note carefully that the spring balance shows a tensile force in the string of 10 N, not 20 N. 12

13 13 Next take a metal ring of about 3 cm in diameter and where the finger and thumb held the string, attach the string to the ring and hold the ring in position by placing it over a nail fixed in the wall or in the drawing board or, as shown here, on the apparatus. The spring balance again shows a tensile force of 10 N in the string. 13

14 14 Now attach another length of string to the metal ring and then pass it over a second pulley (at about the same distance as the first pulley from the ring) and tie it to a 10 N weight hanging vertical- ly downwards. Allow the metal ring to move away from the nail so that the ring is free of it. Again the spring balance shows a tensile force of 10 N. We have now quantified the forces involved in our first experiment. This may be stated as follows: first, a vertical force downwards of 10 N; second, a vertical force upwards through the muscles of our first finger and. thumb of 10 N; and third, a tensile force in the string of 10 N. 14

15 15 Here are one or two more examples to illustrate what we have been discussing. At the top we see a man pulling on a rope which is tied to a stake driven into the ground. If the man pulls on this rope with a force of 400 N, his back and back muscles will be conscious of making an effort of 400 N. Suppose he now removes the stake and asks another man to pull with a force of 400 N on the rope, again his back and back muscles would be conscious of reacting with an effort of 400 N. If the second man suddenly drops his end of the rope to the ground, the first man cannot apply a force Qf 400 N to his end of the rope, since he will fall to the ground because there is no equal and opposite force at the other end. It takes two people to make a quarrel and it takes twq equal and opposite forces to make what we understand as Simple Tension. 15

16 16 Here we see that it takes two equal and opposite forces to make what we understand as Simple Compression. A man is carrying a weight Won his head. This weight acts vertically downwards through his body. There is an opposing force acting vertically upwards from the ground through his feet and legs. The man's body is being compressed between the downwards action of the weight and the upwards reaction of the ground. On the right is a man holding a block of wood in his hand. The block is being compressed between the downward action of the weight of the timber and the upward reaction of the muscles in the man's hand and arm. 16

17 17 One of the most important properties about a material is whether it is heavy or light. We think of steel as a heavy material and aluminium as a light one. But what happens when aluminium is com- pared with, say, timber? It would appear, therefore, that calling a material heavy or light is not enough; a more accurate method of comparison is needed. You know that two bricks weigh more than one--in other words, that the weight of a body varies directly as the quantity of matter in it. This quantity of matter is called Mass. We all know that steel is heavier than wood and therefore that if we lift a cubic centimetre of steel it is heavier than a cubic centimetre of wood. This means that the mass of a cubic centimetre of steel is greater than the mass of a cubic centimetre of wood. The word used to express this idea is Density. Dead loads are the self weight of the structural materials and members which make up a building. These dead loads include roof and roof finishes, roof trusses and frames, floors and floor finishes, ceiling finishes, beams, columns, load-bearing walls and parti- tions, staircases, lift shafts, foundations etc. Loads of this kind are present throughout the life of the structure or building, and must be accurately determined before proceeding with detailed calculations appertaining to design. Self weights of materials can be obtained in textbooks and tables which you will probably find in your school or college library. 17

18 18 Density is Mass, that is, quantity of matter per unit volume, and is expressed by the simple formula shown here. By knowing the densities of different materials we are able to compare their self weights or Dead Loads, as they are commonly called. This inform- ation is important because the self weight of a timber beam would be very much lighter than a steel beam of the same dimensional proportions. Dead loads are those considered as the weight of the structure itself and parts of the structure. Superimposed or live loads are all loads other than dead loads. They cover all movable weights in a building including the occupant (human or animal), machines, furniture, fittings, storage materials and goods, non-structural partitions and walls, rain, ice and snow on roofs etc. Since the structure has to support these loads, it is necessary to determine them as accurately as possible in relation to the type or occupancy of the building. Fortunately Codes of Practice and Building Regulations specify the minimum loads in kn.m2 which must be taken into consideration on roofs and floors whether the floor be in an office block, hospital, flats or ware- house. Reference should be made to British Standard Code of Practic CP3, Chapter V, Loading, where all of these are 18

19 19 This slide gives examples of the densities of some common materials You will note that the density of mild steel is 14 times greater than timber and almost 5 times greater than dry sand. In a multi-storey building, other than those used primarily for storage purposes, it is obvious that the superimposed loads will not occur simultaneously throughout the entire building; in such circumstances the superimposed load can be reduced on all floors by proportional amounts from 10%, but not exceeding 40%, with regard to columns, piers, walls and foundations each giving support to one, two, three, four and five or more floors. Snow loads vary but depend upon the geographical location and climate of the country or region where the building is to be sited. 19

20 20 Now I want to talk about another kind of weight or load which we call Superimposed Load. All structural members are required to support loads other than their own dead loads. These are known as superimposed loads or Live Loads. A floor in a house, for example, is required to support the weight of people moving across it, sitting on it and also the furniture which is positioned on it. The amount of furniture may increase or decrease, the number of people may increase quite considerably for a party whereas it may support no people at all on other occasions. Wind loading or pressure on vertical surfaces of multi-storey buildings placed normal to the wind direction is an important consideration in structural design. The active wind force is based upon the geographical location, height and degree of exposure of the building, eg on the coast, top of a mountain or in a naturally protected area where shielding is possible. The wind pressure (p) on a vertical surface of a building is calculated in so many Newtons per square metre which is divided into two parts: (a) positive pressure on the windward side (+) and, (b) negative pressure or suction on the leeward side (-) 20

21 21 The type of structure will detenfline the intensity of the super- imposed loading and the form of the loading. For example, the deck of a bridge will be required to support a moving load, that is, the movement of the traffic across it from one side to the other. The floor of a warehouse storing bags of flour will have to be stronger than a warehouse storing cardboard boxes. The floor of the auditorium in a cinema will have to carry a greater loading of people than, say, the floor of a bedroom in a house. 21

22 22 Here is a table of a few superimposed or live loads which are used in the design of buildings. I would like you to note the differ- ences; for example, the superimposed load on the floor of a work- shop making lightweight goods is 5 times greater than that for a floor in a house and 3 times greater than that for a floor in a general office. There are other loads on structures which have to be considered, such as wind loads and snow loads, but I do not intend to discuss these at this early stage in the course. 22

23 23 Here as a reminder is a list of the loads I have just described. It is important to remember them, since they are the loads to be considered when designing a building. 23

24 24 Finally, here is a section through an office building. You will see the live loading in the building due to the people and the furni- ture and the way the forces generated by these loads are transferred to the foundations and the reactions through the foundations by the subsoil. Similarly, the dead loads of the materials which form the roof, the floors and the walls generate forces which are transferred to the foundations. Again the subsoil sets up reactions through the foundations so that all the forces are balanced. 24

25 QUESTIONS 1 Describe the difference between dynamics, statics and hydrostatics. 2 What is density and how is it determined? 3 (a) Explain as clearly as you can the difference between forces and reactions. (b) A man carries a load of 30 kg on his head. If the man weighs 70 kg determine the amount of the reactions through his feet when he is standing still and when he is standing on his left leg only. 4 (a) A stone column supports a vertical load of 1000 kn including the self weight of the column. If the vertical reaction through the foundation to the column is 1000 kn determine the amount and nature of the force in the column. (b) A rope which is fixed to a pole is pulled by a man who exerts a force of 50 N. Determine the magnitude and nature of the force in the rope. 5 (a) When designing a building, what kinds of external forces have to be considered? (b) Make two lists, one of dead loads and the other of superimposed or live loads. Place the loads in each list in order of magnitude, ie the heavier loads at the top and the lighter loads to the bottom. Try to memorize these loads. 25