CEGE0101 Structural Mechanics Lecture Notes PDF

London’s Global University CEGE0101: Structural Mechanics: Supplementary Lecture Notes for Lecture 2 Katherine Cashell Key items covered:  Understand Newton’s laws  Understand and apply conditions of equilibrium  Draw ‘Free Body Diagrams’ ...

London’s Global University CEGE0101: Structural Mechanics: Supplementary Lecture Notes for Lecture 2 Katherine Cashell Key items covered:  Understand Newton’s laws  Understand and apply conditions of equilibrium  Draw ‘Free Body Diagrams’  Become familiar with concept of Normal Forces, Shear Forces & Bending Moment  Become familiar with the concept of stress and strain under normal loads Suggested reading:  Structural Mechanics by Ray Hulse & Jack Cain - Chapter 1 2.1 What is a Structure? A structure is an assortment of elements that are connected and transfer loads from one point to another. A structure is not a simple collection of independent/individual elements. Figure 2.1: Examples of structures: (a) Tower Bridge, London; (b) St Mary’s Axe, office buildings, London; (c) Wembley Stadium, London; and Illustration of elephant and human skeletons (organic structures) 2.2 What typical loads do structures need to resist? The load all structures need to resist is their own weight. The weight of the structure and any connected parts is a load that is classified as a “permanent load”. In addition, a structure typically needs to resist a number of loads which are not always present, or not constantly present. These are termed “live loads”, and are for example loads imposed by:  People, machinery, vehicles   Environmental actions – wind, snow, water, waves  In addition, on rare occasions the structure may be subjected to large actions from extreme events, like earthquakes, tsunami, cyclones, flooding, volcanic ash etc. Finally, the loads acting on structures may be static (i.e. do not very with time) or dynamic (vary with time). Examples of static loads include self weight, snow and hydrostatic loads from standing water. Examples of dynamic loads are those associated with earthquake ground shaking or rough winds. Note: Such dynamic loads are often idealised as static loads to simplify structural analysis calculations. Figure 2.2: Illustration of typical loads on a building 2.3 What are the performance objectives for structures (and learning objectives for you!): Every structure usually has a specific function (e.g. residential, commercial, industrial, hospital, school, university, stadium, power plant, etc.). When designing a structure engineers must provide this function whilst meeting, (at a minimum), two performance objectives: 1. Structural serviceability under normal/frequent loading (i.e. limited deflection, excessive vibration)  2. Structural safety under extreme loads (i.e. ensure stability and sufficient strength against collapse). This can be achieved through appropriate choices for structural form and materials. Note: In addition to the performance objectives of Safety and Serviceability, in real projects a number of other constraints or performance objectives may shape an engineer’s choice of design and materials. These include considerations of aesthetics, sustainability, economic cost and space and other environmental constraints. Building codes, which regulate structural design in most countries, set out design approaches and equations that aim to help engineers meet these two “performance” states. However, in order to follow these building codes approaches one must: 1. Understand what the loads are that act on a structure, 2. Be able to represent these loads in structural analysis, 3. Know how to analyse a structure under the loads’ actions 4. Interpret the analysis results and choose an efficient structural form/geometry, connections and materials to meet the performance objectives.. ….your learning objectives! …..this is where this module comes in ѡ Ѥ ѣ Ѣ 2.4 Some basic definitions: A force is a vector quantity and must be specific both in magnitude (i.e. size) and direction. In addition to specify the magnitude, we typically use arrows to represent the direction of a force. Mass (࢓ ) is a measurement of the amount of matter something contains. Weight (ࢃ ) is the measurement of the pull of gravity on an object. We measure forces in “Newtons” (N), where 1 N = 1 kg.m/s2 e.g. Weight (ܹ ) [N] = Mass (݉ ) [Kilogram, kg] x Acceleration of Gravity (݃) [meter per second squared, m/s2] A structure is made up of a number of elements. These can be of different shapes and sizes. Three parameters we use to describe the element characteristics are its length and cross section shape and cross-section area. See Figure 2.3. Figure 2.3: Element of length l, with a non- uniform rectangular cross-section. Illustration of how cross sections are drawn. 2.5 Statics Statics is the study of bodies at rest i.e. not in motion. A body is a general word used to describe, for example, a building, a bridge or a structural element. All bodies are subject to the action of forces such as their own self-weight. Static Equilibrium The conditions governing static equilibrium follow from Newton’s Laws, i.e.: a) A body continues in a state of rest or uniform motion in a straight line unless forces act on it. b) The force acting on a body is proportional to the rate of change of momentum (i.e. impulse). This means that the acceleration of an object is dependent  upon two variables - the net force acting upon the object and the mass of the object. c) Action and reaction are equal and in opposite directions. From b, the familiar equation F = m∙a follows, where F stands for force (unit: N), m stands for mass (unit: kg) and a stands for acceleration (unit: m/s2). Static equilibrium concerns bodies at rest. In a body at rest, the acceleration (a) of the body is zero. This means that the total of the forces (F) acting on the body must be zero. This does not mean that there is no force acting, but that the sum of all the forces acting is zero (Σ F=0). Equilibrium considerations can relate to overall equilibrium in a structure – that is, the relation between the applied loads and the reaction – and to the equilibrium of any part of the structure used to determine the internal forces (e.g. axial or shear force) caused by external loads (e.g. wind load or self-weight) and associated reactions. Structures which can be analysed using equations of equilibrium alone are termed statically determinate structures. Equilibrium in 2D systems In a 2-directional system (2D), 3 types of motion can take place: 1. Displacement along the x direction 2. Displacement along the y direction 3. Rotation about the z direction Figure 2.4: Coordinate system If a 2D system is in equilibrium, the sum of the forces which would cause acceleration, corresponding to each of these three types of motion, must be zero:  Sum of forces in the x direction are equal to zero ΣFx = 0   Sum of forces in the y direction are equal to zero: ΣFy = 0  Sum of moments about the centroid (centre of mass) are zero ΣMz = 0 These are termed equations of equilibrium in 2D system. 2.6 What is a Moment? A moment is measured at a point. The moment of a force about a point is the force times the perpendicular distance from the line of action of the force to the point. For example, in the image below, the moment about point A is MA = P.d kNm Figure 2.5: Moment about point A Equilibrium of moments can be achieved if the sum of the moments at a point is zero. See saw example here: Figure 2.6: Equilibrium of moments, on a see-saw Co-Planar Forces These are forces which act in the same plane (e.g. forces in xy plane). If all forces act over the same point, rotation cannot occur, and the set of forces is known as co-planar and concurrent (they act through a single point). However, rotation can occur if the set of forces do not act through the same point, in which case the forces are co-planar, but non-concurrent. Combination of Forces Forces can be combined graphically by vector addition and magnitude calculated through simple trigonometry. By using simple trigonometry, we can describe all forces in terms of their x and y components. We can apply these principles and vector addition to the combination of different forces. By sub-dividing a set of forces into their x and y components and using equations of static equilibrium (i.e. Σ ࡲ࢞=૙, Σ ࡲ࢟=૙, Σ ࡹ ࢕=૙), we can calculate unknown forces for equations of equilibrium to be satisfied. 2.7 External forces External forces are applied to the boundary of a structure. This includes externally applied forces (actions) as well as the forces that are applied by the supports to restrain the structure (reactions). External forces can be characterised as either normal forces, shear forces or bending moments, depending on their direction of action with respect to the element axis and cross section. A normal force acts perpendicular to the cross section of the element. As the normal force acts along the element axis, it is also commonly called an axial force. Normal forces induce compression or tension in elements. Typical sign convention sees compression characterised by a –ve sign and tension by a +ve sign. Shear forces act along the surface (or tangent to the surface) under consideration. This is a force acting in a direction parallel to a surface or to a planar cross section of a body. Shearing forces are unaligned forces pushing one part of a body in one specific direction, and another part of the body in the opposite direction. This is different to compression, which occurs when the two opposing forces are pushing into each other at the same point (i.e. they are not offset), resulting in compressive stress. When a structural member experiences failure by shear, two parts of it are pushed in different directions, for example, when a piece of paper is cut by scissors. Bending moment is a moment that causes the element to bend. In the above image of a cantilever with point load P, P and RA are shear forces and MA is a bending moment. 2.8 Internal forces Internal forces are those applied on a portion of the structure by the rest of the structure. You can only see these if you “cut the structure”. The internal forces acting within the system are in opposite directions to each other and the forces cancel out. They are represented by the axial, shear and bending moment diagrams, that will be introduced later in the course. Stresses and strains The internal forces induce internal stresses in the materials of the structural elements. Both the material properties as well as the element form and geometry play a role in how effective the structural element can resist the applied forces. The stress is calculated by dividing the internal force by the cross sectional area of the element: Where you have a normal force acting on the element, the stresses developed are called “normal stresses”. If we take the above example, as the normal stress increases in an element, the element will tend to deform through elongation. We characterise the deformation using the parameter of Strain: Strain is calculated as the integral of the elongation divided by the length of the element: The rate of strain increase with stress is a property of the material used in the element, If we plot the evolution of strain with an increase in stress, we obtain a stress strain diagram. An example of such a diagram is presented here below: It can be seen that as the stress is increased from zero, the stress strain relationship is linear until the yield point is reached. The slope of the stress-strain curve is the Modulus of Elasticity of the element, and is calculated as follows: The yield point flags the end of the elastic response of the element. If the element is unloaded at this point. However, if the element is loaded to beyond the yield point and then unloaded, residual deformation will be seen. When the stress reaches the maximum value, this indicates that the maximum strength of the element has been reached. Beyond this maximum strength point, the element cannot take any further loading, but will instead deform for a constant (or decreasing) load. The element continues to deform in a stable way until it starts to rapidly lose strength. This is called the ultimate point and the element is considered to have failed.