MECHANICS OF CIVIL ENGINEERING STRUCTURES A b a s i c s g u i d e t o t h e c i v i l e n g i n e e r i n g s t r u c t u r e s c o u r s e A U T H O R N O R M A H B I N T I J A I N U D I N M O H A M M E D A Z M I B I N L A D I
Authors Normah Binti Jainudin Mohammed Azmi Bin Ladi POLITEKNIK KUCHING SARAWAK MINISTRY OF HIGHER EDUCATION KM22, JALAN MATANG, 93050 KUCHING, SARAWAK. Phone No. : (082) 845596/7/8 Fax No. : (082) 845023 E-mail : [email protected] Website : http://www.poliku.edu.my/ Copyright © 2022 Politeknik Kuching Sarawak eISBN: 978-967-2953-63-0 All rights reserved. No parts of this publication may be copied, stored in form or by any means, electronic, mechanical, photocopying and recording or otherwise or by any means for reproduced without the prior permission of Politeknik Kuching Sarawak. National Library of Malaysia Cataloguing-in-Publication Data Normah Jainudin MECHANICS OF CIVIL ENGINEERING STRUCTURES : A basics guide to the civil engineering structures course / AUTHOR NORMAH BINTI JAINUDIN, MOHAMMED AZMI BIN LADI. Mode of access: Internet eISBN 978-967-2953-63-0 1. Structural engineering. 2. Civil engineering. 3. Government publications--Malaysia. 4. Electronic books. I. Mohammed Azmi Ladi. II. Title. 624.1 Published by: Politeknik Kuching Sarawak Ministry Of Higher Education
Preface The experience of teaching the Mechanics of Civil Engineering Structures course for the Civil Engineering Diploma program has sparked the author's idea to produce a more focused e-book publication as a facilitator of understanding theoretical explanations and problem-solving. The writing found in the e-book is related to the polytechnic syllabus found in the course Mechanics of Civil Engineering Structures where the main topics involving the calculation of loads in beams and the determination of gradients and deflections in supported beams are easily adapted according to the requirements of the diploma program. The content of this e-book is appropriate according to the syllabus requirements from time to time because it is based on the fundamentals of structural mechanics theory. This book is actually aimed at additional references to strengthen students' understanding in addition to other references that can be obtained in the market. This book is suitable as an additional reference to the requirements of the courses available at the polytechnic according to the level of knowledge and understanding of the students. With the hope that this book can help to increase the understanding of students, especially those who lack understanding of the content published in other reference books on the market. The authors would like to express her/ his appreciation and gratitude to the individuals who were directly involved and indirectly involved in the process of writing this e-book. A million thanks to the Head of the Civil Engineering Department, Mr. Che Zaidi bin Che Hassan, Head of Program Diploma in Civil Engineering, Mrs. Ledia Anak Angul, and the Head of Program Diploma in Engineering Building Services, Mr. Redzuan Safri bin Abdul Rahman. And also, for the students, thank you for your support and encouragement to the authors to produce this e-book. The author hopes that this e-book can be used by all. Thank you. Normah binti Jainudin Mohammed Azmi bin Ladi Jabatan Kejuruteraan Awam Politeknik Kuching Sarawak 93050 Kuching, Sarawak Email : [email protected] [email protected]
Abstract The structure is one of the most important elements in building construction. Safe and economical beam design in terms of beam’s cross section and reinforcement bar, accurate analysis is required to facilitate the process of beam design. Mechanics of Civil Engineering Structures presents the material needed by practicing engineers engaged in the analysis of civil engineering structures. This book focuses on the beam to give some knowledge regarding properties of materials, the concept of equilibrium forces, shear force and bending moment diagram, the concept of stress and strain in beam, second-moment area, slope, and deflection of the beam. Besides that, in this book is loaded with examples and exercises that are necessary covers knowledge of facts and basic principles of types of forces, the strength of materials, and the behaviour of loaded structures. Six chapters cover topics including an introduction to mechanics of structures; equilibrium forces, shear force, and bending moment; direct stress; bending stress in beams; shear stress of beam and slope and deflections of the beam due to symmetrical bending. This book is a valuable guide for civil engineers needing basic analysis in understanding the concept of the solution to find the required value for the next step in designing the building structures.
CHAPTER 1 INTRODUCTION TO MECHANICS OF STRUCTURES 1.0 Introduction 1 1.1 Definition of Mechanics of Structures 1 1.1.1 External and internal forces 2 1.1.2 External forces 2 1.1.3 Internal forces within structures 6 1.2 Forces of gravity, pressure, and reaction 9 1.2.1 Gravity force 9 1.2.2 Pressure 10 1.2.3 Reactions 11 1.3 Structures in Civil Engineering 11 1.4 Supports, Reactions, and Direction in Structures 14 End of Chapter Problem 16 CHAPTER 2 EQUILIBRIUM FORCES, SHEAR FORCE, AND BENDING MOMENT 2.0 Concepts of Static Equilibrium 19 2.1 The Relationship Between Force and Reaction 19 2.2 Beam Definition 20 2.2.1 Simply supported beam 22 2.2.2 Cantilever beam 25 2.2.3 Overhanging beam 27 2.2.4 Continuous beam 27 2.3 Types of Beam Supports 28 2.3.1 Pinned supports 29 2.3.2 Roller supports 32 2.3.3 Fixed supports 33 2.4 Statically Determinate and Statically Indeterminate Beams 34 2.5 Types of Load 36 2.5.1 Point load or concentrated load 36 2.5.2 Uniformly distributed load 37 2.5.3 Moment load 38 2.6 Draw Free Body Diagram 38 2.7 Values and Directions of Vertical Reaction, Horizontal Reaction, and Moment 42 2.8 Illustrate Shear Force and Bending Moment in a Beam 43 End of Chapter Problem 52 CHAPTER 3 DIRECT STRESS 3.0 Introduction 60 3.1 The Relationship Between Direct Stress and Direct Strain 60 3.1.1 Effect of axial force on direct stress and direct strain 61 3.1.2 Definition of direct stress and strain 62 3.1.3 Calculating the cross-sectional area, direct stress, direct strain, and deformation of materials on the prismatic bar and sectional bar 63 3.2 Hooke’s Law 68 3.2.1 Stress-strain diagram 68 Table of Contents
3.2.2 Characteristics of the material 69 End of Chapter Problem 73 CHAPTER 4 BENDING STRESS IN BEAMS 4.0 Introduction of Bending Stress 80 4.1 Understand the Basic Knowledge of Bending Stress in Beams 80 4.1.1 The effect of bending moment in a beam 81 4.1.2 Centroid and second moment of area of a section 83 4.1.2.1 Centre of gravity 84 4.1.3 Second moment of area @ moment of inertia 90 4.2 Bending Stress Formula 97 4.3 Bending Stress Diagram 97 End of Chapter Problem 107 CHAPTER 5 SHEAR STRESS OF BEAM 5.0 Introduction 108 5.1 Understand the basic knowledge of shear stress in bolt or rivet in the beam 108 5.1.1 Shear stress, shear strain, modulus of rigidity in bolt or rivet, and shear stress in the beam 113 5.1.2 The effect of shear stress on a loaded beam 116 5.2 Shear Stress Formula Applying in Bolt and Rivet and Beam 122 End of Chapter Problem 132 CHAPTER 6 SLOPE AND DEFLECTION OF DETERMINATE BEAM 6.0 Introduction 134 6.1 Slope and deflection of beam 134 6.2 Determination of slope and deflection by using Macaulay Method 138 6.2.1 The Moment Equation of Macaulay 138 6.2.2 Macaulay’s method for different types of loading 141 6.3 Moment Area Method 157 6.3.1 Another method of determining the slopes and deflections in beams is the area moment method, which involves the area of the moment diagram 157 6.3.2 Slope and deflection of simply supported beam –Moment Area Method 159 6.3.3 Slope and deflection of a cantilever beam 170 End of Chapter Problem 177 References
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 1 CHAPTER 1 Introduction to Mechanics of Structures GENERAL SUMMARY This topic covers the basic knowledge of structural mechanics, the definition of structure in civil engineering, types of forces, support reactions, and beams. 1.0 Introduction Mechanics of Structures equips students with knowledge of facts and basic principles of types of forces, the strength of materials, and the behaviour of loaded structures. This course provides exposure to loaded structures on direct and shear stresses, slope, and deflection. The domination of the cognitive domain brings intellectual force and student thinking from the lower to the higher level which includes knowledge level and understanding level to application level in solving problems that involve calculations. In this approach, all concepts learned can correlate with everyday life's structural behavior. 1.1 Definition of Mechanics of Structures Mechanics –research about moving objects (forces that can move the objects) Structures – solid bodies that are made up of various parts to form a particular shape Mechanics of structures’ is a research/analysis upon characteristics and structure’s behaviour when load or force is applied. Rationally, when an object is loaded, it will change depending on its magnitude and its direction. Mechanics can be broadly divided into two branches; Statics and Dynamics. Statics deal with the bodies at rest whereas dynamics involve studies related to bodies in motion. 1.1.1 External and Internal Forces Measuring Forces A force is a push or pulls that tends to cause an object to change its movement or shape. It tends to change the state of rest or motion of a body. Force is represented in magnitude and direction so it is a vector quantity.
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 2 a) Magnitude, Direction, and Location The actual effect of a force on a structure depends on: • the magnitude, or size, of the force (the bigger the force’s magnitude, the stronger it is and the more effective it will have on a structure) • the direction of the force • the location where the force is applied When drawing forces, the force is represented by an arrow. The different-sized arrows tell us a little about the magnitude, direction, and location of the forces in a diagram. b) The Newton The standard unit for measuring force is called a Newton (N). One Newton is the amount of force needed to hold up a mass of 100g. Source: https://whyfiles.org/170skyscraper/images/structure_illo.gif Figure 1: The forces are represented by an arrow 1.1.2 External forces External forces on structures are stresses that act on a structure from outside the structure. Gravity is one such force, acting on all things all the time. Impact forces (things that
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 3 collide with the structure) are another type of live load. External forces produce internal forces, or stresses, within the materials from which the structure is made. These internal stresses can change the shape or size of a structure and are called deformation. This deformation can lead to repair of the damage to the structure, or failure of the structure. a) Centre of Gravity The centre of gravity is the specific point where the structure’s mass is evenly distributed around. The force of gravity acts on all parts of the structure and if all parts are evenly distributed around the centre of gravity, then the structure will be stable. Engineers need to locate the centre of gravity of a structure to stabilize the structure. By locating the structure's centre of gravity, an engineer can tell if the structure is stable or unbalanced. To increase the stability of a structure you can increase the width of the base compared to its height and move the base closer to the ground Source: https://discover.hubpages.com/education/Aircraftflightcontrols From Figure 2, a rod AB is applied with external forces, P with magnitude P (newton). Figure 2: A rod AB is applied with external forces, P.
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 4 WHAT DO YOU THINK??? How do you find to find the centre of gravity for the shapes below?
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 5 b) Symmetry Symmetry is a balanced arrangement of mass occurring on opposite sides of a line or plane, or around a centre or axis. The force of gravity on either side of the centre point of this line is the same. c) Load The load is an external force on a structured d) Static and Dynamic Loads A static (dead) load is a permanent force, acting on a structure. This includes the weight of the structure itself and the non-moving parts it supports. A dynamic (live) load is a changing, or non-permanent force acting on a structure. This includes the force of the wind and the weight of things that are in, or on a structure. e) Supporting the Load Different kinds of structures are designed to withstand different loads and forces. Different bridges are built for different purposes. Type of Bridges Beam Bridge • Most common bridge used • Flat beam supported at each end Truss Bridge • Lightweight, but strong bridge made of trusses (triangle-shaped frames) along its sides hangs between two ends (towers) that hold it up. • Smaller cables attach the roadway to the hanging cables
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 6 Suspension Bridge Arch Bridge • Is designed to withstand heavy loads. • Roman aqueducts are good examples of this type of bridge 1.1.3 Internal Forces within Structures Internal force is an inner force in the structure to bear the load applied. The force acts as the opposition force of the external force. There are 4 types of internal forces: a) Axial force – Compression and Tension b) Shear force c) Bending force d) Torsion Figure 3 (a): Internal forces in trusses Figure 3 (b): Direction of forces
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 7 a) Axial force i) Compression Compression forces crush material by squeezing it together. Compressive strength measures the largest compression force the material can withstand before it loses its shape or fails. Figure 4 (a): Compression forces ii) Tension Tension forces stretch material by pulling its ends apart. Tensile strength measures the largest tension force the material can withstand before failing. Figure 4 (b): Tension forces b) Shear force Shear forces bend or tear a material by pressing different parts in opposite directions at the same time. Shear strength measures the largest shear force the material can withstand before it rips apart.
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 8 Figure 4 (c): Shear forces by pressing different parts in opposite directions c) Bending force The algebraic sum of all moments created by acting or reacting forces about any section is bending moment, while calculating bending moment forces may be considered either the left-hand side (LHS) or right-hand side (RHS) of a section. Figure 4 (d): Positive BM Concave curve Figure 4 (e): Negative BM Convex curve Figure 4 (f): The beam without a hinge tends to bend
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 9 d) Torsion Torsion forces twist a material by turning the ends in opposite directions. Torsion strength measures the largest torsion force the material can withstand and still spring back into its original shape. Figure 4 (g): Torsion forces twist a material 1.2 Forces of gravity, pressure, and reaction 1.2.1 Gravity force Where gravity = 9.81ms-2
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 10 1.2.2 Pressure Pressure is the ratio of force to the area over which that force is distributed. In other words, pressure is force per unit area applied in a direction perpendicular to the surface of an object. Pressure may be measured in any unit of force divided by any unit of area, the SI unit of pressure (the newton per square meter, N/m2) is called the Pascal (Pa) after the seventeenth-century philosopher and scientist Blaise Pascal. A pressure of 1Pa is small; it approximately equals the pressure exerted by a dollar bill resting flat on a table. Figure 5: The area of rectangle and cylinder is applied with force, F
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 11 1.2.3 Reactions If an object applies the load mg on the surface on contacted axis, there will be the same reaction, R acting on that object. The reactions are acting oppositely as the load is applied. The unit of reaction is the newton, N. Figure 6: The object is applied to the surface 1.3 Structures in Civil Engineering The structure is defined as a system of interconnected members assembled in a stable configuration and used to support a load or combination of loads. The load can have vertical and lateral effects on the structural components. The structural members are connected by providing different types of joints and supports. For example; a house building is a combination of various structural members such as slab, column, beam, and roof truss.
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 12 Img Source: Education connect Source :https://www.viator.com/en-CA/tours/Kuala-Lumpur/Petronas-Twin-Towers/d335-91672P62
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 13 Source: https://www.reddit.com/r/drunkbuildings/comments/av2fk7/krzywy_domek_is_an_unusually_shaped_building_in/ Krzywy Domek is an unusually shaped building in Sopot, Poland Source : https://en.wikipedia.org/wiki/Metropol_Parasol The Metropol Parasol in Spain
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 14 1.4 Supports, Reactions, and Direction in Structures There are THREE (3) types of support that are very common: a) Roller support The roller support is capable of resisting a force in only one specific line or action. The roller can resist only a vertical force or a force normal to the plane on which the roller moves. A reaction to this type of support corresponds to a single unknown figure. Figure 6 (a): Roller support and its direction b) Pinned or hinged support The hinged support can resist force acting in any direction of the plane. Hence, in general, the reaction at such support may have two components, one in the horizontal and another in the vertical direction. To determine these two components two equations of statics must be used. Usually, at the hinged end, the beam is free to rotate but translational displacement is not possible. (The hinged and roller supports are also term as simple supports.) Figure 6 (b): Pinned or hinged support and its direction
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 15 c) Fixed support The fixed support is capable of resisting force in any direction and is also capable of resisting a couple or a moment. A system of three forces can exist at such support (i.e., two components of force and a moment). Figure 6 (c): Fixed support and its direction The supports, reactions, and direction in structures can be simplified in Table 1. Table 1: A simplified of supports, reactions, and directions in structures SUPPORT REACTIONS DIRECTION TOTAL OF UNKNOWN Roller Ry or Fy y direction only 1 Pinned @ Hinged Ry and Rx @ Fy and Fx x and y direction 2 Fixed @ Built-in Ry, Rx and M @ Fy, Fx and M x, y, and moment 3
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 16 END OF CHAPTER PROBLEM Problem 1 Define ‘Mechanics of structure’. Answer
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 17 Problem 2 List and sketch four (4) types of internal forces. Answer Problem 3 Fill in the table below with the correct answer: SUPPORT REACTIONS DIRECTION TOTAL OF UNKNOWN Roller Pinned @ Hinged Fixed @ Built-in
CHAPTER 1 – INTRODUCTION TO MECHANICS OF STRUCTURES Mechanics of Civil Engineering Structures 18 Problem 4 Define reactions, force, and pressure. Answer Problem 5 An internal force is a force that acts from within the structure. With the aid of illustration, list FIVE (5) internal forces in Civil Engineering. Answer
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 19 CHAPTER 2 Equilibrium Forces, Shear Force, and Bending Moment GENERAL SUMMARY This topic introduces the students to the equilibrium principles and the relationship between forces and reactions. It also covers the calculation of shear force and bending moment for a loaded beam. The lab work covers the determination of shear force and bending moment in a beam. 2.0 Concepts of static equilibrium Static equilibrium takes place when all the forces acting on an object are balanced and the object is not in motion about the relative plane. An object which is in static equilibrium is unable to move. This is because all the forces which act on it compensate for one another. 2.1 The relationship between force and reaction Force is a quantitative description of the interaction between two physical bodies, such as an object and its environment. Force is proportional to acceleration. In calculus terms, force is the derivative of momentum concerning time. Contact force is defined as the force exerted when two physical objects come in direct contact with each other. Other forces, such as gravitation and electromagnetic forces can exert themselves even across the empty vacuum of space. The concept of force was originally defined by Sir Isaac Newton in his three laws of motion. He explained gravity as an attractive force between bodies that possessed mass (gravity within Einstein's general relativity doesn't require force). Force is a vector. The SI unit for force is the newton (N). One newton of force is equal to 1 kg * m/s2 . According to Newton's third law, for every action force, there is an equal (in size) and opposite (in direction) reaction force. Forces always come in pairs - known as "action-reaction force pairs." Identifying and describing action-reaction force pairs is a simple matter of identifying the two interacting objects and making two statements describing who is pushing on whom and
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 20 in what direction. The figure below shows examples of the relationship between force and reaction. Figure 2.1(a): The baseball forces the bat to the left; the bat forces the ball to the right. Together, these two forces exerted upon two different objects form the action-reaction force pair. Figure 2.1(b): The baseball pushes the glove leftwards; The glove pushes the baseball rightward Figure 2.1(c): Bowling ball pushes pin leftwards; Pin pushes bowling ball rightward. Figure 2.1(d): Enclosed air particles push balloon wall outwards. Balloon wall pushes enclosed air particles inwards
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 21 2.2 Beam definition A beam is a structural member which is primarily subjected to a system of external loads that act transverse to its axis. The forces in the longitudinal direction and twisting moments about the longitudinal axis may act in addition to transverse loading. A beam is therefore different from a bar in tension or a bar in compression because of the direction of loads acting on it. A beam has a characteristic feature that internal forces called shear forces and the internal moments called bending moments are developed in it, to resist the external loads. Many shafts of machines act as beams. The beams may be straight or curved. The actual installation of a straight beam may be vertical, inclined, or horizontal. But, for convenience, the beams discussed here will be shown in a horizontal position. For the beams, the distance (L) between the supports is called a span. Figure 2.2: A free body diagram of the beam with shear force and bending moment diagram
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 22 2.2.1 Simply supported beam Simply supported beams are supported at each end only. A diagram of a simple beam supported at each end is shown in Figure 2.1.2(a). If downwards pressure is applied to bend would occur in the middle of the beam. Figure 2.2.1 (a): A diagram of a simple beam supported at each end
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 23 Figure 2.2.1 (b): Example of beams shape
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 24 Source: https://www.excelcalcs.com/calcs/repository/Trial/UDL-on-simply-supported-beam.xls/ Figure 2.2.1 (c): Example of the simply supported were a kid as a uniformly distributed load Figure 2.1.2 (d): Roller support and pinned support in the simply supported beam
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 25 2.2.2 Cantilever beam Cantilever beams overhang their supports. The cantilever section of the beam cannot support the same loads as the back span which is the section of the beam between the supports. A diagram in Figure 2.1.2 (e) shows a simple beam supported at one end and free at the other end. For this beam, the bending occurs between the support. Figure 2.2.2 (a): A simple beam supported at one end and free at the other end
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 26 Figure 2.2.2 (b): Balcony Figure 2.2.2 (c): Car porch Figure 2.2.2 (d): Cantilever bridge Figure 2.2.2 (e): Cantilever balcony and cantilever room FIXED SUPPORT FREE END FIXED SUPPORT BALCONY ROOM FIXED SUPPORT
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 27 2.2.3 Overhanging beam A beam that is freely supported at two points and has one or both ends extending beyond these supports is known as an overhanging beam. Figure 2.2.3 (a): The pictures above show examples of overhanging beam 2.2.4 Continuous beam A continuous beam is a structural component that provides resistance to bending when a load or force is applied. These beams are commonly used in bridges. It has more than two point supports along its length. These are usually in the same horizontal plane and the spans between the supports are in one straight line. A continuous beam has more that are required to provide equilibrium, and deformation behaviour under load is also considered when determining support reactions. As a result, the continuous beam is statically indeterminate.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 28 Figure 2.2.4 (a): Examples of continuous beams 2.3 Types of beam supports The three common types of connections that join a built structure to its foundation are; roller, pinned, and fixed. A fourth type, not often found in building structures, is known as simple support. This is often idealized as a frictionless surface). All of these supports can be located anywhere along a structural element. They are found at the ends, at midpoints, or any other intermediate points. The type of support connection determines the type of load that the support
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 29 can resist. The support type also has a great effect on the load-bearing capacity of each element, and therefore the system. The diagram illustrates the various ways in which each type of support is represented. A single unified graphical method to represent each of these support types does not exist. Chances are that one of these representations will be similar to local common practice. However, no matter what the representation, the forces that the type can resist are indeed standardized. Figure 2.3 (a): The illustration of support types Figure 2.3 (b): The reactions resist at each type of support 2.3.1 Pinned supports A pinned support can resist both vertical and horizontal forces but not a moment. They will allow the structural member to rotate, but not to translate in any direction. Many connections are assumed to be pinned connections even though they might resist a small amount of moment in reality. It is also true that a pinned connection could allow rotation in only one direction; providing resistance to rotation in any other direction. The knee can be idealized as a connection that allows rotation in only one direction and provides resistance to lateral movement. The design of a pinned connection is a good example of the idealization of reality. A single pinned connection is usually not sufficient to
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 30 make a structure stable. Another support must be provided at some point to prevent the rotation of the structure. The representation of a pinned support includes both horizontal and vertical forces. Figure 2.3.1 (a): The examples of pinned supports Pinned connections are confronted daily. Every time a hinged door is pushed open a pinned connection has allowed rotation around a distinct axis; preventing translation in two. The door hinge prevents vertical and horizontal translation. If a sufficient moment is not generated to create rotation the door will not move at all.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 31 Figure 2.3.1 (b): Typical pinned supported and Illustration connections connection (metal) What do you think? Have you ever calculated how many moments are required to open a specific door? Why is one door easier to open than another? THINK ABOUT IT!!
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 32 2.3.2 Roller supports Roller supports are free to rotate and translate along the surface upon which the roller rests. The surface can be horizontal, vertical, or sloped at any angle. The resulting reaction force is always a single force that is perpendicular to, and away from, the surface. Roller supports are commonly located at one end of long bridges. This allows the bridge structure to expand and contract with temperature changes. The expansion forces could fracture the supports at the banks of the bridge structure was "locked" in place. Roller supports can also take the form of rubber bearings, rockers, or a set of gears that are designed to allow a limited amount of lateral movement. Roller support cannot provide resistance to lateral forces. Imagine a structure (perhaps a person) on roller skates. It would remain in place as long as the structure must only support itself and perhaps a perfectly vertical load. As soon as a lateral load of any kind pushes on the structure it will roll away in response to the force. The lateral load could be a shove, a gust of wind, or an earthquake. Since most structures are subjected to lateral loads it follows that a building must have other types of support in addition to roller supports.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 33 Figure 2.3.2 (a): Typical roller supported connection (concrete) 2.3.3 Fixed supports Fixed supports can resist vertical and horizontal forces as well as a moment. Since they restrain both rotation and translation, they are also known as rigid supports. This means that a structure only needs one fixed support to be stable. All three equations of equilibrium can be satisfied. A flagpole set into a concrete base is a good example of this kind of support. The representation of fixed supports always includes two forces (horizontal and vertical) and a moment. Figure 2.3.3 (a): Typical fixed support/connections Fixed connections are very common. Steel structures of many sizes are composed of elements that are welded together. A cast-in-place concrete structure is automatically monolithic and it becomes a series of rigid connections with the proper placement of the reinforcing steel. Fixed connections demand greater attention during construction and are often the source of building failures.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 34 Figure 2.3.3 (b): Typical fixed supports connection (metal) 2.4 Statically determinate and statically indeterminate beams The structure is an assemblage of several components like slabs, beams, columns, walls, Foundations, and so on, which remain in equilibrium. It has to satisfy the fundamental criteria of strength, stiffness, economy, durability, and compatibility, for its existence. It is generally classified into two categories as Determinate and Indeterminate structures or Redundant Structures. Any structure is designed for the stress resultants of bending moment, shear force, deflection, torsional stresses, and axial stresses. If these moments, shears, and stresses are evaluated at various critical sections, then based on these, the proportioning can be done. Evaluation of these stresses, moments, and forces and plotting them for that structural component is known as analysis. The determination of dimensions for these components of these stresses and proportioning is known as design. Determinate structures are analyzed just by the use of basic equilibrium equations. By this analysis, the unknown reactions are found for the further determination of stresses. Redundant or indeterminate structures are not capable of being analyzed by the mere use of basic equilibrium equations. Along with the basic equilibrium equations, some extra conditions are required to be used like compatibility conditions of deformations, etc to get the unknown reactions for drawing bending moment and shear force diagrams.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 35 Examples of determinate structures are: simply supported beams, cantilever beams, single and double overhanging beams, three-hinged arches, etc. Examples of indeterminate structures are fixed beams, continuous beams, fixed arches, twohinged arches, portals, multi-storeyed frames, etc. Table 1: Differences Between Determinate and Indeterminate Structures No. Determinate Structures Indeterminate Structures 1 Equilibrium conditions are fully adequate to analyze the structure. Conditions of equilibrium are not adequate to fully analyze the structure. 2 The bending moment or shear force at any section is independent of the material property of the structure. The bending moment or shear force at any section depends upon the material property. 3 The bending moment or shear force at any section is independent of the cross-section or moment of inertia. The bending moment or shear force at any section depends upon the cross-section or moment of inertia. 4 Temperature variations do not cause stress. Temperature variations cause stress. 5 No stresses are caused due to lack of fit. Stresses are caused due to lack of fit. 6 Extra conditions like compatibility of displacements are not required to analyze the structure. Extra conditions like compatibility of displacements are required to analyze the structure along with the equilibrium equations.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 36 2.5 Types of load 2.5.1 Point load or concentrated load A load that is acting at a particular point on a structural element is termed a point load or concentrated load. This can be on the x-axis, y-axis, or inclined to the x-axis as shown in the sketch (Figure 2.1.5 (a)). Figure 2.5.1 (a): Types of concentrated load; F1 = Point load in y axis, F2 = Point load inclined to x axis, F3 = Point load in x axis Figure 2.5.1 (b): The illustration of a loaded beam
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 37 Figure 2.1.5 (c): The free body diagram of the beam where Fc is point load due to column and RA and RB are the reaction produced by columns A and B. 2.5.2 Uniformly distributing load When numbers of point loads are acting of the same magnitude throughout a certain span of the beam then it is called a Uniformly Distributing Load (U.D.L). Self-weight of the beam is an example of a uniformly distributed load.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 38 2.5.3 Moment The moment of a force about a point is the product of force and its perpendicular distance from the point about which the moment is taken. Unit of the moment: Unit of force is ‘N’ and distance is ‘m’, so: MOMENT = Nm Couple: When two, unlike parallel forces, are acting on a body that is equal in magnitude; then these unlike, parallel, equal and non-collinear forces form a couple • The distance between two forces is called a couple of arms • Since forces are equal and opposite R = 0 • Moment of couple = Pa 2.6 Draw a free body diagram Free-body diagrams are diagrams used to show the relative magnitude and direction of all forces acting upon an object in a given situation. A free-body diagram is a special example of a vector diagram. These diagrams will be used throughout our study of physics. The size of the About Q = Force x lever arm = P x a Q = Pa {anticlockwise moment} Q1 = P1 a1 {clockwise moment}
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 39 arrow in a Freebody diagram reflects the magnitude of the force. The direction of the arrow shows the direction that the force is acting. Each force arrow in the diagram is labeled to indicate the exact type of force. It is generally customary in a free-body diagram to represent the object by a box and to draw the force arrow from the center of the box outward in the direction that the force is acting. An example of a free-body diagram is shown on the right. The free-body diagram above depicts four forces acting upon the object. Objects do not necessarily always have four forces acting upon them. There will be cases in which the number of forces depicted by a free-body diagram will be one, two, or three. There is no hard and fast rule about the number of forces that must be drawn in a free-body diagram. The only rule for drawing free-body diagrams is to depict all the forces that exist for that object in the given situation. Thus, to construct Freebody diagrams, it is extremely important to know the various types of forces. If described a physical situation, begin by using your understanding of the force types to identify which forces are present. Then determine the direction in which each force is acting. Finally, draw a box and add arrows for each existing force in the appropriate direction; label each force arrow according to its type. If necessary, refer to the list of forces and their description to understand the various force types and their appropriate symbols. The Free Body Diagram (FBD) is a very useful tool in engineering and drawing one is often the first step in solving many problems. Here you will be shown how to draw Free Body Diagrams of different objects/situations:
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 40 STEP DESCRIPTION Step 1: Understanding the problem You have to understand the problem before you can begin to draw it. You may need to make a few sketches of the situation before you can fully understand what is happening. Once you understand this, you can start to draw some free-body diagrams. Some problems required more than one free body diagram because each free body diagram can only be used for one object in the question, you may need to analyze several objects in a single situation. Step 2: Separate the important object from the rest You know what object you want to draw a free body diagram of, so the diagram should only include this object and none of the other ones. You can simplify this object as much as you want to, for example, a car can often be represented by a rectangle or a beam represented as a straight line. Step 3: Draw already known forces on the object Generally, you have things such as gravity and friction acting on an object, but the source of these forces is not generally shown (well for gravity at least) in the original picture of the question. You need to put these forces onto the diagram first. They are drawn simply as vectors. Step 4: Replace other objects with force vectors Generally, problems will require more than one object, and these objects all apply forces to each other such as the normal force applied to an object from the 'ground'. when adding these forces into the diagram it is important to consider all the forces. Generally, you will cross off one external object from your sketch and then add one or more forces to the FBD to ensure that all external objects have been examined.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 41 Here is an example question: Drawing an FBD In the structure to the right, there are several objects 2 support poles (pink), a beam (blue), and 2 crates (brown). The crates can be assumed to have an evenly distributed density. The hanging crate is supported by wires. Draw a free body diagram (FBD) of the beam. Dimensions on the diagram and magnitudes of forces are not important. Assume that gravity exists but the mass of the beam is negligible. Actual structure diagram Free body diagram
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 42 Solution: 1) To draw the free body diagram of this beam, we look at everything that is applying a force to it or in other words everything that is touching it. 2) Firstly gravity exists in this question so everything has a weight however because we assume the beam has a mass of 0, the weight of the beam can be ignored and so too can the effects of gravity (there is no force due to gravity acting on the beam) 3) Now all we have left to examine is the other objects in the diagram pushing and pulling on the beam. 4) We will assume the force due to the support poles acting as point forces pushing upwards and we will call this Fp1 and Fp2. 5) The top crate is quite wide so it will create a distributed load force which we will call Fcrate 6) The two wires connected to the beam are in tension so they will pull on the beam as point forces, we will call these forces Ft1 and Ft2 Now we are ready to draw the FBD 2.7 Values and directions of vertical reaction, horizontal reaction, and moment The body is said to be in equilibrium if the resultant of all forces acting on it is zero. There are two major types of static equilibrium, namely, translational equilibrium and rotational equilibrium. Formulas Conditions of Static Equilibrium of Non-Concurrent Non-Parallel System Concurrent force system Parallel Force System Non-Concurrent NonParallel Force System
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 43 The sum of all forces in the x-direction or horizontal is zero. The sum of all forces in the y-direction or vertical is zero. The sum of moments at any point O is zero. Important Points for Equilibrium Forces Two forces are in equilibrium if they are equal and oppositely directed. Three coplanar forces in equilibrium are concurrent. Three or more concurrent forces in equilibrium form a close polygon when connected in a head-to-tail manner. 2.8 Illustrate shear force and bending moment in a beam Consider a simple beam shown of length L that carries a uniform load of w (N/m) throughout its length and is held in equilibrium by reactions R1 and R2. Assume that the beam (Figure 2.1.9) is cut at point C at a distance of x from the left support and the portion of the beam to the right of C is removed. The portion removed must then be replaced by vertical shearing force V together with a couple of M to hold the left portion of the bar in equilibrium under the action of R1 and wx.
CHAPTER 2 – EQUILIBRIUM FORCES, SHEAR FORCE AND BENDING MOMENT Mechanics of Civil Engineering Structures 44 Figure 2.1.9: Illustration of a simply loaded beam The couple M is called the resisting moment or moment and the force V is called the resisting shear or shear. The sign of V and M is taken to be positive if they have the senses indicated above.