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Building Drawings. Building symbols. Introduction to Bridge Drawings. Orthographic Projections of given model, Sectional Views. Steel Structure Connections detail. Essentials of Drafting by James D.

Engineering Drawing by N. Computer Applications in Civil Engg. Applied Linear Algebra Vector Algebra: Introduction to scalars and vectors, Vectors in the plane, Scalar and vector products, Lines in R2, R 3 and planes, Spheres, Orthogonal projections, Perpendicular distance from a point to a line and a plane, Vector spaces, Subspaces, Linear combinations, Linearly dependent and Independent set of vectors, Spanning of a vector spaces, Bases of a vector spaces and its applications in engineering and Business.

Matrix Algebra: Introduction to matrices, Matrix operations, Inverse Matrix, Rank of a Matrix, Echelon form of a Matrix and its applications in our daily life situation problems, i-e in line-communication as Air-lines, Telephone-lines, Connecting cities by roads. Determinants: Determinants and its properties, Inverse of a matrix, Rank of a matrix, linearly dependent and independent by determinants.

Linear system of equations: Independent, Dependent and Inconsistent system of equations and its graphical representation, Trivial and non-trivial solutions of homogeneous system of linear equations and its applications as linear models in Business, Economics, Science, Electric Circuits and other branches of engineering.

Solution of linear system of equations by determinants and its applications as Leontief input-output matrix of the economy, Coding and decoding theory. Linear Transformations: Reflection operators, Projection operators, Rotation operators, Shear in x and y directions, Dilation and Contraction. Introduction to node voltage and loop current methods.

Principle of House wiring and Industrial wiring Diode Transistor and simple rectifier circuit. Relationship between Load, Shear and Moment. Theory of simple bending, Neutral Axis, Resisting moments and section modulus.

Singer Mechanics of Materials by E. Beam analysis program for simply supported, cantilever, Overhanging Beam Program for finding resultant of forces and its angle Program for stress analysis of composite bar. Nieto Visual Basic. Engineering Materials Cements, Ceramics, and Refractories: Manufacture, properties, and application of lime, cement, ceramic and bricks Mortars and concrete, Pozzolanic material, effects of various chemicals on cement and concrete.

Methods of protection, strength and test of building stone. Quarrying and dressing of stone. Timbers: Varieties and uses of important timbers, method of seasoning and sawing Decay, seasoning and preservation of timber, laminated materials. Glass and Plastics: Composition, varieties, properties and use of glass, plastic, laminates and adhesive.

Metals: Composition and properties of ferrous and non ferrous metals used in civil engineering. Effect of various heat treatments on the properties of steel and its alloys. Corrosion and methods of corrosion control. Properties of thermal insulation material for use in buildings. Paints and Varnishes: Composition, preparation, properties, test and uses of paints, plasters, varnishes and distempers.

Other Materials: Acoustical material and geo-textiles, properties and uses of asphalt, bitumen, rubber and asbestos, laminates and adhesive. Neville A. Material of Construction, McGraw-Hill. Calculus Single Variable Calculus : Basic concepts of single variable function, Continuous, discontinuous and piecewise continuous functions, Periodic, odd and even functions, algebraic functions, Transcendental functions and its graphical representations, Applications of functions in our daily life situations.

Differential Calculus: Limits and continuity, Interpretation of a derivative, Geometric interpretation, Total differential and its applications in our daily life situations, The use of a table of different type derivatives, Higher order derivatives, Tangents and normals, Approximation of a function at a particular point by Taylor's and Maclaurin's series, Maximum and minimum values of a function, The first derivative test, The second derivative test, Point of inflexion and its applications in business and engineering.

Integral Calculus: Basic concepts of integration, A table of integral formulas, Some rules of integration, Definite integrals, The area bounded by a curve, Integration by parts, Integration as the limit of a sum, Volume of revolution, and its applications in our daily life situations Multivariate Calculus: Basic concepts of multivariate function, Level curves and surfaces, Limits and continuity, Partial differentiation, Geometric interpretation, higher partial derivatives.

Tangent planes, Total differential, Vector functions and its differentiation and integration, The directional derivative, The gradient, Scalar and Vector fields, Normal property of the gradient, Divergence , Curl, Tangent planes and normal lines, Extrema of functions of two variables, Second partials test, Extreme value theorem, Method of Constrained optimization and Lagrange multipliers.

Toff and Mckay, Practical Mathematics S. Rizvi, Engineering Mathematics. Engineering Geology Introduction to Geology Importance of Geology for Civil Engineering Projects Physical properties and identification of common rocks forming minerals Rocks formation and classification: According to the mode of occurrences According to the Silica contents Weather and erosion: Weather classification, fresh, slightly weathered, moderately weathered etc.

Discontinuity classification: Joints, faults and other fractures, micro structural features, such as lamination, cleavages, foliations, spacing of discontinuities as close, wide, medium etc, Description of Rock masses as thickly bedded or thinly bedded Identification of filling in joints, sand clay and breccias etc Color of grains, description with respect to the rock color and identification as a course grained, hardness classification as soft with respect to test Geological classification and identification of Rocks by geological names Identification and subordinate constitutions in rocks samples such as seams or branches of other types of minerals for example, Dolomite, Lime stone, Calcareous sand stone, sand.

Classification of Durability of Rocks in Dry and wet condition with durability test Engineering and physical properties of rocks Geological technical properties of rocks used as building stones, as a decorated stones and as a industrial rocks such as color, luster, streak, specific gravity, water absorption and unit weight etc.

Brief Introduction to structural Geology: Plate Tectonics with respect to the global application, earthquakes, causes of earthquakes, protective measures against earthquakes and zoning of earth quakes in Pakistan Role of geology in selection of sites for dams, reservoirs, tunnels and other civil engineering structures Brief introduction of local geology.

Form work Form work for general in-situ construction, props, bracing and horizontal shuttering platforms. Expansion joints and construction joints Wood work in building construction Other Engineering Projects.

An over view of construction aspects of different types of engineering projects, e. Types of structural members. Types of beams, supports and loads. Stability of structures. Determinate and indeterminate structures, Degree of indeterminacy. Sign conventions for bending moment and shear force etc. Analysis of determinate frames. Classification of co-planar trusses. Methods of analysis of trusses; Method of joints, Method of sections, Graphical method.

Unit load method. Theorem of virtual work for trusses beams and frames. Linear arch. Three hinged parabolic and circular arch. Bending Moment and shear force diagrams.

Influence lines for shear, thrust and moment. Influence lines for statically determinate beams and paneled girders. Influence lines for reaction, shear and bending moment of statically determinate beams and paneled girders. Influence lines for axial forces in trusses. Influence lines for composite structures. Criterion for maximum moment and shear. Absolute maximum bending moment. Stiffened suspension bridges with three hinged stiffening girders.

Shear force and bending moment diagrams. Application of 3-moment equation to the analysis of indeterminate beams Labs Demonstration of various types of structures and supports. Demonstrate the stability of structures using model structures. Determination of the horizontal thrust and maximum bending moment in a three hinged parabolic arches.

Determination of the horizontal thrust and maximum bending moment in a two hinged parabolic arches. Determination of the deflections and rotations in overhanging beams. Demonstration of influence line Investigation of the buckling struts Determination of shear cente.

Wang, C. Hibbeler, R. Structural Analysis, Prentice Hall. Influences Geographical, climatic, religious, social, historical. Qualities Strength, vitality, grace, breadth and scale. Factors Proportion, color and balance. Use of Materials Stone, wood, metals, concrete, composites, ceramics. General Treatment to Plan of Buildings Walls and their construction; Openings and their position, character and shape; Roofs and their development and employment; Columns and their position, form and decoration; Molding and their form decoration, Ornament as applied to any buildings.

Town Planning Definitions Trends in urban growth; Objectives of town planning; Modern planning in Pakistan and abroad. Preliminary Studies Study of natural resources, economic resources, legal and administrative problems, civic surveys and preparation of relevant maps. Land Use Patterns Various theories of land use pattern. Snyter, J. A History of Architecture. The Athlone Press. Mechanical Technology Basics of Thermodynamics Thermodynamic systems, Laws of thermodynamics, Laws of perfect gases Energy equation Internal Energy, Enthalpy and entropy of the working fluids Prime Movers: Internal combustion engines: type, working principle, cycle operation and performance, Steam Engines Steam Turbines Air-compressors Air-Conditioning: Introduction to Air-conditioning and refrigeration.

Heating and cooling load and its calculations, comfort chart, outline of AC systems Lab Practical 1: To study the different components of petrol engine. Practical 2: To study the cooling system of automobile engine. Practical 3: To study the lubrication system of automobile engine.

Practical 4: To study the ignition system of automobile engine. Practical 5: To study the fuel system of automobile engine.

Practical 6: To study the air-intake system of automobile engine. Practical 7: To study the 2-Stroke Reciprocating Engine. Practical 8: To study the vapor compression system. Practical 9: To study different components of refrigeration and air-conditioning system. Practical The layout of boiler room. Practical To study the boiler of the steam engine power plant. Practical To study the steam Engine of the steam engine power plant.

Practical To study the turbine of the steam engine power plant. Practical To study the condenser of steam engine power plant. Differential Equations Ordinary Differential Equations: Basic concepts of ordinary differential equation, General and particular solutions, Initial and boundary conditions, Linear and nonlinear differential equations, Solution of first order differential equation by separable variables and its applications in our daily life situations, The techniques like change of variable, homogeneous, non-homogeneous, exact, non-exact, linear and nonlinear Bernoulli could be used in case of complications.

Partial Differential Equations: Basic concepts, Linear and nonlinear p. Fourier Series: Periodic waveforms and their fourier representations, Calculating a fourier series, Fourier series of odd and even functions, Half range fourier series, Fourier series solution for the above p.

Kreyszig, E. Advanced Engineering Mathematics, Wayne and Erson. Distinction between solids and fluids. Surface tension. Compressibility of fluids. Fluid Statics: Static pressure, Pressure height relationship, absolute and gauge pressure, Measurement of Pressure, Barometer, Bourdon gauge, Pizometer tube, simple and differential manometer, Basic principal of various pressure measuring instruments.

Forces on submerged plane and curved bodies. Buoyancy and Stability of submerged and floating bodies. When distinction between the two units is necessary, the force unit is frequently written as lbf and the mass unit as lbm.

In this book we use almost exclusively the force unit, which is written simply as lb. Other common units of force in the U. The International System of Units SI is termed an absolute system because the measurement of the base quantity mass is independent of its environment. On the other hand, the U. A standard pound is also the force required to give a one-pound mass an acceleration of In SI units the kilogram is used exclusively as a unit of mass never force.

In the MKS meter, kilogram, second gravitational system, which has been used for many years in non-English-speaking countries, the kilogram, like the pound, has been used both as a unit of force and as a unit of mass.

The standard kilogram Primary Standards Primary standards for the measurements of mass, length, and time have been established by international agreement and are as follows:Mass. The kilogram is defined as the mass of a specific platinumiridium cylinder which is kept at the International Bureau of Weights and Measures near Paris, France.

The meter, originally defined as one ten-millionth of the distance from the pole to the equator along the meridian through Paris, was later defined as the length of a specific platinum-iridium bar kept at the International Bureau of Weights and Measures. The difficulty of accessing the bar and reproducing accurate measurements prompted the adoption of a more accurate and reproducible standard of length for the meter, which is now defined as 1 However, irregularities in the earth's rotation led to difficulties with this definition, and a more accurate and reproducible standard has been adopted.

The second is Wood River 5 Jack Plane Oil now defined as the duration of 9 periods of the radiation of a specific state of the cesium atom. For most engineering work, and for our purpose in studying mechanics, the accuracy of these standards is considerably beyond our needs.

Unit ConversionsThe characteristics of SI units are shown inside the front cover of this book, along with the numerical conversions between U. In addition, charts giving the approximate conversions between selected quantities in the two systems appear inside the back cover for convenient reference. Although these charts are useful for obtaining a feel for the relative size of SI and U.

In statics we are primarily concerned with the units of length and force, with mass needed only when we compute gravitational force, as explained previously. In statics as well as dynamics we often need to compute the weight of a body, which is the gravitational force acting on it.

This computation depends on the law of gravitation, which was also formulated by Newton. The gravitational force which the earth exerts on the moon foreground is a key factor in the motion of the moon. Gravitational Attraction of the Earth Gravitational forces exist between every pair of bodies. On the surface of the earth the only gravitational force of appreciable magnitude is the force due to the attraction of the earth.

For example, each of two iron spheres mm in diameter is attracted to the earth with a gravitational force of On the other hand, the force of mutual attraction between the spheres if they are just touching is 0.

This force is clearly negligible compared with the earth's attraction of Consequently the gravitational attraction of the earth is the only gravitational force we need to consider for most engineering applications on the earth's surface. The gravitational attraction of the earth on a body its weight exists whether the body is at rest or in motion.

Because this attraction is a force, the weight of a body should be expressed in newtons N in SI units and in pounds lb in U. Unfortunately in common practice the mass unit kilogram kg has been frequently used as a measure of weight.

This usage should disappear in time as SI units become more widely used, because in SI units the kilogram is used exclusively for mass and the newton is used for force, including weight. For a body of mass m near the surface of the earth, the gravitational attraction F on the body is specified by Eq.

Because the body falls with an acceleration g, Eq. The standard values forg of 9. The true weight gravitational attraction and the apparent weight as measured by a spring scale are slightly different.

The difference, which is due to the rotation of the earth, is quite small and will be neglected. This effect will be discussed in Vol. For example, suppose the mm side of a square bar was measured to the nearest millimeter, so we know the side length to two significant fig-ures. Squaring the side length gives an area of mm 2. However, according to our rule, we should write the area as mm 2 , using only two significant figures.

When calculations involve small differences in large quantities, greater accuracy in the data is required to achieve a given accuracy in the results.

Thus, for example, it is necessary to know the numbers 4. It is often difficult in lengthy computations to know at the outset how many significant fig-ures are needed in the original data to ensure a certain accuracy in the answer.

Accuracy to three significant figures is considered satisfactory for most engineering calculations. In this text, answers will generally be shown to three significant fig-ures unless the answer begins with the digit 1, in which case the answer will be shown to four significant figures. For purposes of calculation, consider all data given in this book to be exact.

DifferentialsThe order of differential quantities frequently causes misunderstanding in the derivation of equations. Higher-order differentials may always be neglected compared with lower-order differentials when the mathematical limit is approached.

For example, the element of volume AV of a right circular cone of altitude h and base radius r may be taken to be a circular slice a distance x from the vertex and of thickness Ax. Consider the right triangle of Fig. If the hypotenuse is unity, we see from the geometry of the figure that the arc length 1X0 and sin 6 are very nearly the same. Also cos 6 is close to unity. Furthermore, sin 6 and tan 6 have almost the same values. These approximations may be obtained by retaining only the first terms in the series expansions for these three functions.

Mathematics establishes the relations between the various quantities involved and enables us to predict effects from these relations. We use a dual thought process in solving statics problems: We think about both the physical situation and the corresponding mathematical description. In the analysis of every problem, we make a transition between the physical and the mathematical.

One of the most important goals for the student is to develop the ability to make this transition freely. Making Appropriate AssumptionsWe should recognize that the mathematical formulation of a physical problem represents an ideal description, or model, which approximates but never quite matches the actual physical situation.

When we construct an idealized mathematical model for a given engineering problem, certain approximations will always be involved. Some of these approximations may be mathematical, whereas others will be physical. For instance, it is often necessary to neglect small distances, angles, or forces compared with large distances, angles, or forces. Suppose a force is distributed over a small area of the body on which it acts.

We may consider it to be a concentrated force if the dimensions of the area involved are small compared with other pertinent dimensions. We may neglect the weight of a steel cable if the tension in the cable is many times greater than its total weight. However, if we must calculate the deflection or sag of a suspended cable under the action of its weight, we may not ignore the cable weight.

Thus, what we may assume depends on what information is desired and on the accuracy required. We must be constantly alert to the various assumptions called for in the formulation of real problems. The ability to understand and make use of the appropriate assumptions in the formulation and solution of engineering problems is certainly one of the most important characteristics of a successful engineer.

One of the major aims of this book is to provide many opportunities to develop this ability through the formulation and analysis of many practical problems involving the principles of statics. Using GraphicsGraphics is an important analytical tool for three reasons We use graphics to represent a physical system on paper with a sketch or diagram. Representing a problem geometrically helps us with its physical interpretation, especially when we must visualize three-dimensional problems.

We can often obtain a graphical solution to problems more easily than with a direct mathematical solution. Graphical solutions are both a practical way to obtain results, and an aid in our thought processes. Because graphics represents the physical situation and its mathematical expression simultaneously, graphics helps us make the transition between the two. Charts or graphs are valuable aids for representing results in a form which is easy to understand.

Formulating Problems and Obtaining SolutionsIn statics, as in all engineering problems, we need to use a precise and logical method for formulating problems and obtaining their solutions.

We formulate each problem and develop its solution through the following sequence of steps. Formulate the problem: a State the given data. Develop the solution: a Draw any diagrams you need to understand the relationships.

Keeping your work neat and orderly will help your thought process and enable others to understand your work. The discipline of doing orderly work will help you develop skill in formulation and analysis.

Problems which seem complicated at first often become clear when you approach them with logic and discipline. The Free-Body DiagramThe subject of statics is based on surprisingly few fundamental concepts and involves mainly the application of these basic relations to a variety of situations. In this application the method of analysis is all important. In solving a problem, it is essential that the laws which apply be carefully fixed in mind and that we apply these principles literally and exactly.

In applying the principles of mechanics to analyze forces acting on a body, it is essential that we isolate the body in question from all other bodies so that a complete and accurate account of all forces acting on this body can be taken. This isolation should exist mentally and should be represented on paper. The diagram of such an isolated body with the representation of all external forces acting on it is called a freebody diagram. The free-body-diagram method is the key to the understanding of mechanics.

This is so because the isolation of a body is the tool by which 7 cause and effect are clearly separated, and by which our attention is clearly focused on the literal application of a principle of mechanics.

The technique of drawing free-body diagrams is covered in Chapter 3, where they are first used. Numerical Values versus SymbolsIn applying the laws of statics, we may use numerical values to represent quantities, or we may use algebraic symbols, and leave the answer as a formula. When numerical values are used, the magnitude of each quantity expressed in its particular units is evident at each stage of the calculation.

This is useful when we need to know the magnitude of each term. The symbolic solution, however, has several advantages over the numerical solution. First, the use of symbols helps to focus our attention on the connection between the physical situation and its related mathematical description. Second, we can use a symbolic solution repeatedly for obtaining answers to the same type of problem, but having different units or numerical values.

Third, a symbolic solution enables us to make a dimensional check at every step, which is more difficult to do when numerical values are used. In any equation representing a physical situation, the dimensions of every term on both sides of the equation must be the same. This property is called dimensional homogeneity. Thus, facility with both numerical and symbolic forms of solution is essential. Solution MethodsSolutions to the problems of statics may be obtained in one or more of the following ways.

Obtain mathematical solutions by hand, using either algebraic symbols or numerical values. We can solve most problems this way. Obtain graphical solutions for certain problems. Solve problems by computer. This is useful when a large number of equations must be solved, when a parameter variation must be studied, or when an intractable equation must be solved. Many problems can be solved with two or more of these methods.

The method utilized depends partly on the engineer's preference and partly on the type of problem to be solved. The choice of the most expedient method of solution is an important aspect of the experience to be gained from the problem work. There are a number of problems in Vol. These problems appear at the end of the Review Problem sets and are selected to illustrate the type of problem for which solution by computer offers a distinct advantage.

Determine the weight in newtons of a car whose mass is kg. Convert the mass of the car to slugs and then determine its weight in pounds. From the table of conversion factors inside the front cover of the textbook, we see that 1 slug is equal to As another route to the last result, we can convert from kg to lbm.

We recall that 1 lbm is the amount of mass which under standard conditions has a weight of 1 lb of force. We rarely refer to the U. The sole use of slug, rather than the unnecessary use of two units for mass, will prove to be powerful and simple-especially in dynamics.

Q Note that we are using a previously calculated result We must be sure that when a calculated number is needed in subsequent calculations, it is retained in the calculator to its full accuracy, This may require storing it in a register upon its initial calculation and recalling it later. We must not merely punch Some individuals like to place a small indication of the storage register used in the right margin of the work paper, directly beside the number stored.

Use Table D Ans. The discrepancy is due to the fact that Newton's universal gravitational law does not take into account the rotation of the earth. The vector D is shown in Fig. See Art. Q A unit vector may always be formed by dividing a vector by its magnitude. Note that a unit vector is dimensionless. Write the unit vector n in the direction of V. The experience gained here will help you in the study of mechanics and in other subjects such as stress analysis, design of structures and machines, and fluid flow.

This chapter lays the foundation for a basic understanding not only of statics but also of the entire subject of mechanics, and you should master this material thoroughly.

A force has been defined in Chapter 1 as an action of one body on another. In dynamics we will see that a force is defined as an action which tends to cause acceleration of a body.

A force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action. Thus, forces may be combined according to the parallelogram law of vector addition. The action of the cable tension on the bracket in Fig. The effect of this action on the bracket depends on P, the angle 0, and the location of the point of application A.

Thus, the complete specification of the action of a force must include its magnitude, direction, and point of application, and therefore we must treat it as a fixed vector. External and Internal EffectsWe can separate the action of a force on a body into two effects, external and internal.

For the bracket of Fig. Forces external to a body can be either applied forces or reactive forces. The effects of P internal to the bracket are the resulting internal forces and deformations distributed throughout the material of the bracket. The relation between internal forces and internal deformations depends on the material properties of the body and is studied in strength of materials, elasticity, and plasticity.

The forces associated with this lifting rig must be carefully identified, classified, and analyzed in order to provide a safe and effective working environment. Principle of TransmissibilityWhen dealing with the mechanics of a rigid body, we ignore deformations in the body and concern ourselves with only the net external effects of external forces. In such cases, experience shows us that it is not necessary to restrict the action of an applied force to a given point.

For example, the force P acting on the rigid plate in Fig. The external effects are the force exerted on the plate by the bearing support at O and the force exerted on the plate by the roller support at C. This conclusion is summarized by the principle of transmissibility, which states that a force may be applied at any point on its given line of action without altering the resultant effects of the force external to the rigid body on which it acts.

Thus, whenever we are interested in only the resultant external effects of a force, the force may be treated as a sliding vector, and we need specify only the magnitude, direction, and line of action of the force, and not its point of application.

Because this book deals essentially with the mechanics of rigid bodies, we will treat almost all forces as sliding vectors for the rigid body on which they act. Force ClassificationForces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a body by a supporting surface.

On the other hand, a body force is generated by virtue of the position of a body within a force field such as a gravitational, electric, or magnetic field. An example of a body force is your weight.

Forces may be further classified as either concentrated or distributed. Every contact force is actually applied over a finite area and is therefore really a distributed force.

Force can be distributed over an area, as in the case of mechanical contact, over a volume when a body force such as weight is acting, or over a line, as in the case of the weight of a suspended cable. The weight of a body is the force of gravitational attraction distributed over its volume and may be taken as a concentrated force acting through the center of gravity.

The position of the center of gravity is frequently obvious if the body is symmetric. If the position is not obvious, then a separate calculation, explained in Chapter 5, will be necessary to locate the center of gravity. We can measure a force either by comparison with other known forces, using a mechanical balance, or by the calibrated movement of an elastic element. All such comparisons or calibrations have as their basis a primary standard.

The standard unit of force in SI units is the newton N and in the U. Action and ReactionAccording to Newton's third law, the action of a force is always accompanied by an equal and opposite reaction. It is essential to distinguish between the action and the reaction in a pair of forces. To do so, we first isolate the body in question and then identify the force exerted on that body not the force exerted by the body.

It is very easy to mistakenly use the wrong force of the pair unless we distinguish carefully between action and reaction. Concurrent ForcesTwo or more forces are said to be concurrent at a point if their lines of action intersect at that point. Thus, they can be added using the parallelogram law in their common plane to obtain their sum or resultant R, as shown in Fig. The resultant lies in the same plane as F 1 and F 2. Suppose the two concurrent forces lie in the same plane but are applied at two different points as in Fig.

By the principle of transmissibility, we may move them along their lines of action and complete their vector sum R at the point of concurrency A, as shown in Fig. We can replace F 1 and F 2 with the resultant R without altering the external effects on the body upon which they act.

We can also use the triangle law to obtain R, but we need to move the line of action of one of the forces, as shown in Fig. If we add the same two forces as shown in Fig. Therefore this type of combination should be avoided. The vector sum of the components must equal the original vector. Thus, the force R in Fig. Furthermore, the vector sum of the projections F a and F b is not the vector R, because the parallelogram law of vector addition must be used to form the sum.

The components and projections of R are equal only when the axes a and b are perpendicular. The two vectors are combined by first adding two equal, opposite, and collinear forces F and -F of convenient magnitude, which taken together produce no external effect on the body.

Adding F 1 and F to produce R 1? This procedure is also useful for graphically combining two forces which have a remote and inconvenient point of concurrency because they are almost parallel. It is usually helpful to master the analysis of force systems in two dimensions before undertaking three-dimensional analysis.

Thus the remainder of Chapter 2 is subdivided into these two categories. It follows from the parallelogram rule that the vector F of Fig. In terms of the unit vectors i and j of Fig. With either of these conventions it will always be clear that a force and its components are being represented, and not three separate forces, as would be implied by three solid-line vectors.

Actual problems do not come with reference axes, so their assignment is a matter of arbitrary convenience, and the choice is frequently up to the student. The logical choice is usually indicated by the way in which the geometry of the problem is specified. When the principal dimensions of a body are given in the horizontal and vertical directions, for example, you would typically assign reference axes in these directions.

Determining the Components of a ForceDimensions are not always given in horizontal and vertical directions, angles need not be measured counterclockwise from the x-axis, and the origin of coordinates need not be on the line of action of a force. Therefore, it is essential that we be able to determine the correct components of a force no matter how the axes are oriented or how the angles are measured.

Figure Memorization of Eqs. A neatly drawn sketch always helps to clarify the geometry and avoid error. For the example shown in Fig. Note that the angle which orients F 2 to the x-axis is never calculated. The cosine and sine of the angle are available by inspection of the triangle.

Also note that the x scalar component of F 2 is negative by inspection. The N force F is applied to the vertical pole as shown. In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis. The axis may be any line which neither intersects nor is parallel to the line of action of the force.

This rotational tendency is known as the moment M of the force. Moment is also referred to as torque. As a familiar example of the concept of moment, consider the pipe wrench of Fig.

One effect of the force applied perpendicular to the handle of the wrench is the tendency to rotate the pipe about its vertical axis. The magnitude of this tendency depends on both the magnitude F of the force and the effective length d of the wrench handle. Common experience shows that a pull which is not perpendicular to the wrench handle is less effective than the right-angle pull shown.

The magnitude of the moment or tendency of the force to rotate the body about the axis perpendicular to the plane of the body is proportional both to the magnitude of the force and to the moment arm d, which is the perpendicular distance from the axis to the line of action of the force.

The sense of M depends on the direction in which F tends to rotate the body. The right-hand rule, Fig. We represent the moment of F about as a vector pointing in the direction of the thumb, with the fingers curled in the direction of the rotational tendency.

The moment M obeys all the rules of vector combination and may be considered a sliding vector with a line of action coinciding with the moment axis. The basic units of moment in SI units are newton-meters N-m , and in the U.

When dealing with forces which all act in a given plane, we customarily speak of the moment about a point. By this we mean the moment with respect to an axis normal to the plane and passing through the point. Thus, the moment of force F about point A in Fig.

Sign consistency within a given problem is essential. For the sign convention of Fig. The curved arrow of the figure is a convenient way to represent moments in two-dimensional analysis.

The Cross ProductIn some two-dimensional and many of the three-dimensional problems to follow, it is convenient to use a vector approach for moment calculations. The moment of F about point A of Fig which agrees with the moment magnitude as given by Eq. We establish the direction and sense of M by applying the right-hand rule to the sequence r x F. If the fingers of the right hand are curled in the direction of rotation from the positive sense of r to the positive sense of F, then the thumb points in the positive sense of M.

We must maintain the sequence r x F, because the sequence F x r would produce a vector with a sense opposite to that of the correct moment.

As was the case with the scalar approach, the moment M may be thought of as the moment about point A or as the moment about the line which passes through point A and is perpendicular to the plane containing the vectors r and F. When we evaluate the moment of a force about a given point, the choice between using the vector cross product or the scalar expression depends on how the geometry of the problem is specified.

If we know or can easily determine the perpendicular distance between the line of action of the force and the moment center, then the scalar approach is generally simpler. If, however, F and r are not perpendicular and are easily expressible in vector notation, then the cross-product expression is often preferable.

In Section B of this chapter, we will see how the vector formulation of the moment of a force is especially useful for determining the moment of a force about a point in three-dimensional situations. Varignon's Theorem One of the most useful principles of mechanics is Varignon's theorem, which states that the moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.

To prove this theorem, consider the force R acting in the plane of the body shown in Fig. We begin by constructing a figure which shows the tension force T acting directly on the door, which is shown in an arbitrary angular position 0.

It should be clear that the direction of T will vary as varies. With the cross-product notation of Eq. Because the couple vector M is always perpendicular to the plane of the forces which constitute the couple, in two-dimensional analysis we can represent the sense of a couple vector as clockwise or counterclockwise by one of the conventions shown in Fig.

Later, when we deal with couple vectors in three-dimensional problems, we will make full use of vector notation to represent them, and the mathematics will automatically account for their sense.

Equivalent CouplesChanging the values of F and d does not change a given couple as long as the product Fd remains the same. Likewise, a couple is not affected if the forces act in a different but parallel plane.

Figure shows four different configurations of the same couple M. In each of the four cases, the couples are equivalent and are described by the same free vector which represents the identical tendencies to rotate the bodies.

Force-Couple SystemsThe effect of a force acting on a body is the tendency to push or pull the body in the direction of the force, and to rotate the body about any fixed axis which does not intersect the line of the force.

We can represent this dual effect more easily by replacing the given force by an equal parallel force and a couple to compensate for the change in the moment of the force. The replacement of a force by a force and a couple is illustrated in The transfer is seen in the middle figure, where the equal and opposite forces F and -F are added at point B without introducing any net external effects on the body.

Thus, we have replaced the original force at A by the same force acting at a different point B and a couple, without altering the external effects of the original force on the body. The combination of the force and couple in the right-hand part of Fig. By reversing this process, we can combine a given couple and a force which lies in the plane of the couple normal to the couple vector to produce a single, equivalent force.

Replacement of a force by an equivalent force-couple system, and the reverse procedure, have many applications in mechanics and should be mastered. The rigid structural member is subjected to a couple consisting of the two N forces. Replace this couple by an equivalent couple consisting of the two forces P and -P, each of which has a magnitude of N. Determine the proper angle 6. Now we are ready to describe the resultant action of a group or system of forces.

Most problems in mechanics deal with a system of forces, and it is usually necessary to reduce the system to its simplest form to describe its action. The resultant of a system of forces is the simplest force combination which can replace the original forces without altering the external effect on the rigid body to which the forces are applied.

Equilibrium of a body is the condition in which the resultant of all forces acting on the body is zero. This condition is studied in statics. When the resultant of all forces on a body is not zero, the acceleration of the body is obtained by equating the force resultant to the product of the mass and acceleration of the body. This condition is studied in dynamics. Thus, the determination of resultants is basic to both statics and dynamics.

The most common type of force system occurs when the forces all act in a single plane, say, the x-y plane, as illustrated by the system of three forces F 1? The principle of transmissibility has been used in this process. Algebraic MethodWe can use algebra to obtain the resultant force and its line of action as follows: The unit vectors i, j, and k are in the x-, y-9 and z-directions, respectively. In solving three-dimensional problems, one must usually find the x, v, and z scalar components of a force.

The choice of orientation of the coordinate system is arbitrary, with convenience being the primary consideration. However, we must use a right-handed set of axes in our three-dimensional work to be consistent with the right-hand-rule definition of the cross product. This is equivalent to the right-hand rule. The dot product of two vectors P and Q, Fig. In either case the dot product of the two vectors is a scalar quantity.

The dot-product relationship applies to nonintersecting vectors as well as to intersecting vectors. Thus, the dot product of the nonintersecting vectors P and Q in Fig. A force F with a magnitude of N is applied at the origin O of the axes x-y-z as shown. The line of action of F passes through a point A whose coordinates are 3 m, 4 m, and 5 m.

Part a. We begin by writing the force vector F as its magnitude F times a unit vector n 0A. The vector M is normal to the plane and is directed along the axis through O.

We can describe both the magnitude and the direction of M by the vector cross-product relation introduced in Art. Refer to item 7 in Art. The vector r runs from O to any point on the line of action of F. As described in Art. The correct direction and sense of the moment are established by the right-hand rule, described previously in Arts.

Thus, with r and F treated as free vectors emanating from O, Fig. Note the symmetry and order of the terms, and note that a right-handed coordinate system must be used. Expansion of the determinant gives where r x F replaces M. The expression r x F-n is known as a triple scalar product see item 8 in Art. As in the case of two dimensions, a couple tends to produce a pure rotation of the body about an axis normal to the plane of the forces which constitute the couple.

Couple vectors obey all of the rules which govern vector quantities. Thus, in Fig. In Art. You should also be able to carry out this replacement in three dimensions. The procedure is represented in Fig. By adding the equal and opposite forces F and -F at B, we obtain the couple composed of -F and the original F.

Thus, we see that the couple vector is simply the moment of the original force about the point to which the force is being moved.

We emphasize that r is a vector which runs from B to any point Solution, a Because F is parallel to the j-axis, F has no moment about that axis. It should be clear that the moment arm from the x-axis to the line of action of F is c and that the moment of F about the x-axis is negative. Similarly, the moment arm from the z-axis to the line of action of F is a and the moment of F about the z-axis is positive.

Helpful Hint Q Again we stress that r runs from the moment center to the line of action of F. Determine the moment about point O of the cable force acting on point A and the magnitude of this moment. We begin by writing the described force as a vector. Q The minus sign indicates that the vector M z is in the negative z-direction. Solution b.

The force of magnitude T is resolved into components T z and T xy in the x-y plane. Solution c. The component T xy is further resolved into its components T x and T y. It is clear that T y exerts no moment about the z-axis since it passes through it, so that the required moment is due to T x alone. The direction cosine of T with respect 0. Specify the two forces F and -F, applied in the two faces of the block parallel to the y-z plane, which may replace the four given forces.

The N forces act parallel to the y-z plane. The direction of M 1 is normal to the plane defined by the two forces, and the sense, shown in the figure, is established by the right-hand convention.

The forces F and -F lie in a plane normal to the couple M, and their moment arm as seen from the right-hand figure is mm. In determining the effect of the force on the shaft at a cross section such as that at O, we may replace the force by an equivalent force at O and a couple. Describe this couple as a vector M. The couple vector is perpendicular to the plane in which the force is shifted, and its sense is that of the moment of the given force about O.

We found the magnitude and direction of the resultant force for the two-dimensional force system by a vector summation of forces, Eq. These same principles can be extended to three dimensions. In the previous article we showed that a force could be moved to a parallel position by adding a corresponding couple.

Thus, for the system of forces F 1? When all forces are shifted to O in this manner, we have a system of concurrent forces at O and a system of couple vectors, as represented in part b of the figure. The concurrent forces may then be added vectorially to produce a resultant force R, and the couples may also be added to produce a resultant couple M, Fig. In dynamics we usually select the mass center as the reference point.

The change in the linear motion of the body is determined by the resultant force, and the change in the angular motion of the body is determined by the resultant couple.

In statics, the body is in complete equilibrium when the resultant force R is zero and the resultant couple M is also zero. Thus, the determination of resultants is essential in both statics and dynamics. We now examine the resultants for several special force systems.

A common example of a positive wrench is found with the application of a screwdriver, to drive a right-handed screw. Any general force system may be represented by a wrench applied along a unique line of action.

This reduction is illustrated in Fig. In part c of the figure, the couple M 2 is replaced by its equivalent of two forces R and -R separated by a distance Thus, the resultants of the original general force system have been transformed into a wrench positive in this illustration with its unique axis defined by the new position of R.

We see from Fig Hence, the resultant consists of a couple, which of course may be applied at any point on the body or the body extended. Helpful Hints Q Since the force summation is zero, we conclude that the resultant, if it exists, must be a couple. Determine the wrench resultant of the three forces acting on the bracket.

Calculate the coordinates of the point P in the x-y plane through which the resultant force of the wrench acts.

Also find the magnitude of the couple M of the wrench. The direction cosines of the couple M of the wrench must be the O same as those of the resultant force R, assuming that the wrench is positive.

We see that M turned out to be negative, which means that the couple vector is pointing in the direction opposite Total Shop Wood Planer Jacket to R, which makes the wrench negative.

The weight W is 5 oz, the drag D is 1. If it is known that the y-component of the resultant is Mastery of this material is essential for our study of equilibrium in the chapters which follow.

Failure to correctly use the procedures of Chapter 2 is a common cause of errors in applying the principles of equilibrium. When difficulties arise, you should refer to this chapter to be sure that the forces, moments, and couples are correctly represented. ForcesThere is frequent need to represent forces as vectors, to resolve a single force into components along desired directions, and to combine two or more concurrent forces into an equivalent resultant force.

Specifically, you should be able to Resolve a given force vector into its components along given directions, and express the vector in terms of the unit vectors along a given set of axes. Express a force as a vector when given its magnitude and information about its line of action. This information may be in the form of two points along the line of action or angles which orient the line of action. Use the dot product to compute the projection of a vector onto a specified line and the angle between two vectors.

Compute the resultant of two or more forces concurrent at a point. MomentsThe tendency of a force to rotate a body about an axis is described by a moment or torque , which is a vector quantity. We have seen that finding the moment of a force is often facilitated by combining the moments of the components of the force.

When working with moment vectors you should be able to Determine a moment by using the moment-arm rule. Use the vector cross product to compute a moment vector in terms of a force vector and a position vector locating the line of action of the force.

Utilize Varignon's theorem to simplify the calculation of moments, in both scalar and vector forms. Use the triple scalar product to compute the moment of a force vector about a given axis through a given point. CouplesA couple is the combined moment of two equal, opposite, and noncollinear forces. The unique effect of a couple is to produce a pure twist or rotation regardless of where the forces are located.

The couple is useful in replacing a force acting at a point by a force-couple system at a different point. To solve problems involving couples you should be able to Compute the moment of a couple, given the couple forces and either their separation distance or any position vectors locating their lines of action. Replace a given force by an equivalent force-couple system, and vice versa. ResultantsWe can reduce an arbitrary system of forces and couples to a single resultant force applied at an arbitrary point, and a corresponding resultant couple.

We can further combine this resultant force and couple into a wrench to give a single resultant force along a unique line of action, together with a parallel couple vector. To solve problems involving resultants you should be able to Compute the magnitude, direction, and line of action of the resultant of a system of coplanar forces if that resultant is a force; otherwise, compute the moment of the resultant couple.

Apply the principle of moments to simplify the calculation of the moment of a system of coplanar forces about a given point. Replace a given general force system by a wrench along a specific line of action. EquilibriumYou will use the preceding concepts and methods when you study equilibrium in the following chapters. Let us summarize the concept of equilibrium This means that its center of mass is either at rest or moving in a straight line with constant velocity.

Introduction Statics deals primarily with the description of the force conditions necessary and sufficient to maintain the equilibrium of engineering structures. This chapter on equilibrium, therefore, constitutes the most important part of statics, and the procedures developed here form the basis for solving problems in both statics and dynamics.

We will make continual use of the concepts developed in Chapter 2 involving forces, moments, couples, and resultants as we apply the principles of equilibrium. All physical bodies are three-dimensional, but we can treat many of them as two-dimensional when the forces to which they are subjected act in a single plane or can be projected onto a single plane. When this simplification is not possible, the problem must be treated as threedimensional.

We will follow the arrangement used in Chapter 2, and discuss in Section A the equilibrium of bodies subjected to two-dimensional force systems and in Section B the equilibrium of bodies subjected to three-dimensional force systems. Omission of a force which acts on the body in question, or inclusion of a force which does not act on the body, will give erroneous results.

A mechanical system is defined as a body or group of bodies which can be conceptually isolated from all other bodies. A system may be a single body or a combination of connected bodies.

The bodies may be rigid or nonrigid. The system may also be an identifiable fluid mass, either liquid or gas, or a combination of fluids and solids. In statics we study primarily forces which act on rigid bodies at rest, although we also study forces acting on fluids in equilibrium. Once we decide which body or combination of bodies to analyze, we then treat this body or combination as a single body isolated from all surrounding bodies.

This isolation is accomplished by means of the free-body diagram, which is a diagrammatic representation of the isolated system treated as a single body. The diagram shows all forces applied to the system by mechanical contact with other bodies, which are imagined to be removed.

If appreciable body forces are present, such as gravitational or magnetic attraction, then these forces must also be shown on the free-body diagram of the isolated system. Only after such a diagram has been carefully drawn should the equilibrium equations be written. Because of its critical importance, we emphasize here that the free-body diagram is the most important single step in the solution of problems in mechanics.

Before attempting to draw a free-body diagram, we must recall the basic characteristics of force. These characteristics were described in Art.

Forces can be applied either by direct physical contact or by remote action. Forces can be either internal or external to the system under consideration. Application of force is accompanied by reactive force, and both applied and reactive forces may be either concentrated or distributed.

The principle of transmissibility permits the treatment of force as a sliding vector as far as its external effects on a rigid body are concerned. We will now use these force characteristics to develop conceptual models of isolated mechanical systems. These models enable us to write the appropriate equations of equilibrium, which can then be analyzed. Each example shows the force exerted on the body to be isolated, by the body to be removed.

Newton's third law, which notes the existence of an equal and opposite reaction to every action, must be carefully observed. The force exerted on the body in question by a contacting or supporting member is always in the sense to oppose the movement of the isolated body which would occur if the contacting or supporting body were removed. Because of its flexibility, a rope or cable is unable to offer any resistance to bending, shear, or compression and therefore exerts only a tension force in a direction tangent to the cable at its point of attachment.

The force exerted by the cable on the body to which it is attached is always away from the body. When the tension T is large compared with the weight of the cable, we may assume that the cable forms a straight line.

When the cable weight is not negligible compared with its tension, the sag Wood River Jack Plane Review Online of the cable becomes important, and the tension in the cable changes direction and magnitude along its length.

Although no actual surfaces are per-fectly smooth, we can assume this to be so for practical purposes in many instances. When mating surfaces of contacting bodies are rough, as in Example 3, the force of contact is not necessarily normal to the tangent to the surfaces, but may be resolved into a tangential or frictional component F and a normal component N. Example 4 illustrates a number of forms of mechanical support which effectively eliminate tangential friction forces.

In these cases the net reaction is normal to the supporting surface. Example 5 shows the action of a smooth guide on the body it supports. There cannot be any resistance parallel to the guide.

Example 6 illustrates the action of a pin connection. Such a connection can support force in any direction normal to the axis of the pin. We usually represent this action in terms of two rectangular components. The correct sense of these components in a specific problem depends on how the member is loaded. When not otherwise initially known, the sense is arbitrarily assigned and the equilibrium equations are then written.

If the solution of these equations yields a positive algebraic sign for the force component, the assigned sense is correct. A negative sign indicates the sense is opposite to that initially assigned.

If the joint is free to turn about the pin, the connection can support only the force R. If the joint is not free to turn, the connection can also support a resisting couple M. The sense of M is arbitrarily shown here, but the true sense depends on how the member is loaded. Example 7 shows the resultants of the rather complex distribution of force over the cross section of a slender bar or beam at a built-in or fixed support.

The sense of the reactions F and V and the bending couple M in a given problem depends, of course, on how the member is loaded. One of the most common forces is that due to gravitational attraction, Example 8.

This force affects all elements of mass in a body and is, therefore, distributed throughout it. The location of G is frequently obvious from the geometry of the body, particularly where there is symmetry. When the location is not readily apparent, it must be determined by experiment or calculations.

Similar remarks apply to the remote action of magnetic and electric forces. These forces of remote action have the same overall effect on a rigid body as forces of equal magnitude and direction applied by direct external contact. Example 9 illustrates the action of a linear elastic spring and of a nonlinear spring with either hardening or softening characteristics.

The representations in Fig. Step 7. Decide which system to isolate. The system chosen should usually involve one or more of the desired unknown quantities.

Step 2. Next isolate the chosen system by drawing a diagram which represents its complete external boundary. This boundary defines the isolation of the system from all other attracting or contacting bodies, which are considered removed. This step is Wood Jack Plane Drawing Llc often the most crucial of all. Make certain that you have completely isolated the system before proceeding with the next step. Step 3. Identify all forces which act on the isolated system as applied by the removed contacting and attracting bodies, and represent them in their proper positions on the diagram of the isolated system.

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