MAE206-L05-Ch3.1-2.Behavior of Cables, Bars, and Springs Equilibrium of Particles in 2-D (1)

MAE 206 Engineering Statics Lecture 5 Equilibrium of Particles Ch 3.1. Equilibrium of Particles in Two Dimensions Ch 3.2 Behavior of Cables, Bars, and Spring Prof. Mary Zadeh Assistant Teaching Professor in Mechanical and Aerospace Engineering Department Fall 2023 NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch.3.1 Page: 2 : y /’ /’ 4 MAE 206 Engineering Statics Ay ‘n = R S R 2, . & ! WA\ i o. 5 A\m ;‘\\_-.sh\ P s & Dijstri dfo(('e —= s ) & stribute \P’/'l, “ ‘\\ 30 Ctural Analvsis a“d N\ NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh o&‘es N EQUiIibrium ofb

h. 3. Lecture 5 Objectives g 4 ®Review free body diagram (FBD) ®Cables and Bars ®pPulleys ® Reactions NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

cnep o . . . . Ch.3.1 Equilibrium of Particles in Two Dimensions Page: 5 ®In mechanics, a particle is an object whose mass is concentrated at a point. It is defined to have zero volume. ® An object that is small compared to other objects and/or dimensions in a problem may often be idealized as a particle. ®|f the lines of action of all forces intersect at a common point, this is called a concurrent force system. An object subjected to a concurrent force system may be idealized as a particle as shown in the figure. NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch.3.1 Free Body Diagram (FBD) Page: 7 ® A free body diagram (FBD) is a sketch of a body or a portion of a body that is separated, or made free, from its environment and/or other parts of the structural system and all forces that act on the body must be shown in the sketch. ®Once an FBD is drawn, application of Newton’s second law Y F = md may proceed. ® |n statics, we seek conditions so that a = 0 (equilibrium) whereas in subjects that follow statics, such as dynamics and vibrations, we seek to determine a and how the motion of a body or structure evolves with time. cutj lw =1501b W =1501b N eand l - i Y 5T _____ Bridget Clyde /Sto kSh t/Alamy Stock Photo (b) (c) (d) NC STATE Let’s GO PACK | UNIVERSITY % @MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Procedure for Drawing FBDs g ® Decide on the particle whose equilibrium you want to analyze. ® magine that this particle is "cut" completely free (isolated) from the structure and/or its environment. = |n 2-D, think of a closed line that completely encircles the point. = |n 3-D, think of a closed surface that completely surrounds the point. ® Sketch the particle ® Sketch the forces = Forces from the environment (e.g., weight). = Forces where a cut passes through a structural member (internal forces). = Forces where a cut passes through a support (support reactions or simply reactions). ®Select a coordinate system and show dimensions. Add pertinent dimensions and angles to the FBD to fully define the locations and orientations of all forces. cut W = 1501b W = 1501b i | \ T \\: = LY 4 r -—"|\—— ® X - ,\\ A E \\\ y 2 00*\1 R R 2007 K g < ' X Bridget Clyde/StockShot/Alamy Stock Photo ' (b) (c) (d) (a) NC STAl E fi @ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch.3.1 PUIIGVS Page: 10 ® A pulley is a simple device that changes the orientation of a cable and hence, changes the direction of the force a cable supports. " |f a pulley is idealized as being frictionless and the cable has negligible weight, then the magnitude of the force supported by the cable is unchanged as it wraps around the pulley. " |f 2 cable with negligible weight is wrapped around several frictionless pulleys, then the magnitude of the force is the same throughout the entire cable. " To idealize a pulley as a particle, such as when drawing an FBD, you can look for the point of intersection of all forces applied to the pulley, or better yet, you can simply “shift” the cable forces to the bearing of the pulley. . | If the pulley has friction, g\ T T¥ u then in general 7} # 7T». - R JT Tz\ If the pulley is frictionless, ' @/ + Tl then Tl = T2 always. Frictionless pulleys (a) (b) NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

. Ch. 3.1 Reactions Page: 11 ® A reaction force, or simply reaction, is a force exerted by a support on a body or a structure. ® Consider yourself as you stand on a horizontal floor, as shown in figure (a). The floor exerts a force on your body that you remain in equilibrium. Thus, the floor exerts on your body a vertical force of magnitude R, and your body in turn exerts on the floor a force of magnitude R in the opposite direction. VW ! AR R (a) (b) (a) (b) Person standing on a horizontal surface. Person standing on a rough incline. NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch.3.1 Structured Problem-Solving Approach Page: 13 ®Road Map: Identify data and unknowns and create a solution strategy. * Modeling: Identify the system and make reasonable assumptions leading to the development of an FBD (or possibly multiple FBDs). A coordinate frame is required. " |n drawing an FBD, a real life problem is replaced by an idealization. " |n modeling, reasonable assumptions are made about what is physically important in a system and what is not, with the goal of developing a model that contains the essential physics while hopefully being simple enough to allow for a tractable mathematical solution. ® Governing Equations: Write the equations of equilibrium ); F= 6, and as needed, additional equations for other physical phenomena pertinent to the problem. ® Computation: Solve the system of governing equations. ®Discussion and Verification: The results are critically examined and interpreted. NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

ch. 3.1 Example 3.1 Page: 14 Consider the structure below where the loading consists of a vertical force P applied to point A. (a) If P=100 N, determine the forces supported by cable AB and bar AC. (b) If cable A B can support a tensile force of 1200 N before breaking, and bar AC can support a compressive force of 1600 N before buckling, determine the largest force P that can be applied. 30° B NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch.3.1 Example 3.2 Page: 16 A small cable car is used to transport passengers across a river. If the cable car and its contents have a mass of 400 kg, determine the forces in cables AC and ED and the force in bar AB. Point A is a pin. NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

. Ch.3.1 Example 3.2 Solution Page: 17 NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch. 3.2 Cables and Bars Page: 19 ®Cables " We generally assume cables to be inextensible and perfectly flexible. " A perfectly flexible cable will immediately buckle if it is subjected to a compressive force. " The cable may have a maximum tensile load limit, beyond which the cable will be unsafe or will fail. ®Bars " We generally assume bars to be rigid so that they are inextensible in both tension and compression. " |n addition, a bar may have maximum tensile load and compressive load limits, beyond which the bar will be unsafe or will fail. - ESSSSSSSSSSSSSsssssssssssssssssssssss§ —>» cable, rope, string, ... NC STATE i@ Let’s GO PACK I UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh

Ch. 3.2 Springs Page: 20 ® A spring is a mechanical device that produces a force when it undergoes a deformation. ®Cables and bars are also springs when their deformation must be accounted for. [ 8 N c STATE i ® Let’s GO PACK UNIVERSITY MAE 206 — Fall 2023 — Lecture 5 — Prof. Zadeh