CIV 2109 Lecture 5: Equilibrium & Support Reactions PDF

1 UNIVERSITY OF GUYANA FACULTY OF ENGINEERING & TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING CIV 2109: Statics & Introduction to Structures- Lecture 5: Equilibrium & Support Reactions  PRESENTATION OUTLINE 2 1.0 Introduction 2.0 Equilibrium o...

1 UNIVERSITY OF GUYANA FACULTY OF ENGINEERING & TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING CIV 2109: Statics & Introduction to Structures- Lecture 5: Equilibrium & Support Reactions  PRESENTATION OUTLINE 2 1.0 Introduction 2.0 Equilibrium of Structures 3.0 External and Internal Forces 4.0 Types of Supports for Plane Structures 5.0 Static Determinacy, Indeterminacy & Stability 6.0 Support Reactions References  1.0 INTRODUCTION 3 A structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples. If a structure is in equilibrium, then all its members and parts are also in equilibrium.  2.0 EQUILIBRIUM OF STRUCTURES 4 In order for a structure to be in equilibrium, all the forces and couples (including support reactions) acting on it must balance each other, and there must neither be a resultant force nor a resultant couple acting on the structure. For a space (three-dimensional) structure subjected to three- dimensional systems of forces and couples, the equations of equilibrium is given by: σ 𝑭𝒙 = 𝟎 σ 𝑭𝒚 = 𝟎 σ 𝑭𝒛 = 𝟎 σ 𝑴𝒙 = 𝟎 σ 𝑴𝒚 = 𝟎 σ 𝑴𝒛 = 𝟎  2.0 EQUILIBRIUM OF STRUCTURES 5 The first three equations ensure that there is no resultant force acting on the structure, and the last three equations express the fact that there is no resultant couple acting on the structure. For a plane structure lying in the 𝒙𝒚 plane and subjected to a coplanar system of forces and couples, the necessary and sufficient conditions for equilibrium can be expressed as: σ 𝑭𝒙 = 𝟎 σ 𝑭𝒚 = 𝟎 σ 𝑴𝒛 = 𝟎  2.0 EQUILIBRIUM OF STRUCTURES 6 The first two of the three equilibrium equations express, respectively, that the algebraic sums of the 𝒙 components and 𝒚 components of all the forces are zero, thereby indicating that the resultant force acting on the structure is zero. The third equation indicates that the algebraic sum of the moments of all the forces about any point in the plane of the structure and the moments of any couples acting on the structure is zero, thereby indicating that the resultant couple acting on the structure is zero.  2.0 EQUILIBRIUM OF STRUCTURES 7 All the equilibrium equations must be satisfied simultaneously for the structure to be in equilibrium.  2.0 EQUILIBRIUM OF STRUCTURES 8 Concurrent Force Systems When a structure is in equilibrium under the action of a concurrent force system, that is, the lines of action of all the forces intersect at a single point, the moment equilibrium equations are automatically satisfied, and only the force equilibrium equations need to be considered.  2.0 EQUILIBRIUM OF STRUCTURES 9 Concurrent Force Systems Therefore, for a space structure subjected to a concurrent three- dimensional force system, the equations of equilibrium are: σ 𝑭𝒙 = 𝟎 σ 𝑭𝒚 = 𝟎 σ 𝑭𝒛 = 𝟎 Similarly, for a plane structure subjected to a concurrent coplanar force system, the equilibrium equations can be expressed as: σ 𝑭𝒙 = 𝟎 σ 𝑭𝒚 = 𝟎  2.0 EQUILIBRIUM OF STRUCTURES 10 Two-Force and Three-Force Structures Throughout this course, we will encounter several structures and structural members that will be in equilibrium under the action of only two, or three, forces. The analysis of such structures and of structures composed of such members can be considerably expedited by recalling from statics the characteristics of such systems.  2.0 EQUILIBRIUM OF STRUCTURES 11 Two-Force and Three-Force Structures If a structure is in equilibrium under the action of only two forces, the forces must be equal, opposite, and collinear. If a structure is in equilibrium under the action of only three forces, the forces must be either concurrent or parallel.  3.0 EXTERNAL & INTERNAL FORCES12 The forces and couples to which a structure may be subjected can be classified into two types: 1. External Forces 2. Internal Forces  3.0 EXTERNAL & INTERNAL FORCES13 External Forces External forces are the actions of other bodies on the structure under consideration. For the purposes of analysis, it is usually convenient to further classify these forces as applied forces and reaction forces. Applied forces, usually referred to as loads (e.g., live loads and wind loads), have a tendency to move the structure and are usually known in the analysis.  3.0 EXTERNAL & INTERNAL FORCES14 External Forces Reaction forces, or reactions, are the forces exerted by supports on the structure and have a tendency to prevent its motion and keep it in equilibrium. The reactions are usually among the unknowns to be determined by the analysis. The state of equilibrium or motion of the structure as a whole is governed solely by the external forces acting on it.  3.0 EXTERNAL & INTERNAL FORCES15 Internal Forces Internal forces are the forces and couples exerted on a member or portion of the structure by the rest of the structure. These forces develop within the structure and hold the various portions of it together. The internal forces always occur in equal but opposite pairs, because each member or portion exerts back on the rest of the structure the same forces acting upon it but in opposite directions, according to Newton’s third law.  3.0 EXTERNAL & INTERNAL FORCES16 Internal Forces Because the internal forces cancel each other, they do not appear in the equations of equilibrium of the entire structure. The internal forces are also among the unknowns in the analysis and are determined by applying the equations of equilibrium to the individual members or portions of the structure.  4.0 Types of Supports for Plane Structures 17 Supports are used to attach structures to the ground or other bodies, thereby restricting their movements under the action of applied loads. The loads tend to move the structures; but supports prevent the movements by exerting opposing forces, or reactions, to neutralize the effects of loads, thereby keeping the structures in equilibrium. The type of reaction a support exerts on a structure depends on the type of supporting device used and the type of movement it prevents.  4.0 Types of Supports for Plane Structures 18 A support that prevents translation of the structure in a particular direction exerts a reaction force on the structure in that direction. Similarly, a support that prevents rotation of the structure about a particular axis exerts a reaction couple on the structure about that axis. These supports are grouped into three categories, depending on the number of reactions (1, 2, or 3) they exert on the structures. The types of supports commonly used for plane structures are shown in the following slides. 4.0 Types of Supports for Plane Structures 19 4.0 Types of Supports for Plane Structures 20  5.0 Static Determinacy, Indeterminacy & Stability 21 Internal Stability A structure is considered to be internally stable, or rigid, if it maintains its shape and remains a rigid body when detached from the supports. Conversely, a structure is termed internally unstable (or nonrigid) if it cannot maintain its shape and may undergo large displacements under small disturbances when not supported externally. Some examples of internally stable structures are shown on the following slide.  5.0 Static Determinacy, Indeterminacy & Stability 22 Internal Stability  5.0 Static Determinacy, Indeterminacy & Stability 23 Internally Unstable  5.0 Static Determinacy, Indeterminacy & Stability 24 Internal Stability All physical bodies deform when subjected to loads; the deformations in most engineering structures under service conditions are so small that their effect on the equilibrium state of the structure can be neglected. A rigid structure offers significant resistance to its change of shape, whereas a nonrigid structure offers negligible resistance to its change of shape when detached from the supports and would often collapse under its own weight when not supported externally.  5.0 Static Determinacy, Indeterminacy & Stability 25 Static Determinacy & Indeterminacy An internally stable structure is considered to be statically determinate externally if all its support reactions can be determined by solving the equations of equilibrium. Since a plane internally stable structure can be treated as a plane rigid body, in order for it to be in equilibrium under a general system of coplanar loads, it must be supported by at least three reactions that satisfy the three equations of equilibrium. 5.0 Static Determinacy, Indeterminacy & Stability 26 Static Determinacy & Indeterminacy Thus, a plane structure that is statically determinate externally must be supported by exactly three reactions. It should be noted that each of these structures is supported by three reactions that can be determined by solving the three equilibrium equations. If a structure is supported by more than three reactions, then all the reactions cannot be determined from the three equations of equilibrium. 5.0 Static Determinacy, Indeterminacy & Stability 27 Static Determinacy & Indeterminacy Such structures are termed statically indeterminate externally. The reactions in excess of those necessary for equilibrium are called external redundants, and the number of external redundants is referred to as the degree of external indeterminacy. In particular, if a structure is statically indeterminate, the additional equations needed to solve for the unknown reactions are obtained by relating the applied loads and reactions to the displacement or slope at different points on the structure. 5.0 Static Determinacy, Indeterminacy & Stability 28 Static Determinacy & Indeterminacy These equations, which are referred to as compatibility equations, must be equal in number to the degree of indeterminacy of the structure. For indeterminate structures, the degree of external indeterminacy is given by: 𝒊𝒆 = 𝒓 + 𝒇𝒊 − 𝟑𝒏𝒓 5.0 Static Determinacy, Indeterminacy & Stability 29 Static Determinacy & Indeterminacy Where: 𝒓 is the total number of reactions, 𝒇𝒊 is the total number of internal forces that can be transmitted through the internal hinges and the internal rollers of the structure. Recall that an internal hinge can transmit two force components, and an internal roller can transmit one force component. 𝒏𝒓 is the number of rigid members or portions contained in the structure. 5.0 Static Determinacy, Indeterminacy & Stability 30 Static Determinacy & Indeterminacy If a structure is supported by fewer than three support reactions, the reactions are not sufficient to prevent all possible movements of the structure in its plane. Such a structure cannot remain in equilibrium under a general system of loads and is, therefore, referred to as statically unstable externally. 5.0 Static Determinacy, Indeterminacy & Stability 31 Static Determinacy & Indeterminacy The conditions of static instability, determinacy, and indeterminacy of plane structures can be summarized as follows: 𝒓 + 𝒇𝒊 < 𝟑𝒏𝒓 : the structure is statically unstable externally 𝒓 + 𝒇𝒊 = 𝟑𝒏𝒓 : the structure is statically determinate externally 𝒓 + 𝒇𝒊 > 𝟑𝒏𝒓 : the structure is statically indeterminate externally 5.0 Static Determinacy, Indeterminacy & Stability 32 Static Determinacy & Indeterminacy The conditions of static instability, determinacy, and indeterminacy of plane internally stable structures can be summarized as follows: 𝒓 < 𝟑 , the structure is statically unstable externally 𝒓 = 𝟑 , the structure is statically determine externally 𝒓 > 𝟑 , the structure is statically indeterminate externally 5.0 Static Determinacy, Indeterminacy & Stability 33 Static Determinacy & Indeterminacy Based on the aforementioned conditions, the first implies that a structure is unstable. For the remaining two conditions, i.e 𝒓 = 𝟑 and 𝒓 > 𝟑. Although necessary, are not sufficient for static determinacy and indeterminacy respectively. In other words, a structure may be supported by a sufficient number of reactions (r ≥ 3) but may still be unstable due to improper arrangement of supports. 5.0 Static Determinacy, Indeterminacy & Stability 34 Static Determinacy & Indeterminacy These Structures are referred to as geometrically unstable externally. One possible arrangement of this can be illustrated with the following truss. The structure has sufficient number of reactions ( 𝒓 = 𝟑 ), and all of the reactions are in the vertical directions and as such translation in the horizontal direction cannot be prevented. 5.0 Static Determinacy, Indeterminacy & Stability 35 Static Determinacy & Indeterminacy 5.0 Static Determinacy, Indeterminacy & Stability 36 Static Determinacy & Indeterminacy Another reaction arrangement is shown with the following beam. The beam is supported by three non-parallel reactions. However, since the lines of action of all three reaction forces are concurrent at the same point A, they cannot prevent rotation of the beam about A. In other words, the third equation of equilibrium is not satisfied (σ 𝑴𝑨 = 𝟎). 5.0 Static Determinacy, Indeterminacy & Stability 37 Static Determinacy & Indeterminacy 5.0 Static Determinacy, Indeterminacy & Stability 38 Example 1: Classify each of the structures shown as externally unstable, statically determinate, or statically indeterminate. If the structure is statically indeterminate externally, then determine the degree of external indeterminacy. 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 3 + 1 − 3 1 = 1 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 5.0 Static Determinacy, Indeterminacy & Stability 39 Example 2: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 4 + 0 − 3 1 = 1 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 Example 3: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 6 + 0 − 3 1 = 3 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑡ℎ𝑖𝑟𝑑 𝑑𝑒𝑔𝑟𝑒𝑒  5.0 Static Determinacy, Indeterminacy & Stability 40 Example 4: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 4 + 1 − 3 1 = 2 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑒𝑔𝑟𝑒𝑒 Example 5: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 6 + 2 − 3 2 = 2 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑖𝑛𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑒𝑔𝑟𝑒𝑒 Example 6: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 4 + 1 − 3 2 = −1 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑢𝑛𝑠𝑡𝑎𝑏𝑙𝑒 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦  5.0 Static Determinacy, Indeterminacy & Stability 41 Example 7: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 3 + 6 − 3 3 = 0 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦 Example 8: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 5 + 7 − 3 4 = 0 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦 5.0 Static Determinacy, Indeterminacy & Stability 42 Example 9: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 4 + 2 − 3 2 = 0 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦 5.0 Static Determinacy, Indeterminacy & Stability 43 Example 10: 𝑖𝑒 = 𝑟 + 𝑓𝑖 − 3𝑛𝑟 𝑖𝑒 = 6 + 6 − 3 4 = 0 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒 𝑒𝑥𝑡𝑒𝑟𝑛𝑎𝑙𝑙𝑦  6.0 SUPPORT REACTIONS 44 Example 1: Determine the reactions at the supports for the beam shown.  6.0 SUPPORT REACTIONS 45 Example 1 Solution: ▪ Free Body Diagram (FBD): ▪ Determinacy: 𝑟 + 𝑓𝑖 = 3𝑛𝑟 ⟹ 3 + 0 = 3 1 ⟶ 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒  6.0 SUPPORT REACTIONS 46 Example 1 Solution: ▪ Equations of Equilibrium: 4 ↻ +, ෍ 𝑀𝐵 = 0, 𝑅𝐴 20 𝑓𝑡 − 12 𝑘 10 𝑓𝑡 sin 60° + 6 𝑘 5 𝑓𝑡 = 0 5 4 ↑ +, ෍ 𝐹𝑦 = 0, 𝑅𝐴 + 𝐵𝑦 − 12 𝑘 sin 60° − 6 𝑘 = 0 5 3 → +, ෍ 𝐹𝑥 = 0, 𝐵𝑥 + 𝑅𝐴 − 12 𝑘 cos 60° = 0 5  6.0 SUPPORT REACTIONS 47 Example 1 Solution: ▪ Solving these equations yields: 𝑹𝑨 = 𝟒. 𝟔𝟐 𝒌 (↗) 𝑩𝒚 = 𝟏𝟐. 𝟕𝟎 𝒌 (↑) 𝑩𝒙 = 𝟑. 𝟐𝟑 𝒌 (⟶)  6.0 SUPPORT REACTIONS 48 Example 2: Determine the reactions at the supports for the beam shown.  6.0 SUPPORT REACTIONS 49 Example 2 Solution: ▪ FBD: ▪ Determinacy: 𝑟 + 𝑓𝑖 = 3𝑛𝑟 ⟹ 3 + 0 = 3 1 ⟶ 𝑆𝑡𝑎𝑡𝑖𝑐𝑎𝑙𝑙𝑦 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑒  6.0 SUPPORT REACTIONS 50 Example 2 Solution: ▪ Equations of Equilibrium: 𝑘𝑁 ↻ +, ෍ 𝑀𝐵 = 0, −𝑀𝐵 + 400 𝑘𝑁𝑚 − 15 6 𝑚 11 𝑚 − 160 𝑘𝑁(4 𝑚) = 0 𝑚 𝑘𝑁 ↑ +, ෍ 𝐹𝑦 = 0, 𝐵𝑦 − 15 6 𝑚 − 160 𝑘𝑁 = 0 𝑚 → +, ෍ 𝐹𝑥 = 0, 𝐵𝑥 = 0  6.0 SUPPORT REACTIONS 51 Example 2 Solution: ▪ Solving these equations yields: 𝑩𝒙 = 𝟎 𝑩𝒚 = 𝟐𝟓𝟎 𝒌𝑵 (↑) 𝑴𝑩 = 𝟏𝟐𝟑𝟎 𝒌𝑵 ∙ 𝒎 (↻) REFERENCES 52 ✓ Hibbeler, R., 2012. Structural Analysis. 8th ed. New Jersey: Pearson Prentice Hall. ✓ Kassimali, A., 2011. Structural Analysis. 4th ed. Stamford: Cengage Learning.  53 QUESTIONS THE END ?